You Can Construct Arithmetic

8/14/2019 You Can Construct Arithmetic

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YOU CAN CONST RUCT ARIT HMETIC

SAY NOBI A TO MATHEP HOBIA

BY

JOHN HAYS





INTRODUC TIO N

Those who Know seldom Care.Those who Care rarely Know.When Know meets Care,And teams with Dare,Then Mt. Constipation will blow! (jh)

Is it smar t to f launt Mathem atical in competence? Peopledon't flaunt reading inco mpetence.

We o we ever y s ignif icant advance in Civ ilization t o

Mathe matics , inc luding wr it ing. T he ea rl ies t kn own wri ti ng

ar ti fact is a w olf bone f ound in Eur ope , d ati ng fr om the per iod

of 30,000 -25 ,000 B. C., sho wing f ifty -fi ve cut s in group s of

fi ve. Certain ly, tal lies pr ovi ded the ear lie st for m of

bookk eep ing. In Su meria , icon s inventoried stor es of whea t,

wi ne, etc. L ater t hese icon s became par t of gene ral w ri ting .We a ll pa y a da il y educa tiona l tax i n buy ing ite ms and

ser vices - - even chil dr en pay tax for toys or candy . Why? Our

col le ge gradua te s ar e s o poor ly tr ained in Mathema tics tha t

they mu st be re-educa ted , and co sts pa ss ed on to us . Do you

enj oy pa ying th is tax ?

I pr ed icted br eakdo wn of Gener al Motor s seven y ear s ago ,

forewar ned by Le ster Thur ow, MIT economi st : "Amer icans

people ar e not used to a w or ld wher e or dinar y pr oduct ion

workers need mathema tical s ki lls." Toyota uses a ju st -i n-t ime

pr oduction syste m, coor dina ting car pr oduct ion w ith demand ,

to a void o ver-s tocking . (GM over pr oduces , sh uts do wn

factori es , la ys of f wor ker s, unti l inventor y decr ea ses

enough.) In Japan, high schoo l g radua tes cou ld a ppl y the

for mu la for the j us t- in -t ime system. W hen T oyota s et up a

No rth C ar ol ina p lant , onl y A mer ican s tudent s on un iver sit ymas ter de gree level could appl y the formula .

But mathema ticians and col lecti vist teac hing ar e al so at

faul t. Mathema tic ian, Richar d Bel lm an (1920 -1984), de veloped

Dy nam ic P r og r am m ing to min im iz e the cost or t ime spent on

Depar tment of D efense pr ojects . (Lob by is ts and pol it ici ans

ended hi s system , but it could be r estor ed i f enough ci ti zens

car ed and dar ed .) B el lman sa id th is a bout s ome of h is



col lea gues , "95% of the Mathe matics Depar tment s of North

Ame rica have opted out of c iv ilization. "

As tr onome r, biolog is t Carl Sa gan (1934 -96), sa id in his l as t

book, "T he Dr eam-H aunted Wor ld": "We'v e ar ranged a global

ci vi lization in which the mo st cr uc ial e lement s ... pr ofoundl y

depend on s ci ence and tec hno lo gy. We have als o ar ranged

thing s so th at no one under stands science and te chnolog y ...a pr escri ption for disa ster . We m ight get away with i t for a

whi le, but so oner or later th is combu st ibl e m ixtur e of

ignor ance and power will blo w up in our face s . ... ." .

Becau se M athema tic s evolved fr om the langua ge and cultur e

and da il y lives of our ancestor s, I cons ider thi s a bas is for

"nor ma ls " to lear n Mathemeti cs , if teac hing i s gear ed to

ind ividual experience. I once taught six street-s mart b oys

about fr action s, dec ima ls , per centa ges -- eac h b y adif fer ent

method , w hich I developed after lear ning something about

eac h b oy's exper ience.

That i s the p lan beh ind th is book: to displ ay man y dif fer ent

modes of lear ning M ath . T his is d ir ected t o con str uct ing al l

the N umber Sys tem s and the ir Ar ith metic s, to o ver come

mathephobia.

It e xtend s a method developed b y the great Iri shmathema tici an, Wi ll ia m R owan Hamil ton (1805 -65), for

comple x number s. (Phy si ci st s toda y kn ow tha t Hami lton 's

tr ans for mation of opti cs to mec han ics , and vice -ver sa,

antic ipa ted quantum theor y, which c hanged our c ivil iz ation.)

This con st ructs dir ectl y, w her eas Ax iom s cons tr uct

ind ir ectl y, w ith " contr actual loophole s" . On li ne y ou find that

the P eano Axi oms f or S tandar d Inte ger s allo w Non standar d

Inte ger s, any one of which is g reater than an y Standar dInte ger .

The Euc lidean G eometr y Ax iom s al low " The Banac h-Tarsk i

Parado x": the moon can be cut i nto fi ve p ieces , refi tted, and

put i n your poc ket. No one kn ows how to do th is . Fuz zi nes s on

putti ng pieces together see m t o a llo w it .

Eac h use r of Hami lton' s cons tr uct ion get s the sa me r esul t,

but can lea rn it in an ind ividua l way.



As noted in C ha p. Five, Mathe mat ics i s a bout P atter ns, a

concer n it shar es wi th e ver y think ing per son .

The Frenc h anthr opolog is t, C laude Lév y-S tr auss (1908 -?) said

he f ound , the w or ld over, among s o-cal ll ed pr im it ives and

among the civ ili zed, the common year ning to feel th at t heir

lives made sen se -- f it a pa tter n . He c ite s as instance of love

of pa t ter n an aborig ine, tr acking ga me acr oss the Ka lahar iDe ser t, wearing onl y a loinc loth , ar med with on ly a wooden

spea r, w ho might r est fr om the noonda y s un in the shado w of

a r ock. Then fr om h is lo inc loth mi ght pul l an embr oider y hoop

and be gin to e mbr oider lit tle red f lowers - - s ome thing he

lear ned fr om a miss ionar y woman.

PRAYER OF A FECKLESS FOOL

Before I go to that Great Playground,To spend that Last Recess,Lord! Make my Life a Pattern,Instead of this mishamy! mashamy! mess! (jh)

I'm concer ned about the America I leave to m y two s ons and

four grandc hi ldr en . W ho el se is doing any thing about th is ? I

chal lenge you t o f ind a book w ith any of t he Knowable s of my

book.

If you car e and dar e, Const ruct Ar ithmet ic , and join me in th isupg rading of our educa tion and it s gradua te s, for our

wonderfu l countr y, f or our dear people, for the futur e of you

and your s.

Thank God ! for pr oblem s wi th s ol uti ons .



CHAP TER ONE: H OW YOU CAN LEAR N in " YOU CAN

CONSTRUCT ARITH METIC , sa y Nob ia to M athephobia "

1. Pr otoT ype s evoke a c las s of so me type . One school

Pr otoT ype is "factoring number s", w ith lea st common

mul ti ple and g rea tes t common divi sor . Proto Type f or a ll

par ti al or dering s i n Mathe matics and in mil itar y andbus ine ss h ier ar chie s .

2. Pr otoLear ning : You kno w Kno wables you didn 't know you

knew unt il r eminded of famil iar P rotoT ypes of these

Kno wables . Sa y, the la tt ice of num ber f actor s , known via

"i nclus ion in , conta ined in, dom ina ted by, subse t of ,

factor of, subor dina te to" .

3. TEA CH YOURSELF : Lear n by Teac hing Your self Math and

other K no Wable s. A ppend ix C.4. Gr ammar and Mathema tics : Gramm atica l Con str aint s

lear ned in schoo l ar e im pl ic it Pr otoT ype s for

Ma the ma t ics , since Mathema tic s evol ved fr om human

langua ge and ever yday ance str al experience .

5. Inheritance : our WILL of Inher itance fr om Ance stor al

langua ge and cultur e, invokes a un ique indi vi dual way of

Lear ning ever thi ng, inc luding Math and Science , via

ind iv idual hi stor y.

6. ACTI VITH M : Cha pter Five, on P atter ns c ite s Al ber t

Einstein (1879-1955) and other s saying muc h of w ha t we

see is con str a ined by m ode of thought - - impo sed i n

ACTI VITH M upon the w or ld . An ins tance develops in a

Math iv ity for C hi ldr en , " Col or ed Mu lti pl ica tion Table s,

Co lor ed Con ser vation Laws ", showing pa tter n s of po w e r s

of tw o and fi ve; mu lt ipl es of n ines and e le ven s, i nvok ingfam il iar A lgor ithm s of "ca st ing out n ines and ele ven s - -

ACTI VITH M of i m po si n g t he dec im al ba se upon the

counting number s .

7. Vector Logic . Standar d Logic is con str ained to the

dec lar atv e mood , ignor ing moods of i nter roga t iv e,

sub junct iv e , im per ati ve, pet it iv e , etc. V ector f or m a t

cor rects th is . In phy si cs , speed i s a scal ar (0 -v ector) ,



explic ated b y a s ing le number ; but ve loci ty i s s peed i n a

spec ifi c d ir ection , so expl ica ted by a two-tup le: t wo

number s, sho wing s peed i n "X " and "Y" dir ect ions .

Si mi la rly, a ll spee c h mood s can encompa ss a two-t uple

whose fir st com ponent i s s p eak er' s m ood, s econd i s

speak er' s sp eec h . This also sa ti sf ie s Cons tr aint of be ing

TR UT H -F UN CTI ONA L : TH E TR UT H -V A L UE O F A CO MP OU ND

STATEM EN T D EP EN DS ONLY ON THE TRU TH -VAL UE OF

CO MP ON EN TS , not O RDE R or CON TE XT or "w ha te ver" . The

moods ar e symbol ized:

o dec lar ati ve , by Bertr and Russell 's "tur n-st ile" s ymbo l:

|-li

o inter roga tive : ?

o imper ative : !

o sub junct ive : % (a s in "Wou ld I were at home .")

o peti tive : * (as in " Please go.")

Vector Logic also res ol ves a famou s 1960 's proble m,

when man y thoght C OM PUTER TRA NSLA TION OF

LANGUAGES could simpl ify publ ica tion of tec hn ical or

l iter ar y- dr ama ti c l i ter a tur e . Bu t onl y lim ited s ucces s was

achie ved. Cr it ics of "ma chine tr ans lation " c ited an

Eng lis h s entence , " Time fl ies ", fr om the Latin, " Temp is

fugit s". But a computer trans lated it a s "mea sur ing fl ight

of cer ta in i nsects ". T his amb igui ty ar is es on ly in scal ar

log ic . V ector Log ic d is ti ngui she s:

o [|- , "Time f li es ."] , for the f ir st (dec lar ative) ver sion .

o [!, "Time flies."] , for the second (imper ative) v ersion.

Vector LOGIC also function s as m e talangua ge of s peak er .Any statement is in t he alethi c mode (True or Fals e). But ,

Ar istot le d is ti ngushed the te mpor al mood , such as

"Geor ge W. Bu sh is Pr es ident of T he U ni ted S tate s" , tr ue

at one t ime , not tr ue at time of th is wri ting . The pri m ar y

M O D E S ar e:



o ALE THI C M OD ES or M ODES OF TRUT H ar e, principa ll y,

neces sar y, po ss ib le , i mpo ss ible , contingent . (For

discu ssion belo w adopt, A, as the a leth ic m ode

oper ator .)

o TE NSE MODES repr esent TEM PORAL aspect s of

sen tences . ( Ari st otle cited a dec lar ation w hich is

TRUE a t a g iven ti me, and FALS E a t other s. )o The DEONTIC MODE dea ls with obl iga t ion , can , may ,

etc. (This is the MODE in ETHI CAL and MORAL

discu ss ion s, as well as LOGIC AL and THEOLOGICAL

ones .)

o The EPIS TEMIC MODE dea ls with ter ms such as

kn ow ing, be lie ving . (For pur pos es of th at di scus sion ,

adopt E, as the ep is tem ic mode oper a tor .)

o The PREFERE NCE MODE se rves econom ic and

pol it ical d is cus si on.

o The FRE E M ODE a llo ws for fic tiona l ima gina tion .

o Etc .

VECTOR LOGIC R ESOLVES "T he Mor ning S tar Par ado x". In

syll ogi st ic for m:

o "T he Shepher d kno ws tha t V enus is T he Mo rningStar ."

o "T he Mor ning S tar is The Evening Star ."

o Ther efor e, " The Shepher d kn ows tha t Venu s is The

Even ing S tar ."

But the Shepher d may not actua lly kno w thi s. The second

line is Aleth ic o r D eclar ati ve. B ut the f ir st and thir d line s

afe E pi ste mic : what is kno wn. And noth ing ju st ifi es Modesh ift , so the Par ado x v ani she s. Othe r amb igu iti es and

par ad oxes could be r esol ved simil ar ly. Vector L OGIC

explic ates cer tain di st inct ions nece ssar y to Lear ning .

8. Condot . You 've seen chil dren's p lay books in w hich they

"connect the dot s" to sha pe a per son or ani mal or

land sca pe or whatever. You need to lear n t o "connect the



dots " to r ealize tha t one Kno wable m ay be Connected to

another , and should not be ignor ed.

9. BYPASS : So meti mes , you can BYPASS a d if ficul t or

unf ami liar pr oble m b y

o tr ans for mi ng i nto a pr ob lem you kno w ho w to s ol ve

o so lving the pr oble m

o tr ans for mi ng s olution into t er ms of ori gina l pr oblem

BYPASS ha s for m:

difficult/impossible/desired task

-------------------------> transform| ^transform to new | |back totask | |terms of

| |original| |task

V------------------------> perform task

A PROTOTYP E of BY PAS S IS se en i n the 1963 fi lm , "T he

Gr eat E sca pe" .

flee across a garded space------------------------->

transform | ^tunnel uphorizontal | |to ground level

to | |escape into woods vertical | |

| |V------------------------> dig underground tunnel

You real ize ho w N ONTRI VIAL th is STRATE GY by

WEBlear ning tha t, in MATHEMATICS , IT DETERMI NES THE

EIG ENVALUES OF A MA TRIX OR MULTIV ECT OR. That, in

Phy sics, IT IS TH E PRI NCI PAL TOOL OF QUANTUM MA TH

FOR FI NDI NG T HE S TABLE STATES O R RADIA TIO N STATES



OF FUN DAMENTAL PARTICLES .

BYPASS i s a favor ite str ate gy of mathe matician s. You

tak e a problem to a m athem atic ian and she/he st ar ts

talk ing about another pr ob lem , whic h mak es y ou think

she /he i sn't listen ing to y ou . But not to w or ry! It should

tr ans for m bac k i nto an an swer to your problem .

10. BNF . If y ou kno w SEMIO TIC S ( Theor y of S ign s), th is will ,

in Cha pter Thr ee, be review ed. If you don't know thi s,

you' ll be intr oduced to it . The subject is her ein c ited to

explain another guidance M ethodolog y of th is book.

o Br iefl y, the su bsyte m of Sem ioti cs labe led as

"S yntact ics " -- yes, th at' s al so in G rammar -- dea ls

onl y w ith re la t ions betw een Sign s , but not the ir

Meaning s or R efer ent s,o The sub sys tem of Se mio tic s la be led "Semanti cs " a ls o

deal s wi th the Mean ings or Referents of S ign s.

o The sub sys tem of Se mio tic s la be led "Pr agma tics "

also deal s with t he Sign User . A par ticu lar sign can

have di fer ent m eaning s for dif fer ent peop le, (You' ll be

sho wn an ins tance wher ein a s ign, r esemb li ng an

eight on its side , has a dif fer ent M eaning for thr ee

dif fer ent pr ofessi onal s -- and ho w thi s amb igui ty is

Pr agma tical ly r esol ved.

The above comment is to pr epar e you for a B YPASS of

those various Se miot ic di st inct ions b y a po werful

for ma li sm whic h has been thor oughl y tested : Backus -

Naur -For m (BNF) . It has the structur e:

<term>:@ <another term>

An in stance :

<USA first president>:@ George Washington



Both side s may h ave meaning for you . But it doe sn't have

to be mean ingful . One side could be i n Russi an, the other

side about topo logica l m ani fold s -- both perha ps unkno wn

to y ou. For, in ef fect. BNF sa ys: " Replace the s ign- string

on the lef t b y the s ign-s tr ing on the right. " A pri mar y

school student can do t his.

11. Ant itone : Bypa ss unf old s from th is dia grammed S tr ate gy

whic h ma y be impl ic i t i n al l of Rea l it y' s pr oces se s . It i s

lacking when Force is der ived from N ewton 's Law s of

Moti on . Bu t tho se Laws y ield Mo mentum , with its

Con ser va t ion of Linear Momentum and i s Ant iton ic .

Der iv ing fr om Ant itone , B YPASS tr anso rms i nto

Amp li fic ati on -- mor e in Ouput t han Input - - whic h

explain s Mac hines and pr ovides us wi th " ri ches ".Lear ning the A nti tone-B ypas s- Ampl ify connect ion teac hes

"the wonder s of Nature".

12. Indica tor-S iga l. In Sem iot ics , Peir ce taught us about

Indica tor , S igna l, Icon , S ymbol , a f our some pr ovi ding

Str ate gie s for Sc ience and Educa t ion . Acr onym ISIS :

I(ndica tor) S(ignal) I(con) S(ymbo l). An indic ator has the

fol lowing t wo-tuple st ructur e, wher e "O" denotes

Ob ser va bi li ty and "I" denotes Inf or ma t ion .13.

<Hi O-Low I, Low O-Hi I>

as in

<pink litmus paper, acid in test tube>

A Signal is an Indica tor under Phy sica l and Lingui st icContr o l :

<Hi O-Low I, Low O-Hi I> ⇒ <Phys.-Ling. control>

as in:



<lightning, electricity> ⇒ <telegraph key, Morse Code>

In past ti mes , fever was classified as a di sea se . Later, it

was real ized t o be a highl y obser va ble sym ptom of a

hid den i nfect ion . So, med ica l s ci ence pr ogressed fr om

so lel y al leviator s of fever (e .g, co ld compr esses) to

conjo in w ith alle vi ator s of i nfect ion , s uch as penec il lin.In a Histor y of a S cience -- Wik ipedia has s ever al -- you

see cita tion s of thi s Str ate gy of Science . Tur ning to the

other ha lf of ISIS, an Icon is a familiar sigm on a

computer Des ktop , a s in w a ste ba sk et for dis car de d fi les .

And a Sym bol is what your read on thi s page, The

Str ate g y of Educa tion is to tr ans for m the Icon into the

Sy mbo l . An instance is the thr ee -finger ta ll y whose shape

became the numer al for thr ee . Also the w or d "f ive"deri ves fr om the wor d " fi st ".

14. The Logic -Ass erb il it y Co mit y (toler ant together nes s) , as

descr ibed in Cha p. Ele ven: The "Ep istemo log y Game"

(W hat A re C oncr etion s? A bs tr action s? I ll ations ?) . This

comi ty m or e cor rectl y desc ribe s Sci entif ic advance s than

noted b y sc ient is ts : expla ins w hy monoton ici ty o f Logic

encumber s cor rection of s ci entif ic pr onoun cement s w henexp er iment su rpri se s . This Log ic-A sse rbi li ty C om ity

explain s how the nonmonotonic Am pl i fy ing of A sserb il i ty

compens ates for Log ic not be ing a ble to lose in t ruth

when it s u se in sc ience is cor rected , a s Rela ti vity Theor y

and Quantic Theor y.(As serbi l it y is As ser t ion sub ject to

ver ific ati on, tha t i s , h ypothes i zed pr edict ion conf ir med or

di sconf ir med .) But the immed iate impor tance of thi s for

your Lear ning is tha t i t "put s you i n the s tand s wi th othernonsc ienti sts to watch scienti st s pla y the ir bran d of the

Epi st emolog y G ame ". It al so al lows you to us e A sserb il ity

to sear ch for Kno wa bles and to build your own Secur it y

Sy stem .



15. The Topo logica l Per pecti ve r evea ls i nher ent V alue s not

appa rent otherwi se . To mathema tic ians , Topolog y i s an

advanced sub ject . B ut many of them fai l to real ize th at,

via it s attr ibute " Connnect ion: How Things A re

Connected" , Topolog y i s omnipr esent in our dai ly l ives .

One Pr oto Type of Topol og y is the J or dan Cur v e (one for m

is a cir cle) whic h separ ates the plane in to In side and

Ou ts ide ; c losed Boundar y betw een . The infant copes with

Topo log y in tr yi ng t o k ick of f b lank ets . Ch il dr en cope

wi th T opol og y in slee ves, t rouser le gs, door s, drawers,

etc. We cope wi th Topo log y in our elect rica l cir cu its ,

eac h f or ming a Jor dan C ur ve wi th i ts Ins ide -Ou ts ide -

Boundar y connect ion. Many ph ys icist s forget, or don't

kn ow tha t the Ge rman ph yi ci st , Gu st ave Kir kfof f (1824-

87) used T opolog y i n devising hi s famou s Laws of

El ectri cal Circui ts . And Topo log y is omn ipr es ent i n our

Lear ning , as in th is book: la be ls , defin it ions ,

explana t ions , va lu ati ons , ag reements , w hic h c lose the ir

contents w ith in a Jor d an Cur ve, bounding u s fr om

ambi guit y, sto len ident it ies , t ri cker y -- you name i t ! . We

can lear n m uch about both En vi ronment and Knowledge

by"look ing for The Topo logica l Condot" . Quer yi ng: "Am IOu ts ide ? In si de? acr oss a Boundar y? " - - "Ins ide of or

Ou tside of The Law? " -- "In side of or Outs ide of the Rule s

of Gr ammar or ofMa thema tic s? " "Have I cr os sed a

Boundar y-C ons tr aint ?" Condot !

16. Dimen si onal Al ge br a ( DA) T he F OR MS of di m ens iona l

anal ys i s can be used as a Alge bra for deri ving FO RMS

fr om ba si c F OR MS . In Cha pter T wenty on Science , you

will see how to der ive equa tions never bef ore s een. It has

long been the cus tom to i gnor e D imens iona l A nal ys is for

any except s pec ial ist s. A consequence, in the nineteenth

centur y, was tha t the Ge rman ph ys icist , Lud wig Bo ltz man

(1844-1906), in der iving "The Mo st Pr oba ble Dis tribut ion



of Molecule s in a System" , fai led to complete the

integration at the end and evalu ate the g iven cons tant,

whic h has the dimen sion s of Planc k's con stant h, Had he

done so, quantum theor y mi ght h ave be gun decade s

ear lier . This is based upon the anal ys is of Gerhar d A.

Blass, T heor et ical Ph ys ic s , pp .236-43) . If the vast s ub ject

of Dimen si onal Anal ys is i s ignor ed, or neglected, it is nosu rpri se tha t i t isn 't used for a D im ens iona l A lg e br a .

17. In "Barb ie D ol l" Math (A ppend ix C) , it s Se mantic

Trans for ma tion sho ws you t hat you kno w, at l eas t, tw elve

mor e M athem atic s syst ems than you thought y ou knew ,

bef or e be ing to ld . Y ou kno w y ou kno w the se syste msbecause you s tud ied the Pr oto Type of these systems

when you lear ned to factor number s in ar i thmet ic .

18. Pr og rammed Lear ning (PL) :

o A Kno wable, s uch as the Euc lidean A lgori thm, i s

outl ined

o the s tudent is given rele vant number s f or t hi s

Al gori thm and attempt s to a ppl y the Euc lidean

Al gori thm to t hese number s

o next, the s tudent look s up the solution in a workbook

or on a laptop - - if cor rect, proceeds to another

Kno wable - - if incor rect , the e rror i s r evea led, and

st udent t rie s another v er sion of thi s Kno wable .

This encour ages the s tudent to be inter acti ve and

pr oceed at own pace . PL is not the sa me a s Pr og ram med

Inst ructi on (PI) of beha viour psycho logi st , B. S . Sk inner .

19. Ho molog y : from ancient Greek mathema tic s, no w re-

appli ed to advance mathema tics, but -- in o rig inal sens e,

the be st for mal ism for lea rning and discover y. Ins tance:

To expla in the great ad vanta ge the Fresne l len s over t he



con vex len s. (The F resnel le ns made pos sibl e to wering

lighthou ses to guide ship s at sea.) So you for mulate this:

20.Frenel lens: convex lens:: stairway: ramp

"T he Fresne l len s compar es to the con vex l ens as a

st ai rway compar es to a r amp." N ew Kno wable s i n ter ms

of old one s. H er e i s the gener al Homo log y for m:

A: B:: C: D

Actua ll y, you 'r e t wice Pr oto -educa ted t o the Ho molog y. It

appe ar s in M ult ip le c hoice tes t ite ms . Say,

legs: human::__: vehicle

You se lect the c hoice wheels a s veh icle pr opell ant, just

as the le gs ar e the hu man pr opel lant . You'r e al so Proto -

educa ted to the Ho molog y i n su ch a famil iar f or m, a s

2: 4:: 3: 6

You' ve u sual ly seen thi s as:

2/4 = 3/6

Yes. Impl ici t in lea rning fr action s i s one of the most

powerful simpl e f ormal ism for lear ning and d is cover y wehave. You need onl y kn ow thi s and u se Homolog y.

21. CYCLES O F DIV ERS E S TRA TEGI ES (CO DS) : Var ious of t he

above Methodo logie s, a s wel l as other "ol ogie s" , for m

Cycles.

ANTIT ONE C YCLE

STRA TEGY: CAUS ALITY -TEL EOLOGY



STANDARD CHANGE------------------------->

GOTO NON-| ^STANDARD | |MINTONECHANGE | MOTION |

V------------------------>

MAXTONE⇓

STANDARD CHANGE------------------------->

GOTO NON-| ^STANDARD | |MINTONECHANGE | REVERSE MOTION |

V------------------------> MAXTONE

22. Flowchar t : In the be ginn ings of the compute r, the

pr og ram was im ple mented b y a p lugboar d, a s used at th at

time by telephine oper ator s. And data was input by ho les

in punc hed , a s used in IBM ca lcul ator s. Mathema tici an

Joh n von Neuman (1903-57) and engineer Her man

Go ldst ine (1913 -2004) cr eated t he inter na l pr og ram for

computer s, a g reat sim pl ica tion . T o faci litate the ir work,

they de veloped the flo wchar t i n 1945-7 . Y ou w il l see, i n

Cha pter 5 , sever al flo wchar ts for impor tant mathema tica l

algor ith ms . Wher e pos sibl e, I wi ll di sp lay a flo wchar t f or

so mething , and b y doing you m ay l ear n.

23. The AS P T riad -- Algor ithm , Str ategy, Pr os thetic

(di scus sed in C ha pter 10 -- pr ovides another Methodolog y.

Whene ver us ing a B YPASS , be aw ar e of it s com it y wi thStr ate gy. A usefu l defin iti on of M ili tar y S tr ate gy i s

fight ing the enem y on g rounds of your choo si ng . In

gener al, S tr ate gy i s deal ing w ith a pr oblem under

condit ions of one's com peten c y . And thi s is what you'r e

doing in Bypa ss.



As noted in C ha p. Ten, an excell ent i ns tance of su ch m ili tar y

STRATE GY is to be seen in the great 1938 fi lm, "Alexander

Nevsk y", dir ected by the master Ser gei Ei sens tein (1898-

1948), wi th an resound ing m us ica l s cor e by the great

compose r, Ser gei Pr okofief (1891-1953) .

On Apr il 5 , 1242, Gr and Pr ince of Novgor od and V lada mi r(1220? -1253) le d h is foot s ol dier s onto the ice of the Lak e

Peipu s, kno wing tha t (on the g round of his choos ing), his

enemie s, the i nvading mounted kn ight s of the L ivon ian br anc h

of Teutonic Kn ight s, wer e mor e he avil y ar mor ed . M any

cracked thr ough the ice and w ere droawned, leaving a

sm al ler for ce h is limited for ce cou ld c hal lenge. The resul t

was a g reat victor y. (And what a Bypa ss !

You may f ind th at Appendix A, "Ar ithme tic and Mo i" , sug gest s

many kno wables about the pur pos e of t hi s book.



Cha pter 2 : WHAT IS INT ENSIO N? EXT ENSIO N? RELA TION?

FUNCTI ON ? OPERATION ?

You lear n tha t the ter m intenS ion (not in tenTion ) is a

ref er ence with Kno w a b le r ef e r e nt s . Its Protot ype is Idea:

in tens ion of a w or d i s the id ea imp li cit i n t he wor d , so

in tens ion can se r ve as pr oxy or su r roga te of the w or d . This

you no w know.

The Pr otot ype of in tens ion . in set theor y , you can under stand.

The in tens ion for mat f or def ining a s et , S, by in tens ion i s S

={x|P(x)} : "set of al l x s uch that pr opo si tion P(x) i s tr ue ,

wher e 'x ' ha s so me mean ing" . This def ines a se t i ndir ect l y ,

tha t is, by pr oxy, (cr oss ing a B oundar y) , a s you kno w. O r

define s a set ind ir ectl y , tha t is, by sur roga te , (cr ossi ng aBoundar y), as you kno w. Or def ines a se t i ndir ect ly (cr oss ing

a B oundar y) , by descr ipt ion , as you kno w.

In Appendix B, y ou can witnes s your self tea ching thes e and

related concept s.

SEQU ENCE-PARALLEL LE AR NIN G INTEN SIO N OF SET

(In se ries connected l ights , if one goe s out, all go out; not so

in par al lel . If one of lear n-s equence i s unkn own, ma y su pr ess

other s in sequence; maybe not in par alle l. )

{x:x in N} MEANS se t of Natur al Nu mber s

ß ß ß

{x:x in Z} MEANS set of Inte ge rs

ß ß ß

{x:x in Q} MEANS se t of Rationa l N umber sß ß ß

{x:x in R} MEANS set of Real N umber s

ß ß ß

{x:x in C} MEANS set of Comple x N umber s

You know tha t, in se t theor y , the extens ion of a s et is a l ist

of me mber s becau se y ou 've se en th is i n the r oll -cal l of a

school class . N oth ing m or e is needed ( no B oundar y cr oss ed ).



SEQU ENCE-PAR ALLEL LE ARNIN G OF EXTE NSI ON

number set MEANS {2, 5 , 11, 87}

ß ß ß

le tter s et MEANS {a , d , g, k, v }

ß ß ß

color set MEANS {red. blue , white , pink}ß ß ß

food s et MEANS {br ead, milk, cor n}

ß ß ß

book set MEANS {no vel, text, chec k}

S-P LE ARNIN G RELA TIO N

relation MEANS marital

ß ß ß

relation MEANS par enta l

ß ß ß

relation MEANS neighbor

ß ß ß

relation MEANS emp loyer

ß ß ß

relation MEANS on look erS-P LEA RNI NG FUNCTI ON

function MEANS subtr ahend

ß ß ß

function MEANS inter ect ion

ß ß ß

function MEANS negation

ß ß ßfunction MEANS dividend

ß ß ß

funtion MEANS dif fer enc e

S-P LEARNIN G OP ERA TIO N

oper ation MEANS addition

ß ß ß



oper ation MEANS division

ß ß ß

oper ation MEANS mult ip lic ation

ß ß ß

oper ation MEANS logari thm

ß ß ß

oper ation MEAN S subtr act ionThat a desc ript ion can f it m any d if fer ent se ts or per son s, o r

w hate ver (many Boundar ies m ay be cr os sed) i s something you

kn ow, becau se y ou saw it a bove. So, in t er ms tak en up be low,

in tens ion i s a one -man y r e la tion : one refe r ence w ith man y

pos si b l e r ef e r e nt s .

You real ize th at, as a s pecif ied se t, the e xten sion of a ter m is

a one -one r e la t ion: one ref er ence, one re fer ent (no Boundar y

cr os sed) .

You real ize the se dist inct ions ar e cri tica l because, a s you

lear ned in s choo l, for t wo thousand year s, the s tandar d

defin iti on of a m athema tica l sub ject v ia AXI OM S , (some you

st udied in schoo l) w hich, a s you real ize, ar e i nten siona l:

mul ti r efe r ent (cr oss ing many B oundarie s) .

You can under stand two crit ical con sequences of the one-

many e xpl ica tion of A XIO MS .

You know how to sear ch the Web f or the Peano Ax iom s

defin ing i nte ger s . And you kno w how to sear ch the web f or a

webs ite sho wing tha t the P eano Axi oms f it nons tanda rd

in teger s , eac h of w hic h i s g rea ter than a st andar d in te ger .

(You' ve hear d of a per son' s iden tit y being stol en , so you cannote the similar it y her ein ,) Y ou, then , under stand th is a s a

cri tica l consequence of thi s one s et of A xio ms re la t ing t o

many d is ti nct i nter pr et a ti ons .

You can also sear ch the Web (sa y, Wiki pedia) for the B anac h-

Tar sk i Par ado x , which s hows th at the E uc lidean Geo metr y

Ax iom s al lo w cutt ing the m oon into f iv e p ieces , putti ng the



fi ve p ieces togethe r, and putting the moon i n your poc ket . No

one kno ws how to do t hi s, but Euc lidean Ax iom s impl y i t can

be done as consequence of (Boundar y) fuz zine ss in describ ing

how piece s can f it together .

(T hink of Axio ms as Contr act s. Law yer s f ind loopho les in

contr acts so th at the contr actee s can vio la te the intent s ofthe contr actor .)

The gener al pr ob lem condots bac k to Eur opean Mid ddle A ges ,

when Scho las tics discover ed Langu age t o be prone to the

oxy mor on (s elf -contr adict ion) . Belie ving tha t "whatever can

be s aid m us t be tr uth -exi stent ", the y recogniz ed den ial of th is

in ep ithet, " im potent G od" .

This cr it ica l f au lt is label led " impr edica ti ve" : pred ica t ing

it se lf: s e l f-r efer encing . This can de ve lop w hen a se t i s not

refer enced dir ect ly , but ind ir ectl y , i n ter m s of another se t . A

famou s P totoT ype of th is i s "Rus se ll 's Par axox" .

The great Bri ti sh mathema tici an-l ogi cian -phi lo sopher ,

Ber tr and Russel l (1872-1970) , f ormul ated in 1901 to chall enge

"the nai ve s et theor y" of German log ician , Go tthold Frege

(1848-1925).

Con side r a se t S contain ing onl y tho se sets not member s of

them sel ves . If S is not a m ember of itself , it i s eligi ble under

the def ini tion ; but to be under the defin iti on i s a

CO NTRADIC TIO N!

Wikipedia notes th at "Russe ll 's Par ado x" led to Göde l' s Pr oof

tha t we cannot u se fir st -or der logic to pr ove the cons istenc y

of Mathema t ics .

We can BYPASS th is const ructi vel y by tr an sf or m ing fr om

Ax ioma tic s to G ener at ics , fr om Indir ect ion to D ir ect ion .

(Pl aton ism e xh ibit s the sa me Ind ir ection : don't tak e

respon si bi li y f or an yth ing i n Mathe ma t ics , but sh ift

respon si bi li ty to s ome "Hea venl y r ea lm " .)



Russe ll s ays he wanted cer ta int y a s other s want r eli gion , but

later real ized th at math cer ta int y i s tautol ogica l ( mer el y a

matter of langua ge) , and be yond thi s ri sks contr adic tion via

any defin it ion tha t can be render ed se lf -r efer encing

(i mpr edica t iv e) by Ind ir ection : us ing one s et to ident ify

another s et . Russel l then gave up m ath and l ogi c and

phi los ophy .

This C ondots to what so me con si der a similar contr adictor y

pr oblem in R el igion : Theod ic y - - if ther e i s a God, ho w can

suf f er ing be ? But , again . I mu st not h ide behind thi s

ind ir ecti on . Facing dir ect ly the suf fer ing and ignor ance I see

dai ly, I kno w w hat I can do and must do , and th is "k eep s id le

hands bus y".

You can under stand tha t, in th is book , the cons tr uct ion of

ari thmet ic B ypas se s A xi oms b y being extensi onal (no

Boundar y cr o ss ed) , by giv ing up i nd ir ection and d ir ectl y

accepting respon si bi li ty . You can under stand th at th is means

no one can use the se method s t o con str uct an ari thmeti c

dif ferent fr om the one i n thi s book.

Wher ea s, th is const ruction be gin s w ith W. R. Ha mi lt on'ssee ming ly Non standar d w ay, yet it invokes the Equ iv alence

Relation of eac h con str ucted syste m, which a llo ws the

resu lt s to be in ter pr eted i n the S tandar d appe ar ance -- the

one supposed ly dec lar ed by Ax iom s in the manner of Mos es

del ivering The Ten Co mmandment s fr om M t. Ar ar at.

The resu lt of th is i nten si onal for mal ism is tha t S tandar d

Arithme tic is often pr o ven nonco str uct iv e l y v i a pr oof b ycontr adict ion (cr oss ing Boundar ies) .

Thus, as you can lea rn on the W eb (say , Wik ipedia) , to pr o ve

pr opos it ion P , you h ypothe si ze non- P (cr os si ng a Boundar y),

and, if th is r ea soning impl ie s a contr ad iction , then (cr oss ing

a B oundar y) the contr ad ictor y of non -P must be va lid, na mel y,

P.



As you l ear n on the Web (W iki) , thi s noncon str uct iv i sm be gan

wi th E uclid 's pr oof b y contr adicti on th at t he squar e r oot of

tw o is not a rat ion of t wo inte ger s ,

Equi va lentl y, the s i de of a s quar e i s incomp atib le w i th the

dia gonal of the s quar e . That i s, s o many un it s of the s ide of

the s quar e w ill never match so many other unit s of thedia gonal of the squar e.

In T he E lement s of G eom etr y of Euc lid (365-275 B.C.) ,

number s ar e accepta ble if the y ar e tr ans la ted, for example ,

in to r e la t ions betw een s e gm ents of l ine s .

You know the d ia gonal of a squar e divi des i t in to the fus ion

of tw o r ight tr iang le s wi th equa l ba se and al ti tude . It' s eas y

to s ee when tw o se gments ar e equal : t he se gments extend i n

par all el , w i th end s ma tc hing - - ob v iou sl y compa t ibl e .

But how do you compar e line se gment s of dif ferent length s to

see if the y ar e compa tib le? S ay, in a tr iang le of base thr ee

unit s, of al ti tude four un its , of d ia gonal fi ve un it s, so tha t, by

"T he Pytha gor ean Theor em" (on li ne), 32 + 42 = 52 = 9 + 16 =

25 . (Two su ch t riangles cou ld fuse i nto a thr ee -by -four non -

squar e rectang le s liced b y thei r d ia gonals .)

You see belo w tha t the above ari thmet ic i s geometr ized by

the d ia gram (in red) of a thr ee -unit se gment (a s i n the ba se of

one tr iang le) . Then y ou d iagram (in blue) a f our-un it se gment

(as in the al ti tude of th at tri ang le). Then you dia gram (in

bla ck) a fi ve-un it segment (as in t he tr iang le 's d iagonal).

|- --- ^-- -- ^--- -| |- --- ^-- -- ^--- -^- -- -| |- -- -^- -- -^- -- -^-- --^-- --|

Since 3 · 4 = 4 · 4 = 12 , you can compar e (be low) four copie s

of the thr ee-un it se gment wi th t hr ee copie s of the four -uni t

se gment :

|- --- ^-- -- ^--- -|-- --^ -- --^ -- -- |-- --^-- --^-- --| --- -^- -- -^-- --|



You note that the se two extend equa ll y, tha t i s, the y ar e

cong ruent . This i s the mean ing of "com mensur able" : T w o

se gment s are com mensur ate if a mul ti p le of one se gment is

cong ruent to a m ult ip le of the other .

You can now see - - because 3 · 5 = 5 · 3 = 15 th at fi ve

instance s of the 3 -se gment ba se is commen sur able w ith thr eeof the fi ve-s egment d ia gonal :

|- --- ^-- -- ^--- -|-- --^ -- --^ -- -- |-- --^-- --^-- --| --- -^- -- -^-- --|- --- ^-- -- ^--- -|

|- --- ^-- -- ^--- -^- -- -^- -- -|- --- ^-- -- ^-- -- ^--- -^- -- -|-- --^ -- -- ^-- -- ^-- -- ^--- -|

From the above note on cong ruence, you real ize the

cong ruence of thes e t wo extensi ons indic ates the

commen sur abil ity of the base and dia gonal of the above

tr iang le. A nd, since 4 · 5 = 5 · 4 = 20 , we can co mpar e fi vecopie s of the four- se gment al titude wi th four cop ies of the

fi ve-s egment dia gonal :

|- --- ^-- -- ^--- -^- -- -|-- --^ -- -- ^-- -- ^-- -- |-- --^ -- --^ -- --^ -- -- |-- --^-- --^-- --^ -- --| --- -

^-- --^-- --^-- --|

|- --- ^-- -- ^--- -^- -- -^- -- -|- --- ^-- -- ^-- -- ^--- -^- -- -|-- --^ -- -- ^-- -- ^-- -- ^--- -|-- --

^-- --^-- --^-- --^ -- --|

The cong ruence of thes e t wo extensi ons in dic ates theinco mmensur abi li ty of the alt itude of any s quar e w ith its

dia gonal .

The above demon str ations ha ve been geometr ic . We can again

write their inter preta tion s in ari thmet ic , in t er ms of

fr action s:

•

in ter pr et " four copie s of the thr ee -se gment " a s thefr action , 3/4 ;

• in ter pr et " thr ee cop ies of the 4- se gment" as the fr act ion,

4/3 ;

• in ter pr et "f ive cop ies of the 3-se gment " a s the fr act ion,

5/3 ;

• in ter pr et " thr ee cop ies of the 5- se gment" as the fr act ion,

3/5 ;



• in ter pr et "f ive cop ies of the 4-se gment " a s the fr act ion,

5/4 ;

• in ter pr et " four copie s of the 5 -se gment" as the fr action,

4/5 ;

• then w e find tha t 4/3 · 3/4 = 1 = 5/3 · 3/5 = 5 /4 · 4/5 = 1.

The ari thmet ical equ iva lence of the fr act ional pr oductscor respond s to the geomet ric commen sur abl y of the ir

cor r espond ing e xten sion s . Then geometri c com mensurb il ty of

se gment s cor respond s to r epr esenta tion of thes e se gments

as a fr action .

However, the "pu zz le of the Squar e" i s tha t the r e la t ion of a

s ide of the Squar e to it s d ia gonal cannot be repr esented by a

fr action .

In Euc lid 's E lem ent s of G eom etr y appea rs a pr oof (a ppa rentl y

due to Hipp ias ) th at t he dia gonal of a un it su qar e is not a

fr action . The cr it ical not ion, in t he proof , i s tha t ever y

fraction can be reduced so tha t both numer ator and

denomin ator ar e not even number s, otherw is e t he com mon

factor of tw o can be div ided out . (Remember ! An even na tur al

number ha s the f or m, 2n , for some n atur al number n, and itssquar e has the for m, (2n) 2 = 4n 2= 2(2n 2). Simi la rly, an od d

na tur al number has the for m, 2n + 1, and its squar e has the

form, (2n + 1) 2 = 4n 2 + 4n + 1 = 2(n 2 + n) + 1 .)

The inco mmensur ab le pr oof pr oceed s a s fol lows :

• Con side r a/b = √2 .

• Then a = √2b

• Squari ng both side s: a22b 2.

• The right-hand side has the for m of an even number

(twi ce some number) , mean ing t hat the l eft -hand numbe r,

a, is an even number .

• To denote i t as an even nu mber , y ou w rite a ≡ c, for some

natur al number c.

• Then you h ave (2c) 2 = 4c 2 = 2b 2.



• Dividing out the common factor of tw o , you have: 2c 2 = b2.

• You notice tha t the left -hand s ide has the for m of an even

number (twice so me number) , meaning tha t the squar ed-

number on the right-hand side is an even number . But we

sa w a bove t hat onl y an even number has an even s quar e ,

hence, number b must be an even number .• We no w h ave the r esul t t hat, i f ther e is a fr action , a/b

su c h th a t a/b = √2 , then i t mu st have the pecul iar for m

tha t numer ator and denom ina tor ar e both even and

cannot be r educed . Ther e i s no s uch number . T he

contr adict ion ne ga te s t he a ss um pt ion th a t the s quar e

root of tw o is a f r action .

• Hence , the dia gonal of a un it sq uar e is i ncommen sur ab le

with e ither of i ts s ide s .

You real ize th at t hi s pr oblem -- " the f ir st cr is is in t he

founda tions of mathem atics " - - when so lved v ia r ea l nu m ber s ,

made po ss ible the d if fer entia l and in te gr al calcu lus and

moder n mec han ics w hose appl ica t ion render ed human s laver y

no l onger co st ef f ic ient . You f ind On li ne th at c hi ldr en can

lear n of th is mask ed in a stor y of a Cand y Mi se r and Ge lves --

in Goog le("cand yfr ont+jonhay s") .

Eudo xus of Cnidu s (cited above) developed a theor y of

pr opor tion s (in Book III of Euc lid 's E lem ent s of G eom etr y )

whic h per mitted ir rat ional number s s uch as the squar e root

of two. T he " axio m of continu ity " of Eudo xu s i ndi cated tha t,

given the propor ti on of t wo ma gnitude , you can a lso gi ve tha t

mul ti ple of one as a mul lt ip le of the other , ensur ing tha t

these magnitudes are commen sur ab le . In par ticu lar , th isaxio m of Eudo xus al lows the pr opor tion betw een t wo spher es

to be compar ed w ith two cub ical str uctur es er ected on the

dia meter of eac h s pher e.

The great Ger man mathemti cian , Richar d D edekind (1845-

1916), refor mulated (in 1872) the idea of Eudo xu s a s the

soca ll ed " Dedekind cut" :



• A cut s epar a te s the r a tiona l nu mber s into two c las se s,

"lo wer" and " upper", s uc h tha t e ver y number of the l ower

c las s i s l es s t han ever y nu mber i n the upper c las s .

• Now, i f a repr esent a ti ve of the l ower c la ss can be

for mu la ted in a fr actiona l r e la tion to a pr esenta t iv e of

the opper cla ss , then the cut its elf is ra tional .

• Ot herwi se , the cut is i r rat iona l .

Dedek ind ther eby fr eed thi s d is ti nction fr om geometr y .

For our present case, w e can as sign to the upper clas s a ll

number s w hose sq uar es exceed tw o, and to the lo wer c la ss

al l number s whose sq uar e ar e l es s t han two . So, the cut i s

the s quar e r oot of t wo.

You can sho w th is b y cons ider ing, to seven digi ts , the

appr oxima tion of the squar e root oft wo:

• (1) 2 = 1 < 2 < 22 = 4;

• (1.4) 2 = 1.96 < 2 < (1.5) 2 = 2 .25 ;

• (1.41) 2 = 1.9881 < 2 < (1.42) 2 = 2.0264 ;

• (1.414) 2 = 1,999396 < 2 < (1.415) 2 = 2.002225 ;

• (1.4142) 2 = 1.9999616 4 < 2 < (1 .4143) 2 = 2 ,00024449 ;

• (1.41421) 2 = 1.9999899 924 < 2 < (1 .41422) 2 =

2,0000182084 ;

• (1.414213) 2 = 1. 9999984 09369 < 2 < (1 .414214 )2 =

2,000001237796 ;

• etc.

You notice tha t

• as you augment the a ppr oxim ation b y one digi t,• it s squar e a ppr oac hes clo ser to two,

• whi le i ts exceeder dim ini she s (antiton ical ly!) down

towar d t wo,

• and you (Ant iton ical ly!) appr oac h the s quar e root of t wo

as the cut.



You lear n tha t the Eudo xian theor y o f p r opor t ions moti vated

ancient Gr eek mathema tic ians to abaudon the d iscontinuou s

or d iscr ete s tr uctur es of a rith metic for the continuou s

st ructur es of geometr y to descri be rela tions betw een

se gment s and such. And, s ince t ime was con sider ed

continuou s , it was also separ ated fr om arithmet ic. You r ea li ze

thi s meant tha t concept s of dyna mica l mec han ics , su ch a sspeed , vel ocit y, acceler at ion, f or ce , etc ., cou ld not be defined

in ter ms of ar ith metic .

You see tha t even T he Fundamental T heor em of A ri thmet ic i s

pr oven by tw o pr oof-b y- contr adict ion ar gument s .

This book onl y pr oceed s con str uct iv el y (cr oss ing no

Boundar ies) .

You kn ow the s cimath i mpor tance of re la t ions , funct ions ,

oper a tion s . You l ear n how to explic ate re lation s, function s,

oper a tion s :

• defin iti on of Relation : number of refer ents of a refer ence .

• gi ven R elation refer ence R, Relation s ar e c la ssi fied by

number of ref er ents coor din ated w ith R. Thus:

o Ru referents a unar y Rel ation o r a ttr ibute, as in "red" ;

o Ruv (uRv) r efer ent s a b inar y Rel ation, a s "ne xt to ";

o Ruvw refer ents a ter nar y R ela tion , as in mar ria ge

cer emony with min iste r R br ide, g room

o Relation s ar e man yÞ R Þm an y , as in many ob ser v e

many

o Relation s ar e man yÞ R Þ one , as in man y voter s

elect ing one of ficia l o Relation s are oneÞ RÞmany , as i n one per son tak ing

censu s of man y people

o Relation s are oneÞ RÞone , as in s pous e i n

m onogam ous s oci ety

o The scope (r ange ) of a Relation input is its Doma in

(an In side)



o The scope of a R e la tion output is its Codoma in (an

Ins ide)

o Doma in of a R elation can d if fer i n type fr om

Codoma in ( Boundar y cr ossed betw een them) .

• Def init ion of Function : a R ela t ion tha t is man y-one or one -

one .

• Doma in of a Funct ion can dif fer in t ype fr om i ts Codoma in(Boundar y cr ossed) , as in the Inventor y Funct ion w hose

Doma in compri se s number s, but C odomai n compr ises

war eho use i tems

• An Oper ation is a one-one Function w hose Doma in is

sa me t ype as i ts Codoma in (no Boundar y cr oss ing)

You kn ow your const ructi ve tool s in this book are funct ions

(wi th perha ps onl y domai n B oundar y cr os si ng) and oper ation s

(no Boundar y cr os si ng) , whic h ar e e xten siona l (no cr ossing of

rele vant B oundarie s), contr ar y to the (Boundar y cr oss ing)

in tens ional Ax iom s of S tanda rd Ma them ati cs . ( Topolog y i n

Relation s! )

A ques tion ari se s. Given the man y-many or one -many

in tens ional -noncon str uct iv e -too ls of Standar d Ar ith metic ,

compounded by a ppeal to the s ame amb igui tie s in A xio ms ,

and a ll the Boundar y-cr oss ing . Does th is, a t lea st in par t,

explain the dif ff icult ie s student s di sp lay i n lea rning

Ar ithme tic ? Dif ficul tie s, many or most of which cou ld be

al leviated b y Medodolog ies which ar e extensi onal ,

const ructi ve , b ypa ss ing , m ini mal boundar y cr oss ing ,

syntact ic , pr agma tica ll y ind iv idual is ti c, pr esenta ble b y ch il d-

friend ly pr og ram ming ?



CHA PTE R 3 : WHAT IS S EMIO TICS ? PEIR CE 'S THEORY OF

SIGNS?

You kn ow fr om the Web tha t Cha rle s Saunder s Peir ce

(pr onou nced " pur se" , 1839-1914) i s Amer ica' s

greate st philo sopher , America 's greates t 19th

centur y mathema tic ian and log ician, the planner ofThe Bur eau of Standar ds , and the cr eator of

Sem iot ics : Theor y of Si gns . (You kno w tha t text s and

reference books , ci ted on the W eb, often w ri te mor e

about the independent s emiot ics i dea s of Frenc h

schola r, Ferdinand Saus sur e (1857-1913). You

under stand t hat, s ince P eir ce's wor ks a gener a t iv e

bas is , th is can be bui lt upon. Bit Sau ss ur e's i s quas i-

axio ma t ic , which doe s not a llo w bu ild ing.

You can under stand using the acr onym, "ISIS" to

enca psul ate what m ay be the f our pr imar y sign s

Peir ce t aught us: I(ndic ator)S( ignal)I(con)S(ymbo l) .

You lear n an I NDI CATOR IS A N ORDER ED PAIR O F

SIG NS.

THE FIRST SIGN IS HIGHLY VISI BLE, LOW IN

INF ORMATIO N CONTENT; T HE SECOND SIG N IS LOW

IN VISI BILITY , HIGH IN INFO RMA TION CONTENT.

(Exampl es of INDI CATOR: LI GHTNI NG I NDI CATING

THUND ER STORM -- LITMU S PAP ER TURNING RED,

INDICA TING LI QUI D A CID IN TEST TUBE .)

YOU lear n a SIG NAL is AN INDICA TOR UND ERPHYSICAL AND LIN GUISTIC CONTROL.

(Exampl e of S IGNAL: TELEG RA PHI C SI GNAL - - under

PHYSICAL CONTROL OF EL ECT RIC CIR CUIT AND

BREAKER K EY; under LING UISTI C C ONT ROL OF

MORS E C ODE. )



You can under stand THE INDICA TOR-SIG NAL

STRATEGY: THE GOAL O F SCIE NCE IS T O SEARCH F OR

SIG NALS AND TRY TO TRANSF ORM I NTO SI GNALS!

You unde rstand th is in a Table of Indica tor s and

SIG N ALS .

INDICATOR-SIGNAL TABLE

IND. 1st

COMP.

IND. 2nd

COMP.

SIGNAL PHYS.

CONTROL

SIGNAL LING.

CONTROLCOMMENT

Lightning Thunderstorm Telegraph Key Morse Code

Telegraphy first

important applicat

of electricity

Symptom Disease Diagnostic tests Instructions

Medievalists used

"semiosis" for"symptom"

Fever Infection Diagnostic tests InstructionsFever formerly

miscalled "disease

Red litmus

paperAcidic liquid Chemical tests Instructions --

AS INDICA TOR S- INT O- SIG NALS .

You know, fr om your computer' s De sktop , tha t an

ICO N IS A SIGN T HAT S UGGEST S O R INVOKE S ITS

MEANI NG O R REFER ENCE.

(Exampl es of ICONS : Trash can f or de leted fi les . Also,

"No S moking !" ICONED a s CIG AR ETTE WITH SLASH

THROUGH IT ; O NOMATOPOEIC WORDS; etc.)

You lear n A SYMB OL IS A SI GN WIT H A RBITR ARILIY

ASSIG NE D M EANIN G OR REF ERE NCE.

(An ICO N is H IGHLY VISIBLE , but with LIMITED

MEANI NG OR REFER ENCE. A SYM BOL ha s L OW

'VI SIBILITY" , BUT UNLIMITE D I NFO RMA TIO N

CONTENT.)



You can under stand THE INDICA TOR-SIG NAL

STRATEGY: THE GOAL O F ED UCATIO N IS TO SEAR CH

FOR I CONS AND TRY TO USE T HE m AS BRID GES TO

SYMBO LS! (Y ou can under stand why th is was the

pr actice of the noted adu lt l iter acy t eac her , Dr. Frank

Laubac h, as de scr ibed i n APPEN DIX A.) You kno w,

from the Web, tha t SYNTACTICS IS T HE STUDY OFRELA TIO N BETWEE N SIG NS WI THOUT THEIR

REF ERE NTS .

You kn ow, fr om the Web, tha t S EMA NTI CS IS

SYNTACTI CS WIT H T HE SI GN REFE RENTS .

You know, fr om the Web (w iki ), tha t P eir ce concei ved

PRA GMA TICS AS SEMA NTIC S WI TH THE SIGN USERS .

You real ize th is means a fr equent er ror is: "T he ir

di sa greement was a dif fer ence of seman tic s. " You

kn ow it i s dif fer ence of pr agma tic s.

You can under stand tha t the neglect or confus ion

about Pr agma tics over looks a powerful us e of i t,

wher eby the one-m an y relati on of C ha p. Two becomes

one-one . You real ize thi s mean s tha t Semant ics

al lows one sign to h ave many r efer ent s. Given the

sign ∞, the referent to the mathe matician is " inf ini ty".

To the meteor ologi st , "haz e". To the ca ttle ranc he r,

"the la zy e ight cattle brand". One sign with thr ee

refer ents .

Thus, the Semant ic for mat,

[sign, reference]⇒[referent(s)]

her ein becomes one -thr ee :

[∞, reference]⇒[infinity, haze. cattle brand]



But Pr agma tic for mat r educe s t hi s to one-one

Rela tion s:

•

• [∞, mathematician] ⇒ [infinity]•

• [∞. meteorologist] ⇒ [haze]•

• [∞, cattle rancher] ⇒ [cattle brand]

(ONE -M ANY R ELA TION S TRAN SFO RME D T O SE VER AL

ONE-ONE RELA TIO NS BY PRAGMATIC S. FROM

COLLEC TIVISM TO INDIVI DUALITY !)

SEQU ENCE -PAR ALLEL LE ARNIN G SYNTACTI CS

(RELA TIN G SIGN S)

SemSign SEQUENCE 1000000000 ⇓ ⇓ ⇓

SemSign SEQUENCe abcdefghij ⇓ ⇓ ⇓

SemSign SEQUENCE 1z2y3x4w5u ⇓ ⇓ ⇓

SemSign SEQUENCE !@#$%^&*() ⇓ ⇓ ⇓

SemSign SEQUENCE ,./?:;"'{[

SEQU ENCE-PAR ALLEL LE ARNIN G OF SEMA NTI CS

(inc lud ing REFE RE NTS)

@ MEANS at-sign

⇓ ⇓ ⇓$ MEANS dollar

⇓ ⇓ ⇓& MEANS ampersand

⇓ ⇓ ⇓% MEANS percent

⇓ ⇓ ⇓* MEANS asterisk



And pr agm atic is shown above.

SEMIO TIC S PACE (HOMOLOG Y-- GROUPABLE :

UNGROU PABLE: : FINITE: INFINI TE)

• SY NTACTI CS (r ela tion s betw een SIG NS onl y)

• SEM ANTICS (SI GN S a s RELA TOR S of REF ERE NC Eto R EFE REN TS)

• PRA GMATICS (al so RELA TE t o SI GN -USER)

• TR ANSACTI CS (al so o ther G ROUPABLE SIGN -

USERS)

• CONSENSIC S (r ela ting w ith UNGROUPABLE SIG N-

USERS)

LEVELS OF LITE RACY A ND ILLITERA CY

• SY NTACTI C LIT ERA CY/ILLITE RACY

o LITERA CY: "I kno w my A-B-C 's and my

numer als! And I can r ead and wr ite them !"

o ILLITERA CY: "I don't kn ow my A -B-C's and

numer als. "

• FORMALI C LITE RACY/LITE RACYo LITERA CY: Pr of icient in count ing,

rudimentar y calcu la tion

o ILLITERA CY: Incompetent.

• SEM ANTIC LITERA CY /ILLITERA CY

o LITERA CY: "I can u se my A -B-C's and

numer als to r ead and w rite them and

calcu late -- as words and number s, t oo! "

o ILLITERA CY: "I can read and wr ite my A -B-C' sand numer als. But I don't unde rstand w or ds

and number s. "

• PRA GMA TIC LIT ERA CY/ILLITE RACY

o LITERA CY: "I kno w how what I r ead affect s

me in societ y!"



o ILLITERA CY: "I wi sh I knew how what I r ead

affect s me! "

o (Ho w does PRA GMA TIC LITERA CY/ILLITE RACY

relate to sel f-e steem ?)

• TR ANSACTI C LIT ERA CY/ILLITE RACY

o LITERA CY: I under stand ho w l iter atur e

relates to other sign -us er s.o ILLITERA CY: Doe s l iter atur e real ly rela te to

other s ign- user s?

o (W hat ha s TR ANSACTI C

LITERA CY /ILLITERA CY to do wi th gett ing

along wi th other s? )

• CONSENSIC -C HOIC E-D ECISI ON

LITERA CY /ILLITERA CY

o LITERA CY: Mak es for DEMO CRAACY.

o I gue ss DEMOCRACY tak es car e of it sel f.

• EX PLA NATORY C ONS ENSUS

LITERA CY /ILLITERA CY

o LITERA CY: Con sens us on THEORY and

EXPERIM ENT i s O NLY "CERTAINT Y" for

SCIE NTI STS .

o

ILLITERA CY: I gue ss SCIE NCE can tak e car eof it sel f.

o (W hat ha s EXPLAN ATORY

LITERA CY /ILLITERA CY to do wi th S TATE OF

SCIE NCE IN SCI EN CE? )

• ONTI C-E PIST EMIC LITERA CY/ILLITE RACY

o LITERA CY: Kno ws enough of ONTOLOGY

(w hat i s real) and EPIST OMEL OGY (HOW W E

KNOW) to ne gotia te con sensu s wi th o ther s.o ILLITERA CY: Doe sn't kno w enough a bout

ONTOLOGY and E PISTE MOLOGY to ne gotia te

consen sus w ith other s

o (W hat ha s ONTI C-E PISTE MIC- CONSENSIC

LITERA CY /ILLITERA CY to do wi th

democr acy?)

• SCIE NC E- CONSE NSIC (IL)LITERA CY



o LITERA CY: Kno ws enough sc ience to

ne got ia te consen su s wi th other s b y

explana tion.

o ILLITERA CY: Cannot do th is.



CH APTER 4: W HAT IS M ETALA NGUAGE? ONTOLOGY?

EPIS TEMO LOGY? A XIOLOGY?

You kn ow, fr om the Web, tha t M ETALA NGUAGE IS LAN GUAGE

TALKI NG ABOU T LA NGUAGE .

You know, fr om the Web, the thr ee s ubs y ste m s o f

M E T A L A N G U A G E ar e:

1. ONTOLOGY: The expl ica tion of WHAT IS REAL;

2. EPIS TEMO LGY: The expl ica tion of WHAT WE CAN K NOW;

3. AXI OLOGY: T he e xpl ica tion of WHAT IS OF VALUE.

You lear n on the Web tha t twent ieth centur y Phy sics

wi tnes sed a cr it ica l d is ti nction betw een O NTOLOGY and

EPIS TEMO LOGY, involving a de bate betw een Alber t Ein ste in

and Neil Bohr concer ned the use of P ROBABILITY TH EORY i n

quantum t heor y. That, by Say ing , "God doe s not place dice! ",

Einstein ar gued tha t the use of PROBABILITY in QUANTUM

THEORY was on ly due t o C URRE NT LIMI TATIO NS I N O UR

KNOWLED GE, perha ps due to HIDDE N CAUSAL V ARIABL ES i n

bas ic phenomena, and. the se i dent ified , PROBABILITIES could

be e li mi nated . H ence, you real ize, E inste in was sa ying tha t

QUANT UM PROBABILITIE S A RE EPISTEMIC , w hil e B ohr , incounter argument , s aid Q UAN TUM PROBABILITIES ar e

ONTOLOGICAL , because any attempt to investiga te this i n a

syste m w ould destr oy the s ystem or change i t cri tica ll y.

You know, fr om the Web, tha t, toda y, most phys icis ts a gree

tha t Eins tein was i ncor rect and tha t B ohr w as cor rect, for the

fol lowi ng r eason.

You know, fr om the Web, tha t E inste in f or mulated a " thought-

exp er iment " w hich wou ld i nvolve cr ea ti on and separ at ion of

"tw in" phenomena s uc h tha t c hanging the S PIN of one

changes the SPI N of the o ther , no ma tter ho w far aw ay it i s .

Ar guing tha t th is is a con sequence of the ONTOLOGICAL

inter preta tion, and tha t it cannot be t rue, E inste in thought he

had won the debate. You know, fr om the Web, thi s "thought



exp er iment " ha s been tr ans for med into actua l e xperi ments

confir ming occur rence of w hat Eins tei n thought was

impos sibl e!

You can under stand tha t the di st inct ion betw een O NTOLOGY

and EPISTEM OLOGY ma y concer n your med ical i nsur ance,

since it 's now pos si ble, fr om DNA evidence & other protocol s,to kno w so me per sons ar e genet ical ly d is posed to so me

disea se o r i mpa ir ment . You l ear n tha t, kno wing t he DNA

tes ts , some in sur ance copan ies have cance lled the m edica l

insur ance of s ome client s and refused to ensur e other

appli cants , upon lear ning of s uch di spo si ti ons .

However, you can under stand t hat the s tati st ic s for in sur ance

ar e NOT based upon what i s kn own about cer ta in peop le, but

about "w hat is out ther e" , in the st atis tica l uni ver se .

So , you r ea li ze tha t compan ies do ing t hi s ar e, ther eby, usi ng

A DOUBLE STANDARD: co llec ting data ONTOLOGICALL Y, but

deter mining ELI GIBILITY for insur ance E PIST EMICALL Y. You

real ize th at, i f cour ts cou ld be made to under stand thi s, su ch

action s by i nsur ance companie s mi ght be reversed . Thus , you

real ize th at di st inct ion betw een O NTOLOGY and

EPIS TEMO LOGY see m to have econom ic sign ificance ineverday life.

HY PERNYM: SEM ANTIC- ONTI C R ELA TION OF RANGING OVE R

HYPONYMS

Some HYPONYMS of t he HYPERNYM R ED ar e scar let ,

car m ine, ver mi l ion , cr im son , rose ate, b lush ing .

Some HYPONYMS of t he HYPERNYM r egularP OLYGOn ar e

tr iang le , s quar e, penta gon, hepta gon, hexa gon, nona go n .Some HYPONYMS of t he HYPERNYM of S URFACE ar e soft ,

har d, sm ooth, rough, w et, dr y .

Some HYPONYMS of HYPE RNYM of TASTE ar e sweet, s our ,

sa lt y, b land , b itte r, s auc y .

You can under stand tha t the role of AXIO LOGY i s exempl ifi ed

in the RULE : DO NOT BASE A DECISI ON ON TH E P ROBABILITY

OF A PROTOTYPICAL EVEN T, BUT U PON THE PRODUCT OF T HE



PROBABILITY WITH THE VALUE OR COST OF T HE E VENT IF IT

OC CUR S. Exa mple : In a fai r l otter y, sel ling 1000 lo tter y

tickets , the prob abil it y of wi nning is 1/000 . Suppose the pri ze

is $500 , then the EXPEC TATIO N is (1/1000)·500 = 50 cent s.

Then you r ea lize tha t the ti cket price should be m or e than 50

cents , "to mak e an y mone y" .

You read on the Web tha t, during the "V ietnam Era", Def ense

Secr etar y Rober t Mac Namar a (w ho had been Head og Gener al

Motor s) vio lated this RULE in ar guing tha t South Vietnam

ARVAN soldier s w oul d def eat the Vietcong f omr No rth

Bi etnam , since ther e wer e ten ti me s as man y AR VAN sold ier s

as Vietcong . That s ome one noted , "W hat ha ppen s i f tho se 10

ARVAN soldier s w on't fight, and tha t one V ietcong fight s like

hel l? "

So you r ea li ze tha t Meta li ngui st ic dist inct ions , su ch as

AXI OLOGY, i nvolve National De fense and our sold ier s " put i n

har m' s way".

You also Web-l ear n tha t AXIO LOGY is often ignor ed, but i s

implicit in reference to AXIO MATICS , since an ax iom is

langua ge of spec ial value.

• you r ea li ze ONTOLOGY (S tudy of "Rea li ty ") shou ld be

for mu lated in the SUBJ UNCTI VE MOOD, because we never

know tha t another descr ipt ion m ay better fit -- another

hypothe si s m ay yield as much or m or e in conf ir med

pr edict ions than present ly accepted one. So , we must

sa y, "If th is wer e the ca se about REALITY , then .... "

•

you r ea li ze th at E PISTE MOLOGY (S tudy of Knowing) m aybe i n the D ECL ARA TIVE mood, since we m ay assume we

kn ow what we kno w about te chnica l s ub jects . (T his may

have to be c hanged .)

• You real ize A XIO LOGY (Study of Value) shou ld be i n

INTER ROGATIVE MOOD since we can onl y a sk what

another per son val ues .



You Web-l ear n tha t mathema tic ian-l og ician -ph il lo sopher ,

Ber tand Russe ll (1872 -1970) noted th at i f he suf fer ed fr om a

toothac he , he would not e xpect other s per sons to sen se the

pain (acr oss a B oundar y) . B ut if he hear d a loud noi se , he

wou ld e xpect t hose other s to hear i t (no s oni c B oundar y in

betw een),

SE QUENCE -PARALLEL LE AR NIN G OF ONTOLOGY

thundering MEANS thunderstorm ⇓ ⇓ ⇓

high winds MEANS windstorm ⇓ ⇓ ⇓

loud sound MEANS explosion ⇓ ⇓ ⇓

water rising MEANS flooding

⇓ ⇓ ⇓moon rising MEANS nightfall

SEQUENC E- PARALLEL LEA RNI NG O F EPIS TEMO LOGY

summons MEANS court appearance ⇓ ⇓ ⇓

hisiren MEANS fire alarm ⇓ ⇓ ⇓

phone bell MEANS phone caller ⇓ ⇓ ⇓

store bill MEANS payment due ⇓ ⇓ ⇓

door-bell MEANS visitorS- P LEA RNI NG AXIO LOGY

receipt MEANS payment ⇓ ⇓ ⇓

bankloan MEANS indebtedness ⇓ ⇓ ⇓

bankcheck MEANS cash return ⇓ ⇓ ⇓

statebond MEANS investment ⇓ ⇓ ⇓

I. O. U. MEANS indebtnessYou real ize th at A great pr oble m w hich str ad dles both

On tolog y and Epi ste molog y i s tha t of "so li pci sm", the cla im



tha t since my mi nd is the th ing I kno w to exis t, then perha ps

you and ever th ing an ima te and al l of "r ea li ty " ar e mer el y

cr ea ti ons of m y ima gin ati on . You Web-lea rn So li pci sm can' t

be pr oven, but also can't be pr oven -- s howing the l imits

(constr aints) of Langua ge and Metalangua ge,

You web-l ear n tha t the B ri ti sh poet , A . E . Ho us man (1859-1936), famou s f or h is 1896 book of poetr y, "A Shr opshir e

Lad", wr ote a poem a bout So li pci sm :

Good folk, do you love your life?And have you need for sense?Observe this knife. Like other knivesIt cost but fifteen pence.I need but draw it across my throat,

And down will fall the sky!And Earth's foundations will depart!And all you folk will die!

You Web-l ear n tha t a cri tica l ONTOLOGI CAL pos sibil it y tha t

"cr ies out " f or the SU BJUN CTI VE mood i s tha t, in WAVE

THEORY (a bas is theor y of Q UANTICS) , we do not kno w th at

the comp le x equa t ion model s waves , or the qua ter nion

equa ti o n m ode ls w a v e s , or the octonion equa t ion m odel s

w a v e s , or some hyper comp le x equa t ion be yond th is m odel s

w a v e s - - or i t tak es a l l po ss ib le hyper comp le x equa t ions to

m odel w a v es , to pr ovide the best fit.

You real ize i t is somet mes dif ficult to know what we real ly do

kn ow. Y ou can under stand t hi s ques tion is pr ovoked by t he

discu ss ion on LA TEN CY in the book , " The Natur e of Phy sica l

Rea li ty, A Phi los ophy of Moder n Phys ic s" , pp . 171-6 , by Henr y

Mar genau.

When you see a b lue flo wer, Mar genau a sk s if the f lower is

blue when no one is l ooking at it. He sa ys thi s co lor ati on i s

not po ss es sed by the flower (attr ibuti ve ) but la tent in the

flo wer unt il evok ed b y an ob ser ver . You r ea li ze th is i s

compar able to the old ques tion , "W hen a tr ee f all s i n a f or est

wi th no human ar ound, doe s it make a s ound ?" It m ay mak e a



vibr ation , but "sound" is w hat tha t v ibration become s within a

nor mal hu man ear .

You real ize one may a ls o co mpar e th is to the tr ansducer tha t

changes the "cont inuous " ener g y of a telephone l ine to the

"di scr ete " ener gy of a computer i n a mode m (acr onym for

"modula te dem odul ate"); or vice ver sa. O ther eva lua tor s i nthe l iter atur e ar e "d is pos it ion s" and " pr opensi ti es ".

Latenc ies , di spo si ti ons , pr opens iti es Usual ly, onl y r hetorica l-

phi los ophica l def ense s appe ar in the liter atur e and on li ne.

With the development of The Web and a ss oci ated s ear ch

engine s and develop ment of Ar tifi cia l Intel li gence, another

meaning of "ontolog y" has become w ide spr ead. To a void

langua ge a buse, perha ps it s hou ld be named " A TYPESYSTEM ".

Wikipedia sa ys of i t: "In compute r s cience and i nfor mation

sc ience , an ontolog y i s a f ormal r epr esenta tion of a s et of

concepts wi thi n a domai n [of d is cour se] and the rela tion shi ps

betw een thos e concept s. It is used to reason a bout the

pr oper tie s of tha t doma in , and ma y be used to def ine the

domain .

Examp le of an onto log y.

OWL ONTOLOGY FOR WORLD WIDE WEB

The data descr ibed by an OWL ontolog y i s i nter pr eted as a

se t of "i ndi vi dual s" and a set of "pr oper ty asser tion s" which

relate the se ind iv idual s to ea ch other . An OWL ontolog y

cons is ts of a s et of ax iom s whic h place cons tr aint s on s et s

of in di vi dual s (cal led "c la sses") and the t ype s of

relation sh ips per mi tted betw een the m. These axio ms pr ovide

se mantic s by a llo wing syste ms to in fer ad ditiona l in for mation

based on the data expl ici tly provided. For e xamp le, an

ontolog y de scr ibing famil ies might inc lude ax ioms s tating

tha t a "has Mother" pr oper ty i s on ly pr esent betw een two

ind iv idual s when "ha sPar ent " i s al so present , and indi vi dual s



of cla ss "Ha sTypeO Blood " ar e never r ela ted via "ha sPar ent"

to m ember s of the "Has Type ABB lood" cla ss . If i t is stated

tha t the ind iv idua l Ha rrie t i s rela ted via " hasMo ther" to the

ind ividual Sue, and tha t Harriet is a m ember of the

"HasTypeO Bl ood" clas s, then i t can be in fer red th at Sue i s

not a me mber of " HasTypeAB Blood".

In theor y, an ontolog y i s a "for ma l, e xpl ici t spec ific ation of a

shar ed conceptual is ation" . A n onto log y pr ovide s a shar ed

voca bular y, w hich can be u sed to m odel a domain — tha t i s,

the t ype of objects and/or concept s th at exis t, and their

pr oper tie s and r ela tions .

On tolog ies ar e used in ar tificia l in tel li gence, the S emanti c

Web, s oftw ar e eng ineering , biomed ical i nfor matics , libr ar ysc ience , and infor mation ar chi tectur e as a for m of kn owledge

repr esenta tion about the world or some par t of it .

FORMAL ONTOLOGY EXPLICA TES ARTIFICIAL INTELLIG ENCE

(AI) " REALITY" via C ONSI STE NC Y C ONT ROL OVER DOMAIN S

AN D SUDOMAIN S O F SUBO NTOLyGIES AND A M ODEL FOR

THEIR DEVE LOPMEN T.

TWO EXPLICA TIVE CONCEPTS OF FORM AL O NTOLOGY AREENDURA NT (a s in an a pp le), WHICH CA N BE PHOTOG RAPHED

and PERDU RANT (as in a pr oces s) , WHI CH CANNOT BE

PHOTOGRAPHED

Some in stance s of the ENDURA NT LOGIN E ar e pear , human,

dog, car , house , sc hool , st adiu m .

Some in stance s of the PER DURANT LOG INE ar e month , t ria l,

se ssi on. seme ste r , examin at ion, va ca t ion, hea ling .Another development ser ving the user s of "onto log y as

kn owledge r epr esention " i s sear ch-pr ocedur es for

"di sa mbigu ation ", fall ing under E pist molog y. W ikiped ia s ays

of th is subj ect: "Disa mbigu ation in W ikiped ia i s the pr oce ss

of resol vi ng conf lic ts in W ikiped ia ar tic le t it les tha t occur

when a s ingle ter m can be associ ated w ith mor e than one

topic , mak ing th at t er m likely to be the natur al ti tl e f or m or e



than one ar tic le. In other w or ds , di sa mbigua tion s ar e paths

lead ing t o d if fer ent ar tic le s whic h could , in princip le , have

the s ame tit le.

"For e xamp le, the word "Mer cur y" can refer to s ever al

dif fer ent t hing s, i nclud ing an ele ment, a planet , an

automobi le brand, a r ecor d l abel , a NASA manned -spacef lightpr oject , a p lant, and a R oman god . S ince onl y one Wikipedia

pa ge can ha ve t he gener ic name "Mer cur y" , unam biguou s

ar tic le ti tle s ar e u sed f or ea ch of t hese topi cs : Mer cur y

(element) , Mer cur y (planet) , Mer cur y (automobi le) , Mer cur y

Recor ds, Pr oject Mer cur y, Mer cur y (plant) , M er cur y

(my tholog y). Ther e must then be a w ay to d irect the r eader to

the cor rect specif ic ar tic le w hen an amb iguous ter m i s

refer enced by l inking , br owsi ng or se ar ching ; thi s is what isknown as d is ambi gua ti on . In thi s case it is achieved u sing

Mer cur y as a di samb igua t ion pa ge .

"Two method s of di samb igua ting ar e dis cus sed her e:

• di samb igua t ion l ink s – at the top of an ar tic le ( ha tnotes ),

tha t refer the r eader to other W ikiped ia ar tic les w ith

similar ti tl es or concepts .• di samb igua tion pages – non-ar ticle pa ges tha t refer

reade rs t o other Wik ipedia ar ticles. "



CHAPTER FIVE : W HAT ARE P ATTE RNS I N LIFE? IN S CIE NCE ?

IN M ATHEMA TICS?

You can under stand tha t "p atter ns " ar e what you mus t look

for your wor k or resear ch and in your dai ly l ife.

You can under stand the F renc h anthr opolog is t, Claude Lév y-

Str auss , when he s aid he found , the wor ld o ver, a mong so-cal ll ed pr im it ives and a mong the ci vi lized , the com mon

year ni ng to fee l th at thei r l ives made s ens e - - fit a pa tter n . He

cites an e xamp le of lo v e of p a tter n .

You can under stand hi s instance of an aborigi ne, tr acking

game acr oss the Kalahar i De ser t, wearing onl y a lo inc loth,

ar med wi th onl y a wooden s pea r, who m ight rest from the

noonday sun in the shado w of a rock. T hen from his loinc lothmight pu ll an embr oider y hoop and begin to embr oider litt le

red flowers -- someth ing he le ar ned fr om a m issionar y woman.

You found online a verse about th is need for pa tter ning .

PRAYER OF A F ECK LESS FOOL

Befor e I go to that G reat Pla yg round ,

To spend tha t Last Rece ss,

Lor d! Mak e my L ife a P atter n,

In stead of thi s misham y! m asha my ! me ss !

You found online tr ue stor ies a bout two men who made

disco ver ies a bout patter ns. You r ea li zed the di sco ver ies

became po ss ib le by the famil ia r pr act ice of expo sing a

"fi gur e" a ga inst a contr ast ing "g round" to mak e the figur e

"s tand out ", as when childr en col lect a f ew s now- cr ysta lsagains t the g round of a bla ck c lo th to mak e the ir geometr ic

patter ns "stand out".

You read tha t, an ar chaeo logi st traveled , as air pl ane

pas senge r, over ter rain he knew a t g round level. From th is

per spect ive, t he ar chaeo logi st sa w, outl ined on the ground

belo w, the buried fr agments of wall s and bu il ding s. L ater ,



dig ger s di sco ver ed ruins d ating fr om the R oman inva si on of

Br ita in (ar ound 120 AD ), duri ng E mper or H adr ian' s reign .

People had lived upon thes e r uin s for centurie s without being

awar e of them . The ar chaeolog is t r ea li zed tha t tho se people

were too close to se e bur ied p atter ns.

You also lear ned tha t, in 1839 , ho w G er man biolog is t,Theodor e S chwann, di sco ver ed (in the k ind of pa tter n -

sear ching b iol ogi st s do , s uch as d ye-s ta ini ng speci mens to

invoke patter ns) tha t the ba si c p lant and an ima l un it is the

cel l . That i t repr oduces (subd iv ide s) dur ing the pr oce ss

cal led "m ito si s" t o e xhib it subp a tter n s , highl ighted agains t

the g round of the dye-st ain . You lear ned tha t the se

subp a tter n s became la be led b y two Greek w or ds : "chr omo "

for "col ored" and "soma" for "bod y", fus ing into the wor d"chr omo some s" . (You lear ned tha t, la ter , chr omo some s wer e

ident ifi ed a s her editar y car rier s, w ith gene s of "DNA". )

You also found on line a ma thti v it y about the pa tte rn th at

number factor become a figur e a gains t the g round of a ten-

by -ten gr id to be co lor ed as a k i nd of " stain ing" .

TEN- BY-T EN G RID F OR COLOR ED NUMBE R P ATTE RNS

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19

20 21 22 23 24 25 26 27 28 29

30 31 32 33 34 35 36 37 38 9

40 41 42 43 44 45 46 47 48 49

50 51 52 53 54 55 56 57 58 59

60 61 62 63 64 65 66 67 68 6970 71 72 73 74 75 76 77 78 79

80 81 82 83 84 85 86 87 88 89

90 91 92 93 94 95 96 97 98 99

TEN-BY-TE N GRID FOR COLORE D NUMBER PATTER NS 100 -199

100 101 102 103 104 105 106 107 108 109



110 111 112 113 114 115 116 117 118 119

120 121 122 123 124 125 126 127 128 129

130 131 132 133 134 135 136 137 138 139

140 141 142 143 144 145 146 147 148 149

150 151 152 153 154 155 156 157 158 159

160 161 162 163 164 165 166 167 168 169

170 171 172 173 174 175 176 177 178 179

180 181 182 183 184 185 186 187 188 189

190 191 192 193 194 195 196 197 198 199

TEN-BY-TEN GRID FOR COLORE D NUMBER PATTER NS 1000-

1099

1000 1001 1002 1003 1004 1005 1006 1007 1008 10091110 1111 1112 1113 1114 1115 1116 1117 1118 1119

1120 1121 1122 1123 1124 1125 1126 1127 1128 1129

1130 1131 1132 1133 1134 1135 1136 1137 1138 1139

1140 1141 1142 1143 1144 1145 1146 1147 1148 1149

1150 1151 1152 1153 1154 1155 1156 1157 1158 1159

1160 1161 1162 1163 1164 1165 1166 1167 1168 1169

1170 1171 1172 1173 1174 1175 1176 1177 1178 1179

1180 1181 1182 1183 1184 1185 1186 1187 1188 1189

1190 1191 1192 1193 1194 1195 1196 1197 1198 1199

TEN- BY-TEN GRID F OR COLORED NUMBE R PATTE RN S 10000-

10099

10000 10001 10002 10003 10004 10005 10006 10007 10008 10009

10010 10011 10012 10013 10014 10015 10016 10017 10018 1001910020 10021 10022 10023 10024 10025 10026 10027 10028 10029

10030 10031 10032 10033 10034 10035 10036 10037 10038 10039

10040 10041 10042 10043 10044 10045 10046 10047 10048 10049

10050 10051 10052 10053 10054 10055 10056 10057 10058 10059

10060 10061 10062 10063 10064 10065 10066 10067 10068 10069

10070 10071 10072 10073 10074 10075 10076 10077 10078 10079









the r emainder . O bviou sl y, 25 and 16 agree i n a d igi tal root of

seven , but do not agree as nu mber s. Repea ting . thi s for m of

chec king on ly te lls when the calcu la tion (ad dit ion,

sub tr action , m ult ip li cation, div ision) i s i ncor rect , so mething

Wikipedia fai ls to tel l you . T hus , modu lar ari thmet ic i s many -

one , just as factor ar ithmet ic -- a suba rith metic of ar ith metic

is many -one , as shown in Chap. 23,

You may a ls o kno w a bout cas ting out e le ven s . (Wikipedia

doesn 't s eem to ment ion t hi s, as of 5 /19/09.)

You note the other main diagonal of the t en-b y- ten g ri d i s

from upper lef t to lower righ t, with number s 0, 11, 22 , 33, 44,

55, 66, 77 , 88 , 99 .

TEN- BY-T EN G RID F OR COLOR ED NUMBE R P ATTE RNS

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19

20 21 22 23 24 25 26 27 28 29

30 31 32 33 34 35 36 37 38 9

40 41 42 43 44 45 46 47 48 49

50 51 52 53 54 55 56 57 58 5960 61 62 63 64 65 66 67 68 69

70 71 72 73 74 75 76 77 78 79

80 81 82 83 84 85 86 87 88 89

90 91 92 93 94 95 96 97 98 99

You note tho se are all mult ipl es of e leven but don't sum to

the s ame numbe r, as in the nine s case !

You ju st lea rned someth ing a bout patter ning! That the

cov er se pa tter n need not be t he same . You f ound the nine s

patter n to be ANTIT ON IC . You not ice tha t the ele vens patter n

is ISO TONIC. Star ting fr om zer o, which can be named the

codig ita l r oot , you dr op down one r ow and shift right -- both

incr ea ses .



However. you may kno w t hat mathema tici ans use an

al ter na t ing s um tha t w or ks for thi s co ver se pa t ter n . Star ting

on the ri ght, ad d th is number ; s ubtr act the ne xt nu mber - -

al ter nating ad di tion and s ubtr act ion. You note th at an

ISO T ONI C pa tter n , when al ter na tel y su mmed , acts as an

AN TIT O NI C p a tter n .

You now al ter na tel y ad ding t he d igi ts of th a t ele ven s

dia gonal , yielding -1 + 1 = 0 ; -2 + 2 = 0; -3 + 3 = 0; etc . You

have al ter na tel y su mmed to t he cod igita l r oot of zer o .

You tes t the Algor ithm of ca sti ng out ele ven s on 16 x 11 =

176 ,

You find +1 - 7 + 6 = 0. Che cks ! An ele ven mu lt iple .

On the other hand , 175 + 5 = 181 and +1 - 8 + 1 = - 6 . Given a

ne gative cod igi tal root, you Just ad d eleven to it : -6 + 11 = 5 .

And you f ind t hat 181 di vided by ele ven yield s fi ve r ema inder .

You may kno w r esul ts s uch as t hese nine s and e le vens

Al gori thm s der ive fr om modula r ar ith metic , in which you

di vi de a ll number s by a fi xed number (la be led " modulu s") , but

keep onl y the rema inder patter n . For mu lt ipl es of t wo andfi ve, or nine and e leven, a s modul i, the a bove " invar iants " ju st

pop out . You under stand about t wos and fi ves patter ns ; the y

ar e subpa tter ns of the tens p stter n impo sed on the count ing

number s. And the s ubpa tte rns for ni nes and e levens ar is e

because they ar e the near est neighbor s of the base ten .

You wonder if ther e i s a subpa tter n for sevens in ba se ten.

Yes, but it is so co mpl ica ted tha t you are bette r of f dividi ng.

You may kno w th at these co lor ed patter ns reveal co lor ed

conser va tion la ws . Since the twos patte rn repea ts ever y ten ,

you can say the tw os pa t ter n i s conser ved by ad d ing ten .



You also find tha t the 2·2 = 4 pa tter n repea ts ever y 10 · 10 =

100 . Equi va lentl y, the four s pa tter n i s conser ved under

ad dit ion of one hundr ed .

The 2 · 2 · 2 = 8 r epea ts ever y 10 · 1o · 10 = 1000 .

Equi va lentl y, ...

And the e ight s pa tter n is con ser ved under ad dit ion of one

thousand . Etc .

You real ize th at c hi ldr en lear ning t hi s will be pr epar ed f or the

conser va tion la ws of ph ys ics - - con ser vat ion l aw s of ener gy,

of l inear mo mentum, of angular momentu m , etc .

You may kno w the se two "ca st ing" Algor ithm s car ry over to

Nu merica l Algebr a. Here, you change fr om ba se t en to base x ,so that the homo logue of 10 - 1 = 9 i n the A lgo rith m i s x - 1,

and the homologue of 10 + 1 = 11 in the Algor ithm is x + 1 .

That i s,

9: 10 :: x -1: x 11: 10: : x+1: x

cas ting -out- nine : decima l base :: ca st ing- out-(x -1): x-base

casti ng-out -ele ven: deci mal bas e: : cas ting -out- (x+1): x -ba seYou know tha t (x - 1) 2 = x 2 - 2x + 1 has factor (x - 1) by

const ructi on, And the coe f fic ient s of (x - 1) 2 = x 2 - 2x + 1 su m

as 1 - 2 + 1 = 0, ind ica ting (se miot ic ind ica to r!) it ha s a f acto r

one les s than the ba se (her e, x), co mpar able to the nines

case (one less than base ten .)

You also kn ow tha t (x + 1) 2 = x 2 + 2x + 1 has factor x + 1 by

const ructi ng. If you perf or m alter na ting su m on i ts

coe ff ic ient s fr om the ri ght , you find 1 - 2 + 1 = 0, so has

factor x + 1 , compar able to the ele ven s case .

Or con si der x2 - 1 . Its coef ficien t s um is 1 - 1 = 0 . So i t has

factor (x - 1) . Its al ter na t ing s um is - 1 - 0 + 1 = 0, so it has

factor (x + 1) . In fact, (x - 1)(x + 1) = x2 - 1.



Another Algori thm is imp licit in a ll thi s, appar entl y not taught

wi th the pr evious A lgor ith ms . It is The "Ac ti vi thm " S tr a te g y ,

meaning tha t kno wledge may co me onl y fr om fal li b le act ion

whic h s om et im e s creates pa tter ns- in -p a tter ns -i n- pa tte rns -

in -. . . , w hen you

• impose a pa tter n , P ( say , ten) upon "w hatever " ( sa y, thecounting number s) ;

• enquir e if pa tter n P ha s subp a tter n s , S (sa y, of nine s and

ele vens -- neighbor s of P.

• If so , then enquir e if subp a tter n s of S ar e conser ved

under a tr ans for ma tion gr oup G (say, adding powers of

ten);

• If so , then la be l thi s as the pa tte r n P con ser ved by gr oup

G (a s de scr ibed i n Cha p. 22) .

If your Figur e of pa tter n P fits the Gr ound of "Rea lti y" , then

you W IN ; otherw ise, so me pr eviou sl y unnoticed Si gnal of

Expecta t ion has been e xposed , which m akes you a winner .

You know tha t a f ami liar example of THE ACTIVITH M i s

impos ing thr ee d imen si ons of s pace and one of t im e upon our

"wor ld" . T hat Alber t Eins tei n (x -y) said of th is, " ... time andspace ar e m odes in w hich we thi nk and not cond iti ons in

whic h we l ive."

You know tha t R. Sear le wrote i n T h e P hi lo soph y of Langu a ge

, 1978 : "[ W]ha t counts as rea li ty . .. a s a glas s of water or a

book or a ta ble . .. i s a matter of what cate gor ies we i mpo se

on the wor ld . .. . Ou r concept of real ity is a matter of our

[l ingui st ic] cate gorie s. "

You know tha t thes e t wo comment s sh ift the per specti ve fr om

ONTOLOGY (w hat is REALITY) to EPIST EMO LOGY ( your

kn owledge) and A XIOLOGY (w hat i s of va lue) .

You know the v alue of di sco ver ing or impo si ng patter ns to

see what you can lear n ther eby.



In Su mmar y, you kno w thi s ha s taught y ou tha t pa tter ns can

reveal the con ser vat ion law s w hich ar e im pl ic it wi thin them !

Cher chez la p atter n!

You kn ow tha t subl im ina l p atter ns can al so occur i n a

per son' s dai ly l ife. A fr iend may not ice tha t thi s per son

usua ll y goes thr ough s ever al beha voria l st eps i n a repea tedsitu ati on, al though unaw ar e of it.

You kn ow tha t ty pe of beha viour was ob ser ved during a v er y

cri tica l period of indus tri al iz ation i n the tw ent ieth centur y,

The pur pose was to in stal l qual it y engineer ing on ma ss

pr oduction l ine s i n factor ies . (T his had emer ged fr om the

work of engineer , Walte r Shew har t (1891-1067), "the father of

qual it y cpntr ol" , after WWI .) A s par t of the pr ocedur e, arandomi zed sa mple of pr oduct s w as dr awn fr om t he

pr oduction -- randomi zed to a void or dinar y patter ning in

se lect ion. B ut s ome senio r eng ineer s pr ootested the "extr a"

st ep of dr awing b y a r andom ta ble , sa yi ng, the y could dr aw

randoml y. It was nece ssar y for jun ior engineer s to con vince

them of pa tter ns in the ir beha vi our b y wr iti ng do wn, ahead of

time, w hat a gi ven s eni or eng ineer wou ld do in drawing a

sa mple , then s how i t to him afterw ar ds .

Qua li ty contr ol engineer ing had a r evoluti onar y ef fect upon

the econom y.

You know tha t the great Frenc h m athem atic ian , J ules Henri

Poincaré (1854-1912), said, "Mathema tics is the a rt of gi ving

the s ame name to dif ferent th ing s. "

Con ver sel y, you might s ay tha t mathema tics is the ar t of

gi vi ng d if fer ent name s to the s ame or similar pa tter ns. You

see the la tter in A ppend ix C, " Barb ie Dol l Math". And y ou see

in vari ous forms be low.

You kn ow tha t the ter m, "figur ate geometr y" , la be ls the

"ar ithmet ic geometr y" orig ina ted by Py tha go ras of Samo s

(579-475 B C) in contr ast to the dia gramma tic geometr y of hi s



teac her, Tha les of Miletu s (624-547 BC) . You kn ow tha t the

Py tha gor ean geometr y was appar entl y figur ated fr om p ebbles

or s uch, ori gin ating the "geometr ic" langua ge of today 's

ari thmet ic - - la bel s such as "squar es", " cubes ", "t riangu lar

number s", etc . ( You kno w that th is, r ather than "the

Democr itean atom ", ma y be ba si s of "r eduction ism" in

sc ience .)

You kn ow tha t the figur ate ori gin and f igur ate explic ation

her ein of so many number concepts man ifests simp licity th at

young c hi ldr en can in ter activ ely compr ehend , a ttr acting them

to m athem ati cs . For, mathe matics i s a bout patter ns.

You kn ow tha t the Py tha gor eans f igur ated squar es of in teger s

by recur sively ad ding od d number s, i.e., nu mber s of the figur e2n + 1, n = 0 , 1, 2, . .. . Thus,

• 0 + 1 = 12;

• 1 + 3 = 4 = 2 2;

• 1 + 3 + 5 = 9 = 3 2;

• ....

In gener al , 1 + 3 + 5 + . .. + (2n - 1) = (n - 1) 2.[1] Y ou kno w that,

in the liter atur e, the od d nu mber funct ion i s defined a s O(n) =

2n + 1, a single ter m or for mula . You know, ho wever, th at 0the

tr iangular number function is wr itten both as a sum, T(n) = 1

+ 2 + 3 + ... + n, and as a sing le t er m or f or mu la, T(n) = 1 /2n(n

+ 1) . For par ity , you can def ine the odd number recur sive sum

as O(d n) = 1 + 3 + ... + (2n - 1). (You kno w thes e con st ructs

wer e pr ecur sor s of ar ith metic , geometr ic, and har monic

pr og ression s, w hich the P ytha gor eans used to con str uct thechroma tic musi cal scale of Wester n musi c.)

Then , y ou find th at, g iven n i n od d and t riangu lar funct ions ,

and in t he recur sion function , R (n) = n + 1, the se jo in wi th the

resu lt , T(n) - T(n-1) = (1 /2n(n+ 1)) - (1/2;n(n -1)) = n. Then you

can wr ite : O(n) = 2n + 1 = n + n + 1 = n + R(n) = T (n) - T(n - 1) +



R(n) . You se e th at, i n fulf il lmen t of Poincaré, you can ad joi n

the thr ee "th ings ": R(n) = O (n) + T(n -1) - T(n) .[2]

You know tha t the recu rsive f igur ation of squar es by odds is

usua ll y demon str ated geometri cal ly, via dots o r s uch, wi th

rhetori cal gener al ization , but r ar el y put i n ari thmet ic f or m.

However, you can sho w tha t, gi ven (n + 1)2

2 = n2

+ 2n + 1 = (n2

)+ (2n+1) , then , t o the sq uar e, n2, ther e is ad ded the ne xt odd

number , 2n + 1, comb ini ng for the next squar e: n2 + 2n + 1 = (n

+ 1) 2.

One may wonder as to why do od d number s adjoin thus , but

even number s do not. You kno w tha t reason i s a patter n often

discu ss ed her ein, c los ur e . The su ms of od d nu mber s for m

both od d and even number s, clos ing on a pr oper s ub set of al lnumber s . However, s ums of even nu mber s close on even

number s, bifur cat ing the s et of a ll number s b y excluding

od ds .

You know tha t cubes recur s iv el y deri ve fr om mo vi ng s um s of

od d number s . Thu s, r ecur sively for med

• 1 = 13;

• 3 + 5 = 8 = 2 3;

• 7 + 9 + 11 = 27 =3 3;

• ....

This l eads to f o rm ing cube s fr om dif f er ence s betw een

squar e s o f consecut iv e tr iangu lar nu m ber s, of the r ecur s iv e l y

for med T(n) = 1/2n(n + 1) for T(1) = 1 , T(0)= 1. And , in gener al ,

recur si vel y for med : n3 = T(n) 2 - T(n - 1) 2 = (1/2n(n + 1)) 2 -

(1/2n(n - 1)) 2.

You know tha t the d if f er ence pa tter n is pr oven by expanding

the gener al for mula : n3 = (T( n))2 - (T( n - 1 )) 2 = (1/2n(n + 1)) 2 -

(1/2n(n - 1)) 2 = (n 22 + 2n + 1) - ( n2 - 2 n + 1 )) = ( n2/4)(4n) - (n2)(n) = n3 . Thus:

•13 = 1 = T (su b 1) 2 - T (sub 0 )

2= 1 - 0 = 1 ;

•23 = 8 = T (su b 2) 2 - T ( sub 1 )2 = 32 - 1 2 = 9 - 1 = 8 ;



•33 = 2 7 = T (sub 3 )3 - T (sub 2 )2 = 62 - 32 = 36 - 9 = 2 7;

•

....

You know the ge ner at ion of cubes by odd num ber r ecu rsion and b y a tr ia n g ula r nu m be r r ec u r s io n impl ies

that one for m is p ropo r t io nal t o t he o the r , with fu r ther c onn ecti ons . You k now t his l eads t o n um be r p o w er s

a s tr ia n g ula r n u m be r s .

For, f rom T(n) = 1 /2n(n + 1 ) and T(n - 1) = 1/ 2n(n - 1 ), you find: T(n) + T (n - 1) = (1 /2n(n + 1 )) + 1 /2n(n - 1) = n2

= O(d n) . [3a]

T(n) - T (n - 1 ) = (1/2n(n + 1 )) - 1/2n(n - 1) = n . [3b]

Before n otin g th e r elation from [ 2] and [ 3], y ou c onside r a sing le r esul t f rom [3a], t o mo tiva te

gen erali zation or e xtende d r elating: T (5) = 1 + 2 + 3 + 4 + 5 ; T(4) = 1 + 2 + 3 + 4. T (5) + T (4) = 1 + ( 2 + 1) +

(3 + 2) + ( 4 + 3 ) + (5 + 4) = 1 + 3 + 5 + 7 + 9 = 25 = 5 2 = O (d 5 ) = T (5) - T(4) = 1 + (2 - 1) + (3 - 2) + (4 - 3) + (5

- 4 ) = 5 .

From [ 3], You find t he o rderin g o f p owers:

•n = T (n) - T (n - 1) ;

•n2 = T (n) + T (n - 1);

•n3 = ( T(n) - T (n - 1) (T(n) + T (n - 1) ) = ( T(n)2 - (T(n-1) )2;

•n4 = ( (T(n) + T(n - 1)) 2 ;

•

n5 = ( (T(n) - T (n - 1))((T(n) + T(n - 1 ))2;

•

n6 = ( (T(n) + T(n - 1 )) 3;

•n7 = ( (T(n) - T (n - 1))((T(n) + T(n - 1 ))3;

•

.... [4 ]

. You k now t hat, by de fining , f or d i f f ere nc es a n d s u m s : D = T (n) - T (n - 1) , S = T (n) + T (n - 1 ), the o r d eri ng o f

p o w e r s be comes:

•n = D ;

•n2 = S ;

•n3 = D S;

•n4 = S 2;

•n5 = D S2;

•n6 = S 3;

•n7 = D S4;

•.... [4A]



And you find, in g eneral: n(2i-1) = D S(i - 1), n (2i) = Si , i = 1, 2,. .. . [ 4B]

Also , you f inf th at [2 ] becomes a recu rs ion o f numbe r po wers : R(n) = O( n) - D . [2A]

You shif t, n ow, t o N um be r P o w er s as O dd Num b er s . T o sho w it can be d one , y ou de fine O(1, n ) =1 + 1 + . .. +

1 = n . Then, this c onn ects:

•n = O (1,n );

•n2 = O (d n );

•

n3 = O (1, n)O (d n );

•n4 = ( O(d n )2 ;

•

n5 = O (1,n)(O( d n )2 ;

•

.... ;

•

n(2i-1 ) = O(i ,n)O (d n)(i - 1 );

•

n(2i) = ( O(n))i, i = 1 , 2,... .

In pa rticula r, (1 + x) n = ( 1 + x)O(1,n) .

Of cour se, t his is l ess "ele gant " t han "powers from t ria ngul ar s", bu t y ou see t hat i t symbo lica lly explic ates

anc ien t Py tha gorean f ig urat ions and i nvokes A Tri ple Rela tion ( Poincaré! ), as follows.

Given:

1.

O(n ) = 2 n + 1 = n + ( n + 1 );

2.T(n) - T(n-1) = n ;

3.R(n) = n + 1 .

You find The Tr iple R elat ion : O(n) = T(n) - T (n-1) + R (n) . Or: T(n) - T(n-1 ) = O (n) - R(n) [5 A].

D(n) + R( n)= O( n) [5B].

O(n )(1/2) = n i [5 C].

D = O (n) (1/2) [5D].

You not e be low that c ub e ro ots c a n b e c a lc u la t ed b y su btr a ctin g o dd n u mb e r s , whose ini t iator is

deter mined v ia " mov in g" tr iang u lar s .

Giv en t he p resent symbolism , th is ca n be opera t iona lly w rit ten : O(d T(n)) = n 3 = DS [5E].



You consi der , n ow, relate d P atter ns as Alg orit hms , w her ein n umbe r r elations n ot recog nized i n t he

liter atur e a re i mpli cit or e quiva lent to reco gniz ed alg ori thms.

Thus, y ou see t hat the o rder ing o f po wers in [ 3 ] h as a l gori thm ic r e lat ions .

Thus, p o w er on e f o r m t ells us y ou c an p ass from f or m , T(n-1) to the ne xt f orm , T(n) by adj o in i ng t he

figura te for num ber n. T his is a n a l g or ith m im p lic i t in th e pa tt e r n o f n u m be r p o w e r s a s t ri a ngu la r n u mb e r s ,

rhetor icall y m ani fested . T hen, power t wo is imp lic i t ly the a lg ori thm t hat sec ond p owers o f numb ers ar e

c al c ul a te d by m o vi n g p a ir e d s um s o f t ri a ngu la r s . Then, power t hree conveys a n a lgor ith m in the lit e ratur e

and onl ine . And , p o w er f o ur is im p lic itl y t h e al go ri th m t h a t t h e sq ua r e ro ot o f t h e f o ur t h p o w e r o f n u mb e r s

is c a lcu l a t ed by m o ving p a ir e d-su ms o f t r i an g ula r n u m be r s . Etc.

You shif t, n ow, t o B in o mia l C o e f fi c ie n t P a tt e r n s .

You know it is k nown, in the liter atur e a nd o nli ne, t hat a tr iang ula r nu mber has a bin omia l co efficien t f orm:

T(n) = C (n,2) = n !/2!( n - 2)! = n (n - 1 )(n - 2) !/2!(n - 2 )!= 1 /2n(n - 1). A lso kn own is th at a d ia g on al o f " T h e

P as ca l T r i an g l e " c o nta in s th e t r ian g ula r n u m be r s , as suc h. Y ou t hen see t hat th e a bo v e r e l a t io ns sh o w th a t

a ll o ther d ia gonals o f The P asc a l T r ian g le can be c onst ructed f rom th is d ia gona l . You know th at it is also

noted i n the l itera ture th at th e Fi b on a c ci N u m be r s a r e p re sen t i n t h e " T r ia n g l e " , s o th ey c an b e d e riv e d

fro m t ria n gu l ar n u mb e r s . T he imp lici t r elation o f Bernou lli n umb ers to "Fer mat num ber s" is k nown. B riefly,

th e " P a sc al T ri a ngle r e pe r to r y ", a s e x plic a t ed i n th e li te ra tu r e , is su bs u m ed i n t h e T r ia n gu l a r r e pe r to r y .

And all even per fect num ber s a re T r ian gula r o f the for m , T(p) , f o r p rim e p. And K. F . Gauss (1777-186 5)

proved the co n jec tur e o f Pi er re Ferm at (1 601 -166 5) that ever y pos it ive in te ger is t he sum o f a t most th ree

tr iang ular num ber s .

You now turn t o D e te r m in a t io n o f S q u ar e a n d C u be R o ots . You k now i t is not ed, i n t he l itera ture an d on line ,

that squar e roo ts can be e xtr acted by sub t ract ing c onsec utiv e ly o rder ed odd n umbe rs .

Thus,

1.144 - 1 = 143 ;

2.143 - 3 = 140 ;

3.140 - 5 = 135 ;

4.135 - 7 = 128 ;

5.128 - 9 = 119 ;

6.119 - 1 1 = 10 8;

7.108 - 1 3 = 95 ;

8.95 = 1 5 = 8 0;

9.80 - 17 = 63;



10.63 - 19 = 44;

11.44 - 21 = 23;

12.3 - 23 = 0 .

So, sinc e 12 su btr actions, o f c onsec utiv el y o rder ed odd n umbe rs, r educes 144 to zero, t he sq uar e root o f

144 is 12 , as may b e c onf irmed by cal cul ation.

Also k nown is a trick ( imp l ici t ly in volving t he d igi tal r oot ) which may c on si d er a bl y r ed uc e th e nu m be r of

requ ired su btr act io ns .

As in th e sta n d ar d s qu ar e r o o t pr o ce du re , t h e t a r g et n u mbe r is m a r k ed o f f i n p a ir s o f d i g its f r om th e ri g ht .

(This is be cause a one -d ig it n umb er h as a squ are o f one or two d ig its , as i n 42 = 16 .)

Thus y ou par se: 1'44. T his t rick n ow reduc es the subtracti ons from tw elv e to 1 + 2 = 3 , i.e ., o ne

subtr action f or the l eft pa ir (01); two subtracti ons f or th e ri ght pai r (12) . A nd t he d igi tal root o f 12 is 1 + 2

= 3.

However, you find n oth ing i n t he l itera ture o r on line to d emo nstr ate, o r e ven c omme nt, that, by th e same

procedur e, y ou ca n e xtr act th e square r oot nu mbe r tw o to any d esir ed n umber o f decima l pl aces.

Dua lly, you can sho w t hat t he p r oc e d ur e f o r c a lc u la tin g c u b es b y m o v i ng t ria n gu la r n um be r s , is a

procedur e for using th is "m oving " p loy t o c a lcu late c ube roots by su btr act ing odd nu mber s .

Thus, g iven 216 , q uic kly find ( by p rodu ct (5)(5)(5) = 125 ) that 216 (b eing g reater t han 125 ) has a cub e r oot

greater than fi ve.

Via t ria ng ul ar n um be r s -- and the fi ve c ube l ower bo und, above -- y ou k now that the subtracti on sho u ld

s ta r t a f te r 1 + 2 + 3 + 4 + 5 = 1 5 odd n umbe rs.

So, y ou be gin su btr action with the sixt eent h odd n umbe r, 2(16) - 1 = 3 1.

You proceed:

1.216 - 3 1 = 18 5;

2.185 - 3 3 = 15 2

3.152 - 3 5 = 11 7;

4.117 - 3 7 = 80 ;

5.80 - 39 = 41;

6.41 - 41 = 0.



So, sub tr act ion o f six c onsec utiv ely o rder ed odd n umbe rs redu ces 216 to z ero . And you con fir m, by

cal cul ation, t hat 63 = 21 6.

(Proof of th e ge n er al c u be r oo t pr o ced ure is i m pli c it i n t h e ab o v e p roo f o f c u b es from d if f e r e n ce s o f

tr iang ular num ber s , and may b e le ft as h omew ork.)

You not e th at a c orrespon ding "trick" p roce dur e i n t his case would n ot r educe steps, since it w oul d in volve

1+ 2 + 6 = 9 s teps t o r eplace the precedur e of si x steps.

You tu rn t o Factor ia ls ( compressed g eom etr ic prog ress ions) .

You know th at f a cto ria ls p r o vid e a b asis f o r w riti ng f o r m u l as f o r m ul a s in n u mb e r t he o r y an d in

com bin ator ics .

You know it is som etim es not ed i n th e l itera ture (a nd onl ine ) t hat tr iang ular num ber s f orm the addit iv e

dua l to th e mu lt ip lic ativ e f acto ria ls i n c onst ruct ing num ber -t heoreti c o r com bin atoric f o rms . You can sho w

this symboli call y: S ince n = D, i t f ollows that: n(n - 1)(n - 2 )...1 = D(D - 1)(D - 2 )...1 .[6]

The l eft side of [1] is symb oliz ed in the liter atur e as n! . Using the S pan ish i nver ted e xc lama t ion poi nt ,

nam ely, "¡", you ca n symbol ize t he r igh t side of [6] as " ¡D", r etaini ng pref ix f ormat t o e mph asiz e th e

dist inct ion. Then [ 6] becom es: n! = ¡ D. [ 6a] T hus , n-factoria l eq uals D-triango rial .

You tu rn n ow t o C om bin a ti o ns (ra t io s o f c om p r es se d G eo m et ric p r o g r e ss io ns ) . T hus, giv en th e

com bin at ional for mula , C(n, r ) = n! /r!(n - r )! , y ou f ind:

•C(n, 1 ) = n!/1! (n - 1 )! = n = D

•C(n, 2 ) = n!/2! (n - 2 )! = 1/2n(n - 1) = T

n-1

•C(n, 3 ) = (1/3)T

n-1(D - 2 )

•C(n, 4 ) = (1/12)T

n-1(D - 2 )(D - 3 )

•C(n, 5 ) = (1/60)T

n-1(D - 2 )(D - 3 )(D - 4)

•A = [ 2T

n-1/r!D(D-1 )]

•...

•C(n, r ) = AD (D-1) (D-2) (D-3 )...( D - r - 1).

You introduc e new not ation: B(-) , d efinin g B(n,r) = D(D-1 )(D-2 )(D-3) ... (D - r - 1 ). And you can writ e: C(n, r ) =

AB(n, r). [7]

Thus, n-comb ina tio ns ar e p ropo rtional to B-comb ina tio ns.

You tur n t o P ermutations (a lso ra tios o f Compressed Geom etr ic P rogressions ). W hat is that t erm "A " in [7]?



A = [ 2Tn-1

/r!D(D-1 )]. A nd 2Tn-1

/r!D(D-1) = 2Tn-1

/r!n(n-1) = Tn-1

/r!Tn-1

= 1/ r! = A .

So [7 ] becomes: C(n,r) = ( 1/r!)B(n,r) . Or,r!C(n, r) = B(n,r ). [8]

Now, th e le ft side o f [7] is f amilia r in com bin atorics: r!C(n-r ) = P(n,r) = n! /(n-r)! , f or p er mutatio ns.

Thus, t he n-comb ina tion eq uals t he B-per mut ation.

In para phr ase of (transla ted ) P oincaré, t he art of m athematics giv es t hr ee nam es to many " different

things " by r ecogn izing patterns. I f stude nts un der stand t his, t hey c an r ealiz e how to lea rn t he n ew from

the f amiliar o ld.

You know th at a p rimar y p atter n typ e in mathe matics is t hat o f the Al gori thm, so ma ny of w hich were

consi der ed.

You know th at Wikip edia bel ieves t his w ord d eriv es f rom a m an's na me by a misu nder stand ing. Al-

Khw ariz mi, P ersian astr onomer an d ma thema tician, wr ote a trea tise i n 82 5 AD , 'On Calcul ation wit h Hind u

Numer als'. It was tr ansla ted i nto La tin in the 12t h ce ntu ry as ' Algor itmi de n umero In dorum', w hic h ti tle

was lik ely i nte nded to m ean 'Algorit mi o n th e nu mbe rs of t he I ndians ', w her e 'Algo ritmi' was th e

transl ator's r enditi on o f the au thor 's n ame; but pe ople misun der sta ndin g th e t itle treated ' Algori tmi ' as a

Latin plural a nd t his le d to th e w ord ' algorithm ' ( Latin ' algorism us' ) c omin g to mean ' calcul ation m eth od' ."

You know th at, f or different in ter pretations o f th e t er m, you can consu lt W ikiped ia. " Algor ithm

chaacter iza tions ", a nd f or so me a lgo rith ms, "A lgorithm e xamples ".

You know it may b e e nlig hten ing t o assoc iate t he n oti on o f Alg ori thm wi th t he d iagram " flowchart", o f

whic h ma ny appear in C hap. 1 9.

Orig in o f flowchar t: I n be ginni ngs o f the c omp ute r, the progr am was imp leme nted by a p lug boa rd, as used

at ta t tim e by t elep hine opera tor s. A nd d ata w as in put by ho les in pun ched, as us ed i n IB M c alcu lator s.

Mathem tici an J ohn v on Neu man ( 1903-5 7) an d en gine er Her man Go ldsti ne ( 1913-2 004 ) crea ted t he i nter nal

program for com pute rs, a great simp lica etio n. T o facil itate t heir w ork, t hey de velop ed t he f lowchar t in

1945-7.

You know th at the fi rst alg orit hm k nown to appear in E uropea n li ter atur e was "Euclid 's A lgo rith m" (in h is

"Elemen ts o f Geomet ry") for cal culating t he GCD of two inte ger s.

As f ollows:

•given nu mber s a < b, t o f ind GCD( a,b ) by Euclidea n Al gori thm:





| | DIVIDE L AST | |

| | DIVISOR B Y I TS | |

--- ---<-- ---| R EMA INDER | ---<---

| |

--- ---- ---- ---- ---- -

To f ind LCM( a, b ): given GCD( a, b ), then ( as sh own i n Ch ap. 1, LCM(a, b) = ( a · b) /GCD( a,b ). This is so

simpl e it isn 't w orth l isting steps o r f lowcharti ng.

You knoW ho w to fi nd the facto rial funct ion N!= 1 ·2·2 ·... ·n:

\====== =/

\ START/

\ /

\_/

|

|

_____|_____

|se lec t N |

|M=1=F |

|_________|

|

|

_ _ | _ ___

| se t |

|F&mi dot t;M|

|_________ |

|

|

| |

/ \

/ \ ______________ / \ / \

/ \ --- YES-->--| PRIN T A S |- ->--/ STOP\

/ \ |AS N FACTORIAL | _ _____

/ \ |______________|

/ \

--- ---- >----- --/ M = N? \- ->--- NO- ----



^ \ / |

| \ / |

| \ / |

| \ / | | \ /

|

| . |

^ | V

| v |

| | |

| _______|__________ |

| | | |

| | set | |

--- ---<-- ---| M + 1 |- --<-- -

| |

--- ---- ---- ---- ---- -

You know ho w t o fi nd the PRIMOR IALS: just as th e FACTORIAL, n! , is P RODUCT of FIRST n

NUMBER S, so th e nth PRIMOR IAL , n# , I S PRODUC T OF THE F IRST n PRIMES, 2,3,5 ,7,. ..

You also k now how to find t he P ARTORI ALs: n## IS PRODUC T OF T HE FIRST n PR IMORI ALS . You

know its nam e de rives f rom DEN SITY FUNC TION o f n## which, P ROVIDING A CANONIC AL P ARTITION

OF (n +1)! -- I T RELATES TO o-NUMB ERS ( Chap. 17 ) as FACTORIAL RELATES t o t -numb ers.

You know th at a p rope rty o f PATTERN S w hich w as n oted in anc ien t ti mes, w hich is n ow widel y

obse rved a nd p ract ised, a nd is ever-pr esen t i n da ily life, is S YMME TRY. W ikipedia e xpl icates t his as

"'patter ned se lf-simil ari ty' t hat c an b e de monstra ted or proved accor ding t o th e r ules o f a formal

system: by geo met ry, th roug h physics o r o the rwise."

You know th e p leasin g 1D symmet ry o f the l ine segme nt; the 2D symm etr y of t he c ircle; t he 3 D

symmet ry o f the spher e; et c. S ymme try is obse rved, dail y, i n l ids f or r ecept acles: roun d, squ are,

even the less symme tric rect angle. T his p rope rty f aci litates c losi ng t he r ecept acle: y ou ne edn 't

retur n l id i n t he o rie nta tio n it cam e of f. You kn ow t hat, c onversel y and inversel y, you may d etec t

pilferi ng o f goodies f rom y our r ecept acle by BR EAKING THE S YMM ETRY via se gment al m arking on

lid coi ncid ing w ith se gmen tal m arkin g on side o f r ecept acle, be low the lid mar king. I f, i n l ast usa ge,

you le ft t he mar kin gs coi ncid ent , bu t la ter fi nd them n ot, y ou susp ect a tampe ring by some one .

(You remem ber t he use, say , i n a James B ond m ovie, o f a t hread a cross a d oor c losin g, t o d etec t

intruder entrance .)



You know th at SPON TANEOUS S YMM ETRY BREAK ING h as, f or se veral y ear s, b een obj ect o f

assiduo us researc h i n P ar ticle P hysics, to provide cri tica l in for mation as to how our univ erse c ame

into b eing and too k shape, T hat's ho w im por tant SYMME TRY a nd A SYMM ETRY ar e,

You know a f amilia r e xamp le o f SYMME TRY-BREAKING an d NE W SU BSYMME TRY is i n a "sp or t" o f

children: t hrowing rocks into a pon d. T he S YMM ETRY of t he w ater sur face -- bec ause g ravity

activa tes w ater t o a ttain equ ile vel ("Water seeks i ts o wn le vel. ") -- is BR OKEN by r ock entrance ,

followed by S YMME TRICAL W AVES as a S UBP ATT ERN.

You know th at RANDOMNE SS r epresents a n p ecul iar kin d o f SYMM ETRY -- no n otable disti ncti on o f

one sect ion f rom an othe r. You a lso k now t hat p leasi ng S YMME TRY is an E MERGENT of SYMM ETRY-

BREAK ING OF RANDOMNE SS -- e vid ence d in obse rvations o f, perhaps, e very o ne, b ut si ngled o ut i n a

book. " Symme try" by J oe R osen.

You know th is is e xpli cated by a symme tri cal p rocess, n amed qu asi-symmet rica lly, as "F u-Te-sa-te -

fu"

You know th at th e g reat Cz ech-Ame rican m athem atician , Kar l Men ger , i n his b ook, What Is

Cal culus? , d escri bes three b asic m a t he m a ti c al m o de ls : FLUENT, TREMB LAND , S ALIENT.

oFLUENT: mod els a steady-st ate proc ess eit her u nchan ging or c hangi ng u n i f o r m l y

oTREMBLANT : osci lla tor y

oSALIENT ( from L atin, "sa ltus ", f or " jump"): j umps f rom one state t o an oth er.

You know th is insp ir ed " The F uTeSatefu Hypot hesis: E ver y process is a F ute Satefu p rocess ",

mea ning it passes from o ne c ond itio n to ano ther : FLUENT ®TREM BLAN T ® SALIENT ®

TREMBLAN T®F LUEN T.

You know th at th e com posit e label , " futesa tefu" , uses t he f i rs t c onson ant, fi rs t v owel o f the t hree

labels -- "fu" f or " fluent ", "te" for "tremblan t", " sa" for "salie nt". S ince " fu" an d "t e" occu r in in i t ia l

and f in ia l posi t io ns , their c onson ants are capit aliz ed in the in i t ia l posi t ion and left l ower-case the

f in ia ls : FuT esatef u.

You know th at most folks have e xperi ence an " ever yday" FuTesa tefu p rocess : st ar ting a n

automo bile . Be fore " sta rting", th e e ngin e is OFF in a stea dy-s ta te , h ence , FL UENT (F u); t urning t he

ign iti on k ey a nd st eppi ng o n sta rter evokes as TREMB LANT ("Ug-u g-u g-ug ", mor e voca l an d

sustai ned i n pre vious g enera tion c ars); th e en gine "catches" , go ing S ALIENT LY (Sa) f rom OFF-st ate

to ON- state; a TREMB LANT in " settli ng d own" ( also ver y n oticeable i n ö lde r"car s); t hen t he eng ine

sett les do wn F LUEN TLY to the r evving fixed f or en gine -idl ing.



You know th at, i n his b ook , Meng er c ites as a m ath-e xampl e o f TREMB LANT, the "Weierstr ass

Functi on", w hich is "co ntin uous e ver ywher e", b ut "differentialble nowhere" .

You know th at th is is, p oten tia lly, a W IN-W IN S TRATEGY. You c lai m "A ll processes ar e F uTesate fu".

If th e pr ocesses your au dito rs obse rve f it th is claim, you W IN. If appar ently, fu rther study sh ows

why a co mpon ents h as bee n o verlook ed, o r reasons w hy a compo nen t is e xclude d. A nd t his r esul ts

from q uer ying w hich, likely, y ou'd n ever though t o f making -- a gain W IN. He nce, WIN -WIN .

(You k now the FL UENT is typi call y CON TINUOUS o r " analog ic" , while th e SA LIENT is

DISCON TINUOUS or "digi tal ". He nce, you've sp ecul ated abou t th e " SaTefu tesa P rocess".

Retur ning t o R osen, he t akes, as example , o f the symmet ry-from-rando mness, b lowing o n a g rass

bla de o r o n th e mo uth o f a hor n,

You blow on a g rass bl ade l yin g on your to ngu e. T he wi nd f lows symmet rica lly on eith er sid e o f the

blade -- a f luent process: t he first " Fu" o f "FuTesatefu". B ut " someth ing " c an " cause" an a sym m e t r y

of f low -- g reater on o ne s i de t han the oth er -- i nvoki ng osc i ll at ions in the f l ow : the " Te" t remb lan t

for ming the seco nd sta ge o f the " FuTesatefu" process. Then t his asymme tr y i nvokes a "j ump" and a

"ragged " t one em its -- the salie nt, "sa ", forming the thi rd a nd m edi al sta ge o f the p rocess. A fter

going t hr ough i ts own oscil lation , or tr embla nt p hase ( the " te" f orming t he f our th stage o f the

pr ocess), i t t hen sta bliliz es int o a " smo oth" t one : th e fi nal fl uent " fu" o f the pr ocess.

The process o f b lowing o n the l ip o f th e bo ttl e is ( presu mably) simi lar. ( Try it. )

Retur ning t o the i mpo rtan ce o f sp on ta ne o us s ym me t r y -b re ak i n g , you know th at W ikip edia says: "In

physics, spo nta neous symm etr y brea k ing occu rs w hen a system t hat is symme tri c with respec t t o

some symm etry group [ Chap.22] goes i nto a vac uum st ate th at is n ot symme tric. W hen that

happens, the system n o lo nger appear s t o b eha ve in a symme tri c ma nne r. I t is a p hen omen on t hat

natur ally occur s in many si tua tio ns. .. .. A com mon e xam ple t o h elp explain this p hen omen on is a

ball sit ting on t op o f a hil l. T his b all is in a co mple tel y symmet ric s tate. H owever, i ts sta te is

unst able : th e sli ghtest pertur bing force will cause the bal l to r oll do wn t he h ill i n som e pa rticula r

dir ecti on. A t t hat po int , symmet ry h as bee n br oken b ecause the dir ectio n in whic h th e ba ll r olled

has a feature th at dist ingu ishes it from a ll o the r directi ons. . ... T he Standar d Model of particle

physics is a th eory of t hree of t he f our known f undame ntal inter actions [ omitting g ravity ] an d th e

elemen tar y p artic les that tak e par t i n th ese in ter actions. These p articles m ake u p a ll visib le m atter

in t he univ erse. T he stand ard m ode l is a gau ge t heo ry of the electr oweak a nd c hromodyn amics . ..

str ong in ter actions wi th t he g aug e g roup W and Z bosons."



As st ated a bove, t his is r ese arched " to pr ovide cr itic al i nfor mation as to ho w ou r un iver se c ame

into b eing and too k shape. " ( From thro wing r ocks in the water t o u niv erse -bui ldin g!)

The f ollowing is a Me thod olog y i n P atter ning.

TRAPPING TH E W ILD W ORD!

This ma thtivity in troduces t eens t o u ni m od a l sea rc h m e th od s . For examp le, y ou a re assig ned t o

deter mine t he b o il in g p o in t o f h y dr o q uin o ne , a c hemica l use ful in p hot og raphy. Giv en th e typi cal

sensit ivity o f most i nstr uments, p articul ar ly t hose a vail able in s chools, t his is o nly measura ble in a

RANGE of measur es, so th at the process of R EACHING BO ILING T EMPERATURE AN D GOING BE YOND

resemb les th e f amiliar "bell curve" used (incor rectl y!) f or gradi ng stu den ts. T he g raph o f suc h a

process is cal led UN IMOD Al, e xpla ined as f ollows.

The w or d MOD E is a gener al term f or "avera ge" . S uppose y ou h ave a 10-da ta sa mple : 5, 7 , 4 , 7, 3, 8 ,

5, 1 , 9, 6 . This samp le h as two mo des -- or two kin ds of "mos test" -- namel y, 5, 7 , o ccuring twice in

the samp le, i n c ontr ast to o nce for the ot her s. T hat is, t his samp le is B IMO DAL ( not UN IMOD AL).

Consi der, on t he o the r ha nd, t he 1 0-d ata samp le: 2, 1, 6, 8 , 4 , 6, 7, 4 , 5 , 6 . These are on e-ti mer s o r

two-ti mer s, e xcept th at 6 is a three-t ime r. Hen ce, t his sa mple has j ust ONE MOD E ( one "m ostest

kin d"): it is UNIMOD AL.

Many measur ing p rocesses and ot her types o f experi ment a l p roced ures ar e UNIMOD AL.

Since RESEARCH CO STS MONEY AND T AKES TIME AN D SOM ETIMES SEVERAL RESEARCHERS, y ou

wish t o MIN IMI ZE T HE PROCESS.

There ar e three f avor ed UN IMOD AL SEARCH P ROCEDURE S:

1.BIN ARY SEARCH;

2.FIBBON ACCI SEARCH;

3.GOLDEN S ECTION S EARCH.

Of th ese, B INARY S EARCH is t he sim plest and , usua lly, the chea pest t o car ry ou t. T his mathtiv i ty

teaches the pr ocedure o f B INARY S EARCH ( a recur siv e proc ess , as disc ussed in Chapte r T wenty-

One) ,

BIN ARY SEARCH requires lab eq u ipm ent : TEACHE R; S TUDEN TS; T WO-COL UMN DI CTION ARY, of 10 00+

pages.



•One t een picks ou t a DEFIN ED WORD ("The W ild W ord") ON A P AGE OF THE D ICTIONARY, sh owing

WORD an d P AGE NUMBE R to teacher. Also not ed is t he o r d i na l n u mb e r of the wor d's p osit ion o n t he

page: th e fi rst defin ed w ord; or the secon d; o r th e u mpte enth .

•Ano the r stud ent , ha ving b een giv en th e c losed D ict iona ry, g uesses the page num ber , t hen the

or deri ng o f the "w ild w ord" o n the page.

BIN ARY SEARCH E NABL ES THE PERSON T O F IND THIS WORD E XACTLY IN 15 GU ESSES! H ow?

Let's say t hat t he D ict iona ry has 1017 pages. Y ou n ote that 1017 < 1 024 = 210 (a bin ar y n umber) .

Then B INARY S EARCH A LLOWS FIN DING T HE PAGE IN TEN GUE SSES.

Havin g th e P age, t he g uesser c oun ts the DEFIN ED WORDS on t he p age, and GUES SES ITS

ORDIN ALITY.

A ty pica l di ctio nar y of t his desc rip tion has, a t most, 32 DEFINED W ORD S T o A PAGE. N ow, 32 = 25 .

So th e word can b e g uessed i n 5 or f ewer gu esses -- t otal ing, 10 + 5 = 15 guesses, f or t he e ntire

pr ocedur e.

As an exampl e, sup pose t he w ord is on p age 849 an d is t he F IFTH D EFINED W ORD o n that pa ge.

Taking t hat 101 7 pa ge fi gure (t he page num ber w oul d be known t o a ll t he st uden ts), and

appr oxim ating i t by the num ber 1024 = 2 10 , the gu esser asks, " Is th e pa ge nu mber greater tha n

512 ?" ( This is h alf of the nu mber 102 4.)

Since it was desig nated abo ve as on page 849 , t he AN SWER is "Y es." Then t he P age Num ber is

between 513 an d 102 4. O f the 512 pa ges, ha lf of that is 256 , and 512 + 256 = 768 . Hence , GUE SSER

(SECOND Guess ): " Is the P age Numbe r g reater th an 768 ?" AN SWER: "Yes."

The R ANGE is n ow 769-1 024 = 2 56 pages. Ha lf o f 256 =128 . And 768 + 128 = 896 . Hence , GUE SSER

(THI RD GU ESS): " Is th e Page Numb er g reater t han 896 ?" AN SWER: "N o."

The R ANGE n ow is 769-8 96 = 1 28 pa ges. Hal f of 128 = 6 4. And 768 + 6 4 = 83 2. Hen ce, GU ESSER

(FOUR TH GUE SS): " Is t he Page Nu mber greater tha n 832 ?" ANS WER: " Yes. "

The R ANGE n ow is 833-8 96 = 6 4 pa ges. Hal f of 64 is 32 . And 832 + 3 2 = 86 4.

GUES SER (FIFTH GU ESS): " Is th e Page Numb er g reater t han 864 ?"AN SWER: "No." T he R ANGE is now

833 -864 = 32 pages. A nd h alf of that is 16 . A nd 832 + 1 6 = 84 8. GUE SSER ( SIXTH GU ESS): " Is th e

Page Numbe r g reater than 848 ?" AN SWER: "Yes."



The R ANGE is n ow 849-8 64 = 1 6 pa ges. Hal f o f t hat is 8. And 848 + 8 = 8 56. Hen ce, GU ESSER

(SEVENTH GU ESS): " Is th e P age Numb er g reater t han 856 ?" AN SWER: "N o."

The R ANGE is n ow 849-8 56 = 8 pages. Ha lf of that is 4. And 848 + 4 = 852 . Hence , GUE SSER

(EIGHTH GUE SS): " Is the P age Numbe r g reater th an 852 ?" AN SWER: "N o."

The R ANGE is n ow 849-8 53 = 4 pages. Ha lf of that is 2. And 848 + 2 = 850 . Hence , GUE SSER (N INTH

GUES S): "Is t he P age Num ber g reater t han 850 ?" ANS WER: " No."

The R ANGE n ow is 849-8 50 = 2 pages. Ha lf of rhat is 1. And 848 + 1 = 849 . Hence , GUE SSER (TENTH

GUES S): "Is t he P age Num ber g reater t han 849 ?" ANS WER: " No." GUE SSER:"Then t he page n umbe r is

849 ." AN SWER: "Yes."

Havin g f ound t he P age Num ber , l et's fi nd the DEFIN ED (WILD) WORD B Y TH E OR DIN ALITY OF ITS

POSITION ON THE PAGE. We sa id t hat i ts ordinal ity is f ifth.

Assumin g a M AXIMUM o f 32 DEFINED W ORD S t o a two-co lumn page, the RANG E is 1-32 . Hal f of that

is 16 . Hence , GUE SSER (F IRST GUES S): "Is t he o rdinali ty greater tha n 16? ÄN SWER: "No."

RANGE now is 1-8 = 8 posit ions. Ha lf of that is 4. Hen ce. GU ESSER ( SECOND GUE SS): " Is t he

or dina lity g reater th an 4?" ANS WER: " Yes." R ANGE is 5-8 . Hal f is 2. And 4 + 2 = 6 . GUES SER (THI RD

GUES S): "Is t he o rdinali ty greater tha n 6?" AN SWER: "Yes."

RANGE is 5-6 . Hence , GUE SSER (F OUR TH GUE SS): " Is the or dina lity g reater t han 5?" ANS WER: " No."

GUES SER: " Then o rdinali ty is 5." ANS WER: " YES."

And , lo okin g on Page 849 , the Guesser f inds th e fi fth DEFIN ED (WILD) WORD.

This w as just for fun. As not ed abo ve, t he same B INARY S EARCH c ould det er mine t he b o il in g p o in t

o f h y d r o q u in o ne , a chemical us eful i n ph otog raphy .

You find a RELATION b etw een ANS WERS T O GU ESS AN D PAGE TO F IND. You wr ite down th e

seque nce o f ANSWERS: Y , Y, N, Y , N, Y, N, N, N, Y .

You RE PLACE "Y" by on e; " N" by z ero, to f ind: 1,1,0 ,1,0 ,1,0 ,0,0 ,1 .

You INV ERT T HAT S EQUENC E, a nd o mit co mmas t o fi nd: 100 0101 011 . A B INARY NUMB ER!

. E VALUATING t he I NVERTED SEQUENC E A S POWERS OF t wo, y ou o btai n a su m in DECIM AL T ERMS : 1

+ 0 x 21 + 0 x 2 2 + 0 x 2 4 + 1 x 24 + 0x2 5 + 1 x 26 + 0 x 27 + 1 x 2 8 + 1 x 29 = 1 + 1 6 + 64 + 256 + 512

= 849 . BIN ARY GUES SES TRAN SLATE AS T HE NUMBER S EARCHE D FOR !



<F

"Tra pping The W ild W ord" is a MA THT IVITY t hat trains stu den ts for USING BINARY SEARCH T O

PERFORM US EFUL MEA SUR ING OR O THER R ESEARCH P ROJECTS!

USEFUL A PPLICATION: A c om mi tte e mus t choose ON E am ong m any P ROPOS ALS o r RE SUM ÉS or

whatever. T hey ca n S EPARATE OUT THE W ORS T HALF OF THE OR IGIN AL SET REJECTED. O f

SMALLER S ET, SEPARATE OUT THE W ORS T HALF AN D RE JECT. Et c. CONT INUING IN THIS WAY, a SET

OF RE ASONABLE SIZE CAN BE RE ACHED SO THAT E ACH RE MAIN ING ONE C AN BE CARE FULL Y

CONS IDERED.

You know th at BIN ARY NUMERA TION (while used by com pute rs is n ot " MODE RN", not pa rt o f "The

New Ma th" . A ctua lly, BINARY COMPU TATION ( bUT NO T BIN ARY NUMERA TION ) is ANC IENT!

BIN ARY COMPU TATION is men tion ed i n Th e B ible . For member s o f the T ribes o f Isr ael l ear ned i t i n

Egypt, d urin g th eir "captivity ".

The t erm " mediation" means " to com e be tween"; i n c ompu tation , " to halv e a n umbe r". An d

"duplatiion " m eans, of cour se, "t o d oub le" .

The A LGORI THM OF M EDIATION AND DU PLATION was used t o MUL TIPLY T WO NUMBER S - - d ifficult to

do w ith Gr eek n umer als or R oma n nu mer als.Giv en, say , d x m :

1.You dup late on d (doublin g it ) while medi ating on m (h alvi ng i t), u ntil the med iation proc ess

reaches o ne ( "the bo tto m").

2.Note : In medi ating ( halving a n o dd n umbe r), su ch as k , y ou m edi ate on ( halve) k - 1, an e ven

num ber : (k - 1) /2 is t he m edi ate in suc h a c ase.

3.Going back t o t he results, Y ou UND ERLINE TH E OD D ME DIATES.

4.You ADD THE DU PLATES A SOC IATE D W ITH ODD M EDIATES . ( You k now t hat, in B INARY

NUMERA TION, t he OD DS would b e r epresent ed by n umer al on e. t he e vens by n umer al z ero .)

5.This SUM is THE P RODUCT OF d x m.

(You n ote the ANT ITONIC form of t his P ROCESS: as one ORDE RING - - DU PLATION -- INCRE ASES, the

other COOR DIN ATED ORDE RING - - M EDIATION DE CREASES. Of cour se, any c alc ula tion or or dered

system is Anti toni c .)

Let's st art with 85 x 2 6, wr itin g th e produc t in this w ay b eca use it shor tens t he p rocess t o ME DIATE

ON T HE SM ALLER ONE ( if one b e sma ller ). The process lo oks lik e th is, UNDE RLING DU PLATES



ASSOC IATED W ITH ODD M EDIATES:

85, 26 - > 170 , 13 - > 340 , 6 - > 680 , 3 -> 1360 , 1. FINI SHED.

Adding the UNDER LIN ED DU PLATES, w e f ind: 170 + 6 80 + 1 360 = 2 110 . Then 85 x 2 6 = 2 110 . (Check

the stan dar d way.)

See how a nci ent B INARY COM PUT ING is!

Onlin e a so urce o f pat tern -f ind ing is data mi ning .

The g eo me tri c p a tt e r n s o f p ol y he d r a ar e explai ned by the following r esu lt, i n t he b eginngs o f

Topolog y.

EULE R CHA RACTERISTIC

As a p ol y hed r o n co nside r th e tet rahe dron . It has four vertices ( V), six e dges ( E), f our f aces ( F). And

al ter nate summ ing yi elds 4 - 6 + 4 = 2 .

Explo ring fu rther, sp lit a face wi th a new e dge , ca using o ne f ace to bec ome t wo. N ow w e h ave 4 - 7

+ 5 = 2 .

Next, spli t an edg e wit h a n ew v er tex, ca using t he o ne e dge to b ecome t wo. W e ha ve 5 - 8 + 5 = 2 .

This is n ot c oinc iden ce b ut a dem onstra tion o f the Surface Euler charac ter ist ic : x = V - E + F , and

the be ginn ing o f a p ro o f o f th e in v aria n ce o f t h e E u l er c h ara cte ris ti c . T his result is known as

Euler's f ormula, as il lustr ated in W ikiped ia. T he sevent een w alpaper g roups ar e also sh own in

Wikip edia .

Even m ore a mazin g th an I sla mic d isco ver y of these g roups is th e f ollowing stor y, whic h a ppear ed

onl ine.

Medi eval I slam ic d esigne rs used e labor ate g eom etri cal t iling p atter ns at le ast 5 00 y ear s b efore

West ern m athem aticia ns develope d th e co ncep t. P hysic ist P eter L u o f Ha rvar d Univ ersity sigh ted

15th-c entu ry t iles t hatfor med so -cal led P enrose g eometri c pa tter ns, cr eated by m athem atician

Roger P enrose i n 19 70s.



CHAP TER 6: W HAT IS IA TROGENIC MEDICIN E?

You Web-l ear n tha t IATROGE NIC M EDICI NE deal s wi th

ME DICAL PROBLEM S C AUSED B Y MEDICAL PROCEDURES . For

example , y ou go into the hosp ita l for an appendectomy and

get s ta ph i nfect ion . (In an episode in the TV ser ies , M*A *S *H ,

sta ph in fection was tr aced to the wooden floor of the

Oper ating Room, w hich had to be r ep laced b y a concr etefloor .)

You Web-l ear n tha t the Gr eek pr efix "i atro-" mean s " se lf" .

So you r ea li ze tha t man y pr oble ms in S CIMA TH ar e cau sed b y

its EXPOSITI ON , TEA CHI NG , and other cor rect ible procedur es

You real ize need for im medi ate attent ion to th is ! And f or other

sug gesti ons of pr ob lems .

You WEB-l ear n tha t, to d ia gnos is one IATROGENIC F ACTOR,

you can intr oduce an ep inome , a referent named after a

per son who taught u s about it. (YoU WEB- lear n Med icine is

ful l of epino mes -- d is ease s named f or t heir dia gnost ician s --

su ch a s "A lzhei mer' s di sea se" , named f or D r. Alz hei mer , who

fir st sati sfactori ly d iagnosed it .)

The epinome you lear ned fr om W ikiped ia is "bu rke", named

for the l iter ar y crit ic and rhetor etician , Kenneth Bur ke (1897-

1993), descr ibed in Wikipedia as l eader of a per specti ve un

Rhetor ic. Bur ke cr it ic ized "gener al ph ilo soph ies ", e specia ll y

when the phi los opher tr ied to e xp lain "ever ythi ng" i n ter ms of

a g iven "cau se" or f actor . (Examp le: a phi lo sopher e xpla in ing

"ever ything " i n ter ms of "the environment " wr ote about the

environment as if i t is al ive and can act as " an agent" .)

You see an e xamp le of bur king i n Ia tr ogenic Ph ys ic s , when

pr ob abil it y becomes a DYN AMIC .

You recogniz e IA TROGENIC MATHEMA TICS as dea ling wi th

mathema tical pr ob lems cr eated by the tea ching of Math, it s

langua ge, etc .:



• Ka rl Menger : "T he langua ge of calcu lu s is so atr ociou s

tha t it mak es the pr ofound look tr iv ial and t he tr ivial look

pr ofound!" ["Gee! You mean tha t ∫xdx = ∫udu = ∫vdv? W hat

theor em' s tha t?" No! T hi s is Log ical replacement , not a

calcu lus theor en .

As another m isunder stand ing, " Sur e, I kno w tha t(dy /dx)(dx/dt) = dy /dt . I lear ned tha t in fr act ions !"

Wr ong! This impl ied theor em look s pr oper ly pr ofound

when view ed in Leibn iz ian nota tion : (Dxy)( D tx) = D ty.

• The pri mar y oper ations of "calcu lu s of one v ar ia ble" ar e

dif fer ent ia t ion and antid if fer enti ation -- not "i nte gration",

whic h is a functiona l , not a funct ion-funct ion ma pp ing ;

"i ndefin ite and defin ite in te grals " invoke "i atr ogen ic"

confus ion.

• Ber tr and Russ el l: "T he thing a bout a v ar ia ble is tha t i t

doesn 't v ar y." You can s ay functand , since it' s the m a p of

a funct ion , just a s we s peak of oper ator and oper an d .

• You kn ow tha t some student s detect what s eem s to be

chea ting in the quas i- axio matic pr esenta tion of

ari thmet ic i n our s choo ls ["Y ou say you can su btr act 5from 3 unle ss you put a funny sign in fr ont of it !" "You say

you can 't d ivide 12 by 5 wi thout r emainder unle ss you put

a funn y s ign betw een 12 and 5! "]

• you r ea li ze th at s ay ing "ad d a top ic" , instead of "ad join a

topic ", mixes ar ithmet ic and rhetor ic, confu sing man y

about mathema tic s.

• You web-l ear n tha t "l ess" is analog ic - - tha t "fewer" is

dig ita l. (E ven " langua ge m aven" W. Saf ir e s tut ter s overthi s) ,

• You real ize th at a great bu zz -wor ds is "aver age", which

means " se t-r epr esentor" : mode(s) , median , ari thmet ic

mean, geomet ric mean , har monic mean , l ogar ithm ic

mean, etc ., a ll aver ages . You can under stand tha t the

wor d " aver age" deri ves fr om Latin "ha var ia" for "shar e" ,



in a pri mi ti ve f or m of Med iter ranean mer chant 's theft -or -

other- lo ss in sur ance.

• You real ize s ay ing "ne gative one", not "minu s one"

confuse s s ubtr act ion w ith or dering in the number syste m.

You real ize r eplacing med ial signs by super script signs

help s: -5 , not -5; and + 3 , not +3 ; etc.

• You WE B-l ear n tha t, not A ri st otle , but C hr ysippu s t heSto ic (280 -207BC) inst ituted the constr aint of biva lenc y

in logic (tha t st atement s ar e on ly t rue or fa ls e ), as cited

in E. W. Beth , "T he Founda tion s of Mathema tics" , 1959.

• You recognis e Q uanti ty ! Qual it y! Qu ib ble ! Quac ker y: th at

"qual it ati ve" is measur ed by typol ogica l and or dina l

sca le s; " quantit ati ve" , by inter val and r atio scale s -- as

noted i n Cha p, 19 .

• You recogniz e IA TROGENC MATH in The "Chal lenger

Disaster R epor t", descri bing the k il ling of astr onaut s and

a s choo l teac her . Good eng ineering mea sur ement s wer e

wasted and b ypas sed by v ague ad min is tr ative deci sion

langua ge. (An engineer was order ed to "tak e of f your

engineer ing h at and put on your adm ini str ative hat and

vote wi th u s for tak e-of f !")

IATROGENIC SCIEN CE

• "We've ar ranged a globa l ci vi lization in which the mo st

cr uc ial elements . .. pr ofoundl y depend on science and

tec hno log y. W e have a ls o ar ranged t hing s so tha t no one

under stand s s cience and tec hnolog y. T his is a

pr escr ipt ion f or d isas ter . We mi ght get away w ith it for a

whi le, but so oner or later th is combu st ibl e m ixtur e of

ignor ance and power wi ll blo w up in our face s . ... ." , Car lSa gan, "The Demon-H aunted Wor ld".

SEQUENCE -PARALLEL N OTING IATROGE NIC MATH

calculus MEANS pebble, stone ⇓ ⇓ ⇓



average MEANS representative ⇓ ⇓ ⇓

less MEANS continuous decrement ⇓ ⇓ ⇓

fewer MEANS discrete decrement ⇓ ⇓ ⇓these kind MEANS this type of typons

variable MEANS fuctand of function ⇓ ⇓ ⇓

minus one MEANS negative one ⇓ ⇓ ⇓

reductio proof MEANS failure to construct ⇓ ⇓ ⇓

food set MEANS {bread, milk, corn} ⇓ ⇓ ⇓

book set MEANS {novel, text, check}

S-P NOTING IATROGENICS CIEN CE

law MEANS uniformity by statute ⇓ ⇓ ⇓

declarative MEANS subjunctive ⇓ ⇓ ⇓

relation MEANS neighbor ⇓ ⇓ ⇓relation MEANS employer

⇓ ⇓ ⇓relation MEANS onlooker

An ia tr ogenic ques tion ari se s. Given the many -m any or

one-man y in tens ionna l-noncon str uct iv e -too ls of Standar d

Ar ithme tic , compounded by a ppeal to the s ameambi guit ie s in Ax iom s, and al l the Boundar y -cr oss ing .

Doe s t hi s, a t lea st in par t , explai n the dif f ficul tie s

st udents disp la y in le ar ning Ar ith metic ? Di f ficul tie s,

many or mo st of w hic h could be a lle vi ated by

Medodolog ies w h ic h ar e extens ional , cons tr uct iv e ,

bypa ss ing, m ini mal boundar y cr oss ing , syn tactic ,



pr agm atica ll y in di vi dual is ti c. pr esenta ble by ch il d-

friend l y pr og ram ming ?



CHAPT ER SE VEN : WHAT IS A SYNTACTIC DICTI ONARY? A

MET HOCOPOEI A?

You real ize a dic tionar y of ma thema t ical te rms can be

defined SYNTACTI CALL Y, wthout mean ings .

You know, i n a typ ical dictionar y, the RELA TION BETW EEN

DEFIE ND UM (ter m D EFIN ED) and DE FINIE NS (ter m DE FINI NG)is SE MANTIC , th at i s. betw een TERM and REF ERE NCE

(MEA NIN G) -- a "semant ic d ict ionar y" -- wher eas a TWO-

LANGUAGE DIC TIO NARY (say , Eng lish-Ger man or Engli sh -

Latin) i s a "syntact ic dict ionar y", mer el y r ep lacing the sign s

of one l angua ge by the signs of the other lagua ge, without

ci tne refer ent s i n ei ther l angua ge.

That i s, you kno w that A T WO- LANG UAGE DICTIO NARY IS(so meti mes or most ly) A SYNTACTIC DIC TIO NARY in w hich A

TERM I N LANG UAGE L IS RE PLA CE D B Y A TERM IN LANGUAGE

K -- A RELA TION ONLY BETWEE N SIG NS : THE SY NTACTIC

RELA TIO N.

But you kn ow tha t A ONE-LAN GUAGE D ICTI ONARY IS A

SEM AN TIC DIC TIO NARY i n whic h A TERM IS R EPLA CED BY A

REF ERE NT: T HE SEMA NTI C R ELA TION .

You real ize th at W HAT WE NEED TO UNDERS TAND

MATHEMA TICS (AND SCI ENCE) ! M ATHEMA TICS IS A S PE CIAL

LANGUAGE, WHICH CAN RELA TE T O AN Y UNIVER SAL

LANGUAGE.

You real ize our textbooks present ENGLISH TERMI NOLOGY

and MATHEMA TIC AL T ERMI NO LOGY -- BUT GUID ANCE IN THE

RE PLA CEM ENT S TEP (TH E CRI TICAL TEA CHI NG S TEP) IS

OMI TTE D!

You real ize A MATHEMA TICAL SYNTACTI C DI CTI ONARY wou ld

cor rect th is egregious omis sion .

You can under stand ACHIEVI NG th is mor e ef fect ivel y than i n

any other way by r esor ting to a POWERF UL DEVICE ("B NF") OF



CO MPUT ER SCIE NCE W HICH H AS WI THSTOOD MANY DE CADES

OF ANALYSIS AND CRITI CISM.h is is BNF (intr oduced i n Cha p.

1): BAckUS NORMAL FORM . renamed B ACK NAUR FORM.

You Web-l ear n tha t in 1955 , an IBM team , headed by compute r

sc ient is t, J ohn B ackus , announced comp leti on of F ORTRAN ,

the s econd (after COBOL) "hi gh-l evel pr og ramming langua ge" .To expla in i t, Bac kus developed a SYNTACTIC PROCEDURE for

DEFI NIN G TE RMS . This became kno wn as "BNF" ("Ba ckus

No rmal For m"). That, late r, i n 1960, an I BM t eam i n Vienna,

Austria , led by Peter Naur, announc ed the de velopment of

another h igh-l e ve l pr o g r am mi ng l angua ge , ALGO L. N aur

modif ied the B ackus N or mal For m and used i t to define ter ms

in ALGOl.

You Web-l ear n tha t Dut ch co mputer s cienti st , E. W. Di jk istr a,

sug gested k eep ing the in it ial s "BNF", and simpl y change the

"N" fr om " No rmal " to " Naur ".

From C ha p. i, you kno w tha t the bas ic for m was a s fol lows:

<langua ge-te rm-1>::= <langua ge-te rm-2>.

You mer el y REPLA CE TER M-O N-LEFT by TERM -ON-RI GHT.

You see you don't have to UNDERS TAND E ITH ER TERM. T he

TERM- ON-LEFT could be i n Russian, spel led by the Cyr il lic

alpha bet , and the T ERM -ON-THE-R IGHT COULD BE IN THE

ADVANCED MATH OF LIE GROUP THEORY. You s imply

RE PLA CE ONE B Y THE OTHER! Pur el y SY NTACTIC -- RELA TIN G

SIG NS , WIT HO UT RE GARD FOR MEA NING OR REFER ENCE.

You kn ow tha t Backus said he u sed the " ::=" CONNECT ORbetw een ter ms because it wouldn't be confu sed with other

nota tion. You kno w man y now dr op one of the colon s, as wi ll

be done her e. Al so, you rea lize the need for the equa l s ign as

a T ERMI NAL on the r ight sude of the f orm, so you c hange the

"=" of the lef t of t he for m t o " @" ". So you rename t hi s BNFA,

for "BACKUS NAUR FORM ADAPT ED" . You kn ow tha t the



Wikipedia ar tic le on B NF lists man y modif ica tion s of it ,

usua ll y for the sak e of a par ticu lar pr og ramm ing l angua ge.

<langua ge-te rm-1>:@ <langua ge-te rm-2>.

You REPLA CE TERM-O N-LEFT (kn own as a "nonter minal) by a

TERM- ON-RIGH T, whic h ma y be a TERMIN AL, sub ject to no

fur ther c hange in an y of the lis t of for ms at a par ticula r s ite.

You rea; ize th is bypa ss es the comp laint of a student w ho

sa ys, "T he Chem Pr of wants the ' ratio of pr essur e-to-

volume of a ga s' . Is th at some k ind of math?"

You can expla in, "You ha ve a mea sur ement of the gas

pr es sur e and of the gas volume . To get thei r rat io , you D IVID E

THE PR ESS URE MEAS UR EME NT BY THE VOLUMEME ASU REM ENT. W hen you hear 'r atio ', th ink ' di vide' !"

In B NF :

<rati o-of -pr essur e-to- vol ume >:@ <pr essur e>/< volume >.

You kn ow thi s can be refor mu lated in B NFA (but w ithout

color s) a s a pur el y mathema t ical r a t io .

<r ati o-of -thr ee-to -f our> :@ 3 /4

You note that ther e ar e no br ackets on t he RIGHT. The

brackets on the right in the "chem istr y" example are

neces sar y becau se of reference to PRESS UR E and VOLUME by

the E ng li sh langua ge of the LEFT . If you put number s i n, you

wou ld dr op the br ackets, as in th is las t case .

Repe ati ng, a " nonter minal " i s a te rm or e xpr essi on h ich needsto be replaced by a ter mina l (on the r ight or in another for m

belo w). Opti ons ar e s ymbo li zed by "| " ("o r"). You kno w some

simple in stance s of BNFA: <etc .>: @ ...

<numer als>; @ 0, 1, 2 , 3, . ..

<na tur al number s (denoted b y numer al s)>; @ 0, 1, 2 , 3, . ..

<pos it ive in te ge rs>: @ +1, +2, +3 , . ..



<n egative inte ge rs>: @ -1, -2, -3, ...

<r ati onal number s (denoted as ratio s of i nte ge rs)>: @ 1/2 , 3 /7,

-5/11, ...

The cha pter t it le ask s, "WHAT IS A METHODOC OPOE IA? "

You Web-l ear n tha t a Methocopoeia is a d atabase of teac hingmethod s m odeled on the Phar mocopoe ia pf or ph ys ic ian s.

You Web-l ear n a phar m acopoe ia i s a m edica l book contain ing

an of fic ial l i st of m edic inal d rugs , together wi th the ir

ef f icacie s and s ide -ef fects , a long with tes ted pr ocedur es . You

you r ealize that ph ys ician doesn 't have to remember all of

thi s. But you can imagine the publ ic reaction if phy sici ans

sa id ther e i s onl y one medic ina l d rug to dea l w ith a gi venmedi cal pr oble m!

You real ize th is is the s itua tion wi th regar d to mos t

mathema t ical pr ob lems . The teac her usual ly kno ws onl y one

and of fers onl y one . It ma y be ef ficac ious for th is s tudent and

"bad m edic ine" for tha t student .

You real ize th is is the p rinc ipal sour ce of IA TROGENIC

MATHEMA TICS -- MATH PROBLE MS C REATE D B Y MATHPROCEDURES , just as IA TROGENIC MEDICIN E concer ns

ME DICAL PROBLEM S C REA TED BY MEDI CAL PROC ED URE.

And you r ea li ze it is the princ ipal cau se of the " lear ning-

cycle s" we go thr ough , betw een e xtr eme s such a s "T he Ne w

Math" and "Ba ck to Basics ".

You may f ind, in educa tiona l liter atur e, many ar tic le s andpa pe rs c laim iing t hat "a sign ificant number of elementar y

school childr en" h ave lear ned SUBT RACTIO N by "the tak e-

away method" ; but a ls o o ther ar tic le s and pa per s cla im ing

tha t "a sign ifi cant number " h ave lear ned fr om " The Austrian

Method" ; e tc.



You Web-l ear n an account of a v olunte r tuto r a t Yor kvi lle

Sett lement Hou se in New Yor k City who used s ix dif ferent

method s f or tea ching fr action s, dec imal number s,

per cent a ge s to 6 dif fer ent s tudent s. After eac h m aster ed

these subjec ts, eac h taught the other s "the way he lear ned" .

You real ize th at A Methodocopaoeia , loaded on a l aptopcomputer could ena ble the teac her -on- the-l ine to find th at

method w hic h su ited the needs and bac kg round of the

ind iv idual st udent . Online tutor ial s could "walk" the tea cher

thr ough any algori thm or pr ocedu re.



CHAP TER EIGH T: THE GRAMMA R O F MATHEM ATICS

From y our stud ies , you real ize th at the e ss entia l att ribute

shar ed by G rammar and M athem atic s is con str aint .

The onl ine Mer r iam -W ebster Di ct ionar y define s "1 a; the act

of being con str ained by the st ate of be ing c hec ked,

restr ained, or compe lled to avoid or per for m so me act ion ; b : aconstr aining restr icti on, agenc y, or for ce. 2 a : repr essi on of

one's own feel ing s, beha vi our , fr om a sense of con str aint .

EMBARRASSM ENT."

Al so s har ed i s the e xperience of a chi ld , in tuit ively, le ar ning

the C ons tr aint s of Gr ammar and the Contr aints of

Mathe matics , fr om counting to s chool calcu lat ion .

You sug ge st th at, bef or e the age of fi ve, a child kno ws the

gramm ati cal dif fer ence betw een "B il ly hit Mar y! " and " Mar y

hi t B ill y,": evidence the child implic it ly the dif fer ence

betw een sub ject and ob ject i n a sentence with a tr ans it iv e

verb , su ch a s "hi t" .

You can under stand tha t the link betw een our a lpha r oot s,

flo w ering in to g ra mmar and numer ic roots can be e xpl ica ted

by a flo wchar t (a dia gram de scibed in Cha p. 5).

You Web-l ear n tha t Mathema tical Langua ge Theor y has

become a ri ch fie ld due to the r esear ch of MIT' s lingu is t,

Noa m C homsk y. That T he P HR AS E S TRUC TURE GRAMMA R of a

SIMPLE SENTEN CE EXAMPL E f orms the TREE -G RAPH:

[SE NT ENCE]/\

/ \

/ \

/ \

/ \

/ \

/ \



[S UB JECT] [P REDI CATE]

/ \ / \

/ \ / \

/ \ / \

/ \ / \

[ARTICL E] [NOUN PHRASE][VE RB PHRASE]

[OBJEC T P HRAS E]| /\ /\ / \

| / \ / \ / \

| / \ [AD VER B] [ VER B] / \

| / \ | | / \

| [ADJ ECTI VE] [NOUN] | | [ARTICLE]

[NOUN]

| | | | | | |

| | | | | | |

| | | | | | |

THE SPO TTE D DOG LOUDLY YELPED

A GREETI NG

Do you s ee how th is become s a F LOWCHART FOR C HIL DRE N

TO TEACH H OW TO HOP OUT A SE NTE NCE?

• CREATE GRAMMA R T YP E-S ETS with va riou s MEM BER S

(one or mor e of eac h S ET TERMIN ATING THE TREE) :

o ar tic les ;

o adject iv es ;

o nouns ;

o adv erb s ;

o verb s .

• STARTIN G FROM " SENTE NC E" POSITI ON , A CHILD HOPS

DOWN A TREE -B RA NC H TO A TERMI NAL ( RED) ;• THE CH ILD CALL S O UT THE TER MIN AL TYP E ( ar t ic le or

adject iv e or verb , etc.) TO T HE CALCUL ATOR;

• CALCULATOR RANDOMLY PIC KS AN ITEM OF THE GIVE N

GR AM MATIC AL T YPE FROM A S ET OF T HE M;

• The CH ILD ANNOUNCE S T HE WORD TO PRI NT ER, WHICH

PRI NTS IT OUT.



FLOWCHART

-----------

\ START /

\ /

\ /

\ /

\ /*

|

|

v

|

|

-------------->-------------*

| / \ *

| / \ / \

^ / \ / \

| / \ __ __ ___ ___ ___ _ / \

_ __ __| ___ ___ /TERMI NAL? \- YES -> -|A NNOUNCE

TYPE| -- >/LA ST \

| FLY AGAIN |- -< - NO-- --- ---- \ / -- ---- ---- ---- -

/TERMI NAL ?\- ---- ---- --- \ / __ ___ ___ ___ \ /

\ /- <- -- |FL Y A GAI N| -< -- NO- --- -\ /

\ / -- ---- ---- - \ /

\ / \ /

* \ /

*

|

YE S|

V

/ \

/ \

/STOP \

-------



CA N YOU SEE H OW TO CREATE TREE -G RAPHS F OR OTHER

SENTE NCE FORMS ?

YOU CAN DO THIS.

CH OMSKY pr oved th at EVE RY TYPE OF PHRASE STRUCTURE

GR AMMAR CORRESPO NDENDE D TO A PARTICUL AR TYPE OF

FINIT E S TATE A UTOMA TON ("mac hine" , actuall y a "l ogine" , adevi ce of F OR MAL LANGUAGEUAGE) .

A FINIT E S TATE AUTOMA TON (a.k .a. FSA, a.k.a . COMPL ETE

SEQUENTIAL MACHIN E) con si sts of :

• A FINIT E S ET, I, OF I NP UTS (homo logous to GRAM MAR-

TYPE MEM BERS) ;

•

A FINIT E S ET, S, OF I NTERNAL STATES (homologou s toGR AM MATIC AL-T YPE S);

• A FINIT E S ET, O, OF OUT PUTS (homologou s to w or ds in

sen tence);

• AN INITIAL STATE, S0 (homologou s to t ree-r oot pos it ion,

"SENTEN CE ");

• A NE XT STATE F UNCTI ON , N (homo logous to TR EE -

BRANCH);

• AN OUT PUT F UNCTI ON , T (homologou s to T ER MIN AL-BRAN CH), "or der ing w or d selec tion" .

Student s i n COMPUTER SCIE NC E use thes e FS A in plann ing

PROGRAMMI NG LAN GUAGES or P ROGRA MS.

You real ize th at t hi s bond of C ons tr aint betw een Grammar

and Mathema tic s exhibi ts fur ther sign ificance in the recent

judg ment th at an ancient Grammar ian of India -- Panini (cir ca

500BC) -- cr eated , f or t he grammar of hi s langua ge,

meta ru le s, tr an sf or mat ions , r ecur s ions th at gi ve h is s ystem

the " ri gor" of the moder n T uring for mal is m, whic h anti cip ated

the co mputer .

Al so, th at Pan ini u ses connect ives so simi la r to the moder n

powerful Ba ckus -Naur- For m (f or e xpl ica ting ter minol og y

syntact ica ll y, w ithout se mant ic r efer ence) . T hat thi s



for ma li sm is now la be led, b y many , the Pan ini -B ackus -Naur -

For m. T hat a h ighl ight of P anin i' s wor k is "Panin i' s Theor em

on R anking of Con str aints ".

You Web-l ear n tha t the Amer ican lingu ist, John J.Mc Car thy , in

hi s book, "Opt ima li ty i n Phonolog y" (onli ne i n Gogle Book s),

pr ovides two f or mu la tions of Pan ini 's T heor em.

"If a cons traint is mor e gener al t han another in the sen se

tha t the se t of i nput s whic h it non vacuous ly appl ies i nclude s

the other 's non vacuous input set , and if the two confl ict on

input s to whic h the mor e s pec ific appl ies non vacuousl y, t hen

the m or e specif ic constr aint mus t do min ate the mor e gener al

one for its ef fect s to be visible i n the g rammar . . ... Intui ti vel y,

the i dea is tha t, if the mor e s pecif ic were lower -r an ked , then ,for any input to which i t non vacuou sl y applie s, thi s ef fect

would be over -r uled by the higher -rank ed constr aint wi th

whic h it confli cts . The ut il it y of th is is tha t it al lows the

anal yst to spot eas y ranking ar gument s. "

"Panin ian Relation . Let S , G be two con str aint s as spec ific to

gener al i n in a PR if, for an y input i to whic h S a pp lie s

non vacou sl y, any par se of whic h S fails G .

Pan ini 's Theor em: Given above. S uppose thes e con str aint s

ar e par t of a hier ar chy CH on so me input i . If G >> S, then S i s

not act ive in i ."

Given the CONSTRAI NT BOND betw een Gr ammar and

Mathe matics , Pan in i' s (Gr ammar) T heor em on Rank ing of

Con str aint s mu st have a simi lar for mul ation in Mathema tic s,

whic h (unnoticed el sew her e, Wikiped ia, in par ticu lar mak es

no r efer ence t o th is sub ject) i s replete wi th Cons tr aint s, as

you r ea li ze fr om your st udie s

MATHEMA TICAL CONST RAI NTS

• The CONSTRAIN TS on the gener al connectiv e of

RELA TIO N: i n it s binar y for mat it can be M AN Y-MANY



(many peop le r ead many book s of a par ticu lar l ubr ar y) or

MANY-O NE (many peop le boar d the bus) or ONE-MANY

(one bus dr iver tr an spor ts many pa ssenger s) or ONE-ONE

(one hus band monomou sl y has one wi fe). The

CONSTRAIN TS on R ELA TION yield two i mpor tant

MATHEMA TICAL "par ts of speec h" :

o CONSTRAINI NG the R ELA TION to onl y be MANY-ONE(many i nputs for one output, or one i nput for one

output) tr ansfor ms the RELA TIO N into it s F UNCTI ON

spec ial iz ation

o the D OMAI N of a R ELA TION or it s F UNCTI ON

spec ial iz ation is the a ttr ibute of thei r I NP UT , while

thei r C OD OMAI N i s the a ttr ibute of the O UTP UT . The

CONSTRAIN T of ATT RIB UTIVE AGREEM ENT of INP UT

AND OUP UT tr an sfor ms RELA TION and F UNCTI ON into

an O PERATION. Order ing in Relat ions invokes the

Relationa l Data base, s o usefu l i n gover menta l and

cor por ate w or k. Or acle is a gian t cor por ation

develop ing R el ationa l D ata base s.

• the L OGIC AL C ONSTRAIN T of C OND ITIO NAL: sta tement P

is necs sar y for s ta te ment Q if q i mp li es P, wher as Q i s

suf fic ient for P • the L OGIC AL C ONSTRAIN T of B ICONDITI ONAL : if, onl y i f

("If f") P <-- > Q

• the L OGIC AL C ONSTRAIN T of D ISTI NGUIS HIN G EQUALITY

fr om A SSI GNMENT

• the M ATHEMA TICAL CONSTR AIN T of DISTI NG UISHI NG

EQUALS fr om EQUIVALEN CE

• the O PERAND CONSTRAINT on Natur al Number

sub tr action tha t SUBT RA HE ND NOT BE G REATER THANMIN UEND

• the O PERAND con str ain t on DIVISI ON of NONZERO

DIVIS OR

• the O PERAND con str ain t on DIVISI ON , i n Natur al Number

and Inte ger System s, tha t DIVIDE ND MUST BE M ULTIPLE

OF DIVISO R



• the O PERAND con str ain t on LOGARIT HM , in Natur al

Nu mber , Int eger , Rationa l System s, th at BASE AND

EXPONENT BE N ONCOPRIM E

• al l of the se O PERA ND C ONS TR AINT S ar e CLOSUR E

CONSTRAIN TS

• pr escedence con str aints on oper ator s.

The la st list ing , "pr escedence constr aints ", i nvokes pau se i n

the l isti ng becau se Wikiped ia de scr ibes "pr escedence" r ules

in for mal isms wi thout u si ng t he la be l "cons tr aint ".

You Web-l ear n tha t compr ehens ib le not ation without

par enthese s was developed b y the P oli sh logic ian , J an

Lukas iew icz (1878-1956), one of man y P olish logic ian s and

mathema tici ans who pr ovided excel lent work in t he 20thcentur y.)

This not ation has become kno wn as " Pol ish not ati on".

Read ing left to right, it place s oper a tor s bef or e oper ands ,

wi thout par enthe ses , yet the or dering can be par sed w ithout

ambi guit y.

In th is not ation, a problem could er roneous ly be wr itten as

+ - 10 , 3 , 2

Since the sub tr action oper ator ( "-") is near est the oper ands

and since it is a binar y oper ation (scop ing t wo oper ands), the

oper ation of 10 - 3 = 7 is perf or med and provides one oper and

(7) r equ ir ed by t he rema ini ng binar y oper ation of ad diti on

("+") , s o the oper ation is erf or med 7 + 2 = 9. (Yes, i t does tak e

longer to explain it, than to perf or m it -- for the f ir st t ine . But

I'l l show you belo w tha t you may a lr ead y kno w - - by usa ge - -

but m ay not kno w you kn ow "Rever se Pol ish notaion" , tha t is,

"suf fix P oli sh nota tion" .)

But the intended problem would be w rit ten in P oli sh nota tion

as



- + 3, 2 , 10

so th at the ad dit ion would be fir st perf or med on t he near est

oper an ds a s 4 + 2 = 5, provid ing one of the two oper ands

scoped by the subtr action oper ation , yie ldi ng 10 - 5 = 5 .

You know abo ut thi s if you have e ver used a hand ca lcul ator ,you f ind it tak es oper ands bef or e oper ator . Thu s, for t hat

"intended" pr ob lem , above, y ou w ould punc h the k eys f or the

oper an ds 10 , 3 , then punc h the subtr act ion k ey, to obtain 7.

because "10 - 3" = 7 . Next, not punc hing the "clea r" key, so

tha t 7 i s st ill "on de ck", you punc h in the 2 key, and now

punc h the ad dit ion key ("+") , to see t he ans wer, 0, since 7 + 2

= 9 .

Do you not ice tha t th is rever ses what was perf or med above i n

Pol is h not ation?

You can now under stand a f or ma li sm in log ic wher ein th is

nota tion e xce ls . (It' s a v ar iant on a game adv er tized i n

ma gaz ines for the Mens a Soc iet y of "ner ds", an appela tion

they ma y accept.)

As noted in C ha pter Se venteen , standar d s tatement log ic is t-

log ic, r estr icted to type . You kno w th is can be ea si ly lear ned

by ass ign ing alpa betic letter s to the oper ation s and oper ands

of t-l og ic, with cons tr aini ng rule s for the " wf f" or wel l- for med

for mu la, co mpar able to the w ell -formed s entence in a

uni ver sa l l angua ge.

You kn ow tha t st ate ment l ogi c concer ns on ly asser tions

whic h, potentia ll y can be deter mined as tr ue or fal se ,

You kn ow tha t the combin ator s of t-logic are:

1. conjunct ion ( "and"),

2. di sj uncti on ("or") ,

3. condit ional ( "if _, then _) ,

4. bicond iti onal ("If _ and onl y i f _ "),



5. ne gat ion ("not _") .

You see tha t the fir st four are b inar y , that i s, oper a te on t wo

sim ple or compound ass er ti ons at a ti me , wher eas the fifth is

unar y , tha t is, oper ates on a s ing le simp le or compound

as ser tion at a ti me .

Contr ar y to Men sa' s mor e co mpl ica ted assignment of letter s

for oper ator s, you a ss ign the vowels , A, E, I, O, U, to denote

these oper a tor s : "A" for "and" , "E " f or " equi val ence or

bicond iti onal ", "I" for "cond it ional " (" if") , "O " for "or" , and " U"

for "ne gation " ( "undoes") . You a ss ign con sonants as oper ands

of sim ple or noncompound pos it iv e sta tements , that i s,

wi thout N EGATION .

You kn ow you can use , for thi s, Pol is h pr ef ix not ation .

You know you can be gin w ith the Con str aint s of th is

DEFI NITIO N:

• reading left to right. The s cope of oper ator is the a llo wed

number of oper ands fol lowing it

• Whatever i s denoted by a con sonant is a t -w f f

• Whatever i s denoted by A , E , I, O , fol lowed by two t -w f f 's ,is a t -w f f

• Whatever i s denoted by U follo wed b y a t -w f f is a t -w f f .

You real ize th is mus t be cons tr ained aclo sur e ru le : Noth ing i s

a t -w f f unle ss it is constr ained by rules 1-4.

You kn ow tha t the constr aining rules given for t-logic can be

eas ily demon str ated for m odus ponent s (a .k.a . va li di ty ofas ser ting the pr ecedent ), the mos t famous of log ical pr oof

rules .

You assign con sonants P, Q as two oper and statements , tha t

is, dec lar ati ve sen tences constr a ined b y st atu s as verif ia ble

as TRU E or FALS E . Proceed ing:



• "If P, then Q" denote s the cond itiona l , "If statement P i s

TRUE, then st atement Q is TRUE". In Poli sh Prefix

No tation, th is i s for mu la ted (wi th " I" for CONDITI ONAL) :

IPQ

• "(If P, then Q) and P " denote s thi s cond itiona l as ser tion

conjuncted ("anded") w ith the st atement tha t P is

ver ifued a s TR UE ; or AIP QP • "If ((If P, then Q) and P) , then Q " denote s th at the

pr emi se , "(If P, then Q) and P ", imp lies its consequent ,

"then Q" i s so; or IAIP QP Q. You kno w an Exampl e of M P:

"If i t is rain ing , then the st reets ar e w et ; it i s raini ng;

then the streets ar e wet"

. You kno w tha t T he s tandar d w ay of PROOF is to CHECK

on pa per , whic h involves UNDERLINI NG the t-WFFS ,spac ing the ter ms , for con ven ience.

Pr oceeding :

o IAIP QPR Þ I A I P Q P Q (by R ULE 2 on CONSONANTS)

o IAIP QPR Þ I A I P Q P Q Þ I A I P Q P Q (b y RULE 3 ON I)

o IAIP QPR Þ I A I P Q P Q Þ I A I P Q P Q Þ I A I P Q P Q (by

RULE 3 on A)o IAIP QPR Þ I A I P Q P Q Þ I A I P Q P Q Þ I A I P Q P Q Þ

IAIP QPQ (b y RULE 3 on I)

o ence, IAIP QP Q = IAI PQPQ .

QED,

On pa per , y ou kno w y ou can o mi t the cha in of st eps above

by UNDERLINI NG an UND ERLI NE, but you can't eas ily

sho w tha t by the computer .

The above log ic appea rs a lso i n Cha pter Se venteen,

"W hat is t-ma th? o -ma th? ", to sho w how the above t-math

(r ecogniz ing onl y type , not or der ) can be extended to an

o-math allowing mu lt ip le tok en s of a s pecif ic ty pe .





t-logicnumers CHAPT ER NINE : WHY IS PHYSIC S C ORRI GIBLE AND

MATHEMA TICS INCO RRI GIBLE ?

You Web-l ear n tha t, in "The W orld of Mathema tic s" , edi ted by

Jame s R. New man, the Austr alian phi losopher . Douglas

Gasking, has an essay stating in the dec lar ative whatt is

stated in the in ter roga tive in the above title.

You real ize G asking meant tha t, if phy sics does not a gree

wi th r ea lity, phy sics is cor rected. But, if mathema tic s does

not a gree wi th real ity , one mathe matical for mal is m is

exchanged for another one.

You can under stand a s imple e xamp le of thi s in "T he Cas e

Wher ein T he A ver age I s Be low Aver age" . In a s ma ll subof ficeof thr ee e mpl oyees , it is found t hat "the aver age" sa lar y is

$25,000 per year, but th at two of the thr ee e mpl oyees ear n

belo w thi s aver age ea ch y ear of the ir emp loyment ther e. T he

"head" recei ves a sa lar y of $ 40 ,000 per year . The typ ist

recei ves $ 20 ,000 per year . The reception is t r ece ives $15,000

per y ear . The year ly s alar y tota l is $ (40 ,000 + 20,000 +

15,000) = $75,000 . The aver age (ar ithmet ic mean) y early

sa lar y is $ 75,000/3 = $25,000 , which p lace s t wo of theempl oyees under the average. You real ize the s eem ing

par ad ox is tha t, wher eas "aver age" should be r epr esenta tive

of the statisti cal uni ver se involved, the ari thmet ic m ean is

not r obu st under e xtr eme s of va lue , s o ma y not be

repr esent ive.You real ize t he repr esenta tive aver age i n thi s

instance is the medi an , whic h, her ein , i s $20,000 , plac ing one

empl oyee above aver age and one be low, a r epr esenta tive

resu lt .

You real ize th at the ma thema t ic s of the a rith metic mean is

not cor rected by r ejecti ng its use . Rather one mathe matics

for ma li sm is exchanged f or another . (So ma them atucs

remain s in cor ri gib le .)



You can under stand tha t the reason so man y people thi nk that

the ar ith metic mean is "the average" is th at, a s the

st atis ti cal uni ver se incr ease s in size, al l other a ver ages --

mode, med ian , geome tric mean , etc. -- appr oac hes the

ari thmet ic m ean in v alue.

You web-l ear n of a s tati stica l problem in the book. Fact s IntoFigur es , by F. J. Mor one y, a Br it is h co mmunic ati on eng ineer .

Mor oney st ated the fol lowi ng pr ob lem, w hich hundr eds h ave

so lved incor rectl y over t he year s.

• A fl ier in a "pr op" plane is flying a squar e cour se, 100

miles on a size.

• He flie s the f ir st 100 -m ile cour se , W -E, at an a ver age

speed of 100 mph .• He flie s the s econd 100 -mile cour se, N-S, at an average

speed of 200 mph.

• He flie s the t hir d 100 -mile cour se, E-W, at an a ver age

speed of 300 mph. He f lies the four th (fina l) 100 -mile

cour se, S -N , a t an average speed of 400 mph, to comple te

the s quar e-cou rse.

• Pr oble m: WHAT IS THE AVERA GE SPE ED FOR THE E NT IRE

SQUARE-COURSE?

You see tha t solution of th is pr oblem requir es onl y A DDI TIO N,

SUBT RACTI ON , MULTIPLI CATIO N, DIVISI ON , all s ubj ects

in tr oduce d by FIFT H G RAD E s tud ies !

You know tha t the typ ica l r espon se pr oceed s a s fol lows .

Thinking it is an ar ith metic mean pr oblen , the for mul ation

unf old s:

PHYSICIST 'S SOL UTI ON B Y AVERA GE SPE ED MET HOD

• Aver age speed is TOTAL D IST ANCE TRQVER SED DIVIDE D

BY TOTAL TI ME I N TRAVE RSIN G.

• What i s TOTAL DIS TANC E TRA VER SED . For a s quar e, 100

miles on a side , or 400 MILE S.



• What i s TOTAL TIME IN T RAVERSI NG ? F ind t ime f or eac h

side -cour se from i ts aver age s peed .

• Fir st 100 -mile cour se, W-E , is tr aver sed at 100 mph,

hence, time is 1 HOUR.

• Second 100 -m ile cour se , N -S, is tr aver sed a t 200 mph --

tw ice a s fast f or s ame distance, so tak es ha lf the time of

the f ir st cour se: so , t im e i s 1/2 HOUR.• Third 100 -m ile cour se , E -W, i s tr aver sed at 300 mph - -

thri ce a s fas t f or s ame dis tance, so tak es one -thir d the

ti me of the f ir st cour se: so, t im e i s 1/3 HR.

• Four th (and f inal) 100 -m ile cour se , S -N , is tr aver sed a t

400 mph -- four times as fast for sa me d istance, so t akes

one-f our th the time of the fir st cou rse: hence, time is 1/4

HR .

• TOTAL TIME : (1 + 1/2 + 1 /3 + 1/4) H R.

• Least com mon denom iana tor of the se fr actions i s 12 , so

we have: (12/12 + 6 /12 + 4/12 + 3 /12) HR = (25/12) HR.

• AVERA GE SPE ED FOR 400 -MILE COURSE IS TOTAL TIM E I N

TRAVERSIN G DIVIDED BY TIME IN TRAVERSI NG: (400

MI)/(25/12 HR) = [(400 )(12)]/25 M PH = (4800/25) MPH = 192

MPH.

You kn ow tha t, since the cor rect ans wer is 192 MPH, the

RELA TIVE ERROR com mi tted i s (250 - 192)/192% -

(58/192)% , a little over 30% r ela tive er ror.

STATISTI CIAN 'S SOL UTI ON BY "ANOTHER AVERA GE "

You kno w ther e is an average tha t yield s the co rrect

ans wer "r ight on the no se". It is T HE HARMONIC M EAN,

denoted h.

You know the ea si es t comput ati on of a har mon mean is is

to us e the Bypa ss of computing the recipr oca ls of the

number s ( inver se s! ), and tr ansf orms th is to a rith metic

mean of recipr oca ls , then tr an sfor ming the ans wer

obtained into i ts revipr ocal (in ver se) as the har monic

mean an swer.



1/h , as ari thmet ic m ean of the aver age speed s , 100 , 200 ,

300 , 400 →1/100 , 1/200, 1/300 , 1/400 , whose leas t

common denomin ator is in the fraction 1/2000

Then you h ave:

1/h = (1/4) (1/100 + 1/200 + 1/300 + 1/400) =(1/4) (12/1200 + 6/1200 + 4/1200 + 3/1200) =

(1/4) (25/1200) = 25/4800 = 1/h.

Then ,

h = 4800/25 mph = 192 mph.

You know thi s sho ws tha t the har mon ic m ean i s thea v er a ge f o r R ATE S .

So you kno w thi s i s another case of real izing an er ror in

stistical "aver age" (e xposed by the phy si ci st 's Average

Speed pr ocedur e), but not chang ing the math, mer ely

exchanging for a d if ferent mathema tica l for ma lism.

But y ou kno w th is is usua ll y not taught i n our s choo ls o r

regular col le ge class es ; and isial ly doe s not a ppear on

standar di zed tes t.

You can under stand th at t he aver age whic h often be st

repr sent s incr ease or decr ease i n economic va lue s (land ,

st. ). You Web-lea rn th at t he ea sie st way to compute a

geomet ic mean i s to u se the B ypas s of tr ansfor ming the

number s i nto thei r l ogar ithm s (in ver ses !) , so lve by

arithme tic mean method , then tak e ant ilog (inverse) of

the ans wer as the geometr ic m ean.

You know tha t the phy si cs of quantu m theor y is. in

seem img par ado x, t he science tha t s eem s mos t to

conf ound "co mmon s ense" , yet has had s ucces se s

inco mpar able in his tor y, with meas ur ements pr ed icted

wi th accur acy inompar able in h is tor y. You also kno w i t



has been commented by some phy sic ists th at these

achie vements r esul ted fr om ig noring "comon se nse " and

mak ing theor y fit e xper iment. (Ph ysi cs i s co rrig ib le!) You

also r ead onl ine t hat quantum t heor y gained , ear ly, a

signif icant number of adher ents -- which cont inued to

incr ea se w ith fur ther find ings - - because of good

exper iment s based upon the spectr oscop y whi ch hadbeen matur ing for decades bef ore "quantu m" was

ment ioned. The real it y of con sensu s tha t the

experi menta tion w as "sound", yet s eemed to contr ad ict

so muc h of "Cla ssical Phys ics", was too mu ch to ig nor e.

This everw hem ing ly s howed phy sics t o be cor rugible . At

the sa me tine , as shown in W ikiped ia, " the m athem atic s

of quantun mec han ics ", m any primar y mathe matical too ls

of Cla ss ica l Phy sic s wer e "secondaried " b y mathema tics

(suc h as matrix t heor y) pr evi ous ly unkno wn to pr acti si ng

phys ic is ts of man y year s of experience .

Im fact, ther e is no con sens us on inte rpr eta ti ons of

quantum t heor y, except the limited one th at "Quantum

Mec han ics is an algor ith m th at pr edict s exper imenta l

mea sur ement wi th great accur acy."

You kno w th at the dif ference her e is Topolog ical : math

and grammar mus t, eac h, be con str ained by i nner

con si sten cy s o one for mal is m may face r evi si on b y being

ran ked b y another f or ma li sm , yet rema in as a constr a int

(wi thin the Boundar y) . But sc ience mus t be con str ained

by an exter nal cons is tenc y, s ubj ect to "r evi si on b y

ex cis ion" .

You kno w that tea cher s mi ght get some good wor k fr om

thei r s tudent s by havi ng the m s ee if the a bove these s

need any cor recti on or emenda tion , other wise expli cating

th is per specti ve muc h, m uch mor e. You also kno w that

thi s appl ies to wr iter s of books . Good thi nking and

writ ing !



CHAP TER TEN: WHAT IS STRATEGY ? WHAT ARE TACTICS ?

You real ize th at, cur rent ly. you find no u seful def init ion of

these ter ms , nor , appar entl y, any in ter es t i n these ter ms .

You know tha t, onlIne , the We bs ter' s Ne w U na bridged

Di cti onar y defin it ion of "s tr ate gy" as " gener als hip ",

sug gesti ng th at on ly a gener al can kno w it s m eaning , andtha t the rest of u s, inc lud ing admir als , ar e not s upposed to

kn ow its mean ing.

You weblear n the s imilari ty of "s tr ate gy" to the L atin ter

"s tr atu s" (for "g round") sug gest s the def ini tion for mi litari st s,

"Str ate gy is meeting the enemy on ground s of one 's

choos ing ", and the defin it ion for nonmi li tar is ts , "Str ate gy is

deal ing w ith a pr oblem under condit ions of one 'scompetenc y."

You WE B-l ear n tha t an e xce llen t i ns tance of such mil itar y

STRATE GY is to be seen in the great 1938 fi lm, "Alexander

Nevsky" , dir ected by the bri lli ant Ser gei Ei sen ste in (1898-

1948), wi th an resound ing m us ica l s cor e by the great

compose r, Ser gei Pr okofief (1891-1953) .

On Apr il 5 , 1242, Gr and Pr ince of Novgor od and V lada mi r

(1220? -1253) le d h is foot s ol dier s onto the ice of the Lak e

Peipu s, kno wing tha t (on the g round of his choos ing), his

enemie s, the i nvading mounted kn ight s of the L ivon ian br anc h

of Teutonic Kn ight s, wer e mor e he avil y ar mor ed . M any

cr acked thr ough the ice and w er e dr owned , l eavi ng a smal le r

for ce his l esser f or ce could cha llenge . The resu lt was a great

victor y. (And w hat a Bypa ss !)

You real ize th at, w hen a teac her does not mot ivate a student

to l ear n a K nowable in a way favor ing her /hi s experience , the

teac her is, unintent iona ll y, den ying Str ate gy to the s tudent .

You lear n, from other act ions cited i n thi s book , the mean ing

of a WIN -WI N ST RATEGY. If other s agree, you WIN. But i f



so me one sug ge sts a cor rect ion or emenda tion , you WI N al so ,

because your pur pose is TO LEA RN.

You know, then , tha t, gi ven th is , you can connect to w hat i s

perha ps the g reatest epi ste mic st rate gy , the one th at t he

Canad ian mathema tic ian, Z . A. Mel zak, taught us i n hi s book,

By pas s , A Si mp le A ppr oac h to Comp le xit y , intr oduced inCha p. F our teen . B RIEFL Y, T o so lve a gi ven pr ob lem :

• you b ypa ss i t

• by transfor ming it i nto a s imilar pr ob lem whic h you kno w

how to solve;

• so lve the tr ans for med pr oblem ;

• THEN tr an sfor m the ans wer bac k i nto ter ms su ita ble f or

the or iginal pr ob lem.

This i s so simpl e i t can be eas il y dia grammed :

bypass difficult/intractable problem --------------------------->

transform| ^transform into a| |answer into

tractable| |form of originalproblem V-------------------------->problem

solve tractable problem You kn ow tha t Cha p. 14 conta ins a lis t of BYFR AMS (dia grams

of Bypa ss es) , skowing how f ami li ar thi s is i n mathe matics ,

sc ience , eng ineering , and da il y life.

You know tha t you lear n ther e t hat bypa ss ing i s not tr iviaal .

That, as noted by Me lz ak, it is, under the la be l " conjugac y" ,

the pr imar y al gori thm f or s ol ving pr oble ms i n quantum t heor y ,

lead ing t o the trans isto r, the laser, and other device s.

You know of Mel zak' s sp ecul ation tha t " Ho min id became

human b y in ter nal iz ing bypa ss" .

You real ize y ou can connect STRATE GY with the concept of

ALGO RIT HM in m athem ati cs , whi ch act iv ate s mathe matics ,

and, on the other hand, to with the concept of PROSTHESIS ,



whic h all human s need in some for m or another . (C ivi lization

is the pri mar y pr ostheti c of humanit y. The dolph in ada pt s so

wel l to i ts environment tha t it ha s no need of such a

pr osthet ic .)

You real ize the se connection s can be d ia grammed :

STRATEGY/\/ \/ \/ \

ALGORITHM/________\PROSTHESISYou real ize th is evokes an agenda for resear ch: examine the

rich mathe matical stor e of Al gori thm s (W ikiped ia exten si vel y

lists algor ith ms) to see if any A lgo rit m s ug ges ts a Str ate gy orPr os thetc; examine Str ategies to se e i f one sug gests a

Pr os thetic . You real ize th is yie lds not onl y car e for the

Di sa bled but also oppor tun iti es to contri bute to Civi lization.



CHAPTER ELEVE N: T HE " EPIST EMO LOGY-GA ME" (W HAT ARE

CONC RETI ON S? AB STRA CTIO NS ? ILLA TIO NS ?)

YoU kno w t hat Cha p. F our explic ated M etalangua ge w ith thr ee

Subs yste ms : On tolog y, Ep is temo log y, and Ax iol og y.

You kn ow tha t thr ee i mpor tant s ubs yst ems of Epi ste molog y

(s tudy of what one can kno w) ar e Concr et ions , Abstr act ions ,and Ill ati ons .

You know tha t Concr et ions ar e what w e obser ve, such as a

bear .

Tou kno w tha t A bs tr action s ar e col lect ions of concr etions , as

in the spec ie of bear . Abstr act ions can thus tr ansfor m into

concr etion s. Peirce, founder of Sem ioti cs (Cha p. Thr e)

for mulated the ter m "il lation" fr om L atin wor d for "i nfer" .

Thus, fr om the Concr etion s of a bear and it s tr acks. On one

occas ion, you kno w tha t, mer el y confr onted wi th tracks in the

mud. So you can i nfer tha t a bear m ade the se tr acks . If a bear

appe ars, you kn ow you can cla im tha t this bear made the

pr evi ous ly obser ved racks .

Thus, you kno w peop le be lie ve, fr om s uch experience s, th at

an I llation can become a Concr et ion. But y ou kno w tha t, in

gener al, th is not po ss ib le , becau se t he alle ged C oncr etion is

too s mall for obser vation or too d is tant in time .

SEQUENCE -PAR ALLEL LE ARNIN G OF CONCRETI ONS

(In se ries connected l ights , if one goe s out, all go out; not so

in par al lel . If one of lear n-s equence i s unkn own, ma y su pr ess

other s in sequence; maybe not in par alle l. )

Concretion MEANS house ⇓ ⇓ ⇓

Concretion MEANS rabbit ⇓ ⇓ ⇓

Concretion MEANS automobile ⇓ ⇓ ⇓

Concretion MEANS little girl ⇓ ⇓ ⇓



Concretion MEANS tall buildingSEQU ENCE -PAR ALLEL LE ARNIN G AB STRA CTI ONS

Abstraction MEANS color

⇓ ⇓ ⇓Abstraction MEANS gender

⇓ ⇓ ⇓

Abstraction MEANS ethnicity ⇓ ⇓ ⇓

Abstraction MEANS legality ⇓ ⇓ ⇓

Abstraction MEANS flavorSEQUE NCE-PARALLEL LEA RNI NG OF ILLA TIO NS

Illation MEANS ancestor

⇓ ⇓ ⇓

Illation MEANS Mammoth ⇓ ⇓ ⇓

Illation MEANS exoplanet ⇓ ⇓ ⇓

Illation MEANS gravity ⇓ ⇓ ⇓

Illation MEANS yesterdayThis Scide bate become s a k ind of "Ep is temo log y Ga me" . Does

thi s scientif ic c la i m " scor e as a Kno wn" ? Doe s tha t cla im ?

Sci enti st s have the ir way of "keep ing scor e" in th is

"Ep is temo log y Game" . T hey peer review pa pers. And tr y t o

dupl ica te an e xper iment tha t a s cienti st per for med and

publ is hed h is data and resu lt s. If other s achieve s imi lar

resu lts, a con sesu s be gins to develop among sc ient ists of

tha t par tici lar science .

But thi s a li ena tes non sci enti st s (puts the m out si de aBoundar y), which may explai n the accepted pub lic ig nor an ce

of sc ience and te chnolog y tha t C ar l Sa gan deplor ed in the

quota tion c ited in Cha pter One .

This i s why you kn ow tha t the onl y pr esentl y kn own Bypas s of

thi s Aliena tion is (as noted in the Methodolog ies of C ha pter

One) Ass erbi l it y Measu ring of hypothes is-



confir m ed/di sconf ir m ed - - pr ed iction s . This is eas y to

under stand and is ca lcul ated on li ne by a chil d- friendl y

computer [Goog le(a sser calc+jonha ys )].

As serbi lit y uniquel y gi ves a nonsc ient is t one ad vanta ge over

a s ci enti st in thi s "Ep is temo log y Ga me" . Scien ti st s say the y

use Log ic in t heir rea soning s. Noth ing wr ong wi th u si ng Log icfor its two a chie vements : Logic cannot l os e an y tr uth put into

it ; and Log ic sho ws th at kn owing one Kno wa ble s ome ti mes

sho w s yo u kno w another impl ied by the for mer . But Log ic i s

Con str ained never to "go bac kwar ds" (and a s Kierkegaar d

sa ys, "Li fe can onl y be under stood bac kwar ds" and B el lman

sho ws opti mal it y is attained by ad jus ting a pr eviou s sta ge to

a l ate r one .

But Logic cannot do tha t: lo se in the sco ring of t ruth ta bles .

This means tha t Log ic i s kn own to be monotonic : to remain

the s ame in t ruth ta ble va lue or i ncr ea se i n va lue, but never

to decr ea se i n tr uth value .

However, bef or e discu ssing the monotonic it y of Log ic - - which

is in compa tibl e w ith the pur pose s of sc ience and of l ife --

le t's con si der o ther l imi ta tion s.

The tr uth ta ble s of logic mer ge a ll t ruth s in one c las s,

equi val ent and in terhc hangea ble; hence, all val idit ies ar e

equi val ent and inter change able ; hence, syn tactic (not

se mantic , pr agma ti c). equi va lent to s ay ing, "If i t is r aining ,

then i t is raining ."

The bes t Log ical method of proof , modu s ponens (La tin: mode

of af fir ming -- a.k .a. l aw of detac hment, ass er tion of

pr ecedent, affi rmation of antecedent) has t he for m: ((P ->Q) &

P) - > Q. It is, by t ruth tables , equ iva lent to P&Q - > Q. In

gener al, equ iva lent to (S1&S2&. .. &Sn)-> Sn .

And it i s a ls o tri vially equi val ent to wr it ing (2 2 - 1) in b inar y

numer at ion to yield al l all one s on last l ine; hence, and-i ng

wi th th is yie lds a one to agree wi th one s in last ter m: pr oof .



Her e's a compari son to monotonic it y of Log ic: ne ver being

able to l os e - - a not ion s cience adopt s with Log ic.

Ima gine the S toc k Exchange if st ocks could "ne ver go down".

The "Bear s" of the Mar ket buy up "fall ing stock", ther eby

maki ng poss ib le, l ater, for the " Bul ls" to set up a "r al ly of

buy ing" . This i s as essentia l to a market - - and to a sci ence --as the food-c hain is in t he Envi ronment.

Sci ence is not monoton ic . You do not hear scient ists saying i t

is. But , if remi nded, they would have to agree t hat the Log ic

they u se is monoton ic.

Becau se the "r evolution s" of Relati vi ty and Qu antum Theor y

and Par tic le Theor y r ele gated n ineteenth centur y phy sics t o

ver y constr ained "Cl as si c" ranks : "belo w the s peed of light ",

or " suffic ientl y above the quantal level". This inter regnum

invoked the g rea te st tec hno logica l c hanges in h is tor y and the

mos t succe ss ful p r edict ion s and nos t accuur ate calcu la tion s

in h is tor y . That's K nowable!

The "team of Log ic and Asserbi lit y" cor rectl y des cribe

sc ient ific change , w hil e putt ing nonsc ienti st s in " the s tand s"

to w atch sc ient is ts p lay t heir par t in " the Ep is temo log y

Game".

We a ll m us t M AKE D ECI SIO NS -- ever y da y. Some mor e t han

other s. In Busines s, in pr ofi t or nonpr ofit mana gement , i n The

Mi litar y, a s par ents , etc.

When undecided , human s seek GUIDANCE i n MAKIN G

DE CISIO NS . The most fr equent mode , perha ps, is pr ayer.Since the 18th centur y, m athem atician s have gi ven s ome

thought to this pr oblem .

One su ch guide was by, not PROBABILIY , but "expecta tion".

Expla ined as follo ws. Suppose in a "fair lotter y", 1000 tickets

ar e sold. Fair ne ss imp lies th at eac h t icket has 1/1000 th

probability of be ing the wi nning tic ket . Suppo se t he Pri ze is



$500 . Then the e xpecta t ion equa ls v al u e t im e s pr oba bil it y , or

$500/1000 = 50 cent s. (To accr ue an y mone y, the ti cket pr ice

mus t be greater than th at expecta t ion ).

Histor ic examp le: the famed phi losopher , Volta ire (1694-

1778), di sco ver ed a town l otter y gi ving muc h mor e p ri ze

money than tota l co st of t icket s. B or rowing fr om fr iend s,Volta ir e bought al l the t ickets and c lai med the Pr ize.

Inves ting it shr ewdl y, he was "fi xed for life", and cou ld de vote

his ti me to ph il osoph y and w rit ing .)

No t pr oba bi li ty , but exp ecta tion as a deci si on-gu ide can be

usefu l in case s similar to the fol lowing .

The probabil ity of a fir e in your neighborhood may be ver y

sm al l. But , if it occur red, the COST to you could be V ERY

GR EAT. Not onl y pr oper ty, but y our "dear one s". S ince the

ne ga t iv e e xpecta t ion of not en suri ng a gain st f ir e is so muc h

extr eme than the C OS T OF A PREMI UM, E NS URING is the best

decus ion.

However, the g reat Swiss mathema tic ian, Danie l Ber nou ll i

(1700-82), disco vered a " par ado x" about expecta t ion .

Con side r a game in which one is of fered two choices :

• Toss a " fair co in" (head on one side , tai l on another) , to

recei ve $1 for eac h H EAD resu lt , but ga me end s w ith a

TAIL r esul t.

• The other c hoice is an outrigh t s um of mone y, s ay,

$50,000 .

What' s the par ado x? Well , RA NDOMNESS DOE S N OT RULE OUTAN "ENDLE SS" RUN OF HEA DS. So , THE EXPEC TATION OF THE

GAME IS INFI NITE . However, v as tl y mor e people ask ed would

tak e the second choice. Dec idi ng again st the g r e a t e r

exp ecta tion .

This s et of f mathe matician s and econom is ts , since that t im e,

toward con sider ation of other GUIDES . Many econom is ts ha ve



won Nobe l Pr izes in E conomic s for writ ing on thi s pr ob lem.

For example , He rber t S imon ( Nobel is t, 1978), s ug ges ted

"satis fic ing" your deci sions . Thus , a y oung woman ma y not

continue to look for the " Pr ince C har ming of her dr eam s" , but

mar ry the " bes t" of her cu rrent sui tor s. (She sat is fi ce on

"w hat's out ther e". ) S imilar ly, an potent ia l investor with a

"nes t egg" m ay not continue to look for the "chance of alifetime", but sat isf ice on the "be st " of t he cur rent of fering s.

Of al l the DE CISIO N-GUIDES so far, onl y EXPE CTATIO N use s

an ar ithme tica l measur e (PR OBABILITY) to RA NGE OVE R T HE

POSSIBILITIE S.

You now lear n another such one -- ASSER BILITY -- which

RELA TES TO LOGIC AL A SSE RTIONS THE WAY P ROBABILITYrelatES TO E VE NTS. And , like P ROBABILITY , A SSE RBILITY is A

ME ASU RE FOR DECISIO NM AKI NG UNDE R CONDITI ONS O F

UNC ERTANT Y! That is, two per sons , gi ven t he same data and

pr oceeding cor rect ly, wi ll ARRIV E A T T HE S AME

CO NCLUSIO N! (W e can pr ove thi s by AUTOMA TING T HE

PROCEDURE , a s a C OMPUTE R P ROG RAM or a CALCULA TOR

PROCEDURE.)

Given th is , you may so meti mes be le ss BE DE VILED BY

DE CISIO NS .

From the verb "ampl ify " deri ves the su bs tanti ve, "a mpl ia ti ve",

jar gon i n the ph ilo soph ical theor y of log ic .

· "Ampl iative" : A gi ven rea soning pr oces s can i ncr ease the

kn owledge a lr eady po sse ssed .

· But thi s i s be lie ved to be i mpo ssible in S T A ND AR D LOGI C ,

whic h REVE ALS "ONLY W HAT IS THERE" . STANDARD LOGIC is

al so MONOTONIC - - REMAINI NG THE SAME OR INCRE ASI NG

ONLY WHE N NE W TRUTHS AR E A DJ OIN ED -- from outs ide .

ONLY A NONM ONOTON IC MEASU RE WHIC H INCRE ASE S A ND

DE CRE ASE S C AN GUID E U S IN DECISI ONS. Log ician s sa y tha t

su ch a COR RE CTIVE is i mpo ss ible .



However, CIVILIZA TION DERI VES FROM AT LE AST TWO

AM PLIA TIVE PROCESSES :

• TRADI NG B ETW EEN HUMANS STARTED TO " SHOW A

PROFI T". T he C ALVINIST outmanue ver ing of THE ROMAN

CATHOLIC BAN ON "USURY" ACC UMULA TED OUR

PRESE NT ECONOMIC SOCIE TY.• CRUDE PREH IST ORIC DE VICE S P ROG RE SSE D I NTO

ME CH ANI CS . By defin it ion, A MA CHI NE is a PROCESS

WHOSE OUT PUT MA GNIFIES OR AMPLIFIES ITS INPU T.

Examp les :

o A LEV ER TRADE S-O FF LEN GTH FOR LOAD: Place the

FULCRUM (balancer) of the LE VER so PART ON

FOR CE- ARM i s THREE TIM ES THE LEN GT H ON LOAD-ARM, and you can RAI SE THREE TIM ES AS MUCH FOR

A GIVEN F ORCE. A LE VER IS AM PLIA TIVE .

o A PULLEY TRADES- OFF ROPE- LEN GTH FOR LOAD: A

PULLEY SYSTEM WITH THREE ROPE-L ENGTHS ACTIN G

ON TH E LOAD RAISE S T HR EE TIME S T HE LOAD F OR

GIVEN F ORCE. A PULLE Y IS AMPLIA TIVE .

o So ar e other MACHIN ES : w hee l-and -axle , inc lined

plane , scr ew, etc.o So al so ELEC TRI NES : computer s, radios , TV s ets,

ta pe and C D pla yer s, etc. .

AM PLIA TIO N in our DECISIO N- MA KING C AN U SHER US INTO A

WILD CAT REV OLUTIO N!

What can Logic do?

Her e ar e t wo INV OLUABLE PURPOS ES of Standar d Log ic

• Logic pr ovides a for m -- T HE CONDI TIO NAL A SSE RTION - -

tha t extends and "d ynam izes" the f or ma lism . This is the

for m "if A , then B" (sy mbo li zed A → B) wher e A, B ar e

dif fer ent A SS ERTIO NS . That is, "if A, then B" says " IF

YOU KNOW ONE THING, THEN YOU KNOW ANOTHER".



• Logic provides a TAUTOLOGY, "modu s ponen s" (MP) --

us ing a C ONDI TIO NAL A SSERTION - - tha t PROVES T RUT H

WHEN PRESEN T. It i s of the for m: "If A, then B; & A is

TRUE. T hen B is TRUE."

Using → for the CONDITIO NAL OPERA TOR and & for

CONJUNC TIO N, M P can be s ymbo li zed thu s:

((A → B) & A) → B.

The tautolog ica l n a tur e of M P can be ea si ly DEM ONSTRATED

in two d if fer ent ways .

• By INDICA TOR or TRUTH TABLES :

o WFir st , wr ite COL UMNS for al l POSSI BLE TRUT HSUBTABLES of ASSE RTION S A, B, ENCO DING TRUE AS

"1" , FALS E A S "0" :

A B

0 0

0 1

1 0

1 1

o A CONDITI ONAL ASSERTIO N (such a s A → B isCONSIDE RED FALSE ONLY WHEN PRECE DENT (her e,

A) IS T RUE and CO NSE QUENT (her e, B) IS FALSE --

thir d ROW in above T able. A C ONJ UNCTI ON - - s uch as

(A → B) & A -- IS TRUE ONLY WHEN BOTH CONJUNCTS

ARE T RUE, four th ROW in Table. So we can compl ete

the S ubta bles for M P, r epea ting thos e co lumn s a bove:

A B A → B (A → B) & A ((A → B) & A) → B

0 0 1 0 1

0 1 1 0 1

1 0 0 1 1

1 1 1 1 1



o The total ity of "1's " in the last SUBTABLE reveal s the

TAUTOLOGICAL na tur e of M P: IT CA NNOT B E

FALSIFIE D.

o The second way to pr ove M P u ses these CLUES: A → B

is EQUIVAL ENT T O "B INCL UDES A" ; and s ayi ng " A is

TRUE" is EQUIVAL ENT T O "A IS NOT EMPTY, a s

FALSIT Y WOU LD BE". If you draw a CIR CLE (orrectang le) f or " A" , putti ng it insi de t he CIR CLE

(rectan gle) for "B" , then a point (or asteri sk) in

CIR CLE (r ectang le) A is NECESSA RIL Y i n CIR CLE

(r ectan gle) B -- a s simple and ob viou s as tha t.o

_________________________________________________Bo |

|o ||

o | __________________________________A |

o | || |

o | || |

o | | *

| |o | | ________________________________| |

o ||

| _______________________________________________|

WARNI NG ! The f or m ((A → B ) & A) → B uses the

CONDITI ONAL OP ERATOR ( → )twice, but in TWO D IFFFERE NT WAYS, whi ch is

IMPLIED BY WHAT IS P LACED

PAREN THETIC ALL Y and WHAT IS NOT. The par t ((A →

B) & A) s ays (A → B)

IS T RUE, and a ls o A SS ERTIO N A IS T RUE. SO, THESE

TWO TRUTHS IMPL Y TRUTH-HOOD

OF ASSE RTION B.



An yone who has s tud ied the for m of a S YLL OGI SM

can under stand the above, b y homo log y. Take the

Pr otoT ype of Syl logi sm s: "A ll men ar e mor ta l.

Socr ate s i s a m an. Ther efor e, S ocr ates i s mor ta l. "

The fir st two ASSERTION S ar e PREMISES of the

SYL LOGISM . The thir d ASSERTIO N is the

CONCLUSIO N of the S YLL OGISM . This is some ti messho wn by the following f orm:

All men are mortal.Socrates is a man. __________________ Socrates is mortal.

No te how th is rese mble s, s ay, a SU M:

32 _ 5

Then you can put M P in a s imilar for m:

A → B A ______ B

That i s, the PREMISE S ar e A → B and A; the

CONCLUSIO N is B. (T his is cr it ica l i n what fol lows .)

Rest ated , M P i s of the for m: "If A, then B; & A isTRUE. T hen B i s TRUE ." U sing → for T HE

CONDITI ONAL OP ERATOR, & for CON JUNC TIO N

OPERA TOR , then MP can be symbol ized thu s:

((A → B) & A) → B.



Now, M P P ROVIDES THE FORMA T F OR FAC (FALLA CY

OF ASSE RTING THE CONSEQU ENT) , fr om which

deri ves A SSE RBILITY .

In per spect ive, MP sh ould be ca ll ed " Tautolog y of

Asser ting T he Pr ecedent" . For, given the

CONDITI ONAL STATEMEN T, (A → B) , MP thenCONJUNC TS ("A NDS") t hi s CONDITI ONAL wi th

AS SER TIO N A, DECLARI NG T HAT A SSERTIO N A IS

TRUE. But A SSSER TIO N A is the PR ECE DEN T ("par ty

of the fir st par t") of CONDITI ONAL , (A → B) , so thi s is

ASSER TIO N OF THE PREC EDE NT, and lead s to a

TAUTOLOGY. -- wi th A TRUT H-TABLE C ONSI STIN G OF

ALL ONES (f our ONES) , m eaning al l TRUES, a s se en

belo w:

A B A → B (A → B) & B ((A → B) & B) → A

0 0 1 0 1

0 1 1 1 1

0 0 0 0 1

1 1 1 1 1

A TAUTOLOGY. And TAUTOLOGIES "ar e not a bout

REALITY (outs ide ONTOLOGY) but a bout LAN GUAGE

USAGE".

But suppose WE INTER CHA NGE A (CONDI T I ONAL

PR ECE DEN T) w ith B (CONDI TIO NAL C ONS EQUE NT,

"par ty of the second par t") in the rest of t he for m.

Then , w e have: ((A → B) & B) → A.

It i s sho wn belo w th at FAC has a TRUT H TABLE (a.k .a.

INDICA TOR TABL E) NOT OF F OUR ONES (as with MP),

but T HR EE ONES and a SIN GLE ZERO (for "FALSE") .



Three "th ings ha ppen" her e:

FAC fai ls (by the SIN GLE Z ERO) to be a

TAUTOLOGY.

This FAIL URE p luc ks FAC out L AN GUAGE and

gives i t RE ALITY potenti al . IT CA N BE FALSIFIED :

EXPEC TED EVEN TS CAN FAIL TO O CCUR. But the SIN GLE FAIL URE MINI MIZES THE

DE VIA TIO N FROM TAUTOLOGY -- which gives FAC

its USEF ULN ESS and (as later sho wn) its POWER

TO GENERATE BO NUSES) .

Her e is the TABL E f or FAC:

A B A → B (A → B) & B ((A → B) & B) → A0 0 1 0 1

0 1 1 1 0

1 0 0 0 1

1 1 1 1 1

You note "0 " i n second Row of the LAST C OLUMN O F

THE TABLE . This i s the " devi ant" -- which g ives FAC

it s REALIT Y potent ial ; but it s SI NG ULARIT YMINIMI ZES DE VIA TIO N FROM TAUTOLOGY.

You note someth ing e lse. T he th ir d C OLUMN (TABLE)

above is THE SAME AS SECOND COL UMN (TABLE)

above. What doe s tha t mean ?

It m eans tha t (A → B) & B) i s TRUT H-E QU IVALEN T T O

B. Hence , ((A → B) & B ) → A is TRUTH-E QUIVALEN T

TO THE CONDITI ONAL statement, B → A ! That i s, W E

ARE G IVEN THE INFO RMA TION O F THE C ONDITIO NAL

STATEM ENT, A → B, and we c lai m to der ive fr om i t the

CONDITI ONAL STATEMEN T, B → A.

We l ear n two "th ings " fr om th is:



The for m B → A i s labeled "CONTRARY" of A → B .

So FAC can also mean "Fal lac y of Asser ting the " .

To mak e thi s for mal ism usefu l, we need to

REPLA CE "A" (IN "A → B") b y A CONJUNCTIO N OF

MA NY ASSE RTIONS o r H YP OTHES ES, tak ing the

form & i H i, with i = 1,2 ,. .. ; and we need to

RE PLA CE B (in A → B ) b y A CONJUNCTIO N OFMA NY OTH ER ASSER TIO NS or PREDIC TIO NS ,

taking the for m, & jP j, with j = 1,2 ,. .. .

Good ! This f inding about Con tr ar y mak es it EASIER to

WRI TE and PROVE A FORM ULA for this

GENE RALI ZATIO N of FAC. We simpl y work w ith (& jP j)

→ (& iH i).

The sim ple for m i s: ((H ⊃ P) & P) ⊃ H.

H ⊃ P : Hypothese s H i mplies pred iction P.

(. .. ) & P: Pr edict ion P is CON FIRME D.

((. ..) &P )⊃ H: This imp li es th at H is (po ss ibl y)

TRUE.

Rela bel li ng t he TABLES for thi s:

H P H -> P (H -> P) & P ((H -> P) & P) -> H

0 0 1 0 1

0 1 1 1 0

1 0 0 0 1

1 1 1 1 1

However, it was noted thi s uses someth ing the

Logic ian l abel s "FALLA CY OF A SSE RTING T HE

CONSEQUENT". However,

FAC FAILS BY JUS T O NE CASE OUT OF FOUR.

That s ing le FAIL URE QUALIFIES IT TO DEAL WIT H

RE ALITY , w her eas NO FAIL UR E (a s w ith MP)

TAKES IT OUT OF REALIT Y A ND PUTS IT INT O

LANGUAGE!





1 1 1 1 1 1 1

THE NUMBER OF R OWS (PO SSIBILITIES)

DOUBLE D, fr om FOUR ROWS to EIG HT ROWS.

THE NUM BER OF P OSSIBLE FAIL URES

REMAIN ED THE SAME, O NR -- NOW ONEPOSSIBILIT Y O UT OF EIGH T OF BEI NG

INCORRECT REASONING!

Cont inuing , if H P RE DICTE D O NE MORE

CO NFIRME D P RE DICTI ON ,

THE PO SSIBILITIE S W OUL D D OUB LE TO

SIXT EEN

with s til l onl y ONE POS SIBILLITY OUT O F

THE SIXT EEN OF INCORRE CT RE ASON G!

"SUC CES S B REEDS SUCCESS! " A s l ong a s THE

NUMBER OF C ONFI RMED PREDI CTIO NS FOLL OW

FROM THE SINGL E H YPOTHE SIS, the "SCO RE " ( in

the E PISTE MOLOGY GAME) get s BE TTE R and

BE TTE R. That' s the " great th ing" about FAC.

And , i ns tead of chec king b y COL UM NS OF

TABLE S, YOU' LL B E GIVEN A F ORMUL A T O DO

THAT IN A LINE O F WOR K. EVEN BETT ER, Y OU"LL

BE DIREC TED TO AN O NLINE CALCULA TOR TO

CA LCULA TE T HE ASSE RBILITY OF Y OUR ENTRI ES.

Revi ewing , fir st the s imp le f or m o f FAC,

pejor ativel y l abel led "falla cy of a sser ting theconsequence" , w hich (unl ike MP and other

tautolog ie s ) g ives FAC the potent ia l of r efe r r ing

to R ea li ty .

Its A SSE RTION S ar e "H, P " -- "H" for

"HYP OTHESIS ", "P" for "P REDI CTIO N" . So it r uns :

((H -> P) & P) -> H . That i s,



(H -> P) : "Hypo these s H implie s the pr edict ion

P" .

(. .. )& P: "Pr ed iction P is CONFIRM ED" .

((. ..) & P) - > H: "T his EVALUATES H as

(pos si bl y) T RUE".

It ha s the T ABLE :

H P H->P (H->P)&P ((H->P)&P)->H

0 0 1 0 1

0 1 1 1 0

1 0 0 0 1

1 1 1 1 1

Notice tha t the fina l Table (Col umn) ha s thr eeof ones (f or "True") and a single z er o ( for

"False") : One pos sib il it y out of four for

IN CO RR ECT RE AS ON IN G .

That can be quant ified by def ining a

M E A SU RE , whic h wi ll incr ease : ASS ERBILITY .

DEFI NITIO N: The BASIS of a TABL E equa ls

the N UMBER of SIGNS it conta ins . Abo ve,BASE is F OU R.

(You r emember t hat BASIS is a POWER OF

TWO (for the pai r, T,F)

DEFI NITIO N: The BALLOT of a C OLUMN

equal s the N UMB ER of ONE S it conta ins .

DEFI NITIO N: The AS SER BILITY of a L OGICALFORM, denoted A(F) , is the RATIO (in i ts

FIN AL COL UM N) of BALLOT to BASI S: A(F) =

BALLOT/BASIS .

Abo ve, A(FAC) = 3/4 .



Then was cons ider ed the ca se w her ein THE

SAME H YPOTHESIS Y IELD ED AN OTHE R

HY PT HESIS, AND IT WAS CO NFIRME D. That

is, one hypothes is, H, wi th t wo pr edict ions ,

P1 and P1. This took the FORM, ((H - > (P 1 & P2)

& P 1 & P 1)) -> H .

Her e are the COLIM NS for thi s:

H P1 P2 (P1&P2

H-

>(P1&P2)

((H-

>(P1&P2))&(P1&P2)

(((H-

>(P1&P2))&(P1&P2))-

>H

0 0 0 0 1 0 1

0 0 1 0 1 0 1

0 1 0 0 1 0 1

0 1 1 1 1 1 0

1 0 0 0 0 0 1

1 0 1 0 1 0 1

1 1 0 0 1 0 1

1 1 1 1 1 1 1

The BASIS for th is i s eight . Agai n, a sing le

zer o. So it s A SS ERBILIT Y i s A(F AC) = 7 /8 >3/4 , an incr ea se of ONE -EI GHTH.

What w oul d ha ppen if, g iven the sa me s ing le

HY POTHESIS , H, another CONFIRME D

PREDIC TIO N was deri ved? B ASE wou ld

DOUBLE , fr om EI GH T to SIXT EEN , wi th st ill a

single zer o, A SS ERBILIT Y W OULD BE COME

A(FAC) = 15 /16 - - AN INC REA SE of ONE -SIXT EEN TH.

You see the ANTIT ONICITY , so far, of this

CHANG E PROCES S: AS T HE ASS ERBILIT Y

INCRE ASE S, THE INCR EMEN T OF CHA NGE

DECRE ASE S. THIS PREVILS FOR ONE-



HY POTHESIS YIEL DIN G MORE CONFIRME D

PREDIC TIO NS.

However, some ca ses r equ ir e i ncr ease s in the

hypothe se s , s o w e need a F ORMUL A f or t he

GENERAL CASE , NAMELY, (((& iH i) - (& jP j)) &

(& jP j))) - > & iH i, wi th i = 1 ,2, ... ; j = 1 ,2, ...

That i s, a number (i) of hypothe se s IM PLY

another number (j) of pr edict ions , which ar e

CONFIRME D; so w e CLAIM " TRUTHNESS" OF

THE HYPOTHESIS SUM.

We need a FORM ULA on the number s i, j in

thi s GENER AL F OR M. For reason gi ven be low

file, th is GENE RAL FORM is l abeled "T he

Ockam Function" , i, j PAR AMET ERS .

FORMU LA -TH EORE M: A(O( i, j)) = 1 - 1 /2 j + 1 /2 i+j .

Try it on the s imp les t ca se, na mel y, O(1 ,1) :

A(O (1, 1)) = 1 - 1/2 1 + 1/2 1+1 = 1 - 1 /2 + 1/4 =

3/4 . Che cks .

Try i t on tha t second case (above), namel y,

O(1 , 2) : A(O (1,2)) = 1 - 1/2 2 + 1 /2 1+2 = 1 - 1/4 +

1/8 = 7 /8 . Chec ks.

The great th ing a bout O(i,j) , is what happens

in the case O(1 ,j ) w her ein j INC REA SES - - tha t

is, THE SAM E H YPOTHESIS Y IELDS MOR E

AND MORE CONFIRM ED PREDIC TIO NS .

A(O (1, j)) = 1 - 1/2 j + 1 /2 1 + j = (2 1+j - 2 + 1) /2 1 + j =

(2 1+j - 1)/2 1+j = 1 - 1 /2 j+1 .

In that L AST FRA CTIO N:

The numer ator remai ns CONSTANT at

one;



The denomin ator INCRE ASES with j;

So thi s fraction "g rows s mal ler and

smal ler ";

For some lar ge value of j, say, j = 100 , it

is "subtr act ing pr act ial ly noth ing fr om

the one"

So , wi th i ncr eased su cces s, AO(1 ,j) --GENE RALI ZED FAC -- ha s A SS ERBILIT Y

APPROACHIN G ONE! a lmo st as good as

the T AUTOLOG Y, MP!

(Students of "Ca lculu s" w il l note

rese mblance to m any "li mi t pr oces ses ". )

In fact , we find the above result as a theor emalong wi th o ther B ONUSES (theor ems) of thi s

FORMU LA .

The ASSER BILITY FORMU LA is: THEOREM

ONE:

A(O(i,j)) = 1 - 1/2j + 1/2i+j.

(Pr oof is on line (Google(pr oof .ht m+jonha ys)) .)

Many T HEOR EMS follo w fr om thi s FORM ULA --

so me M ATHEMA TICALL Y DE RIVI NG

INTUITI ON S of long standing liter atur e - -

SOME VE RY SURPRISI NG. The fir st one

FORMALIZ ES what w as noted above a s

"succes s incr ea ses ".

THEO REM T WO (PR OMI SEDL AN D T A (O(1 , j)) =

1 - 1/2 j + 1/2 1+j , THEN lim(j - > *∞)A( O(1, j)) = 1 .

That i s, "I N THE LIMIT" , A(O (1, j)) IS "AS

GOOD AS A TAUTOLOGY" , su ch as MODUS

PONE NS, the pri mar y " pr over" in LOGIC .



PROOF: As j -> & infin ', the ter ms - 1/2 j + 1 /2 1+j

GO TO ZERO, leavi ng onl y one r emain ing.

Bef or e the ne xt BONUS, a l ittle hi stor y.

W il l ia m of Oc kam , bor n in Ockham , Eng land ,

was a F ranci scan Friar of man y

accompl is hment s.

Long bef or e Gali leo or Newton, Will iam

expressed the es sent ial s of t he ph ys ica l

concept of " iner tia " .

Con sider ed m any -v al u ed l ogi c , now a

spec ial f iel d of LOGIC .

Expr es sed idea s in ter pr eted a s "Ockam 's

Razor : D on't mu lt ipl y enti tie s" -- th at i s,IF A SIM PLE EXPL ANATION O R

HYPOTHESIS (or TI P) DOES A S WELL AS A

MORE COMPLICA TED O NE, DE CIDE UP ON

THE SIMPLE ONE .

Ga li leo used " Ockam' s Ra zor" to def end " The

He li ocentri c T heor y" a ga inst the Pto loma ic

"Geocentric Theor y", because it wasSIMPLE R.

Becau se the Francis cans adv oca ted po ver ty

for the C hur ch and Wil liam ad voca ted shar ing

po w er wi th T he V a t ican C ounci l (a step

adopted by P ope John VII i n 1950), the

Chur ch denounc ed Wi lliam as a "her et ic" ,

saying it would "be no s in to kil l him".Wil liam of Ockam d is appear ed during THE

BLA CK PLA GUE , cir ca 1249-50 .

THEO REM T HR EE (OCK AM-RA ZOR- THEORE M):

A(O (i , j)) > A (O( i + 1 , j) .

That i s, REQUIRIN G ONE MOR E H YPOTHE SIS,

Hi+1 TO Y IELD TH E SAME j P RE DICTI ONS & jP



j, DECREA SES THE AS SER BILITY . That is, THE

SIMPLE R F ORM H AS G REATER ASS ERRBILITY

MEASU RE TH AN THE MORE COMPLI CATED

ONE, FOR THE SAM E P RE DICTI ON

(EX CEPTI ONS).

PROOF: A(O (i, j)) - A( O(i + 1, j )) = (1-1/2j

+1/2i +

j) - (1-1/2 j+ 1/2 i+j+1 ) = (1/2 i+j )-(1/2 i+j+1 ) = (2 -

1)/2 i+j+1 =1 /2 i+j+k+1 > 0, for all i,j >=1 .

"Ockam 's R azor " ha s hi ther to been an

ANSA NTZ: ASSUME D f or P URPOSE . But it

fol lows a s a T heor em of the F or mula .

The following LE MMA (to the next T HE OREM)

sho ws tha t the A SS ERBILIT Y Mea sur e is

OP TIMALL Y SENTITIVE TO P RE DICT ABILITY .

LEMM A: Given a "simple" argument and a

"comp lex" ar gument , r epr esented,

respect ively, by Ocka m-Funct ions A(O( i, j ))

and A(O( i + k, j )), both argument in voking the

same j conf ir med predict ions . Then , A(O (i , j))

> A (O( i + k, j )) .

PROOF : This i s, of cou rse, simpl y another

case of the OCK AM-RAZ OR-THE OREM, a lr eady

pr oven. But the "mar gin of v ictor y" wi ll be

rele vant to the Theor em whi ch follo ws:

A(O(i, j)) - A(O(i+k, j)) = (1-1/2j

+1/2i+j

) -(1-1/2j + 1/2i+j+k)=(1/2i+j) - (1/21+j+k)= (2k

-1)/2i+j+k > 0, for allk > 1.

For large va lue s of k, the compound ar gument

can be "ver y far behind" . But the following

Theor em proves that - - no matter "ho w f ar



behind" -- i t can "a lway s ca tch by mak ing

onl y one mor e confir med pr eduction than the

simpler argument.

THEO REM F OU R (TORTOISE& HAR E-

THEOREM): A(O (i + k, j + 1) > ( O(i ,j )) .

PROOF: A(O (i + k, j + 1) - A(O (i , j) = (1 - 1 /2 j+1 +

1/2 i+j+k+1 ) - (1 - 1/2 j + 1/2 i+j ) = (1/2 j - 1 /2 j+1 ) +

(1/2 i+ j+k+1 - 1 /2 i+j ) = ((2 - 1) /2 j+1 ) + ((2 K+1 - 1)/2i+j+k+1 ) = 1 /2 j+1 + ((2 k+1 - 1)/2 i+j+k+1 ) = (2 k(2 i -

1)/(2 i+j+k+1 ) > 0, for all i,j,k >= 1 .

Thu s, no matter "ho w far behind " i s the

COMP OU ND ARGUMENT WITH M ORE

HYPOTHESES , it can ALWAYS "ca tch up and

for ge ahead" of the SIMPLE R one if it

INV OKES ONE MO RE CONFIRME D P REDICTI ON

THAN THE S IMPLE R O NE. T he " Tortoi se" can

alw ays "catch up and for ge ahead" of the

"Hare".

And the " mar gin" shown above can be

cons ider able a s PARAM ETE R k. It w as t hi s

tha t sug gested the ne xt ( "WILD CAT")

THEO REM, the mos t supr is ing of a ll these

Theor ems.

To pr epar e for this , a deri vivation of an

"a ppr oxim ating " C or ol lar y fr om eac h of

Theor ems 3 and 4.

COROLL ARY (T H. F OUR) : A(O (i , j)) - A( O( i + k,

j)) = (2 k - 1)/2 i +j+k > 1/2 i+j , for all i,j ,k >= 1.

COROLL ARY (T H. FI VE): A(O(i + k, j + 1)) -

A(O (i , j)) = (2 k(2 i - 1))/(2 i+j+k+1 ) > 1/2 j+1 , for all

i,j ,k >= 1.



What ar e the consequences of the se

Cor ollar ie s?

EXAMPLE : Suppose that the compound

ar gument involved in Theor em Fi ve, and its

Cor ollar y, mus t accept twenty INDE PEN DENT

HYPOTHESES to INVOKE THREE CONFI RME DPREDIC TIO NS, wher ea s the "simp ler"

ar gument invol ved ther ein need mak e onl y

THRE INDE PEN DENT HYPOTHESE S T O

INV OKE thes e s ame THREE CO NFI RMED

PREDIC TIO NS. If we subs ti tute i = 3 , j = 3, k =

19 i n the C OROLLA RY of THEOR EM FOUR, we

find tha t its A PP ROXIMATE resu lt is 1/2 i+j =

1/2 1+3 = 1/2 4 = 1 /16 . Then we h ave:

A(O(1, 3)) - A(O(20, 3)) > 1/16,

whic h is signif icant .

Suppose , however, fur ther anal ysis of the

compounD case sug gests a crit ical

experiment deri ved from the hypothese s ofthe co mple x ar gument wi th i ts assoc ia ted

pr edict ion. Suppo se, fur the r, th at thi s

pr edict ion i s obser ved. Then the DO MIN AN CE

seen above "changes side s":

A(O(20, 4)) - A(O(1, 3)) > 1/16,

again sign ificant .

But the RELA TIVE change i s even mor e

sign ificant : 1/16 + 1/16 = 1/8 .

And we may lear n even m or e i f we MEA SU RE

the A SSERBILIT Y c hange wi thi n the comp lex



argument as it advances from thr ee to four

CONFIRME D P RE DICTI ONS:

A(O(20, 4)) - A(O(20, 3)) = (1 — 1/24 + 1

/224) - (1 — 1/23 + 1/223 =< (1/23 — 1/24) + (1/224

— 1/223) = (2 — 1)/24 +(2 — 1)/1/224 =1/24 + 1/224 =(220 + 1)/224 > 1/24 =1/16.

But the BASIS shou ld be i n the P REVIO US UNIVERS E, th at, a

BASIS of tw ent y- thr ee, so we have 220 /223 = 1 /2 3 = 1/8 , as

bef or e.

This A SS ERBILIT Y c hange is so SIG NIFIC ANT th at i t de ser ves

its own s pecif ica tion as a MEASU RE -C ONCEPT.

Def . ONE. Given an OC KAM UNIVER SE, w ith OCKAM

FUNCTIO N, A(O (i , j)) , with A(O( i, j + 1)) , the ASSER BILITY

ME ASU RE resu lt ing when THE CO NJUNCTIVE HYPOTHESIS of

thi s un iver se YIELDS ONE M ORE C ONFI RMED PREDI CTIO N. Let

p denote the numer ator of the A(O(i, j + 1)) - (O(i, j)).

Then P A(O(i, j)) denotes the PREDICTION POTENTIAL of this OCKAMFUNCTION (or its universe) iff (if, and only, if) R(O(i, j)) = r + 1.

The "predictive potential" has been symbolized by "r" and "R"(second letter of "predictive, rather than "p",which might lead toconfusion with the probabiity measure. And this result leads us toanother Theorem.

THEOREM SIX (WILDCAT THEOREM): A(O(i, j)) = 2i.

PROOF: R(O(i, j + 1)) - R(O(i, j)) = (1 - 1/2 j+ 1 + 1/2i+j+1) - (1 - 1/2 j +1/2i+j) = (1/2 j - 1/2 j+1) + (1/2i+j+1 > 1/2i+j) =(2 - 1)/2 j+1 + (1 - 2)/2i+j+1 = 1/2 j+1 - 1/2i+j+1 = (2i - 1)/2i+j+1 .

Then we have r = (2i - 1. And D(O(i,j)) = 2i - 1 + 1 = 2i , as stated.(The "+1" in the definition of the prediction potential is to removethe unsightly "- 1" in the Theorem. But what does this mean?



ANALYSIS SHOWS THAT A CONJUNCTIVE HYPOTHESIS WHICH CANYIELD ONE MORE CONFIRMED PREDICTION THAN PREVIOUSLY DOESTHEREBY INCREASES ASSERBILITY MEASURE. But WHAT PRODUCESTHE PREDICTIVE POTENTIAL? You might expect the DIFFERENCEBETWEEN "BEFORE" AND "AFTER" TO CARRY A TERM RELATED TOTHE PREDICTION PARAMETER, j. But it DOES NOT. It CARRIES A TERMRELATED TO THE HYPOTHESIS PARAMETER, i. THE PREDICTION

POTENTIAL CAME FROM WHAT THE HYPOTHESIS CLAIMED -- which isTHE RISK TAKEN BY THE INVESTER.

Then, this result is A CLARION CALL TO WILDCATTING, professionalor amateur! If one is "willing to take the risk" of "loading" theHYPOTHESES -- a procedure which scientists and philosopherspeoratively label as "ad hoc" -- then the POTENTIAL "waiting to bereleased" -- in the event "the risky hypothesis pays off" -- is ANEXPONENTIAL OF THE TOTAL NUMBER OF "RISKS TAKEN"!

THEOREM SEVEN (NEGENTROPY THEOREM): GIVEN "WILDCAT"THEOREM, THE INCREASED ASSERBILITY HAS THE FORM OF THENEGENTROPY MEASURE IN THE SZILARD-SHANNON THEORY OFINFORMATION.

PROOF: From WILDCAT THEOREM, R(O(i, j)) = 2i. In the SZILARD-SHANNON THEORY OF INFORMATION, the INFORMATION MEASURE islog2c, where c is THE NUMBER OF CHOICES in a given situation. Herewe have log22i = i BITS.

THEOREM EIGHT (NONMONOTONICITY THEOREM): A FAILEDPREDICTION MUST BECOME A NEGATED PREMISE, DECREASING THEASSERBILITY MEASURE, HENCE, NONMONOTONICITY IN LOGIC.(PROOF, obvious.)



CHAPTER TWEL VE : WHAT IS C OGNITIV E D ISS ONANC E?

(An "AXIO LOGY GAM E" ?)

Cogn iti ve di ss onance is a confl ict ive feel ing caused by

simultaneou s im pos it ion of oppos ing act ivator s. Se ver al

dif fer ent oper ator s may i nvoke th is "Ax iol og y Game":

• Langu age : brain scans may sho w tha t hear ingmul ti meaningfu lw or ds can evoke confl ict ive br ain events

fr om contr ad ictor y ideas , rese mbl ing mus ica l di ssonance

fr om oppo si ng tone s; th is can be By pas sed by langua ge

whic h is UNIVALENT - - w ith one meaning .

• Per petr ator : br ainw ash ing of a per son's v al ue s ystem by

inst il ling cer tain a tti tudes and bel iefs in a per son (often

confl icti ve) t o contr ol the per son 's thought -pa tter ns and

beha vior• Demogog y :

• Self -Impo si ti on : ad dict ion can change the v al ue s ystem ,

engendering acts not witnes sed bef ore ad dic tion

• Acc ident :

• Disease :

• Per sonal it y Do minance : (m ir ror neur ons , belo w)

Un ivalenc y is dif ficu lt to a chie ve, g iven pr esent a tti tudes

of exper ts and t eac her s and pub li sher s. Pr oto Typica l of

thi s atti tude is the follo wing.

You know tha t, although ther e i s m uch teac hi ng of

phonomes , deal ing wi th the sound of wor d component s,

ther e i s on ly one dict ionar y on li ne (and a ppar entl y none

publ ished) dea lng with the m or pheme, the uni t of"wor dines s" . (T hus, "dogs " i s one phoneme and one

sylla ble , but two m or pheme s: "dog+ s" .)

You know tha t one con sequence of th is is the f olo wing

"di ctionar y game" .

You know the Ir is h m athem ati cal phy sicis t, J. L . Sy nge

(1897-1995), invented th is game. (Synge co wrote a book



on anal yt ical d ynam ics , one used in man y uni ver sities .

Synge i s kno wn t o man y mathema tica l ph ys ici st s for

sho wing th at quantum equ ation s in vol vi ng s pi n can be

wr i tten in ter ms of qua ter nion s -- math cr eated b y hi s

fel low Iri shman , Wil lia m R owan Ha mi lt on (1805-1865) .)

You know tha t Sy nge wr ote a del ightfu l l itt le book ofessa ys whic h inc luded an es sa y abo ut the ga me of "Ci rc",

cr eated t o s how the fal lib il it y of l angua ge.

Two o r mor e per son s compete i n "C ir c" , eac h ar med w ith

a cop y of the sa me standar d d ict ionar y, under the

super vi sion of a "r efer ee" . The refer ee choo ses a defined

wor d i n the d ict ionar y th at i s defined onl y i n ter ms of

several synony ms . At " Go ", the contes tants be gin . E achchoos es to look up one of the synon yms, whic h is ther ein

defined in ter ms of sever al synon ym s, one of w hich (w hen

look ed up) is defined in ter ms of sever al s ynony ms , one

of whi ch (w hen l ook ed up) i s ... . The f ir st per son to retur n

to t he or igi nal ly ass igned wor d c rie s out " Ci rc! " (f or

"goi ng in a cir cle") and the game end s.

You kn ow the loop her e ha pp ens becau se y ou sta y in thereal m of wor ds or names . If y ou shif t out si de l angua ge to

the w or ld of r efe r ent s , the pr oces s wou ld a br upt ly end .

You kn ow tha t Cogn iti ve Di ssonance fr om Langua ge

shou ld be of pr imar y cons ider ation in th is tw ent y- second

centur y, whic h some cal l "the centur y of the brain ". You

kn ow thi s depends on great resear ch i n the l ate ninteenth

and ear ly tw ent ieth centur y by a Span ish biolog is t.

You know he has been cal led "T he Man Who 'read the

br ain" .

You know his name was Santia go Ra món y (1852-1924).

You kn ow tha t, in 1906, the fir st Nobe l Pri ze in P hy siol og y

and Med icine was aw ar ded to (Ital ian) C ami llo Go lg i

(1843-y1 926) and S antia go Ramón y C aja l for thei r



resear ch on the human ner vou s s ystem . You kno w tha t, in

par ticula r, G ol gi had ( in 1873) in it ia ted and de veloped the

tec hn ique of s ta ini ng i ndi vidua l ner ve and cel l ti ssue to

illumina te the st ructur e. ( Kno wn as the "b lac k reaction ",

it uses a weak s olution of s ilver ni tr ate - - used in

photog raphy since 1614 -- and is par ti cula rly va lua ble in

tr acing the pr oce ss es and m ost de li ca te ramif ica tion s ofcel ls .) You kno w tha t, in 1887 , R amón y Ca jal began u se

of th is te chnique , l ead ing to hi s discover y of the neur on

as the oper ati ve ele ment of t he br ain .

La ter he s et out car efull y to e xplor e the finer aspect s of

the br ain. With h is r educed s ilver ni tr ate tec hn ique he

demons tr ated neur ons and thei r connect ions so eas il y.

His in tr oduction of hi s gold chlor ide- mer cur y bi chlor idetec hn ique to demon str ate as tr ocytes was a monumenta l

contribu tion as was h is work on degener ati on and

regener ati on of t he ner vou s syste m l ater . He w il l be

remember ed a s a w or ld famou s neur opa tho logi st .

Ca jal 's i ma gina tion w as fir ed by the idea tha t the ner vous

syste m i s made up of bil li ons of separ ate ner ve cel ls .

Cajal 's work led to the con clus ion tha t the bas ic unit s ofthe ner vous system were repr esented by indi vidual

cel lul ar e lement s (w hic h Walde yer c hri stened as

"neur ons " i n 1891). This conc lusi on i s the moder n basic

princ ip le of the or gani za tion of the ner vous system .

Also, Caja l defined "the law of d ynam ic polar iz ation, "

st ating t hat the ner ve ce ll s ar e po lar ized, recei vi ng

in for mation on the ir cel l bodie s and dendr ites , andconducting infor mation to d istant loc ations thr ough

axons , which tu rned out to be a bas ic princ ipl e of t he

function ing of neur al connect ions . Cajal a ls o m ade

fundamental obser vation s on the develop ment of the

ner vous s ystem and i ts reaction to injur ies (h is volu me

"De gener ation and R egener ation of the Ne rvous Syste m"

tr ans lated and ed ited by R. M. May , London, Oxford



Un iver si ty Press, 1928, has been r e-edi ted by J. D eFel ipe

and E.G. Jones , O xfor d Un iver si ty Pr ess , 1991) .

GOLGI disco ver ed the too l used by Caja l in his s tud ies

and pr ovided outs tanding contribut ion s in man y fie lds of

cel l bio log y and of pa tho log y, and impor tant contribut ion s

al so on the st ructur e of the ner vous s ystem (such, forexample , the descrip tion of br anc hes g iven of f b y the

axon, of d if fer ent t ype s of neur ons , of glial ce ll s) .

You kn ow tha t great a chie vement s have been m ade i n

brain scann inng, but appa rentl y lit tle has been done

about the pr oblem of Cogn it ive Dis sonance fr om

Langu age .

Neur ologica l of recent y ear s have pr ovided e vidence of

mir ror neur ons , promoting imi tation of another ani mate,

par ti al ly adopt ing another 's val ue s ys tem .

A mir ror neur on is a neur on which f ir es both when an

ani mal acts and when the ani mal obs er ves the same

action perf or med by another an ima l (e specia ll y by

another an ima l of the s ame specie s. Thus, the neur on"mir rors" the beha vior of another ani mal , as though t he

obser ver w er e it se lf act ing . T hese neur ons have been

dir ect ly obser ved in pri mates , and ar e be lie ved to exi st in

humans and other s pec ies includ ing bir ds .

In human s, br ain act iv ity con si st ent w ith mir ror neur ons

has been f ound in the pr emotor cor te x and the in ferio r

parie tal cor te x .

Some sc ient ists cons ider mi rror neur ons one of the mo st

impor tant find ings of neur oscience i n the l ast decade.

Among them is V.S. Ra mac handr an, w ho bel ieves the y

might be ver y impor tant in imitation and l angua ge

acqui si ti on .



However, desp ite the popu lari ty of t hi s fie ld, to date no

plau si ble neur al or computa tional mode ls have been put

forw ar d to descr ibe how mi rror neur on act iv it y suppor ts

cognit ive funct ions su ch as imita tion .

The function of the mi rror syste m i s a subject of much

specu la tion . M any resear cher s in cognit ive neur oscienceand cogn iti ve p sycholog y con si der th at thi s sys tem

pr ovides the phy sio l ogica l mec han is m f or t he per ception

action coupl ing (see the co mmon cod ing t heor y) .

The se m irror neur ons m ay be impor tant for under stand ing

the a acti ons of other people , and f or l ear ning new sk i ll s

by i im it at ion .

Some resear cher s a ls o s pecu la te tha t m ir r o r s y s tem s

may s imu la te ob ser ved action s, and thus contr ibute to

the theor y of mind sk il l s , w hi le other s rel ate m ir ror

neur ons to l angua ge ab i li ti es .

Com mon cod ing c la ims tha t per ception and act ion contr ol

shar e repr esent ation s. An idea in it iated byA mer ican

psychol ogi st Will ia m J ame s. Amer ican neur ophy siol ogi stand Nobel pr ize w inner Roger S per ry. Sper ry sa ys the

per ception–act ion c ycle is the bas ic function of the

ner vous system .

You can onl y exhor t neur olog is ts and br ain scanner s to

work to ame li or ate co rrectib le i ns tance s of Cogni ti ve

Dissonance . B ut you can look to ways of avoid ing CD by

one-man y/ many -m any re fer ences . In Cha p, Thr ee , i t' ssho wn how to avoid one for m of the man y refer ent

pr oblem by u si ng the ne glected or misunder stood

pr agm atic for mal ism , Ther e, given a s ing le sign tha t

in vok es a d if fer ent re ference for thr ee ex per ts , this one -

one by the pr agm atic for mal ism . This can be become a

pri mar y cor rect ive for CD by sho wing ho w m any -r efe r ence

can be a voided .



One way is, for any t echnica l concept aleady assigned an

"ever yday" la be l, to rela be lled by a neolog ism scr ambled

fr om the exi st ing l abel .

Mathe matician s and s cienti st s fr equentl y i nvoke th is

ambi guit y.

Thu s, "Natur al Number s" (w hat wou ld Unn atur al Number s

be?) could be r ela bel led " Ntr l Nu mber s" .

Then the pr agma t ic for mat can explain in S tandar d ter ms.

[Nt rl N umber s]Þ[N atur al Nu mber s]

Also, " Intger s" could be renamed "Nt grs" w ith pr agm aticexplainer for other s.

[Nt grs]Þ[Inte ger s]

Then rename " Rationa l N umber s" (not cr az y number s) as

"Rtnl Nu mber s", with pr a gm atic expla iner for other s.

[Rtn l N umber s]Þ[Ra tiona l Number s]

And "Real Number s" r enamed "R l Number s" w ith

pr a gm atic expla iner for other s.

[Rl N umber s]Þ[R ea l Nu mber s]

Fina ll y, "Comp lex N umber s" renamed "Cmp lx N umber s"

with pr a gma tic ex plainer for other s.

[Cmp lx Number s]Þ[Comp lex N umber s]

With thes e perha ps non sug gest ive name s s ome Cogn it ive

Dissonance may be mini mized.



A good example of w her ein a "consc iou s" comment may

sub li m inal l y i nvoke C ognit ive D issonce.

A famil ia r com ment i s, " tha t's a s sur e as th at 2 + 2 = 4. "

Ot her w or ds have the sound of "2 ", name ly, "to, too" . And

"pl us " s ug ges ts "ad joi n", w hile "equa ls " sug gests

"e gal ity ", and "4" s ug gest s gol f.

So , does "2 + 2 = 4" become , sub liminal ly, "too , ad join to

egality golf"?

CHAP TER THIR TEE N: THE M YSTER Y O F HIST ORY

Since Inher itance fr om your pr edeces sor s v ia H ist or y is one

of your Lear ning Methodo lig ies , i t' s unf or tun ate th at His tor y

is a my ster y to so m any adul ts , teens , and c hi ldr en . I s tha t

the f au lt of h is tori ans ?

You may kno w co mments b y thr ee i mpor tant B ri ti sh wr iter s,

about the ignor ance of h is tor ians or the ir ignor ing h is tor y.

You WE Blear n Br it is h Rebecca West (1892 -1983) was a fine

writer of novels and nonficti on. You kno w her monumental

book, Bl ac k Lamb and Gr ey Falcon - - over 1200 pages was a

tr avelogue and compendiu m of hi stor y, i nc lud ing a

descr ipt ion of West and her bank er husband tr aveling in

Yugo sl avi a bef or e WWII . You m ay kno w th at, after WWII , Wes t

cover ed the tr ial s of Bri ti sh citi zens who fled to Ge rmanyduring the War to broadcast propa gan da to po ssible Brit ish

listener s. You Weblea rn West wr ote ar tic les about the ir tr ial s,

whic j a ppear ed in The N ew Yor ker Ma gaz ine , and wer e

col lected in her book, T he Meaning of Trea son . You Weblear n

tha t her peer s cal led W est "the bes t r epor ter of our time ".



It m ay inter es t y ou that, man y year s late r, West was

in ter view ed on PBS b y Bi ll Moyer s. That M oyer s asked West

how sh e f ound ti me, i n her bu sy life, to co ver The Nur ember g

Trial of N az i mil itari st s.

That W est ans wer ed, "I fel t I had t o. H istori ans ar e s uch liar s,

you kno w!"

You may kno w th at, a few year s l ate r, on PBS, Rober t McNei l,

then w ith the McNe il -Lehr er News R epor t, in ter view ed the

famou s no velist, John Le Car ré (1931- ), author of "The S py

Who Came in F rom the Col d", " Ti nker, Tai lor , So ldi er, Spy" ,

and other spy novel s -- who said, "Histor y i s t he l ie on whic h

histor ian s find consen sus ."

You may kno w of a th ir d op inion by a wr iter on histor ians ,

tha t of the great B ri ti sh dr ama tis t, Geor ge B er nar d Shaw

(1856-1950). You may kno w Shaw w rote the pla y, "Pygma lion",

ada pted as the pri ze winning musi cal , fi lmed in 1964 , " My Fair

Lady ". You WE Bl ear n th at an ear ly play of Shaw i s "T he

Devil 's Dis cip le" which w as abl y fi lmed in 1959 , sta rring Bert

Lancaste r, K irk Doug la s, and Lawr ence O livier, about the

be ginning of our Ame rican Revolut ion. You WEBlear n t hatOli vie r p lay s B ri ti sh Gener al B ur goyne , who is to ld tha t the

reinf or cement s he need s w on't be avai lable because some

one in t he War Of fice fai led to rela y hi s ur gent reques t. You

may r emember hi s commenton on th is: Burgoyne says,

"H isto rian s wi ll tel l the usua l lie ."

OUR ALPH ANUMERI C R OOTS

"T he task of the educa tor i s to m ake the chi ld 's spi ri t pa ssagain wher e it s forefather s have gone, movi ng rapidl y thr ough

cer tain stages but surpr essi ng none of them ." Henri P oincér e

(ma the matician who - - a year bef or e E in ste in' s "Relativ ity"

pa per -- described Relativ it y at " The Sain t Lou is World 's Fair "

whic h Judy Garland and her f ami ly v isited at end of the 1944

fi lm, "Meet Me in St . Loui s") .



You ar gue the T HESIS: H uman s best appr ecia te the value of

so me ar ti fact b y inter act ively go ing thr ough t he sta ges to its

development . If childr en wor k thr ough our "al phanumeric

roots ", the y may bet ter under stand and appr eci ate the

sho rtcuts of our AL PHABET and NUMERA TIO N,

ALP HA ROOTSYou sa y childr en sh ould be s hown a char t of the stages from

crude dr awing s to an alpha bet, and keep it po sted on wal l

during al l session s.

• picto gr ams (i.e., icon s of concr ete or abstr act ideas ; fir st

appe ar ed on cavewall s)

• ideo gr ams : combin ations of p ictog r am s , but often run

together ; examp le, Ch ines e i deog rams for ëye"and"water "to r epr esent "tear" ; al so in Me sopotam ian

cuneif or m i nscr ipti ons , Eg ypt ian hier og lyphic s, and al l

Ch inese char acter s ar e l ogog rams

• logo gr ams : ideo gr ams se par ated i nto words; may be

par tial ly sound -reference d as in the Sumerian langua ge t o

repr esent people 's name s

• s y lla bar y : symbol s repr esent sy lla b le s , not wor ds;

appa rentl y fir st developed by Sem ite s & Phoen ici anscir ca 1700 BC ; ada pted i n Old Hebr ew, Cypr iote, and

Per si an s cr ipts . A s y l la bar y for the Cher okee l angua ge

was developed in the la te 19th centur y b y Sequo ya (1776-

1614).

• alpha bet : symbo ls r epr esent phoneme s (unit s of sp ok en

sound) ; thus Ameri can Engli sh has about 44 phonemes ;

fir st came con sonants , then vowels l ater ; f ir st fu ll y

alpha bet ic langua ge was Gr eek

NUMERIC R OOTS

You sa y childr en sh ould be s hown a char t of the stages from

tal lies t o an Hindu -Ar abic Dec ima l N umer ation , and keep it

pos ted on wall dur ing al l se ssions .



• To count , our ances tor s used such materia l

repr esenta tion s of quanti tie s as ta ll y cuts i n bone or

ivor y or stone; cla y tok ens ; tal ly s t ic ks; knots in cor ds,

pe bble s in ba gs ; mar ks on w al ls ; wooden beads on a wi re ;

etc.

• Tal lying is a ppar entl y t he ear li es t for m of wr it ing , a s in

tha t wolf menti oned above. Cer tainl y, tal l ies pr ovided the

ear lie st f or m of bookk eep ing .

• In later Neol ith ic times a ppear ed tok ens : clay

r epr esenta tion s o f ty pes of count a b le of pr oduct s o f

ag ri cultur e or cr aft. . From these wer e a b str acted s ign s

for words , another instance of wr iti ng der iv ing f r om

pri mi t i ve ma the ma t ics .

• Dif ferent numer ic system s developed in var ious cu ltur es.

The Gr eek numer als spr ead wi th the E mpir e of A lexander

the G reat. But the se w er e often replaced wi th Roman

Nu mer al s wi th the extens ion of The Roman E mpi re.

• The Hindu s be gan de veloping the pr ecur sor s of the

numeri c s ymbo ls w e pr esent ly use, c ir ca 3r d centur y BC .

The se nu mer als wer e absorbed and mod ified in the

extensi on of I sl amic r ul e. But H i ndu-Ar a b ic num er a ls

were res isted in Eur ope for so me t ime . An y mer chantfound w ith the se i nscri ption s was l ia ble to char ge s of

rel igi ous her esy or po li tica l in trigue . Yet one w ould have

to go to the be st uni ver sities i n Ital y t o mu lt ip l y wi th

R om an nu m er als , and onl y at the University of Bologna

wou ld y ou l ear n how to div ide wi th Roman nu mer als .

• Leona rdo of Pi sa (a.k.a . Fibonacc i) (1175? -1250 ?)

intr oduce d them into Eur ope in 1201 w ith his arithmet ic

book, Liber Abacc i . They "caught on" by fac il itati ng thegreat per iods of Trade and N avi gation .

You kn ow you can 't ba lance y our c heckbook or do y our

inco me tax wi th Roman numer al s tha t preceded Hindu-Ar ab ic

numer at ion . You remember an extant letter , wr itten to an swer

a m er chant f ather , who wished to tr ain his s on for tr ade. He

was tol d th at, to lear n ad ding and s ubtr act ing i n Roman



numer ation, man y loca l school s wer e suf ficient . To lear n to

mul ti pl y i n Roman nu mer ation , a f ew co lle ges cou ld teac h

thi s. But , to l ear n to di vi de i n Roman nu mer ation , onl y one

unver sity in Eur ope cou ld t eac h h is son -- the Un iver si ty of

Bo logna. You know tha t's ho w d if ficu lt ca lcul ation w as in

Roman nu mer ation ! You kn ow in tr oduction in Eur ope of Hindu -

Arabic numer ation m ade possible the s ucceeding Age ofTrade and The Age of Navig ation , lead ing to d iscover y of thi s

continent .

You know the e vent in it iating deci mal numer ation was

publ ica tion , i n 1201, of "L iber Abac i", b y Leonar do of Pi sa

(1175? -1250) -- not to be confused with Leona rdo da Vinci ,

who lived m uch later . You kno w of Leonar do of P isa , a ls o

cal led "Fibonacc i" ("Son of Fibon"). But uou know tha thi stor ioan s w ri te l itt le o r l itt le or noth ing a bout t hi s event

whic h changed ci vi lization!

Pr oto- Mathem atics con si st s of some of the kno wable s of our

"pr im it ive" ance stor s which l ate r became for mal ized a s

mathema tics .

The Pr oto- Maths ci ted in the liter atur e, such as ta ll yi ng, ar equas i- mathema tical - - "half -way ther e" . But ther e m us t h ave

been mor e rud imentar y kn owables bef or e thes e "qua si 's" .

To count , our ances tor s used su ch mate ria l repr esenta tion s

of quanti tie s as ta ll y cuts i n bone or ivor y or stone ; clay

tok ens ; tal l y s ti cks ; knot s in cor ds , pe bb le s in ba gs ; ma rks

on w a ll s ; w o oden bead s on a w i r e ; etc.

Tal lying is a ppar entl y t he ear li es t for m of wr it ing . A wol f

bone found in Eur ope , da ting fr om the per iod of 30,000 -25,000

B. C ., shows f ifty -f ive cut s in group s of fi ve. Ce rtain ly, tal l ies

pr ovided the ear li es t f or m of bookk ee ping .

You know why some f inancia l inves tments ar e cal led

"s toc ks" . A st ock is a stou t s ti ck. Thus, a st ockade i s a f ort

or p ri son bor der ed b y tal l sti cks, dr iven into t he ground ,



edged togethe r, to keep domes ti c an ima ls i n, other s out .

Cattle or other ani mal s fenced i n by st icks con st ituted f a r m

stock .

When a c lient gave m oney to a broker to invest, the broker

br oke a s tick in t wo piece s, keeping par t for himself , the

other par t g iven to the i nvestor . W hen t he inves tor retur nedto c lai m ca pital or inter est , he i denti fied hi mself by fitti ng h is

br oken st ock t o th at of the i nvestor . Le gend has i t tha t " Wall

Str eet" be gan under a chestnut tree on lower M anha ttan

Is land . T he tr ee pr ovided a pl enitude of sticks to r epr esent

stocks.

This w as one for m of a ta l ly st ick to repr esent a count of

so mething . We l ear n in schoo l to ta ll y counta bles .

Once onl y a small e li te could read, write, and calcu late.

Il li ter ate cler ks used not ched ta ll y st ic ks to inventor y coin s

or bar s of gold or silver, etc . Thus one notc h touc hed

repr esented a co in counted . W hen al l notc hes of a sti ck had

been ass igned , t he sti ck was la id aside , and a new ta ll y sti ck

accounted for mor e coin s. This was once the pr ocedur e in

The Exchequer -- the Eng lish tr easur y, so- cal led becauseorig ina ll y, of fic ial s sat at a ta ble cover ed with a c hec ker ed

cloth .

Long bef or e Dicken s' time, liter ate cle rks of T he E xchequer

ceased to use tal ly st icks . In 1724 , tr ea sur y of fic ial s

commanded t hat tal lies no longer be used , but the y long

remained valid.

Sa id Dickens , ".. . i t took unt il 1826 to get the se st icks

aboli shed . In 1834 . .. ther e was a cons ider able accumu la tion

of them . ... [W]ha t was to be done w ith such wor n-out wor m-

eaten , r ot ten old b it s of wood ? The sticks w er e hou sed i n

Westmi ns ter , and it wou ld n atur all y occur to any i nte ll igent

per son tha t nothing could be eas ier than to allo w them to be

car ried aw ay f or f ir ewood by the mi ser able peop le w ho l ived





You know tha t soon the se spr ead over E urope. That the

Dome sda y B ook i n Eng land lis ts mor e than five t housand su ch

mills.

However, you kno w th at these event s w er e obscur ed by the

"B lac k Pla gue" ( "B lac k Dea th") i n the m id dle of the 13th

centur y. You kno w thi s ep idemi c a ppar entl y de velopedbecause of wi despr ead ki ll ing of c ats , which k il led the rats,

whic h car ried the l ice, which car ried the d is eas e. You found

onl ine, to the hymn, "Let All Mor ta l Fles h K eep S ilence" a

descr ipt ion of how "noca t" caused thi s pla gue .

nocat pipes the rats through our Heartland,Ravaging our fields and our stores!

nocat and the rats pipe the Black Plague,Blooming forth in boils -- bloody sores!nocat, rats, and Plague -- they pipe Apocalypse Four:FAMINE, PESTILENCE, DEATH, AND WAR!

You kn ow tha t. in so me sector s of Eur ope, 50% of the

populace died fr om the pla gue. That, in man y towns , ther e

sur vived no b lac ksm ith s, car penter s, cooper s, or other

ar ti san s.

Becau se tr aining had been or al among i lliter ate m en, the

Pope now allo wed tr ansl ation of "pa gan" tr aining texts . Given

both of these di spen sation s (to monk s and f or tr aining) ,

scholar s now be gged per mission to trans late l iter atur e and

phi los ophy . And thi s l ed t o " The Rena is sance" - - be ginn ing

wi th an ignor ed indu str ia l revolut ion.( Do y ou wonder that

muc h of his tor y i s a myster y? )

The Amish of pr esent -da y Pennsy lvan ia remain in the

tec hno log y of the me chanica l revo lut ion . (You see that i n the

1985 fil m, W i tnes s , st ar ri ng H ar ris on F or d and Kell y McG il lis.

You know tha t tha t Civi l War (War Between the S ta te s) might

be con sider ed as A War of Indus tria l Revolut ion s, s ince the

Conf eder acy r emained (wi th the Amish) in The Mec han ical

Revolut ion, w hile the spr ead of The Thermodyna mic



Revolut ion in the Nor th allo wed the great pr oduct ion of

resour ce s th at contributed to the U nion w inn ing the War.

You know tha t the Ther mod ynam ic Industr ial R evoluti on w as

followed by The E lectr ica l Indu str ial R evo luion and T he

Electr onic Indust ria l Revo lut ion , and we ar e no w i n the m id st

of a Nano -Photon ic Indu str ial R evo lution . Also, You kno w th atall thi s has been neglected by our his torian s!

You kn ow thei r s our ces contr adict the hi stor y they w rite . You

kn ow tha t even s cienti fic hi sto rian s cr edit Isaac Newton

(1642-1727) with the r ot ational equa tions of mechan ics . Yet ,

as you kno w, reco rds s how the se equa ti ons didn't appear

unti l decades after New ton's dea th, in the works of Swiss

Leonha rd E uler (1707-83) , one of the five or six g reate stmathema tici ans of a ll t ime s, and the most prol ific

mathema tici an in h is tor y.

You know, but hi sto rian s often fail to empha si ze, tha t Eu ler

founded two v ast fie ld s of mathema tics : topo log y (w hic h

inc ludes geom etr y as a s pec ial case) and combin ator ics , the

math beh ind the repr esenta tion of our republ ic , and the

choi ces of our commer cial ma rk ets . You kno w tha t, not onl yar e Ameri cans ig nor ant of Leonar d Eu ler , but neither do t hey

kn ow of the great contr ibuti ons of Swi ss mathema tici ans in

the 17th and 18th centuri es : Jacob Ber nou ll i (1664-17- 6) and

Joh anne s B er noul li (1667 -1748), whose mathe matics f ounded

"vari ational anal ys is" , w hich achie ved mor e in phy sics than

the N ewton ian m ethodolog y. Dan ie l B er noul li (1700-1782), son

of Johann es, taught tha t hea t i s t he mot ion of mo lecule s ; and

his Ber nou ll i Pr inci p le explai ns flight of air cr aft.

You kn ow tha t, in the popular 1949 film, " The Thir d Man" , the

char acter H ar ry Lyme (pla yed by O rson Well s) sneer s at Swiss

histor y as one of "cuc koo cloc ks" .

And you kno w t hat sc ient is ts bear some respon si bi li ty for

thi s. You kn ow one fail ur e involves that famou s A mer ican,



Ben jami n F rankl in (1706-90). You know sour ce s cr edit

Frank lin with founding two vast sc iences i n phy si cs . That

Franklin founded m e teor olog y by not ing tha t so me s tor ms

t rave l . You kno w tha t, as mo st pri nter s of the ti me, Frankl in

publ ished a news paper. T hat, to get new s, he went often to

the b ig Far mer' s Mar ket in Ph ilade lphi a. That, one day , hi s

con ver sation s wi th v isit ing far mer s made h im real ize tha t ast or m recentl y v isiting Ph il adelph ia r esemb led a st or m

obser ved ea rlier in wester n P enn sy lvan ia. T hi s idea -- th at

st or ms tr avel -- eventual ly led to the development of s ynopti c

char ts of the continent being drawn s everal time s dai lyy.

You know the other field tha t Frankl in f ounded was tha t of

elect ric it y . Not wi th th at sil ly and danger ous key on kite

st ri ng explo it ! Y ou'v e read th at the f amous mathema tica lphy sicist , Sir Edmund Whittak er, in h is book, A Histor y of the

T heor y of Aether & E lecr oma gnet is m , says (p . 53) th at

Franklin founded T he Law of Conser vation of Electr ic

Chhar ge, by not ing t hat elect ric char ge i s ne ver los t . If i t

di sa ppear s one place, it r ea ppear s at another .

You kn ow tha t Whittak er cr edit s the Br it is h-A mer ican

sc ient is t, Joseph Pr ie st ley (1733 -1804), w ith the otherdi sco ver y founding the s cience of electr ic ity : the in ver se

squar e la w of electr ic fie ld , s imi lar to Newton' s in ver se

squar e la w f o r g r a v it y . (It was g iven it s pr esent

mathema tical form by Cha rles- August in de C oulo mb (1736-

1806), hence is gener al ly kno wn as " Cou lomb' s Law" .) Y ou

kn ow tha t Frankli n he lped h is fr iend Pr ies tle y di sco ver this

law b y shar ing an i mpor tant obser vation: if you put char ge

inside a meta l hemi spher e -- sa y, via a Ley den jar, su ch aswas used then to e xtr act e lectr ic char ge -- then elect rric

char ge disa ppear ed w ith in, but was found ar ound the rim of

the he mi spher e.

You kn ow cr edit for thes e d iscoveri es should be w ide spr ead.

The Smith sonian Museum shou ld have a regula r e xhib it to

mak e the publ ic know about the se a chie vement s of F rankl in.







That D ANTIT ONE S and CANTIT ONE S m odel va riou s pr oces se s.

You real ize th at, as pr ototype , the D anti tone can be simpl y

explained to chi ldr en or t een-a ger s in ter ms of da il y

pr oces ses .

You may kno w th at Amer ican mathe matit ion, Norber t WIE NE R

(1894-1954), in h is book , C yber net ic s (1948 ) , descibe s the

Dant itone i n di scu ss ing CLIM BIN G UP OR D OWN STAIR S. That

in climb ing, T HE MAXT ONE IS T HE SET OF RISERS UPSTAIRS ;

THE MINT ON E IS THE D IST AN CE F ROM THE T OP. (In

descend ing, the role s ar e rever sed .) Y ou r ea li ze tha t eac h

RISE R CORRESP ONDS TO A UNIT DIS TANCE FROM THE TOP.

Obviou sl y, the N UMBER OF RISER S O F THE S TAIRS IS

BO UND ED, so THI S I ND UCES THE NUM BER OF D EC REASES .

YOU Realize th at find ing a sock in a dr awer, or a folder in a

fi li ng ca binet is DAN TIT ONIC .

You WEBlear n tha t, as Pr ototype , the C ANTIT ON E ca b MODEL

THE LIMIT P ROCESS in CALCUL US . (You r ea li ze tha t th is can

be compar ed with the calcu lu s exp lic ation in C ha pter 16 .)

You real ize th at, i n a (CAN TIT ONIC) LIMIT PROCESS:

• the O RDE RIN G is, to r epea t, CONTIN UOUS or ANALOGIC ,

not D ISC RETE or DIGITAL, a s in the case s above;

• the M AXT ONE cons is ts of the INCRE ASIN G SEQUENCE OF

TERMS ;

• the MI NTONE cons is ts of "di st ance fr om the l imi t" ;

• by CHOOSIN G an EPSIL ON-DIS TANCE fr om LIMIT L, the

MAXTONE IS TE NTATIV ELY BO UNDED;• thi s INDUCE S a DE LTA- BOUND on THE MINT ON E of

SEQUENTS.

• the S PE CIFICA TIO N ALLOWS RE PEA TED CHOICES ON THE

MAXTONE , NE CESSA RIL Y INDU CIN G BOUND O N THE

MINT ON E.

• This " real zes" THE LIMIT of THE SEQUENCE .



• This M OD EL can be ada pted f or a ll LIMIT P ROC ESSE S I N

ANALYSIS .

You real ize th at one of the mo st impor tant Ant iton ic

Pr oces se s was disco ver ed ear ly in the hi stor y of scimath.

Py tha gor as is suppo sed to h ave di sco ver e tha t. when you

subd iv ide a fi xed str ing and pluck one su binter va l, the pi tchof the tone i ncr ease s whi le the l ength of the s tr ing remai ns

constant . The maxtone is tonal p itc h; the m in tone is l ength of

st ri ng subin ter va sl pluc ked .

As to B ypas si ng, you WEBl ear n th at C anadian mathema tici an

mathema tici an, Z. A. Me lzak, taught us about th is in his book,

"Bypa ss es , A Simp le Way to Cope w ith Comple xity ". Me lz ak

sa ys "Man evolved b y lear ning to cope wi th co mple xi ty. .... Itmay even be th at the b ypas s pr inci ple , in it s var ious aspect s,

was so s ucces sfu l a mean s of cop ing with comple xity th at the

evolvi ng ho min id ended up b y in terior izing it , and so became

man. "

You lear n tha t A B YPASS has the for m:

difficult/impossible/desired task------------------------->

transform: | ^transform backpossible | |to terms of

or easy or | |original taskdesired | |

task| |V------------------------>

perform task

You real ize ho w N ONTRI VIAL th is ST RATEGY by W EBlear ningtha t, in MATHEMA TIC S, IT DETERMI NE S T HE EIGENVALUES O F

A MATRIX OR M ULTIVE CTOR. That, i n Phy sics, IT IS THE

PRI NCIP AL T OOL O F QUANTUM MATH FOR FIN DIN G THE

STABL E S TATES OR RADIA TION STATES OF FUNDAMENTAL

PARTICL ES.



Doe s con juga t ion r e la te anti tone and bypa ss ? T he s tr a te g y

kn own as "The C onjuga cy Pr inc iple ", or the mathe matical

oper ation kno wn as con juga t ion , is the pr imar y eval uator of

exp er iment s in quant ic theor y. Perh aps be gan in g r oup theor y

(Cha p.22) wher e conjugac y appl ies to sub g roups .

Given group G, wi th e lement s d, e, . .., s. t . d-1

is the in ver se ofof d, with conca tena t ion a s group oper ation . And cons ider the

resu lt :

ded -1 = fIf the resu lt , f is also i n the g roup , then e,f ar e s aid to be

m utua l con juga te s . As an equi val ence re la t ion (w ith

pr oper tie s of re f le xiv it y, sym metr y, tr an si ti v it y ), con jugac y

par ti tion s a g roup i nto equi val ence c la ss es .

Con jugac y ha s the graph :

f-------------------------> | ^| |

e| |ded | || |V------------------------>

deThe Pr incip le o f C o mjugac y appl ies i n dif fer ent f orms in

dif ferent f iel ds and as "bypas s" (the name and not ion of

Canad ian mathema tic ian, Z . A. Mel zak) pr ovide s perha ps the

mos t powerful mean s we have for inventi on and infor mation

resear ch.

Appl ied to ma tr ices and vector s , it deter mines the

eigen value s and e ign vector s of a gi ven vector . These are

"fi xpoint s" of the vecto r: conjuga tion a ppli ed to them res ult s

in no change . The resu ltant ma tri x ha s al l zer o entr ies except

for the dia gonal , so tha t the " deter minant" of the matrix is

eas ily calcu lated.



When the vector r epr esents a wave function in quant ics , this

resu lt s in pr oba bi l i t ie s for the meas ur es of the w ave vector ,

pr oviding the pri mar y r esul t i n quantic s.

You may kno w Me lz ak taught us i n hi s book th st HOMOLOGY

IS A SPE CIAL CA SE OF BYPASS i n the f or m in the for m of the

theor y of pr opor tions ; in t he for m of kennings i n Ang lo -Sax onliter atur e; etc.

You kn ow you " fle sh out" thi s ans atz :

• extensi vely desc ribe bypas ses in the tool -mak ing and

sur vival tact ic s of ani mal s and hom in ids ;

• use HOMOLOGY to TRANSF ORM THE SE EXT RIN SIC

BYPASSES INT O INTERI OR BYPASSES ;

• sho w how thi s pr oce ss develops it s the pur est f or m in

LOGICO-MATHEMATICS;

• find homol ogie s betw een non-s ci entif ic bypa ss es and

sc ient ific one s.

You kn ow tha t, in Cha p. Nine , the ted ium of di rectl y

calcu la ting a har monic mean i s br t he Bypa ss of computi ng

the r ec ipr ocal s of the numer s (in ver ses !) , and tr ansfor ms thi s

to a rith metic mean of rec ipr ocals , then tr ans for ming the

ans wer obta ined into it s r ec ipr ocal (in ver se) a s the har monic

mean an swer.

You how you a ls o lear ned in Cha p. 9 th at the ted ium of a

geometr ic m ean is Bypas sed by tran sfor ming the number s

in to the ir logar ith ms (inverse s!) , so lve b y ari thmet ic m ean

method , then tak e ant il og ( inver se) of the ans wer a s the

geometr ic m ean.

Another "c la ss ic" Bypas s is i n the anc ient " Came l Pr oble m" .

An Arabic man is riding a ca mel acr oss a deser t expanse ,

when he encounter s a novel sight. Three young A rabic m en

ar e fier cel y ar guing , s ur rounded by 17 came ls . D ism ounting ,

the s tranger was tol d the proble m. Their father had died ,



leaving (as the ir onl y r ea l i nher itance) the se 17 ca mel s. Now,

the e ldes t son was to recei ve ha lf of the came ls; the second

son , one-th ir d of the came ls ; the younge st son , one-n inth of

the ca mel s. Problem : how cou ld the y thus div ide the 17

camel s?

The str anger adjo ined hi s camel to the col lect ion, mak ing i t18 came ls. T hen, the str anger appor tioned 9 (= 1/2(18))

camel s to the e lde st son ; 6 (= 1/3(18)) came ls to the 2nd son;

2 (= 1/9(18)) came ls to the youngest son. Having s olved the

problem and a ssua ged the ir argument, the s tr anger mounted

his own came l and rode aw ay.

You kn ow tha t thi s pr oblem (extending the quantit y to obtain

a s ol uti on, then r etu rning to t he or igi nal quanti ty) has i tscounter par t in a v ast su bject of mathem atic s -- linear

pr og ramm ing - - and tha t an algor ith m f or one of it s pr ob lem

types changed histor y.

After Wor ld War II, Berlin was an "island" sur rounded b y the

Soviet -domi nated Eas t G er many , and Ber lin was also

par ti tioned into W est and Ea st Be rlin . In 1948 , the Soviet s

tr ied to force A mer ican, B ri ti sh and F renc h for ces out ofBe rlin by bloc kading land route s t o the se ctor s eac h of thes e

power s occupied . T his B er lin B lo ckade was thw ar ted (unti l its

aban donment in Sept ., 1949) by a mas sive a ir litt of food , fuel ,

and other suppl ies needed by Berliner s. The succes s of this

air lift , with a limited number of a ircraft, was primar ily due to

car eful plann ing u si ng a mathema tical too l, l inear

pr og ramm ing , finding so lut ion s t o i ts problem s by mean s of a

s im ple x a lgor ithm developed by an Amer ican mathe matician ,Geor ge D antz ig .

The mathema tical pur pose of l inear pr og ramm ing i s to f ind a

sub set of num ber s fr om a pr escr ibed s et of num ber s w h ic h

MA XIMIZES or MINIMI ZES a g iv en pol ynom ial (a lge br aic) for m .

A repr esent ati ve ca se is kno wn as "The Di et Proble m": ho w to

pr escr ibe a d iet whic h wi l l M AXIMIZE NO UR ISH MEN T w hi le



MINIMI ZIN G CO ST . In the Be rlin Bl ockade ca se, a giant flight

plan sh ould MA XIMIZE TH E SU PPL Y LOA D F LOWN w hil e

MINIMI ZIN G the a ir cr aft and per sonnel in vol ved . Typ ical ly,

constr aints on the pr oblem ar e f or mu la ted as a se t of

pol ynomi al in equal iti es , w hi ch g raph a s a s ector in an n-

dimens iona l r eg ion , w her e n i s t he number of con str aint s .

Dant zig 's s imp lex algo rith m i ter at iv e ly "w hitt les " the

rela tion sh ip do wn to a so lut ion .

After the succe ss of Dant zig 's work became kno wn to the

mathema tical wor ld and so me of the gene ral pub lic , it

became kno wn th at a Sovi et m athem atician , Leonid

Kantor ovi ch (1912 -86), had ea rlier obtained thes e r esul ts . Bu t

Kanto ri vi ch's math w as ignor ed , after being cr it ici zed,

because it s eemed in confl ict with Marxi st dogma .

As in the Leontief ca se , m ath was used w ith su cces s. And,

again, the la ck of publ ici ty for s uch method s left the publ ic

ignor ant tha t math w as of any po li tica l use , and students

were lef t w ithout su ppor t.

BYPAS S can (as with any pr oceSs ) GRAPH the s ta ge s of a

MAT HEMA TICAL DE RIV ATIO N , maing it ea sier to under standand lear n, a s in the followi ng impor tant DERIV ATIO N.

BYPASS TO Q UADRATIC FORMULA

need this be monic ax2+bx+c=0------------------------->

remove | ^result:leading| |x2

constant| |+ (b/a)x| |+ c/a (1)| |V------------------------> divide equation by "a"

Need for m w ith liter als on other side fr om " x" ter ms. S tar t

wi th m onic for m, (1), and ad jus t.

x2 + (b/a)x + c/a = 0

------------------------->



literals | ^result:in term| |x2

"c/a"| |+ (b/a)x =| |- c/a (2)| |V------------------------>

subtract "c/a" from both sides

Need l inear equa tion fr om quadr atic. star t with model ofperf ect squar e for m.

(x + k)2 = x2 + 2kx + k2 = 0-----------------------

linear| ^transform (1) toterm | |(x+b/2a)2 +is | |(b/a)x +"k"| |b2/4a2

| |= 0 (3)V------------------------> "2k" matches "b/a" in (2)

(3) dif fer s fr om (2) b y ext ra ter m. Star t wi th (2) and ad jus t to

(3),

(x+b/2a)2+(b/a)x+b2/4a2=0-----------------------

extra| ^(x + b/2a)2=term is| |b2/4a2

"(b/2a)2"| |- c/a (4)| || |V--------------------->

add term Left-s ide of (4) is squar e-r oota ble , but r ight -s ide need s

common denomin ator . S tar t with (4) and adju st.

(x+b/2a)2+(b/a)x+b2/4a2=0-----------------------

least| ^(x + b/2a)2=common| |(b2-4ac)/4a2

deniminator| |(5)is| |

"4a2"| |V--------------------->

convert to common denominator



Can obtain a l inear for m by tak ing squar e root of both sides .

Star t with (5) and ad jus t.

(x + b/2a)2=(b2-4ac)/4a2

-----------------------becomes| ^x + b/2alinear| |= ±(√b2-4ac)/2a

by| |(6)square-| |rooting| |

V---------------------> perform square-root

Want quadr atic f or mu la on ri ght=s ide .

x + b/2a = ±(√b2-4ac)/2a-----------------------

trans- | ^x = - b/2aform | |±(√b2-4ac)/2a| |standard form | |(7)| |V--------------------->

subract b/2a from both sidesThus, gener al quadr ati c equa tion con ver ted t o quadr atic

for mala by seven B YPASSE S.

You unde rstand the se BYGRAMS: Bypas s Di agrams;Humanizing Hominid --------------------->

Hominid| Z. A. Melzak ^Hominid develops| says "Hominid |becomesexternal| may have become |HumanBypasses| human by interior-|thru(such as| izing bypassing" |Bypassingsome of | |those | |below) | |

V--------------------> Interiorizes Bypassing

_____________________________________________________________________________

How to GLIDE along--------------------->

From | ^Restoreupright| |stable





Sharing thoughts another--------------------->

Transform| LANGUAGE AS ^Heard wordsthoughts | BYPASS |transformed

into words| |into thoughts| |in hearer V-------------------->

Speak words_____________________________________________________________________________

Measuring inaccessible object--------------------->

Measure| ^shadow (b)| |Proportion:

cast by| |a:b::x:c or shadow-stick| |a/b = x/c

of known| |hence,length (a):| |x = ac/b tangent of| |

shadow-triangle| |(Erastothenes estimated /|a | | the diameter of the

/_| | | Earth by a similar b V--------------------> Bypass)

Measure shadow cast atsame time by object(say, pyramid) /|another tangent / |x

/__|c

_______________________________________________________________________________

(Arab riding camel in de-Solve problem with insu- sert sees 3 Arab brothersficient structure quarreling, amid 17 ---------------------> camels. Seems their father

Puts| ^Re- willed elder 1/2 camels;his| |claims 2nd, 1/3 camels; youngestcamel| |his 1/9 camels. Division pro-with| |camel vokes quarrel. Stranger

other| |& de- adjoins his camel to camels| |parts theirs. Assigns elder 1/2

| | (18) = 9 camels; 2nd, 1/3 | | (18) = 6 camels; youngest,V--------------------> 1/9(18) = 2 camels. 9 + 6 Apportions camels + 2 = 17. Stranger re-



claims his camel, departs._____________________________________________________________________________

Reach up or work at high places--------------------->

Put| ^Climb down

ladder| |ladder &in place| |remove it| |V-------------------->

Accomplish Task ______________________________________________________________________________

Flee across guarded space--------------------->

Dig| ^Emerge As in the

film, "Thetunnel| |from Great

Escape"out of| |tunnel

sight of| |beyond guards| |guardsight

V--------------------> Flee underground out sight

________________________________________________________________________________

Walk on broken leg --------------------->

Splint| ^When leg broken| |heals,

leg| |remove| |splintV-------------------->

Walk on splint-supported leg _________________________________________________________________

_______________Eat meat/fish/vegetables procured raw --------------------->

Put in| ^Eatcooker| |food

| |V-------------------->

Soften "food" for mastication ________________________________________________________________________________



Dye Easter egg, leaving undyed portion --------------------->

Rub| ^Remove (Similarfor batik

"undyed"| |wax dyeing)with wax| |

| |

| |V--------------------> Immerse egg in dye

_______________________________________________________________________________You know that the BYPASS of CLIMBING STAIRS shows the link between CAUSATION and TELEOLOGY:the increase of risers is CAUSAL; the decrease of distance from top is TELEOLOGICAL.

You know a Bypass can beome an ANTITONE just by collapsing. You know this about AMPLIFICATION:

• From the verb "amplify" derives the substantive,"ampliative", jargon in

• the philosophical theory of logic.• • "Ampliative": A given reasoning process can increase the

knowledge• already possessed.• • But this is believed to be impossible in STANDARD LOGIC,

which • REVEALS "ONLY WHAT IS THERE". STANDARD LOGIC is also MONOTONIC -- REMAINING

• THE SAME OR INCREASING ONLY WHEN NEW TRUTHS ARE ADJOINED

-- from outside. ONLY • A NONMONOTONIC MEASURE WHICH INCREASES AND DECREASES CAN GUIDE US IN

• DECISIONS. Logicians say that such a CORRECTIVE isimpossible.

• • CIVILIZATION DERIVES FROM AT LEAST TWO (CORRECTIVE) AMPLIATIVE PROCESSES:

• •



o TRADING BETWEEN HUMANS STARTED TO "SHOW A PROFIT". TheCALVINIST

o outmanuevering of THE ROMAN CATHOLIC BAN ON "USURY" ACCUMULATED OUR PRESENT

o ECONOMIC SOCIETY.o o CRUDE PREHISTORIC DEVICES PROGRESSED INTO MECHANICS. By o definition, A MACHINE is a PROCESS WHICH MAGNIFIES

OR AMPLIFIES ITS INPUT.o Examples:o

o

A LEVER TRADES-OFF LENGTH FOR LOAD: Place theFULCRUM (balancer) of the LEVER so PART ON FORCE-ARM is 3 TIMES THE

LENGTH ON LOAD-ARM, and you can RAISE 3 TIMES AS MUCH FOR GIVEN FORCE. A

LEVER IS AMPLIATIVE. A PULLEY TRADES-OFF ROPE-LENGTH FOR LOAD: A PULLEY

SYSTEM WITH 3 ROPE-LENGTHS ACTING ON THE LOAD RAISES 3 TIMES

THE LOAD FOR GIVEN FORCE. A PULLEY IS AMPLIATIVE. So are other MACHINES

(wheel-and-axle, inclined plane, screw, etc.). So also ELECTRINES

(computers, radios, TV sets, tape and CD players, etc.).

You know that a Bypass can be transformed into an

amplification by TRANSFORMING INPUT

OF A MACHINE OR PROCESS INTO ITS OUTPUT.

A MACHINE IS A DEVICE ACTING UPON INPUT FORCE OR TORQUE (ROTATIONAL



"FORCE") BY ANTITONICALLY AMPLIFYING IT INTO OUTPUT FORCE OR TORQUE

You know the linking between ANTITONE, BYPASS, AMPLIFICATION can be formulated as an acronym,

AMANBY" AM(PLIFY)AN(TITONE)BY(PASS).



CHAP TER FIT HTE EN: WH AT IS A MACHIN E? AN ELECT RINE ? A

LOGIN E?

You know tha t, regar ding a "co mputing machine" , you' ve

hear d people sa y, "I don 't s ee how a mac hine can do th at."

And you r ea li ze a mer e m achine can't do our comput ing .

You may kno w th at you co mpute with electr on ics . Thatelectr on ics i s electr ic ity contr o ll ed, in i t s flo w, by e lectr ic ity ,

not b y a m ec hanica l de vi ce . Also tha t , m os t of the t ime , it

doesn 't compute ,

So thi s tel ls you tha t you compute w ith an electr ine .

You may kno w th at a ma chine is a device w hose m otion

trans for ms under the tr ans for ma t ion la w , for mul ated b y

Ga li leo (1564-1642) .

Essentia ll y, thi s sa ys the mot ion of a machine is af fected by

the m otion of i ts bac kg round , a s the ca se of a bo at s peeded

by the water flo wing i ts way, but slowed b y a cur rent

oppos ite to i ts mot ion. S im ilar ly, with tai lw ind s or head winds

for plane s.

You may kno w th at electr on ics i s the f lo w of e lectr ici ty under

contr ol of e lectr t ici ty , not by me chanica l device s,

This te ll s you th at an elect rine is an electr on ics de vice w hic h

is not a f fected by mot ion of it s ba ckg round . You m ay kno w

tha t it s tr an sf or m ati on l aw is the Lor entz- E ins tein

Trans for ma tion Law .

You then kno w th at a computer is an elect rine .

You may kno w th at another m isuse of "ma chine" occur s in

speak ing of a " Turi ng Ma chine" , whi ch A lan Tur ing (1912 -

1954) concei ved pr ior to t he fir st computer . You kno w t hat

thi s is not a device . You may kno w t hat a " Turi ng ma chine " i s

descr ibed entir ely in te rms of langua ge. So it i s a l ogi ne,

fr om the Greek wor d "l ogos " for "di scour se".



CH APTER SIXTEE N: WHY IS K ISSIN G THE S EC RET OF

CALCUL US AND THE SECRE T O F THE WHEEL ?

You kn ow tha t the la bel "ca lcul us " i s mislead ing. That it

deri ves fr om the Gr eek for "pe bble " or "stone ". (T hat, in

medi cal jar gon , a gal lstone or k idne ys tone is a calcu lus.)

That i t deri ved mathe matical connota tion from the method by

whic h ill iter ate Gr eek shepher ds inventor iedthe ir sheep flo ck.When eac h sheep went out of the gate of the fold to go to be

her de d to pa stur e, the shepher d put a pebble as tok en in a

pouc h k eep a t hi s wai st . When her ding ea ch sheep bac k

thr ough the gate of the fold , the shepher d r emo ved a pebble

fr om h is pouc h.

You know the f ollo wing demons tr ation der ives fr om the l es son

taught to Isaac N ewton (1643-1747) by his mathema ticspr ofesso r, Isaac B ar row (1630-1677) , a t C ambr idge U niver si ty.

Br iefl y, you kno w th at B ar row showed New ton the

sign ificance of A GEOMET RIC (not t rigonomet ric !) SEC ANT OF

A CU RVE TRA NSF ORMI NG INTO A TANGENT TO THE C URVE.

You kn ow tha t the SEC ANT (in ri sing up a semi-circle arced

upwards) touc he s the cur ve tw ice: on the " left " and inanother po int on the "r ight" of the semicir cle. You kn ow tha t

the L EFT P OI NT ri ses RIGHTWARD on the ar c, w hil e the

RIG HT P OINT rises LEFT WARD on the ar c, so th at t he points

(Ant itonica ll y! as in an y LIMIT P ROCESS) MOVE TOWARD

EACH OTHER.

You know tha t, When SE CANT BECOMES TANGENT a t top of

ar c, these POIN TS UNIT E I N A SIN GLE POIN T. Mathe matician scal l thi s po int an "oscul ation" (fr om Latin , for'k is sing "), so

the T ANGENT JUST KISSES THE CURVE.

YOU kn ow tha t th is r isi ng en graphs w hat is i nvolved in

ME ASU RIN G SPEED .

You know you can thi nk of the "left point " a s the P OSITIO N

OF A PARTICLE (or auto mobi le) a t " the be ginn ing of



meas ur ement" ; and the "r ight point " a s POSITIO N AT A LATER

TIME . The DIS TANC E T RAVER SED DURI NG the TIM E I NTE RVAL

(LEN GTH DIVIDED BY TIME) IS T HE AVE RAGE SPEE D O F

PARTICL E (car) . B ut THE I NS TANTANE OUS SPEED IS DESIRE D.

You kn ow THAT IS R EP RESE NTED BY THE "KIS SIN G POI NT ".

You kn ow you can st ate thi s a lge br aica lly:

• DENOTE INITI AL POSITI ON AS x1;

• TERMIN AL POINT A S x2;

• THE DIS TANC E I NTE RVAL AS ∆x = x2 - x1;

• similarly, THE TIME INTE RVAL A S ∆t = t2 - t1;

• the R ATIO A S ∆x/∆t = (x 2 - x 1)/(t 2 - t 1).

This ha s been for mal ized a s the DIFFER ENTIA TIO N PP ROCESS

AS THE TIME INTE RVAL G OES TO ZERO: LIM (∆t -> 0) ∆x/∆t =

D tx, REPR ESE NTI NG THE TANG ENT POINT ("k issing poin t") b y

"the deri vative of x w ith respect to t". ThUS , the LIMIT

PROCES S, for example , TR ANSFO RMS AVERA GE SP EED INTO

INSTANTANEOUS SPEE D.

You know tha t Bar row also taught N ewton how to MEASU RE

THE AREA UNDER A CURVE, l eading to the i nte gral ca lculu s

in another "ki ssing" proces s.

You know tha t "bar graphs " can be bu ilt fto m the bas e up to

the top of the se mi cir cle . Ineac h ba r, the lef t po int ri se s

upward above the ar c, whi le the r ight poin t of t he bar

descend s down to the arc. You kno w tha t, as the bar s are

decr eased in width, the lef t and righ t po ints of eac h bar ar e

moving *ant iton ical ly) toward eac h other . You kno w tha t,

when eac h bar tr an sfor ms into a line se gment, ju st touc ing

the ar c, the se se gment s fi ll in onl y the ar ea ubder the ar c,

and the tota l s um of the ir lengths is, in the limi t, a mea sur e

of the ar ea under t he semicir cle and repr esent s the i nte gral

calcu lus .

You know tha t, if st udents were taught thi s, for mul ated s o

long ago by I saac Bar row, they would realize tha t the y kn ow



the e ssenti al s of the calcu lus , but did kno w that the y kno w

these kno wable s.

An y "nor ma l" teen- ager can under stand the str ate g y of the

dif fer ent ial calcu lus and the s tr ate gy of the in te gr al calcu lus .

But the har d par t i s lear ning al l the deta il s of the tact ic s of

eac h s ub ject so the se can be ca lcul ated i n a ppli ca tions toso lv e u seful pr oble ms .

You know why ki ssing i s the s ecr et of the wheel, but you

don't kn ow tha t you knew unt il you ar e told .

To be gin , Y ou can ima gine a wheel resti ng on a s urf ace . Y ou

can th ink of the w hee l a s a s tack of c ir cle s, eac h wheel just

ki ssing the su rface i n one po int per c ir cle.You know tha t a w hee l or any m ater ia l ob ject moving acr oss a

su rface i nvokes fr ict ion . But y ou kno w th at ea ch cir cle

invokes mi nimal fr icti on , which i s why the w hee l pr ovide s the

mos t e f fic ient pr opul si on of a veh ic le .

You now kn ow thi s is w hy ki s s i ng i s the s ecr et of the w hee l .



CHAP TER SEVE NTEEN : WHAT IS T-MATH? O-MATH?

You know tha t st andar d se t theor y deal s onl y w ith what ma y

be ca lled "t-s ets" : se ts con str ainted onl y in ty pe (k ind), not in

or der (de gree) -- in f act , pr e cluding or der .

You know these are P rotoT ype s of t-m a them a tic s .

You al so kn ow tha t, for se ts of f ac tor s of number s, thi s

appli es onl y to num ber s w hich ar e la be led "s quar e-fr ee" . s uch

as 30 = 2 ·3·5 , contain ing ea ch pr ime factor onl y once . You

kn ow tha t the se can be ca ll ed " t-nu mber s" .

You kn ow tha t, to encompa ss Natur al Nu mber o r Inte gral

Arithme tic , thi s must be e xtended to o-nu m ber s , such as 12 =

2·2 ·3·5 , whic h contain s two tok ens of pri me f actor 2.

You kn ow tha t thes e ar e Ptoto Types of o-mathema tic s.

sYou kno w t hat, i n the 1960 's , the " mu lti se t" concept

developed as a hybr id , ma pp ing s et elements i nto in te gers.

But the o-set nonh ybrid ly extends the st anda rd for ma lism.

You kn ow tha t the moti vation f or m ult iset s was the need , in

computer programm ing, to symbol ise mul tip le tok ens of anoper ation or oper an d, T his is de scr ibed on pp . 411 -12 of T he

Ar t of C omputer Pr og ram ming , V. 2 , by Dona ld Knuth .

Standar d statement l og ic i s t-logic , constr ainted to type onl y.

You know thi s can be eas il y lear ned b y assigni ng alpa betic

le tter s to the oper ations and oper an ds of t-logic, and

dec lari ng rule s for the " wf f" or wel l- for med for mula ,

compar able to the wel l- for med sen tence in a uni ver sa llangua ge.

You kn ow tha t st ate ment l ogi c concer ns on ly asser tions

whic h, potentia ll y can be deter mined as tr ue or fal se .

You kn ow tha t the combin ator s of t-logic are:

1. conjunct ion ( "and"),



2. di sj uncti on ("or") ,


4. bicond iti onal ("If _ and onl y i f _ "),

5. ne gat ion ("not _") .

You see tha t the fir st four are b inar y , that i s, oper a te on t wo

sim ple or compound ass er ti ons at a ti me , while the fifth isunar y , tha t is, oper ates on a s ing le simp le or compound

as ser tion at a ti me .

You use the v owe ls , A, E, I, O, U, to denote thes e oper a tor s :

"A" f or " and", " E" for "equ iva lence or b icondi tiona l" , "I" for

"condi tiona l" , "O" for "or" , and " U" for " ne gation" ("undoe s").

You use the con sonants denote the oper and ss of sim ple or

noncompound pos it iv e s ta tements , that i s, w ithoutNE GATIO N.

You kn ow you can use , for thi s, Pol is h pr ef ix not ation (as

in tr oduce d by Jan Lukas iew icz (1878-1956) Y ou kno w th at a

hand ca lcul ator u se s "r ever se or postf ix Pol ish": put in the

number s and then the oper ator .

You know you can be gin w ith th is D EFINI TIO N:

1. Read left to right . S cope of oper ator is a llo wed nu mber of

oper an ds f ol lowing it .

2. Whatever i s denoted by a con sonant is a t -w f f .

3. Whatever i s denoted by A , E , I, O , fol lowed by two t -w f f 's ,

is a t -w f f .

4. Whatever i s denoted by U follo wed b y a t -w f f is a t -w f f .

You kn ow thi s must be co mpleted via clos ur e ru le : Nothing isa t -w f f unle ss des igna ted by r ule s 1- 4.

You know tha t he def ini tiona l rules given for t- logic can be

eas ily demon str ated for m odus ponent s (a .k.a . va li di ty of

as ser ting the pr ecedent ), the mos t famous of log ical pr oof

rules .






4. bicond iti onal ("If _ and onl y i f _"),

5. aug ("augment of tok ena ge").

6. ne ga t ion ("not _") .

You kn ow to e xtend the DEFINITI ON:

7. Read left to right . S cope of oper ator is a llo wed

number of oper ands fol lowing it.

8. Whatever i s denoted by a con sonant is an o-wf f .

9. Whatever i s denoted by A , E , I, O, fol lowed by two o-

wf f 's, is an o-wf f .

10. Whatever i s denoted by Y fol lowed by an o-wf f i s an

o-wf f .

11. Whatever i s denoted by U follo wed b y an o -wf f is ano-wf f .

12. nothing is a o-w f f un les s i t is design ated b y Rule s

ONE to FIVE .

You kn ow you can demons trate o-l og ic v ia o-wf f of

independent deci si ons : pick, stor e, ne ither , both.

Given O O B C Y B (ne ither):

o O O B C Y B by R ul e T wo on Cons onants

o O O B C Y B by Rul e F our . (p ick, Y )

o O O B C Y B by Rule T hr ee. (s tor e, O)

o O O B C Y B by Rul e T hr ee

It i s an o -w f f . (both, O)



CHA PT ER EIG HT EEN : WHAT IS PH ILOSOP HY ? (an "ONTOLOGY

GAME"?)

You WE B-l ear n the w or d "ph il osoph y" has the pr efix sp el led

"phi " because it s or igin , and th at of similar pr efi xes , is in the

Gr eek alpha bet , w hich is dif fer ent fr om the Latin alpha bet on

whic h much of our Engli sh langua ge i s based -- des cending

fr om R oman cu ltur e. That thr ee letter pr efix , "phi ", or igi natesfr om a single l ette r, " phi ", in the Greek alpha bet -- wr itten

thus : j. Bu t y ou kno w that, in " phy sics ", "phi " i s wr itten as

"phy ".

You al so lear n tha t the pr efix "p sy" in "psycholog y" is spe lled

wi th a single G reek l ette r, " ps i" , wr itten thu s: y.

You bel ieve th is shou ld be taught in ever y e lementar y schooland repea ted later on.

You lear n tha t the word "phi losophy " is attributed to the

ancient Greek mathema tic ian and phi losopher , Pytha gor as,

who lived ar ound the per iod 580 to 500 BC . That t he wor d

"phi lo soph y" means " Love of W isdom" fr om the G reek " phi lo"

for "lo ve" and " soph ia" for "w isdom" .

You kn ow thi s evokes the quest ion as to the m eaning of

"W isdom ". You've hear d of the epi gram, 'I disa ppr ove of what

you s ay, but I wi ll def end to the dea th y our r ight to say it ",

incor rect ly attr ibuted to the Frenc h ph ilo sopher , Volta ire

(1694-1778).

You can as soc iate th at to ler ance with Wisdom, w hile seeking

per ception of an act of TOLER AN T EMP ATHY .

You lear ned of at leas t one act of Wisdom , enacted by a

neighbo r, invol ving two other ty pes of Lo ve. One is Erotic

Love. But the other has a label whose spok en sound

conf ounds two d is tinct t ype s of Love. One l abel is " phi li al ",

wr itten thus and meaning "fr iendsh ip love". Not to be

confused with "f il ial ", written thu s, and meaning "love for a

son ", an offspring . Agai n, a di st incti on betw een the Greek and





You lear n tha t the y hear d of a W ill Dur ant who was l ectur ing

to w or king men on phi lo sophy a t P resb yter ian La bor Temple ,

in lower Manha ttan .

You lear n tha t some of the lectur es had been pub lished by

Juli us Halde man in " Li ttle Blue Book s" , eac h book devoted to

a s ingle phi los opher . And you lear n tha t the se "Li ttl e B lueBook s" nota bl y attr acted reader s fr om T he Inter nationa l

Wor ker s of the Wor ld, a.k.a . IWW . a.k.a . "Wob bl ies ". You kno w

tha t st udents s houl d be taught of the i mpor tant r ole the y

pla yed in ea rly 20th centur y A mer ica. You lear n t hat

Wikipedia , onl ine, w rite s tha t, in 1923 , the y had perha ps

100,000 member s, and m ight com mand suppor t of 300,000

mor e. You lear n the y thought an y gover nment shou ld be

modeled as one big Union , in the style of the med ievaljour neyman worker s.

You lear n tha t the se Wobblie s jou rney ed from far m to far m,

doing sho rt- ter m job s, car rying thes e "L it tle Blue Books ",

eac h f itt ing to a sh ir t po cket, to read at night ar ound a camp

fir e.

You lear n tha t Simon and S chus ter hir ed Dur ant to write "T heStor y of Phi lo soph y" , whic h they pub li shed in 1926 .

You lear n tha t the resu lt was a sen sation , se ll ing mor e than

any nonfict ion book had e ver sold; mot iv ating The Ne w Y ork

Times t o cr eate its Bes tsel ler Li st .

You lear n tha t mathema tic ian Eri c T emp le B el l wr ote "Men of

Mathe matics " for them; T homas Craven wr ote "Men of Ar t";

Ber nar d J affe wr ote " Cr ucib les , The Hist or y of chemi st ry";

etc.

And you l ear n tha t Wi ll Dur ant and hi s wife, A ri el , went on to

wr ite the e leven v olu me to me, "T He Stor y of Civilization" .

You wonder , of the long list in W ikiped ia, ho w many

phi los opher s relate the ir phi lo sophie s to the a bove def ini tion





"In B ur ke's phi lo sophy , soc ia l inter act ion and commun ica tion

shou ld be under stood in ter ms of a pentad , which i nclude s

act, scene, agent, agenc y, and pur po se. H e proposed tha t

mos t soc ial inter act ion and commun ica tion can be

appr oac hed as a for m of dr ama whose outcomes ar e

deter mined b y rati os betw een the se fi ve pentad ic e lement s.

This ha s become kno wn a s the ' dr ama tis ti c pentad' . Thepentad is g rounded in h is dr amati st ic method , w hich see s the

relation sh ip betw een life and the ater as liter al rather than

meta phorica l: f or B ur ke, a ll the world r ea ll y is a s ta ge . B ur ke

pur sued l iter ar y cr it ici sm not as a for mal ist ic enter pri se but

rather as an enter pri se with sign ificant soc io logica l impact ;

he s aw liter atur e as ' equip ment f or l iving ', of fering folk

wisdo m and common sen se to people and thu s gu idi ng the

way they l ived thei r l ives.

"Another key concept for Bur ke is the ter m ini stic scr een -- a

se t of symbo ls tha t becomes a kind of scr een or grid of

intel li gib il ity thr ough whic h the wor ld mak es s ens e to us.

[Onto log y!] Her e Bu rke of fers rhe torica l theori st s and cr it ic s

a w ay of under stand ing the relation sh ip betw een langua ge

and ideolog y. Langua ge, Burke thought, doe sn't simpl y

'r ef lect' real it y; i t al so help s se lect real it y as well as def lect

real it y.

"In h is book Langu age a s Sy mbo li c A ction (1966), Bur ke

defined humankind as a 'symbo l u sing an ima l' (p. 3). This

defin iti on, he ar gued, mean s tha t 'r ea li ty ' has actua lly ' been

bui lt up f or u s thr ough noth ing but our symbol syste m' (p . 5) .

Without our enc ycloped ias , atlas es , and other assor ted

reference guide s, we wou ld know lit tle about the world thatlies bey ond our immed ia te sense experience . W hat we ca ll

'r ea li ty ', Bur ke s tated, is actua ll y a 'c lutter of s ymbo ls about

the pa st comb ined with whatever thing s we kno w ma inl y

thr ough ma ps, ma gaz ine s, new spa per s, and the l ike a bout the

pr esent . . . a cons tr uct of our symbol syste ms' (p . 5) . C ol le ge

students wandering from clas s to class, from Eng lis h



liter atur e to soc io log y to b io log y to ca lcul us , encounter a

new r ea li ty eac h time they enter a clas sr oo m; the cour ses

listed in a uni versity 's cata logue 'ar e in ef fect but so many

dif fer ent t er minolog ies ' (p . 5) . It s tand s t o r eason then tha t

people who cons ider them selves to be Chr istian, and who

in ter nal ize th at rel igi on's symbo l syste m, in ha bit a r ea li ty

tha t is d if fer ent fr om the one of pr actic ing Bud dhi st s, orJews , or Mu slims . T he s ame would hold tr ue for people who

bel ieve in the tenets of fr ee mar ket ca pita li sm or socia li sm,

Freudian p sychoanal ysi s o r J ungian depth p sycholog y, as well

as myst ici sm or m ater ial ism . Eac h be lief syste m ha s it s own

voca bular y to de scr ibe ho w t he wor ld w or ks and w hat things

mean, thu s pr esent ing its adher ents with a s pecif ic real it y."

You lear n tha t Bu rke's co mpla int s agains t ph il osopher s w astha t eac h tr ied to get al l real it y do wn on pa per , usua ll y in

ter m s of a par ticu lar mot iv e, as i f no other moti ve ma tter ed ,

As an ins tance, a ph il osopher expla ined "e ver yth ing" in ter ms

of "the environment " and talk s about the envir onment as if it

is al ive and can act as "an agent" .

You lear n tha t his " rhetori cal books " w er e

• Phi lo soph y of Liter ar y F or m (1939)

• A Gra mmar of Mot iv es (1945)

• A Rhetor ic o f Mot iv es (1950)

• T he Rhetor ic of R el ig ion (1961)

• Langu age a s Sy mbo li c A ction (1966)

• Dr ama ti sm and Devel opment (1972)

• Es say s Towar d a Sy mbol ic o f Mot iv es (2006)

How did the some of the noted phi los opher s deviate from

"phi lo soph y a s love of w isdom "?

• Her aclitu s (535-475B C): exis tence con si st ening of

pri mar y mater ial a gent s, such as fir e and water



• Par menide s of Elea (ea rly 5 th centur yB C): In oppo si tion

to H er aclitus , rep laced the kn owledge of s ense

exp er ience w ith reason ing about tr uth

• Zeno of Elea (490-430 BC) used reductio ad a bsur dum

ar guments to con st ruct pr ado xes th at see med to deny

exis tence of moti on and other s ens or y experience s

• Socr ate s (469 -399BC) : quest for eter na l ver it ies • Plato (428-347): the eter na l v eri tie s in some "hea venl y"

real m

• Aristot le (388 -327): phi losophy as al l knowledge

• Chr ys ippu s of Stoa (280-207B C): most famou s of Sto ics ,

who bel ived i t is vir tuous to reconci le one's w il l to

natur e; i n 20th centur y, Chri sippus 'contr ibuti ons to

pr opos it ional lo gic were fnal ly a ppr eci ated, inc luding hi s

bi va lenc y of st ate ments as be ing onl y t rue and fal se

• Rene ´ Descar te s (1596-1650): "I think , ther efore I am"

• Francis Bacon (1561-1626): "Baconian" scien tif ic

"method "

• Baruch S pino za (1632-1677): "A ll th ings i n natur e pr ocee d

fr om cer tain neces si ty and wi th the ut mos t per fection ."

• Volta ir e (1694-1778) :Volta ir e's focus was the idea of a

uni verse based on r eason and a r espect for na tur e.• Jea n-Jacques Rou ss eau (1712-1778) : F ocus on

sub ject iv it y and intr ospecti on.

• Immanue l K ant (1724-1804): Rea son w ithout experience i s

ill usor y, but e xper ience is subj ecti ve un les ssub sumed

under pur e reason .

• G. W. He gel (1770-1831): Histor y as T hes is, Ant ithe si s,

Synthes is.

• Ar thur Schopenhau er (1788-1860): The Wor ld as Will andIdea

• He rber t S pencer (1820-1903) : E volu tion as the

pr og ressive development of the ph ys ical world, bi olog ical

organis ms, the human mind, and human cul tur e and

soc iet ies .

• Wil lia m J ames (1842 -1910): H is "pr agma tic" theor y of

tr uth appl ied D arw inian ideas in ph il osoph y, m akin



sur vival the tes t of in tel lectua l a s wel l as biolog ical

fitne ss .

• Friedr ich Nie tz sche (1844-1900): Nih il ism, the bel ief th at

nothing has an y im por tance and tha t life lacks pur pose .

• Geor ge S aunde rs P eir ce (1859-1914) : T heor y of S ign s

unif yi ng O nto log y and Ep is temo log y and A xio log y.

• Henr i Ber gson (1859-1941) : T hought i s a str eam ofconsc ious nes s which intel lect dis tor ts by fr aming in to

concepts .

• Joh n Dew ey (1859-1952) : " Pr agma tis m" as

Inst rumenta li sm : experi menta tion ( soc ial , cul tur al,

tec hno logica l, ph ilo soph ical) could be u sed as a

relati vel y har d-and-f ast arb iter of truth.

• Al fred Nor th Whitehead (1861 -1947): Counter ing the

tr adit iona l ph il osph y tha t "r ea li ty i s t imele ss", W.' s

Pr oces s Ph il osoph y Pr oces s phi lo sophy (or On tolog y of

Becom ing) i denti fie s meta phy si cal r ea li ty with change

and dynam is m.

• Benedetto Cr oce (1866 -1952): The root of r ea li ty i s

immanent exis tence in concr ete exper ience, so C roce

place s aesthet ic s at the f ounda ti on of h is phi los ophy .

•

Geor ge S antay ana (1863 -1952): N atur al is t m eta phy si cs inwhic h human cognit ion, cu ltur al pr actice s, and so cia l

in st itut ion s h ave evol ved s o as to har mon ize with the

condit ions pr esent in the ir environment.

• Ber tr and Russel l (1872-1970) : Sought cer tain ty in

Ma the ma t ics a s other s do in rel igion . But real ized th at

thi s cer ta inty i s onl y tau tolog ical (h is ter m), and an y

atte mpt to ward real it y ri sk ed contr adict ion (h is te rm) ,

so ga ve up Ma thema t ics and Phi lo soph y,



CHAPTER NINET EEN : WHAT IS MATHE MATICS ?

On li ne W ikiped ia sa ys , "Mathe matics i s the stud y of quantit y,

st ructur e, s pace , c hange, and related topic s of pa tter n and

for m. Mathe matician s se ek out patter ns whether f ound in

number s, space , na tur al science , computer s, ima ginar y

abstr act ions , or elsew her e. Mathe matician s for mul ate new

conjectur es and es ta bl ish the ir tr uth by rigor ou s deductionfr om a ppr opri atel y chos en ax iom s and defin it ions ." (You

lear ned war ning s about axio ms in Cha pter T wo.)

You real ize th at t hi s expli cation onl y ta lks a bout m athem ati cs

fr om the out si de; a per son s til l mu st lear n ho w to do

mathema tics ; and , a mong man y other ob jecti ons , it doe s not

sho w you t hat you kno w so me m athem atic s, as is done in

"Ba rbie Do ll M ath" , in Appendix C.

You real ize th at a judic ious way to an swer a quest ion su ch a s

thi s cha pter' s ti tl e i s to f or thri ghtl y attemp t an ans wer t o the

quest ion and face the r eact ions to your ans wer. Y ou rea li ze

thi s is a WIN- WIN str ate gy. If other s accept y ou r an swer, you

WIN . If s ome one cor rect s your ans wer, you a lso Win, becau se

your bas ic pur pose is to lear n.

You lear n tha t the liter atur e shows thr ee F OU NDATIONS FOR

MATHEMA TICS :

• AXI OMA TIC -- deri ving fr om T hale s of Miletus (c .624-

547BC) , exempl ifi ed in E ucli d's E lem ent s of G eom etr y .

• GE NE RATIVE -- deri ving from the pupi l of Thale s,

Py tha gor as (580?- 490? BC) .

•

GROUP-THEORETIC -- deri ving fr om G er manmathema tici an, Fel ix Klei n (1849 -1925).

You real ize the se are especia ll y usefu l in THE TACTIC S O F

MATHEMA TICS . You r ea li ze that the gener atve me thod, sho ws

ho w a co mpr ehensi ve CU RR ICUL UM can be de veloped w hic h

fulf il l s w hat the g rea t Henr i Poincaré (1854-1912) m ay ha ve

meant by say ing , "Ma thema t ics ha s been ar ithmet i zed. " .









her. (Bett y: bangs :: Pur li e :cor nr ows ) O r st ar t with the "side"

of the OTHE R, and t ry to find someth ing compar able wi th

SELF . Thus, w hen J immy t rie s to under stand ho w J uani to felt

when school mate s cal led him a "Sp ic" , Jimmy remember s how

he f elt when schoo lma tes s cal led hi m " a ba star d"

(Ji mmy :ba star d::J uanito :s pi c).

You lear n, in Cha p. Twenty -Thr ee, th at ho molog y shows use

of fr action s, or ratios, to expl ica te dif f er ent do main s o f

exis tence on thi s ear th . That th is deri ves fr om a ne glected

"science " w rit ten up long ago by Gal ileo Ga li lei (1564-1642) .

You lear n, onl ine, th at H . H. Pattee raises the ques tion about

'T he Phys ics of Sym bol s: Br idging the Epi st emi c C ut" , w ri ting

of "the gener al epi ste mic proble m: how to bri dge thesepar ation betw een the obser ver and the ob ser ved , the

contr oller and the contr ol led, the kno wer and the kno wn, and

even the mi nd and the br ain. Thi s notori ous epi ste mic cut has

mot ivated phi los ophica l di spute s f or m illennia , especia ll y the

pr oblem of con sciou sne ss th at onl y r ecentl y has be gun to be

tr eated as poss ib ly an empi rica ll y decida ble pr ob lem (e .g. ,

Shear , 1997; Taylor, 1999). My second que stion w as whether

bridg ing the ep istem ic cut cou ld e ven be ad dr essed in ter msof phy sica l l aws. "

In Cha p. Twelve, you ar e concer ned tha t the rhetor ical

langua ge of muc h math provokes COGNITIVE DIS SONAN CE

wi th nonm ath langua ge. But the SYMBO LIC langua ge of math

usua ll y cons tr ain s t hi s.

You lear n tha t the eminent m athem atical -lo gic ian, H ask ellCur ry (in hi s Founda t ions of M athem atica l Logic ), s ays (p . 8) ,

"T her e ar e t wo mai n t ype s of opin ions i n regar d to the natur e

of mathema tic s. We sh al l cal l these conten si vism and

for ma li sm." You lear n tha t fo rmal i sm (H il ber t is the mai n

spok esman for thi s viewpoint) regar ds MATH AS A " GAME"

FORMALIZ ED OR TAKING SHAPE BECA US E O F RULE S. Take

away the r ules of Chec ker s or C He ss and w hat is l eft?



No thing two people can a gree on; noth ing s o tha t two peop le

wi ll feel the y ar e " talk ing about the s ame th ing" . B ut you

kn ow thi s not so with your FORMALI C C ONC EPTI ON OF

MATHEMA TICS , for CONTENSIVIS M a ppar entl y means tha t

RULE S H AVE BE EN APPLIED TO " SO MET HIN G" : TAKE AWA Y

THE RULE S, and T HE RE 'S STILL "SOME THI NG" PEOPLE CA N

AGREE ON.

FORMALI C R ULES AP PLY TO PATTER NS . TAK E AWAY THE

RULE S, AND P EOPLE CAN STILL AGREE ON "SEEI NG" THE

SAME R ESI DUE PATTE RN!

You real ize th at t hi s mean s th at PROTO- MATH W AS ALWAYS

THER E - - A S LONG AS H UMANS OR HOMI NID S S ENSED

PATTERNS , EVE N AS BASIC AS PER CEP TIO NS AN D "THI NK S" .SO, WE 'RE ALL PROTO- ED UC ATE D I N M ATHEMA TICS. A ND

THIS I S H OW WE C AN HE LP C HIL DRE N!

You WE B-l ear tha t childr en can be taught ari tmet ic oper ation s

by flowchar ts. That Ad di tion can be perf or med by T he

Counter Trans fer A lgori thm , the way compter s w er e

pr og rammed for th is :

• tr ans fer count s fr om one counter s tor age to another ;

• when fir st counter i s empty , the count i n the s econd i s

thei r s um:

That a flo wchar t d ia gr am s an A lgor ith m or a pr ocess . Thst

the f lo w char t for thi s ad dit ion oper at ion is:\=======/\START/

\ /\_/||

_____|_____ |Setup 1st||AND 2nd || ADDEND || BASKETS |-----------



||/ \/ IS\---------YES----

>-----/\/ 2nd \

/ \

/ADDEND \/STOP\------->-------/ BASKET \-->---NO-----

------^ \ EMPTY? / || \ / || \ / || \ / |

| \ / || . |

^ | V | v || | || _______|__________ || |TRANSFER 1 COUNT- | || |ABLE FROM ADDEND | |------<-----|BASKET 2 TO ADDEND|---<---

| BASKET 1 |--------------------

You ind ica te an e xamp le: 4 + 7 = 3 + 10 = 2 + 9 = 1 + 10 = 0 +

11 = 11

You know tha t AN ANTIT ONIC D IAGRAM (Cha p. F our teen) FITS

THIS A LGORIT HM :

^ || |

MAXTONE: | | MINTONE: DECREASINGINCREASING TRANSFERS| | REMAINDERS IN

| | ADDEND BASKET 2| V

And chi ldr en can be taught <S UBTRA CTIO N< i>b y T he

Sta ir step A lgor ithm :

• one child s tand s on a st air step as MINUEND

• a s econd c hi ld st ands on a lower sr airstep as

SUBTRAHEND



• they de scend i n uni son o r i n cadenc e

• when SUBT RA HEND c hil d is at bottom , the nu mber of

ri ser s to MI NUEND chil d i s the DIFF ERE NC E

The flo wchar t f or th is Sta ir step Al for ith m i s\=======/\START/\ /\_/|V

________|_____________ |MINUED CHILD on STEP||SUBTRAHEND CHILD on ||LOWER STEP ||____________________|

|V |/ \

_______________________ / \->--------Y-----|REPORT ZERO

DIFFERENCE|--->---/\/ ARE \

----------------------- / \/ THEY \ /\ / \

--->--/ on SAME \-->---N-----/IS\/ STOP \

^ \ STEP? / /SUB-\--------

| \ / /TRAHND\| \ / /CHILD AT\| \ / / BOTTOM? \| | -------------| | /

| v /| | /| | /| ____|_________v____ | |EACH DESCENDS ONE |-----<|STEP, IN UNISON |

--------------------You ind ica te an e xamp le: 11 - 4 = 10 - 3 = 9 - 2 = 8 - 1 = 7 - 0

=7



A child can alw ays tr ansfor m a sub traction into one s he/he

under stand s.

You remenber th at s ta ir step m odel of the AN TIT ONE was

noted i n Cha p. Four teen .

You real ize th at c hi ldr en can be t aught mu lt ip lic ati on by T he

Table -of-C olu mns -& -R OWS A lgor ithm

The oper ation of mu lt ip lic ati on i s for mu la ted as:a · b = c .

De scri ption :

• mak e a colu mn of a dots

• mak e b copie s of th is

• count number of dot s as P RODUCT

You know how to FLOWCHART THE TABL E M ULTIPLI CATIO N

ALGO RIT HM\=========/\ START /\ /\ /\_/|

v|----------------------------------------|[LEFT MULTIPLIER] X [RIGHT MULTIPLIER]||______________________________________|

|v

___________|_______________ |CONSTRUCT COLUMN OF LEFT ||MULTIPLIER COUNTABLES |

------------------------|v

__________|___________ |MAKE COPY OF LEFT |

------------->|MULTIPLIER-COLUMN || |BESIDE 1ST COLUMN || ----------------------| |



| V ^ || /\| / \| /DOES\| /NUMBER\| /OF COL- \

| /UMNS EQUAL\ _____________ _______________ ----<---NO--------\RIGHT MUL-/--->----YES---|COUNT TABLE|----

>---|PRINT PRODUCT|\TIPLIER?/ -------------

---------------\ /

|\ /

v

\ /|\/

/ \ / \ /STOP \ -------You can DIVIDE WIT HOUT R EMAIN DER , sa y, 12 ÷ 4 BY "ROW-TO-TABLE ALGORIT HM"

1. For DIVIDE ND of TWEL VE, y ou con str uct A ROW OF

TWELVE COUNTABLE S:

* * * * * * * * * * * *

2. Given DIVISO R of FOUR , you REDIST RIB UTE THE T WEL VE

COUNTABLES in C OLUMN S OF EQUAL C OUNTABLES ,

1. st ar ti ng w ith FIRS T C OLUMN of F OUR (r emo ving FOUR

fr om O RIGINAL ROW) :2. * * * * * * * * *3. *4. *

*



5. Then SECOND COLUMN of FOU R (r emo ving FOUR fr om

ORIGI NAL R OW):6. * * * * * *7. * *8. * *

* *

9. Then THI RD COLUMN of FOU R (find ing th is le avesZE RO R EMAIN DER in R OW):

10. * * *11. * * *12. * * *

* * *

2. You then COUNT COL UM NS : THREE - - SO PRIN TS 12 ÷ 4 =

3.

You know how to Divide , wi th R ema inder , sa y, 14 ÷ 4

1. For DIVIDE ND of F OU RTEEE N, you con st ruct a ROW OF

FOURTE EN COUNTABLE S:

* * * * * * * * * * * * * *

2. Given DIVISO R oF F OU R, you R EDIS TRIB UTE thes e i nto

COL UM NS OF E QUAL COUNTABLES ,

1. st ar ti ng w ith FIRS T C OLUMN of F OUR (r emo ving FOUR

fr om R OW):2. * * * * * * * * * * *3. *4. *

*

5. Then SECOND COLUMN of FOU R (r emo ving FOUR fr omROW):

6. * * * * * * * *7. * *8. * *

* *

9. Then THI RD COLUMN of FOU R (r emo ving t hi s le aves

remainder of TWO in ROW) :







(30, 60) = 30 and GCD(30 , 90) = 30 , hence, GCD(30 , 60) =

GC D(30 , 90) , but 60 ≠ 90 .

You real ize th at t hi s al lows a "fr ee" for ma li sm of ar ithmet ic

cons is ti ng of "fr ee natur als or fr eena ts" and " fr ee in te ge rs or

frin teger s.

You real ize th is may be over look ed as a resul t of over looking

the v al ua ble resour ce of ind ica to r ta bles (number ver sion s of

tr uth ta ble s of sta tement log ic) . You se e th at, in the ca se of

factor s of 30 - 2 · 3 · 6, with pos sibl e occur rence s f or eac h

pri me (a bsent, present: 0, 1 ), t hi s yie lds 2<S UP3< sup> = 8

independent pos si bi li ti es , so t he indic ator s for occur rence of

it s pr ime s can be assi gned a s the b inar y not ati on fr om zer o to

seven , respect ively:

0 0 00 0 10 1 01 0 01 0 11 0 01 0 1

1 1 01 1 1

The ind ica tor f or a compos ite number is one when all its

factor s ar e pr esent on that r ow, otherw ise zero.

Then you f ind t he follo wing ind ica tor ta ble. You note th at, i f

i(x) denotes tabula r i ndic ator of a rela tor , then i(LCM(a , b)) =

MA X(i(a) , i (b)) and i(GCD(a , b)) = MI N(i (a), i(b)) . Then, you

find :

TABLE 1

1 2 3 5 6 10 15 30 0 0 0 0 0 0 0 00 0 0 1 0 1 1 10 0 1 0 1 0 1 1



0 0 1 1 1 1 1 10 1 0 0 1 1 0 10 1 0 1 1 1 1 10 1 1 0 1 1 1 11 1 1 1 1 1 1 1(1) (4) (4) (4) (6) (6) (6) (7)

You note a bal lot can be assigned t o count the nu mber s of

ine s in eac h colu mn. This

is sho wn at base of eac h co lumn . ( you not ice no bal lot for 2,

3, 5

count s. ) Y ou note al so

the d is tr ibuti on of t he ba llo t o ver the ranks of th is Table:

1, 3, 3 ,, 1 -- a familiarbino mia l pa tter n

Since the sub system of ar ithme tic i s mu lt iv alen t , you kno w

tha t this i mp li es th at t he

ind ica to r ta ble can be extended to compr ehend al l ba llot s,

one to seven, by acqur ing

factor s with bal lot s two, thr ee , and fi ve. (You see th is a s a

nonstandar d comp let ion

compar able to the standar d comple tion tha t shif ts fr om

rational to real number s.)

You

find a "r ationale" for th is i n another t ype of comp let ion.

Emu la ting a B ASIS pl oy in s et -theor et ic topo log y, you f ind to

the A TOMI C (P RIME) BASIS ,

{2, 3, 5} ,

you ad joi n MIN (1) for the bas is

{1, 2, 3, 5} . You then find tha t

LCM. appl ied to

thi s extended bas is yie lds :

LCM{1, 2, 3, 5} = {1, 2, 3, 5, 6, 10, 15, 30}.



You find al l the f ac tor s of 30 ar e obtained

mer el y from oper ator LCM ! But what about oper ator GC D?

Comp let ion i nvokes th is oper ator .

For ca lcul ating LCM, GCD of factor s, you assign al ter native

labels,

respect ively, MAX(),MIN() appl ied to the ir assoc iated ind ica tor s. B ut,

for con venient labeling of an INDICA TOR (TABL E 2 , belo w),

you

use such for ms as 2V3 for

thei r LCM and 2 ^ 3 , for thei r GCD ; etc .

From T able 1 , you f ind tha t MIN( i(2), i(3)) = 0 , 0, 0, 0 , 0, 0, 1 ,

1,

tha t is, 2^3

has ballot 2. Similarly, 2^5 ha s bal lot

2: MIN(i(2) , i(5)) = 0, 0 , 0, 0, 0 , 1,

0, 1. Also 3^5: MIN( i(3) , i (5)) = 0, 0, 0 , 1, 0, 0 , 0, 1.

What a bout LCD's of thes e new ly obta ined e lement s? You f ind

tha tMA X(MIN(2 , 3) , MIN(2 , 5)) =

i((2^ 3)V(2^5)) = 0 , 0, 0, 0, 0 ,

1, 1, 1 , for B=3 . And

MA X(MIN(2 , 3) , MIN(3 , 5)) =

i((2^3)V(2^5)) = 0, 0, 0, 1, 0, 0, 1, 1,

for B = 3 . And MAX(MIN(2 , 5) , MIN(3 , 5)) =

i((2^5)V(3^5)) = 0, 0, 0, 1, 0, 1, 0, 1 for

B = 3 .

You now have bal lot s 1, 2, 3 , 4, 6, 7 . What of

B = 5 ? You recoup thi s, and

also a " new" ele ment w ith B = 4.

You apply dual it y to t he resul ts in t he pr evious par agraph-but-

one to find



MIN(M AX(2 , 3) ,

MAX(2, 5)) = i((2 v3)^(2v5)) =

i(6^10) = 0, 0 , 0, 1, 1 , 1, 1, 1 , for

B = 5 .

MIN(M AX(2 , 3) , MA X(3, 5)) =

i((2 v3)^(3v5)) = i(6^15)

= 0 , 0, 1, 1 , 0, 1, 1 , 1, forB = 5 .

MIN(M AX(2 , 5) , MA X(3, 5)) = i((2 v5)^(3V5)) = i (10^15)

= 0 , 1, 0, 1 , 0, 1, 1 , 1, for

B = 5 .

You find one mor e resu lt: MIN(M AX(2 , 3) , MAX(2, 5), MAX(3,

5)) = i((2V3)

^(2V5)^(3v5))

= i (6^(10V15)) = 0, 0, 0 , 1, 0, 1 , 1, 1, for

B = 4 . You

note th at thi s resu lt is d if fer ent fr om tha t for 2v3v5 .

You denote, for

For compr ess ion , (2^3)v(2^5) ≡ x;

(2^3)v(3^5) &eqi v; Y;

(2 ^5) v(3^5) &eqi v; Z . You decide to label the oper ands of thecomple ted oper ator s

as "subdom inants ". You then find the se r esults .

TABLE 2: SUBD OMIN AN TS OF 30 (ADJ OIN ED WIT H 6

INDIC ATOR)

2^3 2^5 3^5 X Y Z 6^10 6^15 10^156^10^15 6

0 0 0 0 0 0 0 0 00 0

0 0 0 0 0 0 0 0 10 0

0 0 0 0 0 0 0 1 00 1

0 0 1 0 1 1 1 1 11 1



0 0 0 0 0 0 1 0 00 1

0 1 0 1 0 1 1 1 11 1

1 0 0 1 1 0 1 1 11 1

1 1 1 1 1 1 1 1 1

1 1 (2) (2) (2) (3) (3) (3) (5) (5) (5)(4) (6) (BALLOT)10 ^15 is

the e xcept ion) of Table 2 ar e

factor s or subdo minant s of 6.

You note Table 2

has ranks 1, 2, 3, 4, 5, 6, 7 , so tha t r ank v al ue of an e lement

is

it s ba llo t v al ue, which i s

not the case i n Table 1. You note also

tha t the distr ibut ion of ele ments over the se r ank ings is

1, 3, 3 , 4, 3, 3 , 1, clea rly not

a b inom ial pa tter n. You find

no a lgori thm in the l iter atur e to deter mine the number of

subdo minant s of the factor s of n.

you kno w tha t, in a book on

in for mation retrie val (G. Sa lton , Autom atic Inf or ma t ion

Or gan iza tion and R etr ie va l ,

McGr aw -H ill, 1968), y ou f ind

the H asse d ia gram of st andar d factor s of 30 as "descr ipt ion

set" and the Ha sse dia gram of

the f or mer wi th s ubdom inants ad joi ned as "r equest space".

You note resemb lance of the se resu lt s to w hat i s la beled , in

the l iter atur e, a "fr ee"

st ructur e, s uch as a fr ee a lge br a . You then label the Natur al

Nu mber s thus extended as

"fr eena ts" ("fr ee n atur al s") and the Inte gers as "fri nte gers"

("fr ee inte ge rs") , i n the



sen se of the mathema tici an is fr ee to u se al l oper ator s of a

syste m to comp lete it .

You kn ow the e xi stence of "The Fundamenta l Theor em of

Ar ithme tic " w hich st ate s t hat "a

number can be f actor ed in to a pr oduct of pri me number s i n

onl y one way (except f or or der oflist ing). " D o the above r esul ts v iol ate it ? You kno w thi s not so

if the i mp li ci t pr oper ty

of tota l or der ing of pr oduct is made expl ici t i n tha t caveat

"except f or or der ing".

You lear n tha t mathema tic s is a pri mar y r esour ce in cr eating

mi liar y commun ica tion code s to

outmaneuv er an enem y, or to decode enem y codes . You lear n

tha t an anc ient examp le of thi s

a " graves ide rid dle" of a f amous anc ient G reek

mathema tici an,

The Greek (pos si bl y a H el len ized Ba byl onian)

mathema tici an, Diophantus of A le xandria (c ir ca 200/214 -

284/298 A .D .)

, ha s been ca lled "T he F ather of A lg ebra". But , as s ome one

sa id about another sub ject, " Algebr a ha s many father s". a l-

Khw ar iz mi (c ited in another f ile) cou ld be cal led "T he Is lamic

Father of Algebr a", so we'l l cal l Diophantu s " The GreekFather of Algebr a". W e al so credit Diophantu s w ith a va st

fie ld of Mathe matics , whi ch ha s found g reat appli cation i n

Phy sic s, for e xamp le, in C rystal log raphy -- a s we'l l see be low.

We kno w l itt le a bout the l ife of Diophantus , except f or an

Al gebr aic Riddle quoted i n The Greek Antho log y .



To cite The En cyc loped ia B ri ttanica , "G reek ANT HOLOG IA

HE LL ENIKE, al so cal led PALA TINE ANTHOLOGY, a col lect ion

of Gr eek ep ig rams, s ongs , ep ita phs , and rhetor ical exer ci se s

tha t inc ludes about 3 ,700 s hor t poem s, mos tl y wri tten i n

ele giac couplet s. Some of the poems wer e w ri tten a s ear ly as

the 7th centur y B C, other s a s la te a s AD 1000 .. ..T he l iter ar y

value of the Antho log y lie s in the dist inct ion and char m ofperha ps one-s ixth of the w ho le. For the rest, i t pr eser ves a

good dea l th at i s of hi st orica l in ter est ; it i llu str ates the

continu ity of Gr eek liter atur e f or a lmo st 2,000 year s, because

the l ates t i nclu si ons in it ar e, in l angua ge, st yle, and fee li ng,

not too dif fer ent fr om t he ear li es t inc lu si ons . The A ntholog y

also had a per sis tent and cons ider able inf luence on later

liter atur e."

Some of the pas sa ges in T he Greek Antho log y read l ike

epita ph s on g rave s tone s , as in the fol lowing :

"Ye c hil dr en of the o x, ho w wrong of you to ki ll Her monax,

the s tr ayi ng ba by boy. The poor c hi ld, i n the i nnocence of h is

hear t, went to y ou th inking you were bees , and y ou proved

wor se than viper s. In stead of gi vi ng h im a daint y feas t youdr ove y our mur der ous stings into him, bitter bees , contr ar y in

natur e to y our s weet gifts ."

(T he grave cita tion s of The Greek Antho log y in sp ir ed t he

Ame rican long poem, Spoon Riv er An tholog y , by Edgar Lee

Mas ter s.)

The pas sa ge a bout Di ophantus pr esent s (in tr ansl ation) aRid dle about the phase s of hi s life:

God g ranted hi m to be a bo y for a s ixth par t of hi s l if e, and

ad ding a tw elf th par t to thi s, He clothed hi s cheek s wi th

do wn; He l it h im the l ight of wedlo ck after a s eventh par t,

and f iv e year s after hi s mar r ia ge H e granted h im a son . Al as !

la te- bor ne wr etched c hil d; after atta in ing the mea sur e of half



his father's l i fe, chi l l Fate took him. A fter con sol ing his gr ief

by th is s ci ence of number s for four ye ar s he ended his lif e .

Let

• x D ENOTE the age at dea th of D iophantus ;•

• 1/6x DENOTE b oyhood ;

•

• 1/12x D ENOTE y ou th;

•

• 1/7x DE NOTE batche lorhood end ing i n mar ria ge;

•

• 5 y ear s after ma rria ge , a son w as bor n;•

• le t 1/2x + 4 D ENOTE per iod fr om f ir st fatherhood to

Diophantus ' de ath.

•

• Then we have: 1/6 x + 1/12 x + 1/7x + 5 + 1 /2x + 4 = x.

In sol ving , t he l eas t co m mon denom ina tor of these number s(6, 12, 7, 2) is 12 x 7 = 84 . Then ,

• 1/6(84) = 14 y ear s (b oyhood) ;

•

• 1/12(84) = 7 y ear s ( youth);

•

• 1/7(84) = 12 y ear s (b atche lorhood to mar ria ge);•

• 5 y ear s after ma rria ge a s on ; s on lived ha lf of father' s

life, 1/2(84) = 42;

•

• 4 y ear s la ter , dea th of Diophantu s (a ppar entl y b y

suicide).



CH ECKING (bac k to ARI THM ETIC !): 14 + 7 + 12 + 5 + 42 + 4 =

21 + 17 + 46 = 38 + 46 = 84 . A NSWE R: 84 ye ar s of l i fe .

Mathe matician s today s tud y a vas t and often d if ficul t f iel d of

MATHEMA TICS kno wn as

"D iophant ine A nal ysis" . The latter wor d can be mislead ing,

since mathema tici ans often u se

"anal ys is" for " Dif fer ent ial and Inte gral Cal culu s" , "O rdinar y

and Par tial Dif fer entia l

Equa tions ", and other fie lds r equ iri ng cont inuous too ls , su ch

as "limit pr oces se s" . But thi s

"Diophantine " s ubj ect dea ls w ith "the di scr ete " or

"di scont inuou s" , s o is an extensi on of

NUM ALGE BRA .

DIO PHANTIN E EQUATIONS: Let f(x 1, x2, ..., x n) be a pol ynom ial

in x1, x2, ..., xn w ith INTE GER C OE FFICIEN TS . It is D iophant ineIF SOLUTIO NS MUST BE INTE GR AL.

Linear Examp le: 4x 1 + 6x 2 = 24 . A so lut ion is x1 = 3, x 2 = 2 .

(For a LINE AR EQUATIO N, a1x1 + a 2x2 + . .. + anxn = b, to be

INTEG RALL Y SOLVABL E, b m ust B E DIVISI BLE B Y gcd(a 1, a2,

... , a n) -- as y ou f ind i n the a bove e xamp le, wher ein gcd(4, 6) =

2 divides 12.

(L inear D iophantine equa tions have been usefu l i n model ingchem ical cr ys tal s.)

Cond ition s have been f ound for SOLVABILIT Y of h igher de gree

Diophantine equa tions

In 1912 , the great G er man mathema tic ian, Davi d H ilber t (1862 -

1943), gave a L ist of Prob lem s to be Sol ved -- one



of which w as A GE NERAL SOL UTI ON F OR DI OPHA NTI NE

EQUATIO NS . Eventua ll y, i t was PROVEN THAT

NO SUCH GE NERAL SOLUTIO N CA N EXI ST!

RE tur ning to the s ubj ect of mi litar t cod ing of messa ges, and

appli ca tiom of mathema tic s

to t hi s sub ject .You WE B-l ear n tha t

coding was ori gina ll y appli ed to component s of langua ge unt il

the i ntr oduct ion of (nume rica l)

cyphering in anc ient ti mes i n many countr ies . That an

impotant ad vance i n dec ypering was made in

"Med ieva l t imes". Sa ys Wik ipedia , "It was proba bl y rel ig ious ly

mot ivated textual anal ysis of

the Q ur 'an which l ed to t he inventi on of t he fr equenc y

anal ysis tec hn ique for br eaking

monoalpha bet ic s ubs ti tuti on cipher s by

Of recent t ime s, W iki pedia sa ys, "U nti l the 1970s , secur e

cr yp tog raphy was lar gel y the pr eser ve of

go ver nment s. T wo event s h ave s ince br ought i t squar el y in to

the pub li c doma in: the cr ea t ion of

a pub lic encr yption standar d (DE S); and the in venti on ofpubl ic - key cr yp tog ra phy ." See W iki pedia

for eplana tio of " DES " and "pub lic key encr ipti on". W ikiped ia

sa ys cr yp tog rapher no w rel y upon

the R SA (acr ony m of it s thr ee in ventor s) a lgori thm .T he RS A

algor ith m i nvo lv es thr ee s teps :

key gener ation , encr ypt ion , and decr ypt ion . .

RS A i nvolv es a pub li c k ey and a pr iv ate ke y. T he pub lic key

can be kn own to e ver yone and i sused for encr yp ting me ssa ge s. Mes sa ges encr ypted w ith the

publ ic key can onl y be decr ypted u si ng

the pr i va te key. " T he k eys for the RSA a lgor ith m ar e

gener ated the fol lo wing w ay :

• Choo se t wo di st inct pri me nu mber s, p and



• q

• [their pr oduct i s a " se mip ri me; no m ethod no w kno w can

factor the s em ipr ime of tw o

• many -d ig it number s]

•

• Compute n = pq

• as the modu lus [Cha p. F iv e] for both the publ ic and

pri va te k eys

•

• Compute [Eu ler' s] totien t: ϕ (n) = (p-1)(q-1)

•

•

• Choo se an in te ger e s uc h tha t

• 1 < e <

• ϕ (n)e and ϕ (n) ar e copri me

• (G CD=1) , then e i s re lea sed a s the pub li c k ey e xponent

•

• Deter mine d (u si ng m odular ari thmet ic) w hic h sa ti sf ie s

the

• cong ruence rel at ion de ≡ 1 ( mod{ ϕ (n)},

• and use d as t he pr iv ate key

The Condot of Gr ammar -Ma the ma t ics ha s receiv ed sp ecia l

a ttent ion fr om pr oponent s of const ructi v is m to

replace nonst ructi v ism (pr oof by contr ad iction)in ma th- log ic, nota bl y in in tui tion is ti c (a .k.a .

const ructi ve , M ar ti n -Löf ) t ype t heor y, a

log ical s ys tem -se t theor y bas ed on the pr inc iple s

of mathema t ica l con st ructi v is m, intr oduced

by Per Mar t in- Löf , a Sw ed is h m athem ati cian and ph il osophe r,

in 1972. Intu iti oni st ic type theor y [ TT] i s based on a

spec ifi c i somor ph is m betw een pr opos it ions and ty pes : a



pr opos it ion i s ident ified w ith the t ype of i ts pr oofs.

T his i dent ific ation is u sual ly ca ll ed the

Cur ry– How ard i somo rphi sm , orig ina ll y for mu la ted for

pr opos it ional lo gic and s impl y ty ped l ambda calcu lus .

Type Theor y e xtend s th is ident ifi ca t ion to

pr edica te logic by

in tr oducing dependent type s contain value s. [or der !] T T

in ter nal iz es the in ter preta t ion of in tu it ion is tic logic

pr oposed by Br ouw er, He yt ing , and K olmogor ov, the so

cal led BH K inter pr eta tion .

T he t ypes of T T p la y a s imi la r r o le i n se t theor y

but funct ions def ina ble i n TT ar e a lw ay s computa ble .

T he Cur r y– Ho wa rd cor respondence: two s eem ing ly- unr ela ted

fam il ie s of f or mal is ms , pr oo f system s and m odel s of

computa t ion ar e s tr uc tur all y the s ame type of object .

T hu s, a pr oof is a prog ram, the for mula it pr oves is aty pe f or the pr og ram .

Mos t i nfo r mal l y, th is s ta tes th at t he retur n ty pe of a funct ion

(i .e . , the ty pe of va lues r etur ned by a function) is analogou s

to a logica l theor em, s ubjec t to hy pothese s cor respond ing t o

the

ty pes of the ar gument v alues pa ss ed to the function ; and the

pr og ram to compute tha t funct ion i s analogou s to a pr oo f oftha t

theor em.

Th is s et s a f or m of lo gic pr og ramm ing on a ri gor ous

founda tion: pr oofs can be repr esented as pr og rams , and

espec ial ly a s la mbda te r ms , or pr oofs can be run .



T he cor respondence ha s been the st ar t ing po int of a l ar ge

spect rum of new r esear ch after it s d isco ver y, lead ing i n

par ti cular to a new cla ss of for ma l sy ste ms de signed to act

both a s a pr oof ca lculu s and a s a t yped functiona l

pr og ramm ing

langua ge. T his inc ludes Mar t in - Löf's i ntu iti oni st ic ty pe

theor y and Coquand's C alcu lus of Con str uct ion s, t w o

calcu li i n w hic h pr oofs ar e r e gular ob jects of the discour se

and in w hic h one can st ate pr oper tie s of pr oofs the sa me w ay

as

of any pr og ram . T h is fie ld of r esear ch i s usual l y r efe r red

to a s moder n ty pe theor y -- "moder n" to d is tingu is h fr om

Ber tr and R uss el l' s ea r l y tw ent ieth -centur y t ype theor y to

avo id the R us se l l Parado x.

TY PES inc lude

• tuple s of r ea l num ber s

•

• mixed na tur al s and r ea ls

•

• nul l, un it , Bool ean, al lo wing ne ga tion

•

• uni ver se for eac h t ype



CHAPTER` TWENTY: WHAT IS SCIE NC E?

You real ize th at t he teac hing in our school s about SCI ENCE,

and the publ ic unde rstanding of i t, suf fers fr om the f act th at

no s cienti st or phi losopher of science has made an attempt to

dec lar e a clear one-s entence def ini tion of the sub ject . One

consequence i s the c lai m b y "T he Schoo l of Soc ial

Con str uct ivism" tha t A NY MATH EMA TICAL SYSTEM IS JUST ASOCIAL CON ST RUCTI ON -- compar able to liter ar y cri ti ci sm .

You kn ow then th at a cor rect ive of th is i s to dar e to

commun ica te su ch a defin it ion and hope f or r esponse s.

You bel ieve th at a f easi ble for mul ati on i s: S CIEN CE con si st s

of the four sta ges:

• explana tion

• meas ur ement (a s in Cha p. Nineteen)

• pr edict ion

• retr odict ion

You kn ow tha t pr edict ion concer ns what can be confir med

about futur e e vent s, w her eas r etr od iction concer ns w hat can

be conf ir med about pas t e vent s. (As examp le, Wikiped ia c ite s

anomalou s perihe li on of planet Mer cur y, di sco ver ed in

ninetenth centur y, retr odic ted by E in ste in' s Gener al

Relativity.)

You ar gue tha t the def ini tion di spo se s of man y

counter argument s. T hat the m easur ement r equ ir ement

el im ina tes ar guments of -- no e xperi mental mea sur ement s

ther e. And thi s def init ion el im ina tes "Cr eation S cience" ,

whic h of fer s explation, but none of t he other requir ement s of

the a bove defini tion .

You know sc ient ists may pr ovoke s ome people because thei r

"pos it ivist" asse rtion s ar e a s empha tic as tho se a sser ted

about rel ig ion. The his tori an, Jacque s B arzun , descr ibed

asser tion s of so me scient is ts a s being as fanatica l as any i n

rel igi on. Those who wou ld agree w ith thi s j udgment may see







• ar ti cul ated a spect rum of osci l la tion s of the

electr o ma gn et ic f iel d ;

• thi s spectr um i ncluded not onl y the l ight spect rum but

pr edicted radio waves, detected after M axw ell 's dea th b y

He inr ich He rtz (1857-1894) .

When Kelv in fir st hear d about H er tz' s work, he denounced i tas a hoax, which inti mi dated mos t phy si ci st s fr om

invest iga ting it , unt il Ita li an ph ys ic ist, Gu il le rmo Mar coni

(1874-1937), broadcast acr oss the At lanti c. To the end of his

lif e, Kelv in sai d he didn 't under stand ho w r ad io w or ks --

appa rentl y because he w asn't pr ovided w ith mechanica l

devi ces to acti va te i t.

At so me level, sci enti st s usual ly r esor t to a "b lac kbo x" .

In ph ys ica l sc ience , the atom was once a blackbo x to explai n

chem ical r eact ions . Pr og ress i n "qui ck & penetr ating"

photog raphy provi ded obser va bles w hich so me accept as

atom s, plac ing t hem out side the bo x. A tom ic con st ituent s,

su ch a s electr on s, p r oton s, neutr on s wer e blac kbo xes, but

col li d er photo s sho w events i nter pr eted as "bouncer s" o r

"explo si ons " of these , perha ps plac ing the m out si de t he box.But quar k cont ituent s of pr otons , neutr ons , mesons , ar e a t

so me per specti ve s til l blac kbo x.

Electr odynam ic s was "rid" of " inf ini tie s" by a blac kbo x f il led

with Feynman d ia gram s of p ro longed inter act ion .

Biolog y once had a bla ckbo x of cel l s . Then of cel lul ar

const ituent s . DNA i s out of the box (concr et ion). But bio log y

continue s to h ave a b lac kbo x of evolut ionar y e vent s of the

pas t ( il la tion) -- and s imil ar pr oblem s confr ont geolog y.

C. S. Peir ce (whose sem iot ics i s outl ined in Cha p. Thr ee)

taught u s about thr ee d is ti n ction s, t w o of w h i c h ar e c i ted i n

th i s cha pter' s ti tl e , cr it ical to epi ste molog y (" stud y of w ha t

we kno w") :



• concr etion s : dir ectl y obser va ble (su ch a s a par t icu lar

tr ee );

• abstr act ions : for mu la ted clas se s o f concr et ions

• i ll a t ion s , from Latin "to in fer"

Ber tr and Russel l said , "W hene ver po ss ib le , log ical con str uct s

ar e to be s ubst ituted for inf erred ent iti es ." T he "good ies " inbla ckbo x ar e i nfer red enti tie s . The best l ogica l cons tr uct s

ar e m athem atica l equa ti ons . When they l ink I NP UT to

OUT PUT, we ar e s atisf ied.

In Se mio tice s , Peir ce taught us the Indica tor . Si gnal , Icon,

sym bol , whic h pr ovide us with Str ate gie s for Sci ence and

Educa tion. ISIS : I(ndica tor) S(ignal) I(con) S(ymbo l). An

ind ica to r ha s the t wotup l s tr uctur e, for "O" denot ingobser va bi li ty and "I" denoting . in for ma tion

<Hi O-Low I, Low O-Hi I>

as in

<pink litmus paper, acid in test tube> A s igna l i s an i ndic ator under ph ys ica l and l ingui st ic contr o l :

<Hi O-Low I, Low O-Hi I> ⇒ <phys.-ling. control> as in: <lightning, electricity> ⇒ <telegraph key, Morse Code> In past times , fever was classified as a di sea se . Later , it was

real ized to be a highl y obser va ble sym ptom of an under l yi ng

ifection . So , medi cal sc ience pr og ressed from so lel y

al leviator s of fever (e.g , cold compr esse s) to conjo in th iswi th a lle vi ator s of in fection -- the greatest prog ress was fr om

use of pen ici ll in i n twentieth centur y. T hus : <fever, infection> ⇒ <penicillin, doctor's care>

In the Sci ence of Optics , Ibn a l- Ha ytha m (kno wn i n as

Al hacen or Alhaz en i n Wester n Eur ope) 965 -1040, is r egar ded

by man y as the "father of moder n optic s" (Wik ipedia) , due to



his for mul ation of geomet rica l optic s . He developed the be st

optica l devi ce of t hat time, the Cam er a Ob scur a , In one for m,

a bo x wi th a per atur es on oppos ite side s al lows light r ays to

pas s thr ough the box onto a pa per scr een for mini ng a color ed

ups ide ima ge accur ate i n per spect ive meas ur ements of the

pr e-i ma ge. T hi s then f or med an opt ical i ndic ator :

<Camera Obscura, light ray> In the 18th centur y, a mir ror was used to inver t the ima ge to

the per specti ve of the pr e-im age. This m oti va ted t he

development of photog raphy , when chem ical s wer e

disco ver ed whic h "f ixed" the ima ge , inaugur ating t he signal

sta ge .

<Camera Obscura, light ray> ⇒ <photochemicals, photographersettings> In c he m i str y , the 9th centur y chemi st , Jabi r i bn Hayyan

(kn own as "Geber " i n Eur ope), cons ider ed b y many

(Wiki pedia) to be "the father of chemi st ry", b y intr oduc ing a

syste matic, experi mental la bor ator y resear ch. He invented

the be st chem ica l of those time s, the alemb ic , a chem ical

st ill composed of twu retor ts (g la ss s pher ical vessel s w ith

long down-po inting nec k) connected by a tube; placed overthe f ir e, l iquid s can be separ ated . With thi s, he ana lyzed

many c hemi cal subs tances , co mposed la pidarie s (gems tones) ,

distingu ished betw een alkal i s and ac ids , and manuf actur ed

hundr eds of medi cina l dr ugs . Thi s then f or med a c he m ica l

ind ica to r :

<alembic, substances>

The Russi an c hemi st , Dmitri Mende lee v (1834-1907), in 1869 ,developed the per iod ic ta b le of chem ical e lements to

explic ate recur ring ("per iodic ") tr end s in the pr oper tie s of

the e lement s . This inaugur ated the i>s igna l s ta ge . <alembic, substances> ⇒ <good lab research, period. table> In Me c hanic s , ar ises the Ho molog y. mass: mec han ics ::

elect ric char ge: electric ity . Mas s i s r esi s tance to



acceler at ion and is pr opor t ional to weight . This is impl ici t in

Ne wton's Law s o f M ot ion for Mec han ics . They s tate:

1. Law of Iner tia : ever y bod y w il l per si st in i ts st ate of r est

or un ifor m m otion un les s acted upon by a For ce.

2. For ce equals mass time s acceler ation.

3. For ever y act ion ther e i s an equal and oppo si te r eact ion.

The ancient scale pr ovides a mec han ical indic ator

<scale, masses> Newtonian Laws then pr ovide f or the signa l sta ge :

<scale, masses> ⇒ <mechanical measurements, Newtonian Laws> The concept of "Ant itone" , intr oduced i n Cha pter One , and

discu ss ed i n Cha pter F our teen , e xpl ica tes sc ient if icpr oces ses .

An anti tone coor dina tes an i ncr ea sing or dering -- Maxtone -

wi th a decr eas ing or der ing (M intone) s o tha t a bound on one

or dering coo rd inantl y i nduces a bound on the other or dering .

Cl imbi ng sta irs is ant iton ic: the number of ri ser s ascended is

MAXTONE; di stance from the top is MINTONE. Similar ly,find ing a so ck in a dr awer or f il e i n a f ile ca binet . (Sci math in

your dai ly life.

The Ma the ma t ical Pr otot ype is the hyperbo li c equa tion :

xy = 1

A Pr otot ype of Mec han ics is the mac hine : a dev ice act ing

upon input f or ce or input tor que (r ota t iona l f or ce) by

anti tonica ll y ampl yi fing it i nto output f or ce or tor que .

One of the bas ic mac hine , kno wn as the l ever anti tonica ll y

"tr ade s of f length for input f or ce". (A boy sitt ing on l ong ar m

of a teeter- totter can balance an adult on the shor t end of the

teeter- totter lever.)



The Law of the Le v er is:

Load arm • load force = effort arm • effort force.Similarly with another ba si c ma chine kno wn as the pul le y ,

whic h ant itonica ll y "tr ade s of f ropelength for in put f or ce" .

Al l m achine s anti tonica ll y "mak e so me k ind of tr adeo ff".

Mec han ics :

Loader•loadforce=amplified-efforter•effortforceAn ancient Protot ype is the f ix ed vibr at ing str i ng , attri buted

to P ytha go ras (bor n betw een 580 and 572 BC , died betw een

500 and 490 BC ). The MINTONE i s l ength of st ri ng v i br a ti ng .The MAXTONE is pitc h of s ound .

The anti tone is imp licit in pr oce ss s of moder n ph ys ic s, su ch

as quant ics . which un ified opti cs and m ec han ics .

The bas ic equa ti on of opt ic s i s νλ = c, which i s anti tonic . A

pri mar y equa tion of me chanic s i s im pl ic itl y anti tonic : T = K +

P, wher e T is tota l ener gy ; K is kinet ic ener gy ; P i s potentia l

ener g y . Let T = log S; K = log J; P = log O. Then, T = K + P ⇔log S = l og J + log O ⇔ S = J O, another anti tonic form. The

obser va ble , momentu m , can be der ived from k ineti c ener g y , K

= 1 /2(mv 2): p = m (2K) 1 /2 = m(2(T - P)) 1/2 . The obser va ble,

wavelength , λ, can be der ived from the opt ical equa t ion . The

mi xtur e of the opt ical and mec han ical anti tones resu lts in the

Planc k-E in ste in Law : E = hν, recast anti tonica ll y a s Eλ = c h,

And the (a lready anti tonic ) de B rog lie Law is pλ = h.The above was discu ssed in Cha pter One about using

Dimen si onal A lge br a to der ive phy sica l l aws. But these were

all kno wn. You no w see how to use D im ens iona l A lg e br a to

deri ve Law s never bef or e seen. Can the y r ea ll y be Law s of

Phy sic s? Yes, becau se Law s of Ph ys ics ar e u sed to der ive





The mode of hea t tr ans mi ssi on th at cor rectl y i s radia tion

use s an electr oma gnetic med ium , similar to light , but of a

dif f er ent fr equenc y . That mislabel led device i n house s,

school s, stor es , etc. , is defin ite ly not an e lectr oma gnetic

tr ans mi tter - - in stead , a con vector .





B(B( B(B(B(B( B(Adam)))))) ) = Methu sal eh , the 7th gener a t ion

be ga t fr om Adam .

You real ize the be ga t-funct ion e mul ates ARIT HME TIC' s

succe ss or- function , wher ein S(_) , wher e S(n) = n + 1, that i s,

the s ucce sso r of number n i s number n and one m or e . Thus ,

S(0) = 1; S(1) = 2 , or (embed ded) S(S(0) ) = 2 . Thus,S(S(S(S(S(S(S(0))))))) = 7 . T he se venth succes sor of z er o i s

se ven , ju st a s M ethusa leh i s the s eventh be ga t of A dam .

You real ize th at, w hene ver y ou put sen tences in side

sen tences you ar e RE CURSI NG.. You W EB-l ear n t hat MIT

mathema tical l ingui st Noa m C hom sk y hy pothes ized th at

humans gener ate langua ge r ecur sivel y, rebutt ing beha vior al

psychol ogi st B. S . Sk inner' s cla im tha t human s lear n langua geby t ria l- and-er ror ass oci ation s . Chom sk y showed t hat

as soc ia t ion is m is modeled pr ob ab il is ti cal l y by mul ti ple -or der

Mar kho v c hain s . A cho sen wor d dete rmines the pr oba bi li t i es

of the words th at f ol lo w . Mathema tica l p sycholog is t Geor ge

Mi ller found t hat (gi ven a chos en w or d), "on the aver age" ,

four option s exis t for the g ramma t ica l c ate gor y of the next

w o r d . In a f ir st -or der M ar kho v cha in , a chi ld must lear n 4 · 4 =

42 = 16 ass oci ati ons (four conte xts times four ne xt w or ds) .For a second -or der cha in (deter min ing "od ds" on the ne xt t wo

words), 4 · 4 · 4 = 43 = 64 ass oci ati ons ; etc. As conte xt

incr eases , the number of lea rned as soc ia t ion s Q UAD RUPLE S.

Con side r a gain tha t s entence of the ta lk y l itt le g ir l) :

"Dor oth y, w ho met the W icked W itch of the Wes t i n Munc hkin

Land wher e her wicked w it ch s ister was ki ll ed, liquida ted her

wi th a pai l of water ." Sub ject and pr edic ate of "Dor oth yliqu ida ted her w ith a pai l of water " ar e separ ated by an

eighteen -wor d clause -- i nvok ing an eighteenth -or der M a r kho v

chai n of 418 = 68 ,719,476 ,736 ass oci ati ons s. Lear ning to

speak such a sentence b y Ski nner's t ria l- and-er ror mode l

requir es nea rly 69 bil li on ass oci ati ons . The child should live

so long!



You know recu rsion by t he compound inter est your money

accr ues i n a s avings account in a bank by compound inter est .

Your saving s account grows by recur sion.

The Wik ipedia expli ca tion s hows th at t he Recur sion function

feeds bac k in to i tse lf. B ut it mu st not do s o refle xi vel y for

thi s invokes cir cu lar log ic and will br eakdo wn a ca lcul ator orcomputer .

You know tha t Recur sion u ses s tep s as s imple to follo w as

fol lowing a food r ecipe and rese mble s the recu rsion in

langua ge (a bove), in ser ting se ntence into s entence :

• st ar ti ng w ith a funct ion, S(-) - - kn own as "T he Succe ss or

Function" -- w ith a functand n - - kn own as "a natur al

number" -- so t hat you h ave S(n)

• you kno w thi s pr oceeds by outputt ing a functand, S(n) =

n+1

• after w hich the functand becomes a recu rsi ve funct ion,

S(n+1) = (n+1) +1

• fol lowed by nonr ef lexiv e repeti tion S((n+1)+1 ) =

((n+1 )+1)+1

•

fol lowed by nonr ef lexiv e repeti tion S(((n+1) +1)) =(((n+ 1)+1)+1)+ 1

• etc.

You also know thi s sequence mu st have a spec ifi ed beginn ing

• whic h you assign the pr oper name 0 ("zero")

• so the Succes sor funct ion output s a functand, S(0) = 0+1

• after w hich the functand becomes a recu rsi ve funct ion,

S(0+1) = (0+1) +1

• fol lowed by nonr ef lexiv e repeti tion S((0+1)+1 ) =

((0+1 )+1)+1

• fol lowed by nonr ef lexiv e repeti tion S(((0+1) +1)) =

(((0+ 1)+1)+1)+ 1

• etc.



You also kn ow tha t the other output s can be gi ven pr oper

names , so y ou ha ve:

• S(0) = 0+1 = 1

• S(1) = 1+1 = 2

• S(2) = 2+1 = 3

• etc.

Tou lear n th at, i n computer sc ience . a pr oblem is s ubdi vided

in to sma ll er pr ob lem s of the sa me t ype (pa tter ns in patter ns)

and recur s ion is appl ied to solv ing the s ubpr oble ms . This i s

dyna mic pr og ramm ing , developed i n the 1950 's by

mathema tici an, Richar d B el lman (1920-84) , O ne pur pose was

to min im iz e co st s of De fense Depar tments pr o jects

contr acted to cor por a tion s .

The Be ll man E qua ti on per for ms recur si on on a "po li cy

function" , whic h is dis counted, for the t im e de lay , in the

manner of bus ine ss dec is ion s . (Wiki : "Discount Yield =

Char ge" t o D ela y Payment for 1 year/ De bt Lia bibi li ty.")

Be ll man' s Pr incip le o f opti mal it y st ate s tha t (Wiki) " if the

pol icy function is opt ima l for the i nfin ite su mmation , then i t

mus t be the case tha t - - w hatever the init ial state and [f inal]

deci si on ar e -- the remain ing [sub]deci si ons m us t con sti tute

an opt ima l pol icy wi th r egar d to the s tate r esul ti ng fr om tha t

fir st deci si on (a s e xpr essed by the Be ll man equa tion) ."

You lear n tha t a pr oces s used , for some pur poses , to replace

Recur si on is Iter at ion . Bo th r epea t (loop) a par t of an

Algori thm wher ein the l ogi cal st ructu r i s con stant but the

da ta changes . In Recur s ion , a loop is run for a spec ifi ed

number of times (spec ified number of repeti tion s). In

Iter ati on , a kno wn v alue or gues sed va lue of the des ir ed

computi on is run a s in it ia l v alue" ; i n loop ing (r epea ting) , the

computa t iona l r esul t of fir st run replace s the ini tia l va lue ;

etc. ; unti l pr og ram mer i s sa ti sf ied wi th output and end s the

it er ation .



RE CURSIV E A RIT HME TIC0 1 2

3 4 5 6 7

8 9 10 11 12

13 14 15 16 17

18 19 20







mul ti pl ica tive M No id con ser ving al l mu lt ipl icati ve pr oper tie s.

Suc h a s tr uctur e i s a "Ri ng", her ein a Rng .

The Rtn l Nu mber s al so for a , ul ti pl ica tive G rp. This double

Grp i s kn own as a "Fie ld" , her ein a Fld.

You kn ow tha t chil dren can be taught the G rp concept via

"the cr eeping ba by g roup". You kno w cr eep ing is on hands

and feet , bel ly of f the f loor , (Cr awling scr apes t he be ll y.) You

real ize th at the G rp oper ands ar e cr eeps and the G rp

oper ation is conca ten ation , one cr eep f ol llo wed b y another

creep:

• you l ear n tha t cr eep conca ten ation for ms a Gr poid

• you l ear n cr eeping is assoc iati ve, so y iel ds a Sem ig rp

• you r ea li ze a long pause in cr eep ing (d if ficult with

pr atfall) is the Gr p D nty , tr ansfor ming the cr eep Sem ig rp

in to a cr eep Mno id

• you r ea li ze th at the Nver s cr eep pr ovi des Gr p Nvr s for

eac h cr eep , tr ans for ming the cr eep Mno id into a cr eep

Gr p

(You kno w t hat, p rior to master ing the Mvrs cr eep , a ba by

cr eeps in to a cul -de -sac and y ells to be rescued . You kno w

tha t, after m as tering the Nver s cr eep -- watch out ! -- the ba by

wi ll be "e ver yw her e" .) M athema tic s in The Ba by Wor ld!

You lear n tha t the Swis s cogn iti ve ps ycholog is t, Jean P ia get

(1896-1980), wrote abo ut the cr eep ing G rp, but did not use it

to t eac h m athem ati cs .



CHAP TER TWEN TY-T HR EE: W HAT IS M EAS UR E T HE ORY?

Yoi W EBlear n tha t, unti l found ing of T he N a t iona l B ur eau of

Standar ds , ther e pr evai led what you m ay ca ll " The Right to

Measur e". This meant tha t any s tor ekeeper cou ld s ay w hat

was a ga ll on of mi lk , a pec k of pot atoes , a y ar d of cloth , etc.

The onl y k ind of mea sur es which w er e not open to the

st or ekeeper' s FIA T wer e dig ita l measur es deri ving fr omcounting -- say, a do z en e g g s .

You WE Blear n tha t T he N a tiona l B ur eau of S tandar d s was

es ta bli shed in 1970 , on a p lan ad apted fr om th at s ketc hed by

Char le s Saunder s Peir ce (our greates t 19th centur y logic ian -

mathema tici an, our greatest phi lo sopher a lthough incor rect ly

assoc ia ted wi th "the pr agma tic phi lo sophy "). B ut P eir ce was

never g iven of ficia l cr edit . Toda y it is cal led T he N a tiona l

Inst itute of S tandar ds and Technolog y .

You WE Blear n tha t the se metr o logi st s (meas ur ement-

sc ient is ts ) , as wi th ph ys ics and eng ineering pr ofessor s in the

uni ver sit ies , ig nor e the inte grity of "The T heor y of

Measur ement Sca le s" , as ini ti ated i n 1940 by H ar var d

psychophy sic is t, S . S. Ste vens .

You lear n tha t a Ste vens' m eas ur em ent s ca le i s defined i n

ter ms of tha t ma them atica l tr ans for ma t ion le av ing the s ca le

in var iant (thu s, per muta t ions for a ty pe-s cal e ); and a ls o

defined by clear listing of the "per missib le statis tics" of a

given mea sur ement sc ale (thus , the average for t ype s i s the

mode , or most fr equent -- not the ari thme tic mean , since "you

can't ad d apples and or anges" , except as ite ms of t he fr uit

type).

You kn ow an example of s tati st ica l abuse i n "the bel l cur ve"

of IQ Tes t S cor es. Can two people said to have IQs of 60 eac h

accompl is h al l a per son said to be of iQ 120?

The list of the se sca les , in t he Table be low, lead s off w ith a

dist inct ion S tevens did not mak e of a nom ina l m easur e .



Reason : cri tica l thing s and event s shou ld be uniquel y named

-- pos tul ating "The F ir st Law of Mea sur ement : LET ALL YOUR

NAMES BE UNIV OCAL! "

If you read thr ough T he R epor t of the Pr es ident ia l

Com mi ssion O n the S pace-Shutt le Cha llenger D isas ter (1986) ,

the r epor t to Cong ress about the Chal lenger D isa ster of 1981-- whi ch k il led 8 a str onaut s and a school teac her -- you 'l l see

why. Betw een t he lines , you d iscover tha t the good

meas ur ements of engineer s wer e WAST ED when the dec is ion -

maki ng (as to tak e-o ff) was k icked up sta ir s to be

discombob il ated by the buz z-w or ds of mana ger s and

beaucr at. For e xamp le, infor mation at one point star ts out,

"A ll jo int s ar e leak c hecked to a 200 ps ig sta bil iz ation

pr essur e, fr ee of contami nation i n the s ea l ar ea and mee t O -ri ng squeez e requir ements ." But th is appa rentl y was

tr ans la ted as " The fr eni sl aw i s ri gbik and the mome raths

outg rabe." And note tha t inves tig ating comm ittee me mber and

Nobe l phy si ci st , Richar d Feyman , di sco ver ed th at those O-

ri ngs , when dunk ed in ice-w ater , would not pass tho se

"squeez e requir ements ", which had been conducted a t higher

temper atur es - - and cou ld allo w gas to escape and explode !

(T he one engineer among the f inal dec is ion -mak er s was

or der ed to " Take of f your engineer' s hel met and vote with us

for tak e-of f !")

The PO INT : STANDARD S S ET UP FOR FE DE RAL ENF ORCEME NT

REMAIN SUBJECT TO POLI TICAL WRANGLI NG. But some one

them TRANSLA TE AS M EAS UREME NTS ! Trans for m FIA T into

MEASU REM ENT and nay -sayer s can be put in the company of

The Roman C atho li c C hur ch tr yi ng t o mak e Ga li leo say th at

the ear th doe s not move around the sun.

THE MORE THAT NORMS ARE SUPPORTED BY MEA SUREM ENT

THE MORE SECURE THEY ARE!



(I'v e long had an inter est this, ha ving been a m ember of T he

Ame rican As soci ati on of Q ual it y Con tr ol Engi neer s , and

having taught qual it y contr ol engineer ing m any time s. )

TABLE OF MEASUREMENT SCALES (after S. S. Stevens)

SCALETRANSFORMATION

GROUP

ALLOWABLE

STATISTICS

EXAMPLES

Nominal Identity Mode(s) Proper names; technical terms

Typological Permutation Mode(s)Taxonomy of plants and

animals; etiology of diseases

Ordinal Isotonic Median; range

Moh mineral hardness scale;

IQ; Richter-Kanamori

earthquake scale

Interval LinearArithmetic mean;

variance

densities;Celsius & Fahrenheit

temperature scalesRatio Similarity all statistics weight, length, etc.

We'r e g rowing par tia lly deaf due to en vir onmental noi se s,

espec ial ly ar ound ai rpor ts , at rock concer nts , and due to

publ ic boombo xes . The sa fety r ules are ba sed on the decibe l

sca le , de vi sed accor ding to instr umenta l respon se to s ound .

Ste vens cr eate a sone mea sur e ( ONLIN E), based upon human

respon se to s ound, w hich would be better for us , since smal lchanges in t he dec ibel scale ma y be equ iva lent to lar ger

changes in t he sone s cale. It was because of challenges to

his so ne scale that S tevens cr eated hi s "T heor y of

Measur ement Sca le s" .

You know a " measur ement pr ob lem" in w hich these

dist inct ions become cri tica l, reso lv ing an a ppar ent par ado x.

You are famil iar w ith the jar gon : " Equal s ad ded to equa ls

gives equa ls "; "Equal s sub tr acted fr om equals g ives equal s" ;

"Equa ls t ime s equal s gi ves equa ls "; " Equal s di vided by equal s

gives equa ls ".

Cor rect for number s. But you kn ow thi s is on ly condi tiona lly

for meas ur es , as in t he te mper atur e scale , usua ll y mea sur ed

in FAhr enheit or C enti grade de grees .



You know tha t the fr eeezing of water is 32 ° on the Fahr enh eit

Scale and 0 ° on the Centi grade Sca le . Y ou kno w th at the

boi li ng of water is 212 °F; and 100 °C . H ence , y ou find that,

betw een boi ling and freezi ng, the Fahrenheit range i s (212 -

32) ° = 180 °, w hil e the Cent ig rade range is 100 °.

Then you h ave the de g ree ratio of 180/10 = 1 .8 = 9 /5. You canuse th is r atio to con ver t temper atur e T on the Centi grade

Scale to T on the Fahr enheit S cale , since 9/5 TC° + 32 ° = TF °.

You can us e the inverse ratio (5/9) to con ver t t emper atur e T

on the Fahr enhe it S cale to T on the Cent igrade S cale , since

5/9(T - 32) ° F = T°C.

Thus: 50 °F = 10 °C; 68 °F = 20 °C ; 86 °F =30 °C; 104 °F = 40 °C ;

etc.

Now, you find why " equal s di vi ded b y equal s" fai ls her e. Since

68 °F = 20 °C , and 50 °F = 10 °C , to sa y "Equa ls d iv ided by

equal s gives equa ls", would her e become "68/50 equal s 2/10 "!

Incor rect! What goes wr ong? Ho w can the l eft fr action, w hich

was greater than one, equa l the ri ght fr act ion, which i s less

than one ?

You know tha t the exp lana tion der ives from a Theor y of

Measur ement Sca le s of Ste vens.

If your pr oblem involved a RATIO mea sur e, the "equa ls

divi ded b y ..." would "wor k" ; however, being an I NTE RVAL

MEA UR E, th is ob viou sl y doesn 't . T he LI NE ARITY , y = ax + b, is

impl ic it in the F-to -C tr an sfor mation r ule . You note the

constant ter m b, whic h equal s 32 i n the F . Sca le, but 0 in theC. scale .

RATIO ME ASU RE rel ates to the s im i lar it y equa ti on : y = ax ,

wi thout I NTE RVAL CO NSTANT, b.

You kn ow tha t DIFFE REN CES O F INT ERVAL MEASU RE S WILL

PASS T HE RATIO T EST, s ince th is subtr act s out the INTERVAL



(LINE AR) CO NSTAN T. Thus: (104 °F - 68 °F) /( 86 °F - 60 °F) = 36/36

= (40 °C - 20 °C) /(30 °C - 10 °C) = 20 /20 = 1. QED

You know tha t man y metr ical abuse s would be EXPO SED b y

imple menting the condit ion of NO ME ASU RE WIT HOUT A

PREDIC TIO N! Any one clai ming to ad d I.Q. Scor es (to bu ild a

"Be ll Cur ve") would have to predict tha t he could find two"lo w" I . Q. s cor er s who , co mbined , w ould best a "h igh"

scor er !

You know tha t the excell ent, but now, neglected , concept --

di scu ss ed i n Cha p. N ineteen -- can teac h st udents how to

lear n or di sco ver. In the homolog y, Fresnel len s: con vex

len s: :st air s: r amp , the s tudent can l ear n tha t the F resnel lens

accompl is hes opt ical ly what a con vex len s does i n greaterdimens ion s. A con vex l ens , in a ligh t-hou se, to ma gnif y light

for a g reat di stance , w oul d topp le o ver. The opti mal

dimens ion s of the F resnel le ns -- due to s cienti st Augus t-Jean

Fresne l (1788 -1827) -- made pos si ble the magnicent light -

house s of the nineteenth centur y, which gu ided so many ships

at sea.

Wher ea s t hat the a bove homolog y s er ves to teac h, you kno wtha t the homolog y, 1/2 = 11/? moti vates d is cover y. (you kn ow

the r esemb lance of thi s la st ca se po int s to the use of

homolog y i n "mu lt ipl e c hoice tes ting ", although i t is doubtful

tha t man y teac her s or tes t-mak er s ar e awar e of i ts im pl ic it

pr esence .)

You kn ow tha t the homolog y can sho w use of fr act ions , or

ratios , to explic ate d if f er ent dom a ins of e x is tence on t hi sear th . And how th is deri ves fr om a ne glected "science"

wr itten up long a go b y Galileo G al ile i (1564-1642).

You know tha t "T he W or ld of Mathema tics" , edited by James

R. New man, contain s an a rtic le b y evolut ionar y s tati st ic ian,

J. S. H al dane (1892-1964) , ent it led " On Being the R ight Size" .

Ha ldane note s tha t a human f all ing fr om a n ine -st or y bu lding



would h ave bones cr ushed. B ut y ou note the s ur pr ising

comment tha t a mous e w ould reac h a cer ta in ter minal

veloci ty and f loa t down, to land with a minor bump . Wikipedia

explain s thi s i n ter ms of a squar e-cube la w : "The s quar e-cube

law (or cube-s quar e l aw) is a p rinc iple , dr awn fr om the

mathema tics of pr opor tion , tha t is appl ied i n engineer ing and

bio mec hanic s. It w as f ir st demon str ated in 1638 in Gal il eo' sTw o N ew Sc iences . It s tates : W hen an ob ject unde rgoes a

pr opor tiona l incr ea se i n s i z e, i t s new v olu m e i s p r o por tiona l

to t he cube of the m ul tip li er and it s new su rf ace ar ea i s

pr opor tiona l to the sq uar e of the mu lt ipl ier ." You find :

v2 = v1(l2/l2)3, A 2 = A 1(l2/l2)2

wher e v denote s volume, l denote s length (for any lengthmeas ur e), A denotes surf ace are; sub scr ipt 2 denote s new

dimens ion ; s ub scri pt 1 denotes origi nal dimens ion , This

princ ip le a pp lie s to a ll so li ds .

You kn ow tha t the squar e-cube law for the f all ing bod ies

st ates a r atio of surf ace ar ea contacted> to ma ss acted upon

by gravi ty . You kno w tha t tje s hift fr om human to mouse is a

decr ease in d imen si ons wher eby the volume mul tip li erdecr ease s " fas ter" than the s urf ace area of a ir pushed aw ay

in fa ll ing . So t he denom ina tor her e decr ea ses " fas ter" than

the nu mer ator , so tha t the m ouse slows down in fall ing to a

cer tain ter mina l veloci ty, and floa ts the r est of the way down

You kn ow tha t the squar e-cube law also implies tha t the

giant s of our legends cnnot exis t on the ear th. The "cube"

st ill per tain s to volume . But the "squar e" per tain s to musc le-power , since, as you kno w th is depend s upon the s quar e of

the m usc lle le gth . So , as the mul tip li er incr eases , the volume

(wi th ma ss) wi ll incr ease " fas ter" than musc le t o mo ve. The

le gendar y gi ant w ould shatte r h is legs , ju st by t rying to mo ve

one step .



You know tha t the lar gest an ima ls on ear th are whale s, whic h

mus t be suppor ted b y the bouy an cy of water . You kno w tha t

the l ar ges t of t he dinosaur s had to live in s wamp w ater .

Ga li leo used h is princ ipl e to es timate the maxi mal he ight for

a tr ee . His es ti mate was onl y cor rected when the se qoyahs of

Ca li for nia wer e d is cover ed.

You kn ow tha t Ga lileo' s "Two New Sc iences : concer ned the

la w of f a l li n g bod ies and thi s squar e-cube law which i s lii tl e

taught e xcept in cour se s for engineer s and bio mec hanic s.

Haldane noted tha t, while gravi ty i s not the proble m to the

mous e th at is is to humans , the problem of surf ace tens ionof

l iqu ids i s a g reater pr oblem for the mous e than for the human,

A mou se, in cr awli ng out of a pool , car ri es of f on its body an

amount of water equal to its own w eight, A fl y fal ling into

water wou ld h ave to dr ag out man y ti mes i ts own weight . So

Ha ldane co mments that a fl y, in tr yi ng to get a s ip of water

tha t a human i s i n lean ing over a clif f to p ick a wi ld flo wer.

So you f ind the homo log y, gravity: human: : su rface ten si on:

mous e.

You kn ow tha t on the moon , a human weigh s one-n inth of

what she /he doe s on ear th . Y ou kno w th is in sp ir ed a sc ienth -

fict ion wri ter s to descr ibe teen-a ger s on the moon to go i nto

the l ar ge c hamber w her e o xygen w as created f ir the m oon-

dweller s. T her e, they cou ld j ump out and floa t down war d in

the b il lowing a ir, jus t a s the mouse floa ts on ear th .



CHAPTER TWE NT Y-FOUR :W HAT IS YOUR AGENDA FOR

CONST RUCTI ON OF ARI THMETIC ?

• You kn ow AN ARIT HMETIC CO NSISTS of OPERATIONS AND

OP ERANDS (Cha p, T hr ee), needed because :

• The pr oblem of ted ium in da il y pr actice begs f or

sho rtcuts .

o The tediu m of tal lying , shor tcut b y Numbering via

Recur si on.

o The tediu m of combin ing counts , sho rtcut b y

oper ation of Ad dit ion v ia R ecur sion on N atur al

Nu mber s as oper ands .

o The tediu m of combin ing addend s, shor tcut by

Mul tip li cation via Recur si on on ad dend s a s oper an ds.o The tediu m of combin ing mul tip li cands , sh or tcut by

Exponenti ati on via R ecur sion on Mu lt ipl icand s as

oper an ds.

o You real ize th at R ecur sion al lows un li mi ted

sho rtcutti ng beyond Exponenta tion, but no demand

for thi s.

• You real ize need of Inver se for eac h oper ation , becau se:

o tacti cal ly, for the Lear ning P roces s, In ver ses for eac ha v erif ica tion of ca lcul ation ;

o str ate gica ll y, to COMPL ETE THE GROUP (Cha p. Two) ,

since ari thmet ic s o va riou sl y scr amble s the

proces ses , you need to kno w tha t V ALUE IS

CONSERVED UND ER THESE V ARI OUS

TR ANSFO RMA TIONS. (You can't depend upon The

Tooth Fair y for thi s. Nor on your puri ty of hear t.)

Pr oviding an inver se for any oper ation comp lete s it sgroup , as you lear n, yieldi ng a Con ser vation Law .

• You lear n thi s a bout existence of an i nver se . A gener al

oper ation o, should have a lef t-and -ri ght i nver se o' such

tha t a o b = c imp li es th at c o ' b = a and c o' a = b.



• You real ize an inver se exis ts i f the oper ati on i s WELL-

DEFI NE D ("cance lla ble"), pr ovid ing f or right in ver se or

left in ver se :

o You lear n tha t, if a o b = a o x impl ie s tha t x = b , then

oper ation o ha s a right in ver se .

o You lear n tha t, if a o b = x o b impl ie s tha t x = a , then

oper ation o ha s a left in ver se .o You lear n tha t, if oper ati on o is com muta tive, then

right and left inver ses fuse in to a single i nver se.

You lear n tha t the above provides an Agenda for const ructing

the N umber System s in th is book, together w ith the ir

Ar ithme tic s.







The tediu m of repetit ion s a ls o a ri se s fr om ad di t ion , as in 3 +

3, implicit ly " double counting" . This pr ob lem is sol ved b y the

oper ation of mu lt ip lic ati on def ined by t hir d recu rsion on t he

second recur si on or recur sion on ad dit ion : a • 1 ≡ 1; a • S(b) a

• b + a. Thus, a • S(1) ≡ a • 1 + a, or a • 2 = a + a ; a • S(2) ≡ a •2 + a, or a • 3 ≡ a + a + a; etc . Thus , the pr oble m of t hir d

recur si on become s a pr ob lem of second recur si on, w hich i s a

pr oblem of f ir s t r ecur s ion . And com muta t iv i ty and

as soc ia t iv i ty ar i se for mul tip li ca t ion , der iv ab le fr om t he

recur si ve defin iti on of m ul tip li ca t ion .

Her e al so ar ises the di st ribut iv e l aw : a • (b + c) ≡ a • b + a • c,

whic h is der iva ble fr om the recur siv e defin it ion .

The tediu m of repetit ion s a ls o a ri se s w i th m ult ip li ca t ion , asin 3 ⋅ 3, implicit ly " double mul tip li cation" . This pr ob lem is

solved by four th recur si on on th ir d recur si on: recur s ion on

mul ti pl ica t ion , as exponentia tion : b0 ≡ 1 ; bS(p) ≡ bp + 1 = b p • p .

Thus, 3S(0) ≡ 3 0 • 3 , or 31 = 1 • 3 = 3; 3 S(1) = 3 1 • 3 = 3 • 3 = 9, or

32 = 3 • 3 = 9; etc.

However, component rela tions d if fer f or th is case , fr om

pr evi ous cas es . Ad di t ion and mu lt ip lic a ti on ar e commut ati vebut e xponenti ation i s not . Counter example : 23 = 8, but 32 = 9.

This f ailure ha s a b ig consequence, sho wn belo w.

This expl ica tion suf fices for ari thmet ic, pr ovid ing it s pri m ar y

oper a tion s .

Sum mar y: 1s t r ecur sion yie lds count ing (number) . 2nd

recur si on yield s ad dit ion. 3r d r ecur son yie ld s mul tip li cation .

4th r ecur sion yields exponenti ati on. These ar e "d if fer ent

levels of recur si on" whic h ar e " repr esenta ble dimens iona ll y" ,

whic h aids our thi nking in der iv ing ar ithmet ic ru les i n

dif f er ent nu m ber s y s tem s . Think of 1s t r ecur s ion as of

"di mens ion zero" or " dimens ion of the po int repr esent ing

number" . Think of 2nd recur s ion for ad diti on as of "di mens ion

one" o r "d imen si on of the l ine or se gment or row", a s in (* * * )



+ ( * * ) = (* * * * * ) . Think of thir d recur si on f o r mu lt ip lic a ti on

as of "di mens ion two" or "d imen si on of a grid or ta ble wi th

rows and co lumns ". C er ta intl y, the for mal is m kno wn as

"Ca rtes ian product" ( R X C , for rows and column s) creates

such a g rid or ta ble , and mode ls multipl ication. Fina lly, think

of 4th r ecur sion for expo nent ia t ion as of "d imen si on thr ee" ,

as in t he jar gon for number cubes , su ch a s 23 = 8. And theCar te si an pr oduct , A X B can be e xtended to A X B X C. We'l l

see be low how th is he lps our th inking about ar ithmet ic rules

in dif fer ent number syste m.

We no w an swer our fir st quest ion , " Why so many number

syste ms?" A nswer : T o provide an in ver se oper at ion for eac h

pri mar y oper a t ion . Thi s is rar el y e xpla ined .

• We cer ta inl y need inver ses for chec king our ca lcul ation .

• Another reason for having an in ver se for eac h oper at ion i s

tha t "the group can be comple ted" and "th is con ser ves

the s tr uctur e under tr ans for mation s" (such a s tha t in t he

pr evi ous it em) . H er e ar e the s ta ge s f or de velopment for a

group :

o groupoid f or med b y conca tena t ion of oper a tion s : forgi ven oper ation o, the conca tena tion of one s uch

oper ation follo wed b y another s uch is equ iva lent to a

single such oper ation .

o semigroup b y as soc ia t iv i ty of oper a tion s : o ( o o) = (o

o) o, tha t is, the as soc ia ting of oper ationa l in stance s

is ir rele vant. T his pr ovides for accomp li shing any

oper ationa l demand.

o monoid w ith ident it y ele ment , pr ovid ing f or s ta si s i noper ationa l pr ocedu re and for id entif yi ng an inverse .

The appli ca tion of the i denti ty oper ations to a

st ructur e le aves i t unc hanged.

o gr oup co mpleted by i nver se for ever y oper and of

syste m , th at i s, an ele ment comb ined oper ationa ll y

wi th i ts in ver se is equi va lent to the id enti ty element .

o At leas t t wo benefits fr om group comp letion :







if f (a + d) = (b + c) . Also , in langua ge usefu l la ter , we say

"th is con ser ves defined dif ference". Wit hout th is limitation

"Ntr l Nu mber s ar e not conser ved under su btr action" . Thus, 3

2 = 1, a defined dif fer ence ; but, for 2 3 = w, the nu mber

w cannot recur as a Ntr l Nu mber . So, we s ay "the Nt rl Number

syste m i s not c losed under s ubtr act ion", or "not conser ved

under s ubtr act ion", and is "not tota l, but par tial ". Thus ,sub tr action can onl y become tota l in a new , mor e

compr ehens iv e number syste m .

Also, a · b = a · w implies w = b, so mu lt ip lic ati on ha s a r ight

in ver se . If a · b = w · b im pl ie s tha t w = a, then mul ti pic at ion

has a left in ver se . And mul ti pl ica t ion is a lso com muta t iv e , so

the t wo in ver ses fu se into a s ing le inver se , di vi s ion -- denoted

symbol ical ly by p i s dividend , r is div isor , w i s quotient .) T hi syie lds the equ iva lence ru l e f or def ined quot ients : a ÷ b = c ÷ d

if f a · d = b · c . The inequ iv alence r ules can be der ived fr om

thi s.

Howevever, div ision is defined for Nt r l N umber s i f , and onl y if ,

the d iv iso r i s non zer o and the d iv idend is a mu lt ip le of the

div isor , a limitati on e xpl ica ting "a defined quot ient" (DD ) f or

"par tial con ser vation of Nt rl N umber s, or par tial c lo sur e thi ssyste m under di vision". T hus , 12 ÷ 3 = 4, is a defined

quotient , but, for 10 ÷ 3 = w , the number w cannot recur a s a

Ntr l Number . So, the Ntr l Nu mber s ystem is not c lo sed - - i s

not con ser ved - - under d iv is ion , and div ision can onl y be

r ender ed total i n a new , m or e com pr ehen si v e nu m ber s y s tem .

Now, exponentia tion can be sho wn to be w el l -defined , h av ing

both r ight and l eft in ver ses . However, as noted above,exp onent ia t ion i s not com muta t iv e , so ha s tw o di st inct but

par ti al in ver se s: l ogar ithm and r oot extr action . To s ee thi s,

write be = p for base b; for exp onent e; for power p. Then ,

log bp = e , wher eas (p) 1/e = b . Now, for copr ime b, e, the

number p cannot be ra t iona l . And . in anc ient t ime s, the

ir rati onal it y of √2 was kno wn. Hence , the N tr l N umber syste m

is not c los ed - - i s not con ser ved -- for logar ith m or root



extr act ion and eac h can become t otal on y i n a new , mor e

compr ehens iv e number syste m .

Ther e is a gener al method for in ver sing . You cr ea te a new

number sy stem fr om 2 -v ector s o f component s fr om the

im med ia tel y lo wer syste m . Then the oper ationa l r u les f or the

new s ys tem ar e m odeled on the defined oper an ds of thelo wer s ystem , tha t i s , tho se f or w hic h the par tia l oper ation

becomes tota l under s ome res tri ction . In the new s ystem , the

rest ric tion of the lo wer s ystem by par tia li ty of an inver se i s

bypa ss ed i n new ( vector) s ystem by an oper ation a lr eady

total i n the lo wer system, s o a par t ia l oper ation of the lo wer

syste m become s total i n the e xtended sys tem , as you see in

the ne xt Cha pter .

You kn ow tha t the es sent ial s of th is Nt gr cons tr uct ion can berefor mu la ted in BNFA:

<etc.> :@ ...

<Ntrls>:@ 0, 1, 2 , 3, . ..

<ad diti on oper ator> :@ + <ad dit ion of Nt lr s>:@ <Ntl r> + <Ntlr> =

<Ntlr>

<subtr act ion oper ato r>: @ al lowed <sub tr action oper ation> :@ <Ntl r>

<Ntlr> = <Ntlr> iff

Subtr ahend not greater than Minuend

<mul tip li cation oper ator>: @ • <mult ip li ca tion>: @ <Nt lr> • <Ntlr

> = <Ntlr>

<di vi sion oper ator> :@ ÷

<for ma l di vi sion oper ati on>: @ <di vi dend> ÷ <nonz er o di visor> =

<quotient>

<di vi sion oper ati on>: @ <Nt gr> ÷ <non zer o N tg r> = <Ntgr> with

pos si bl e N tg rs remainder<exponent ia tion oper ation>: @ <Ntg r bas e> <Ntg r e xpo nen t> = <Ntg r

power>

Logarith m and rot e xtr act ion ar e not TOTAL f or N tg rs.

You note that ea ch BNFA tr ansf or ma t ion , above, is (a s

pr evi ous ly stated), pur el y syntacti c - -< no i>semantic

refer ence. You note that, with re fer ence to langua ge , th is



remain s syn tactic pr ovided BH FA defin iens and defin iendum

may be thought of as be longing to dif fer ent l angua ges . (For.

as pr evious ly noted, a l exicon is pur el y s yntact ic, one

langua ge i nto another , wi thout men ing or r efer ence .) B ut i f,

now, def inien s and def iniendum ar e r ead as ter ms in one

langua ge, as i n your uni ver sa l l angua ge of Engli sh , then y ou

ar e now be ing i ntr oduced to new ter minol og y (as , above,intr oduction to s ome pr evious ly unkno wn per son), and eac h

BN FA str ing become s a s eman tc tr an sfor mati on!

And you s ee tha t your two me thods of lear ning -- r eminder of

Pr otolear ning and BNFA con ver ge!



CHAPTER TWEN TY-SI X: YOU CAN CONSTRUC T NTGR

ARI THMETIC

In Cha p. 12, on "Cogn iti ve Dissonance" , it 's explained tha t

mathema tici ans invoke unin tended refer ences by na ming a

new te chnica l ter m by an "ever yday" ter m, su bl im ina ll y

mi xing di spar ate refer ent s. Her ein th is i s avoided by

renaming tho se encounter ed. The Number System s have beenunique ll y renamed, As sho wn above, the Inte ger s have been

renamed " Nt grs", with a pr a gma tic exp lainer : [Nt grs]⇒[Int eger s]

YOU kn ow oper ation s ad dit ion, mu lt ip lic ati on and

exp onent ia tion ar e TOTAL in Ntr l Number s; equi va lentl y, N tr l

Nu mber s clos es under these oper ations ; equi va lentl y, the

ad dit ion, mu lt ipl ication , expo nent ia tion of Nt rl Nu mber s is aNt rl Number ,

YOU kn ow Oper ation s sub tr action , d iv ision . l ogar ithm , and

root e xtr action ar e not TOTAL, but P ARTIAL ("s omet ime s

car ry th ru" , constr ained) in N tr l Number s; equi va lentl y, N tr l

Number s does not clos e under these oper ations ; equi valentl y,

the d if fer ence, quotient , logar ithm , or root oper ations of Nt rl

Nu mber s do not alw ays yie ld a Nt rl Nu mber ,

So , you kno w t hat one or mor e number s ystem s, eac h w ith an

ari thmet ic, mu st be const ructed i n or der to render TOTAL

these PARTIAL oper ation s. B ut y ou kno w i t is simpl er t o

total ize one oper ation a t a t ime . Hence , you be gin wi th

sub tr action , to con str uct an "e xten si on" of Nt rl Number s in

whic h subtr action is TOTAL,

You know you h ave an indic ator (constr ainor) in t he Ntr ls how

to pr oceed . In the s ubtr act ion, a - b = c, number a i s the

MIN UE ND, nu mber b is the SUBTRAHEND, number c is the

DIFFER ENCE .



You kn ow tha t the DIFFER EN CE is a N tr l Number if f

(constr aint) T HE S UB TRAH END IS N OT GREATE R T HAN THE

MIN UE ND.

You kn ow thi s cons tr ain s a m in ia tur e ar ithmet ic (g rammar)

within Ntr l Nu mber s in w hic h subtr act ion i s TOTAL (wi thout

tha t par t icula r con str aint) . So , you kno w s ubtr act ion i sdefined within the lim itation (the Boundar y) of above.

However, you kno w th is constr aint pr ovi des a DEFI NED

DIFFER ENCE (DD) for Lear ning what eac h oper ation must be

(h ow constr ained), to conser ve DD. That i s, th is pr ovides a

MODEL (gramma r) for oper ations in a new number system in

whic h subtr action is (uncons tr ainedl y, w ith in Boundar y)

TOTAL.

You kn ow tha t the defini tion of subtr act ion in t er ms of

ad dit ion pr ovide s the equ iva lence r ule - - usefu l l ater - - for DD ,

of (a b) = (c d) if f (constr aint) (a + d) = (b + c) .

You know the gener al for m, her e, i s: DO1 o D O2 = D O3, wher e

DO denote s "defined oper and" and o denotes an in ver se

oper at ion .

You procee d to the se Ar thmeti c R ul es (con str aint s on

number s) .

You know the l iter atur e descr ibes w hat y ou may cal l "match-

mi x" for mats in l inear alg ebr a in expl ica tion of sym metr ic

and ant is ymmetr ic bi l inear pr oducts

• bi li neari ty :(aw 1 + b w2 ,w') = a(w 1,w') + b(w 2,w') ; and

(w,cw '1 + dw' 2) = c(w , w '1) + d(w ,w' 2) • anti sym metr y: (w,w') = (w' ,w) .

You read "sy mme tric " a s "match", "ant is ymme tric /sk ew-

symmetr ic" as "mi x" , " bi li near pr oduct" a s "bi nar y ar ithmet ic

oper ation ". Then you obtain the model for d is coveri ng inverse

ari thm et ic oper a t ion s fr om pr m ar y one s .





You can chec k th is by number s: (10 3) • (9 5) = (10 • 9 + 3

• 5) (10 • 5 + 3 • 9) = 105 77 = 28 ; agreeing w ith (10 3)

• (9 5) = 7 • 4 = 28 . Chec ks.

You note be low the orig in of the see ming ly "weir d" law of

s ign s .)

For div ision of DD , its l im it at ions (sta ted above) pr ovide for

defin iti on of d iv ision in ter ms of DD. Given (a b) ÷ (c d) =

(e f) if f (a b) = (c d) • (e f) . The con str aint of not

di vi di ng by zer o mean s c ≠d. The constr aint of di vi dend be ing

mul ti ple of div isor for na tur al number and DD mean s (a b) =

n(c d) , hence, a ≠ b .

Then

n(c d)÷(c d)=(ef).(n1)(c d)=(e f), e ≠ f. (cn + d) - (c + dn) = (e f), cn+ d ≠c + dn.

Fina ll y, (a b) ÷ (c d) = (cn + d) (c + dn) , a DD for the

division oper ation.

For exponentia tion , you h ave be = p, for base , b, for exp onent ,

e, for power , p, all a s DD of na tur al s , with tota li ty . But, as

wi th n atur al s, the ir inver se s of logar ith m, root extr act ion ar e

onl y par t ia l , so mus t be vector -e xtended to total it y .

You now deri ve a new number system with total s ubtr act ion

via 2 - vector s of na tur al number component s with oper at ional

ru les ba sed upon clo sur e f or DD. You e xchange DD for a 2

-v ector of na tur al number component s . You se e th at a b

becomes [a,b] . You kno W th is induce s the oper ati onal ru les

for these 2 - vector s of na tur al s , whic h mu st adher e to the

na tur al number cons tr aint s inher ited by the ir na tur al number

components .

• Equi valence: [a,b] = [c,d] if f (constr aint) a + d = b + c.

The inequ iv alence s fol low fr om th is .



• Ad di tion : [a,b] + [c ,d] = [a + c, b + d] . (You note the

ma tc h f or m a t .) Since eac h component is a n atur al

number , it follows that ad dit ion i s a tota l oper ation for 2

-vector s of na tur als . The com muta t iv e and a ssoci a t iv e

pr oper tie s of ad di tion follo w fr om t hi s defin it ion.

• Subtr action : [a,b] [c,d] = [a + d , b + c] . (You note the

mix f or ma t, contr as ting wi th m atch f or ma t of ad di tion .)Since eac h component is a n atur al number , it follows that

sub traction is a tota l oper at ion for -v ector s , as not so fo r

na tur al number s . You h ave ac hie ved your pur po se i n thi s

const ructi on . This tr ans f or m s s ubtr act ion of Nt r l

Nu mber s in to ad dit ion of Nt r l Nu mber s as vector

components - - and ad dit ion i s al lo w ed !• total Ntrl subtraction• -------------------------> • transforn | ^subtraction by• to vectors | |allowed addition• of Ntrls | |of Ntrls yields• | |correct• | |difference• V------------------------>

perform operation

•

Mul tip li ca tion: [a,b] • [c,d] = [a • c + b • d, a • d + b • c] .(You note the ma tc h f or ma t .) Si nce eac h component is a

Ntr l Nu mber , it fol lows t hat mul ti pl ica tion is a total

oper at ion .

The commut ati ve and as soc ia ti ve l aws of mu lt ipl ic ation

fol low from th is defin iti on.

• Division: [a,b] ÷ [c,d] = [a ÷ c , b ÷ d] , provided b ≠ 0, c ≠ 0

(constr aint) and fir st components ar e m ult ip les of second

components (constr aint). T hi s second cons tr aint mean s

division of 2-v ector s i s onl y a par tia l oper ation , as with

na tur al number s .

• The equi val ence r e la t ion r educe s 2- vector s of Nt r l

Nu mber s to t hr ee bas ic f or ms :







divisor> = <quotient>

<di vi sion oper ati on>: @ <Nt gr> ÷ <non zer o N tg r> = <Ntgr>

wi th po ss ible Nt grs r emainder

<exponent ia tion oper ation>: @ <Ntg r bas e> <Ntg r e xponen t> =

<Ntgr power>

Logarith m and rot e xtr act ion ar e not TOTAL f or N tg rs.

You note th at ea ch BNFA tr ansf or ma t ion , above, is (a s

pr evi ous ly stated), pur el y syntacti c - -< no i>semantic

reference. You note that, with re ference to langua ge , th is

remain s syn tactic pr ovided BH FA defin iens and

defin iendum m ay be thought of as belong ing to d if fer ent

langua ges . (For. a s pr evious ly noted , a lexicon i s pur el y

syntact ic , one la ngua ge into another , without m ening orreference.) But if, now, defin iens and defini endum ar e

read as t er ms in one l angua ge, as in y our un iver sal

langua ge of Eng lish, then you ar e now being intr oduced

to new ter mino log y (as , above, introduction to s ome

pr evi ous ly unkno wn per son) , and eac h BN FA st ri ng

becomes a se mantc tr ansfor mation!

And you s ee tha t your two me thods of lear ning -- r eminderof Pr oto -lea rning and B NFA con ver ge !



CH APTER TWEN TY-SEVE N: YOU CAN CONSTRUCT RTN L

NUMBER ARI THM ETIC




mi xing di spar ate refer ent s. Her ein th is i s avoided by

renaming tho se encounter ed. The Number System s have beenunique ll y renamed, As sho wn above, the R ational Number s

have been r enamed "Rtnl Number s", with a pr agma t ic

explainer : [Rtn l N umber s] ⇒[Ra tiona l N umber s]

You now const ruct another number system fr om 2 -v ector s of

Nt g r s (a s 2 - vector s of na tur al s ) with oper a tiona l r u les ba sed

upon c lo sur e f or def ined quot ients , DQ. You e xchange a DQ

with a 2-v ector of Ntg r component s . Thus , a ÷ b bcome s [a,b] , provided s econd component is non zer o .

You know thi s inter change induces the oper a t iona l r ule s f or

these 2 - vector s of Nt g r s , which m ust adher e to the Ntrl

Nu mber con str a ints on thei r N tg r l co mponents . YOU know you

have:

•

Equi valence: [a,b] = [c,d] if f a • d = b • c . This m eans ,wher e fir st component of a 2 -vector is a mu lt iple of the

fir st ( a ÷ b = c), t hat [a, b] = [a ÷ b, b ÷ b] = [c , 1] . The

con ver se of thi s i s ea si ly compr ehended. You kno w t hat

the i nequi va lences follo w fr om thi s equ iva lence r ela tion .

• Ad di tion : [a,b] + [c ,d] = [a + c, b + d] . (Ma tch for mat.)

Since eac h component is an N tg r , it fol lows that ad dit ion

is a total oper at ion for 2 -v ector s o f N tg r s . You kno w tha t

the com muta t iv e and a ssoci a t iv it y p r oper t ies of ad d ition

fol lo w fr om th is defin iti on .

• Subtr action : [a,b] [c,d] = [a c, b d] . (Ma tch.) Since

eac h co m ponent i s an Nt g r , you kno w i t follo ws tha t

sub traction is a tota l oper at ion for 2 - vector s of Nt g r s .

• Mul tip li ca tion: [a,b] • [c,d] = [a • c , b • d] . (Ma tch.) Since

eac h co m ponent i s an Nt g r , you kno w i t follo ws tha t

mul ti pl ica tion is a total oper at ion for 2 -v ector s o f N tg r s .







You know tha t a dec ima l fr action - - taking the for m of

mant is sa .char acter is ti c - - i s one whose char acter is tic is

a po wer of ten . You know tha t the char acteri st ic has an

in ter va l of dig it s whi ch r epea t end les sl y -- as in

0.33333 .. . You know tha t all dec imal fr act ions can be

con ver ted to or dinar y fr action s or rationa l number s. (The

method s f or hand ling all poss ib le case s can be found a tGoogle(standar d ta sk s+j onhay s).)

So , the ar ith metic of dec ima l fr act ions become s, after

con ver sion to fr action s, tha t of the ari thmeti c of

fraction s, or rational number s.

You can cons tr uct , via logar ith m , a number whic h is not

rational : log ba = c , for a,b copri me (no common factor s)yie lds output c which i s i r r a t iona l . You kno w tha t th is ,

along with kno wledge of the ir rat ional it y of the squar e

r oot of tw o , ind ica tes need of a number syste m i n w hic h

logar ith m i s t otal and a number sy ste m i n w hic h root

extr act ion i s total -- two m or e number systems .

The Rtn l N u m ber S y ste m A ri thm et ic i s an Ad dit iv e Gr p

and a Mul tip li ca t iv e Grp , a structur e known as a F ld(Cha p. 22) .

Many m ea sur a ble s , such as sound have such a wide range

of va lue s tha t it is con venient to con ver t t o the smal le r

logar thmic scale , tur ning e xponenti ation in to

mul ti p l ica t ion , as in the loga rith mic -dec ibel sc ale for

sound .

You know you can denote the essent ials of thi s rational

number const ruction in B NFA:

<etc.> :@ ...

<numer als>; ;= 0, 1, 2 , 3, . ..

<Rtn l Number>: @ <Nt gr numer ator> /<<non zer o Ntg r

denomin ator>|fr act ion

lt ;ad dition oper ator> :@ + <ad dit ion of Rtn l Number s >>: @





CHA PTER TWE NT Y-EIG HT: YOU CAN CONSTRUCT R L N UMBER

ARI THMETIC




mi xing di spar ate refer ent s. Her ein natur athi s is avoided by

renaming tho se encounter ed. The Number System s have beenuniquel y renamed, As shown above, the Real Nu mber s have

been renamed "R l Number s", with a pr a gma tic ex plainer : [Rl

Nu mber s] ⇒[Rea l Number s]

You know the need for THE RL NUMBER SYSTE M i s explained

GEOMETRICALL Y (to measur e the dia gonal of a sq uar e),

instead of the need with in A RIT HME TIC to pr ovide an i nver se

for exp onent ia t ion .

You know exp onent ia t ion i s not com muta t iv e , so ha s tw o

di st inct par ti al in ver se s in N tr l, Nt gr l, Rtn l Nu mber s ystem s ,

namel y, logari thm and r oot extr act ion, eac h requir ing it s own

number sy stem to become t otal .

You tur n f irst to render ing logari thm total . (A Scot , John

Na pier (1550 -1617) invented a for m of l ogar ithm s by rela tingari thmet ic pr ogres si ons -- such as 2, 4, 6, 8 , 10, .. . -- with

geometr ic pr ogres si ons -- such as 2, 4, 8, 16 , 32 , ... . The

disco ver y tha t ancient Ba bylon ian s p rie st s used the se

pr og ression s s ug ges ted the y may h ave conceiv ed of

logar ith ms . The Frenc h mathema tic ian, P ier re-S imon La place

(1749-1827), who contr ibuted s o muc h to mathema tical

as tr onomy , cla imed tha t invention of logari thms doub led the

lif e of m athem atica l as tr onomer s, by simpl if yi ng the ircalcu la tion s becau se l ogari thm s - - By pas s! -- tur n

mul ti pl ica t ion s into ad dit ions , d iv isi ons i nto sub tr action s,

exp onent ia t ions i nto mul tip li ca t ion s .

Up to now, our oper at ion s h ave been f inite . But , to dea l w ith

the r ea l nu mber f o r ma t , you kno w y ou'l l need a tr an sfin ite

oper at ion , namel y l im it , gener all y encounter ed onl y i n



calcu lus or anal ys is. (You kno w the l abel "tr ansfin ite" mean s

tha t you can go bey ond an inf ini te f or m , and use i ts fin ite

exp ress ion to fur ther pur pose. T hat is, it involves one l imi t

proces s fol lwed by another one, bef ore fur ther math.)

You know a R tnl Number has two ba sic f or ms, the fr actiona l

for m v ia the so l idu s , and the dec ima l e xpans ion v ia thedeci mal poin t .

You know, f or e xamp le, tha t the number 1/3 = 0 .33333. .. . = 0. 3, displa ys the fr act ional for m on the l eft, and the dec ima l

exp ansion on t he right . Y ou kno w th at the o verl ine ind ica tes

tha t a number or number inter val is repea ted without end . For

example , 1/11 = 0.09090909. .. , so can be w ri tten

1/11 = 0.09

.

The deci mal expansi on expli cates both the rational number

syste m and the new system you need to con struct.

You kn ow tha t two u seful ter ms expl ica te the deci mal

exp an sion you see above. You kn ow the s ube xtens ion bef orethe dec ima l po int is labelled the char acter is tic ; the

sube xten sion after the dec ima l po int is labelled the mant is sa .

In the case of the expansi on of 1/3 , the c har acteri st ic i s 0.

The mant issa is 3.

You know tha t The deci mal expans ion of a rationa l occur s

because the repea ted par t -- the manti ssa in the example - -

denotes a geometri c s er ies , that i s, a se rie s cons tr ucted byrepea ted m ul tip li ca t ion of an ini tia l number by a constant

number . You se e th is for 1/3 b y refor mu la t ing it - - not w ith

denomin ator 3 -- but one which i s a po wer o f the dec imal

base : 1/3 = 3 /10 + 3/100 + 3/1000 + . ... = 3 /10 + 3/10 2 + 3 /10 3 +

... + 3 /10 n + . .. ., wher e "n goes to inf inity ".











CHAP TER TWEN TY-NINE: YOU CAN C ONSTRUCT CMPLX

NUMBER ARI THM ETIC

You kn ow tha t In C ha p. 12, on "Cogn itive Dissonance", it 's

explained tha t mathema tic ians i nvoke un intended r efer ence s

by nam ing a new tec hn ical ter m b y an "e ver yday " te rm,

sub li minal ly m ix ing dispar ate refer ents . Herein natur ath is is

avoided by renami ng tho se encounter ed. T he N umber Sys tem shave been un iquel y r enamed , As sho wn above, t he Co mple x

Nu mber s have been r enamed "C mplx Number s", w ith a

pr a gm atic expla iner : [Cmplx Number s]⇒[Co mple x N umber s]

The great Ir ish m athem atic ian , W illiam Rowan Hami lton ,

(1805-65) d is liked t he standar d for mat of C mpl x N umber s :

R1 + (√-1)R2

wher e R denotes a Rl Number . He said i t is not a s um such as

3 + 4.

Hamilton then cr eated the concept and label of vector ,

wr iti ng com p le x num ber s a s 2-v ector s of r ea l num ber s .

In so doing , he cr eated the f ormat for con str uct ing a l l

st andar d Nu mber S yste ms and their Ar i thmeti cs as 2-v ector s

of a s impler syste m , be ginn ing w ith the Na tur al Nu mber

Sy stem and i ts Ar ithmet ic .

The oper ation s for Cmp lx Nu mber s are:

• Equi va lence : [r 1, r2] = [r 3, r4] if f r1 = r 3, r2 = r 4. The

inequ iv alence s can be der ived fr om thi s.

• Ad di tion : [r 1, r2] + [r 3, r4] = [r 1 + r3, r12, r4]. (Ma tch)• Subtr action : [r 1, r2] - [r 3, r4] = [r 1 + r4, r12, r3]. (Mix)

• Mul tip li ca t ion : [r 1, r2] · [r 3, r4] = [r 1 · r 4 - r 2 · r 3, r1 · r3 + r2 ·

r4].

• Division : [r 1, r2] ÷ [r 3, r4] = [(r 1 · r 2 + r 3 · r4)/ r23 + r2

4, (r 1 · r4 -

r2 · r4)/r2

3 + r 24] .



Please note Hamilton' s mul ti p l ica t ion ru le , which i mp li es i2 =

-1 , in connection with something you may f ind on ly her e and

onl ine in Google("r edux+jonha ys "). That thi s pr oduct r u le is

the pr oduct ru le f or N tg rs wi th su m tu rned to a d if fer ence .

Al so, s ome thing el se you may f ind on ly her e and in

"r edux+jonha ys ", ba sed upon the concept of "modul " (not to

be confu sed with "modu le" !), found in "Intr oduct ion to NumberTheor y" by Oystein Ore, p. 159. A modu l i s a s tr uctur e c lo sed

under s ubtr act ion. Or e note s:

• The Nt r l Nu m ber s ar e not clo sed under s ubtr act ion .

• The Nt grs , R tn l s, R l s , and C mpl x N umber s a re c losed

under s ubtr act ion, so eac h for ms a modul . The

ima g inar ies f o rm thei r o wn m odul .

However, Or e does not note th at Cmp lx N umber s f or m a

bimodul . We can see thi s b y wri ti ng the dif fer ences in the

va riou s number system s with the fir st component in red ,

second in blac k .

• Subtr action of v ector s for Ntr ls : [a, b] - [c, d] = [a + d, c +

d]. Pl ease note tha t, in the dif fer ence ter m (on ri ght) , the

color s ar e mi xed .• The sa me i s found f or the sub tr action of Nt gr s a s vector s :

[ a , b] - [c, d] = [a· d - b · d].

• Si mi la r r esul ts ar e found f or the sub tr action of Rtn ls as

vector s : [a, b] - [c, d] = [a · d - c · d, c2 + d2].

• Subtr action for tr ansf in ite vector s of R l Number s is best

explained by an example : 13/10 - 9 /10 - > 1.3 - . 9 = 0.4 or

[1, 3] - [0, 9] = [0,4] , mi xing 1s t component of 1 st number

wi th 2nd component of 2nd number : not a modu l .• Subtr action for "r ea l" par t of Cm plx vector s : [a , 0] - [c,

0] = [a - c, 0] . P lea se note same separ ation of co lor s: a

modul , un lik e al l above cas es :

• Subtr action f or " ima g inar y" par t of C mp lx v ector s : [0, b] -

[0, d] = [0, b - d]. Pleas e note sa me s epar ation of color s

as in pr eviou s case , al so a m odul .









unit , i = √-1. Hamil ton r ea li zed tha t an extensi on of th is might

rota te a d irected p lane se gment , and be ga n tr ying to

for mulate it as a dir ected span in te rms of 3 un it s, i nvolv ing

another un it , j2 = 1, but j≠ i. Af ter y ear s of fai lur e, Ha mi lton

real ized he must use four un its , 1, i, j, k, wher e k2 = 1 , but k

≠ i, k ≠ j, la be li ng th is the qua ter nion . Later, William Kingdon

Cl if ford (1845 -79) developed the octon ion with eight unit s ,and it was real ized t hat these "h yper comp lex number s" cou ld

be fur ther extended.

A r ecur si o n gener a te s h yper com p le x num ber s :

• The 1 -h yper comp le x number ha s the f or m: h1 = r 1 + r2i,

wher e r1, r2 ar e r ea l nu mber s and i j i s a new unit such

tha t i2

= 1 .• The 2 -h yper comp le x number (qua ter nion) has the for m: h2

= c 1 + c 2j, wher e r1, r 2 ar e r ea l nu mber s and j ≠ i i s a new

unit su ch tha t j2 = 1 . From the pol ynomi al for m c1 = r 1 +

r2 i and the pol ynomi al for m c2 = r 3 + r4i, we f ind (by

sub st itut ion) : h2 = c 1 + c 2j = r 1 + r 2i + r3j + r4ij, for ij = k , ji

= k, k ≠ i, k ≠ j , but k2 = 1.

• The 3 -h yper comp le x number ha s the f or m: h3 = q 1 + q 2l,

wher e l is a new un it , l ≠ i , l ≠ j, ≠ k , but l2 = 1. From thepol ynomi al for m q1 = r 1 + r2 i + r3j + r4k and the pol ynom ia l

f or m q2 = r5 + r 6i + r7j + r8k, we f ind (by sub st itut ion) : h3 =

r1 + r2i + r 3j + r 4k + r 5l + r6li + r 7lj + r 8lk = r1 + r2i+ r3j + r4k +

r5l + r6m + r 7n + r 8o, for new noncom muta ti v e un it s , li = m,

lj = n , lk = 0 s uch tha t the y ar e dis ti nct fr om previous

unit s and ea ch squar ed equa ls ne gati ve one.

• In gener al , the ( n + 1 )-hyper comp le x number can be

gener ated fr om the n-hyper comple x number .

These h yper comp lex syste ms ar e par t of the Ar i thmet ic of

Cl if for d N umber s (a .k.a . C lif for d Al ge br a, mu lt iv ector s,

geom etr ic a lge br a) . The b im odu l char acter of the comp le x

number means tha t the ploy of hid ing t he vector f or mat of

in teger s , rati onal s, r ea ls b y sign s can no longer work, and the

vector for ma t mu st now be m anipu la ted "in the open" .



Besides having the usual eqi va lence and inequ iva lence

rela t ion s and oper a t ions of ad d iti on, su btr action ,

mul ti pl ica t ion , d iv is ion , exponenti ati on, lo gari thm, r oot

extr act ion f o r number s and (w her e rele vant) f o r v ector s , ACN

has two spec ial v ector pr oduct s (outer pr oduct and

mul ti pr oduct) relating to the vector inne r pr oduct , found in

st andar d vector theor y (w hich wi ll be char acter ized be low).These t hree pr oducts are usua lly prsented in an " advanced"

way, but can be der ived on a level of h igh s choo l alg ebr a,

cum a s m idgi n of t rigono m etr y , as follo ws.

DERIV ATION OF VECTOR INNE R P RODUC T FROM THE LA W OF

COSI NE S

Fir st , w e need to ind ica te the r ota t ion oper a to r , R, def ined by

a m atr ix in cos θ, sin θ:

cos θ - sin θ R ≡

sin θ cos θ ,We can now state a princi ple (di st ingu ish ing sca lar s fr om

vector s ) to be i nvoked be low.

Scala r-V ector Pr incip le: a sc alar is a st r uctur e in var iant for θ

= 0 (or R as the i denti ty oper ator) , wher eas a vector i sinvar iant w hene ver deter minant |R| = +1.

Let u s la be l the usual pr esenta tion of a t riang le as a sca lar

tr iang le to dis tingu is h w hat a ppear s when the thr ee s ide s

become d ir ected , that i s, vector s , su ch th at one s i de i s t he

vector su m of the other t wo s ide s . We label the latter as a

vector tr iang le . The sca lar tr iang le satisfie s a Law of

Co si nes . Behold ! Trans for ma tion of the sca lar tr iang le in to

the v ector tri ang le tr an sf or ms the Law of Co si nes in to the

vector inne r pr oduct . (T he standar d pr ocedur e is to deri ve

the Law of C os ine s fr om i nner pr oduct .)

We l abel the s ca lar tri ang le s ides a s a, b, c , with ∠ (a,b) = θ =

C. We then have the Law of Cos ine s:



c2 = a2 + b2 - 2ab cos C (1)

We no w tr an sfor m the sca lar tr iang le in to the vector tr iang le

and indic ate vector s by under scor ing . We have: c = b - a (2)

We w ish to i nter pr et (2) i n te rms of (1) by tr an sf or ming the

sca lar tr iang le in to the vector tr iang le .

1. The squar e s i n (1) sug ge st squar ing in (2) .

2. But cos C = cos θ i n (1) sug gest s mor e than a "pol ynom ial

product". We cons ider a v ector pr oduct , denoted •, as in a

• a.

3. We then have (fr om equa tion (2)):4.

c • c = (b - a) • (b - a)

Or,

c • c = b • b - a • b - b • a + a • a (3)

5. We can now match (3) to (1), tr anf or m ing the Co sine Law

in to a vector pr oduct b y:

1. We dec lar e • to be com mut a ti v e with a • b = b • a ,

and we can col lect 2(a • b) in (3).

2. We dec lar e a • b = |a| | b| cos θ. θ = ∠(a, b) to match

cor respond ing te rms in (1), (3) .

3. We f ind th is vector pr oduct to be s ca lar b y T he

Scala r-v ector Pri ncip le (a bove), w her ein θ = 0.

2. Si nce th is i s pos si ble f or a ll ca ses , we h ave clos ur e, and

a v ector i nner pr oduct .

When u • v = 0, vector s u, v ar e per pendicu lar , that i s,

or thogonal , al lowing for defin it ion of an or thogona l

vector bas is .

We no w l ear n tha t inner pr oduct i s not s uf fici ent f or our

pur pose s.



For a real vector space , an inner pr oduct , <-,-> , sat isf ies four

bas ic pr oper tie s . Let u, v, w be vector s and α be a scal ar ,

then:

1. <u + v,w> = <u,w> + <v,w> .

2. <αv, w> = α<v, w> .

3. <v, w> = <w, v > 4. <v, v> ≥ 0 , being zer o onl y for v = 0. A v ector s pace

together w ith an inner pr oduct on it i s cal led an in ner

pr oduct space . Thi s defin it ion a lso a pp lie s to an abstr act

vector sp ace over any pr oper bas is . Exa mple s ar e:

o The real number R wher e the i nner pr oduct is <x , y > =

xy .

o The Euc lidean Space , R n

, wher eo o <(x1, x2, ..., xn), (y1, y2, ..., yn)> =

x1y1 + x2y2 + ... + xnyn.Henr i Car tan (1904, -) note s tha t a po int in n-dimen si onal

Euc lidean Space can be defined as a set of nu mber s

(coodina tes), [x 1, x2, ..., x n] su ch tha t the d is tance of thi s

point , [x] to the O ri gin, [0, 0, . .. , 0] - - equi valen tl y, the se lf -

innner -pr oduct , <x, x> of vector x to 0 -- is given by the

fundamental for m (p lease note pos it iv i ty ) x • x = x1

2 + x22 + ... + xn

2.However, pseudo -Euc l idean s pace s (a s in rel a ti v is ti c s pace -

t ime ) ma y requir e another pr oduct for m.

DE RIV ATION OF OUT ER PRODUCT OF V EC TORS (AN D

MULTIVECT OR S)

Two vector s ar e i nter depende nt if f one is a m ult ip le

of the other : either they ar e par all el or par t of the

sa me r a y . In such ca se, the ir inner pr oduct i s zer o .

Given two i ndependent v ector s , with nonz er o inner

pr oduct , as in the follo wing case :





syste m -- we can solve (3) for C1, and (4) f or C2, both

in ter ms of C3:

C1 = [(a2 b3 - a3 b2)/(a1 b2 - a2 b1)]C3. {5a)

C2 = [(a3 b1 - a1 b3)/(a1 b2 - a2 b1)]C3. {5b)

We can now mak e " the c lever c ho ice" ment ioned

above. By dec lari ng C3 = (a 1b2 - a 2b1), we can

el im ina te C3 fr om (5a), (5b). Then we have:

C1 = a2 b3 - a3 b2 (6a)

C2 = a3 b1 - a1 b3 (6b)

C3 = a1 b2 - a2 b1 (6c)

A f or m such as (a 1b2 - a 2b1) rese mble s the s tandar d

commut ator for mat r ice s or oper ator s : [A, B] = A B -

BA. In thi s, the " car ri er s" com mute ( inter change) , but

no subs crip t or inde x is involved. T he s ame bracket

for m can be wr itten as [k 1, k<S UB .2< s ub>] = k1k2 -

k2k1 , wi th no "car rier " com muta t ion , onl y sub scr ipt

commut ation .

But thi s doe s not match (a 1b2 - a 2b1 ), which ha s both

car ri er and su bscr ipt commu ta t ion . So we e xtend the

bracket to accomoda te both:

[a, b]ij = (ab - ba)ij = ai bj - biaj = -[b, a]ij (7)

This extended br ac ket is implicit in Ad vanced

Ca lculu s , R. C reighton Buck:

• u = f(x, y) , v = g(x, y);

• du = f1dx + f2dy ; dv = g1dx + g2dy ;

• dudv =(f 1dx + f2dy)(g 1dx + g 2dy = (f 1g2 - f2g1)dx dy =

[f, g] ijdxdy











• We i ntr oduced shar p-pr oduct simpl y to s ee its

output .

• We d id not simpl ist icly add inner product to outer

pr oduct to obtain a new pr oduct , a s is usual ly

done in the liter atur e, often ax iom atica ll y.

• Rather , we cas t the vector-no mia ls in the pa tter n

of alg ebr aic po l yno mia ls and emu la ted the la tter

pr oduct .

• We then di sco ver ed an output p a tter n which is

the s um of i nner and outer pr oduct .

• We ther eby di sco ver ed infor mation not pr esent in

the l iter atur e -- tha t th is oppo ses the patte r n of

hom ogeneit y- s y m m etr y to the p atter n o f

heter ogeneit y- a nti s ym m etr y .

And plea se note th at, on the l evel of high school

alg ebr a (wi th a smidgin of trigonometr y) :

1. we der iv ed the inner pr oduct b y tr ans for ming a

sca lar tr iang le in to a vector tr iang le coor dina ted

wi th tr an sf or ming the Law of C os ine s in to inner

pr oduct ;

2. we der iv ed the outer pr oduct by sol ving a s ystemof linear equa tions obta ined b y appl ying inner

pr oduct to vector s;

3. we der ived outer pr oduct by obta ining the su m of

a hom ogeneou s-s y mm etr ic s ubpol ynom ial and a

heter ogeneo us- anti sy mm etri c s ubpol ynom ial and

as soc ia t ing thi s pa tter n w ith tha t of a new

mul ti pr oduct .

We r ep lace the oper ationa l sym bol , #. Now,

mul ti pr oduct is often denoted simply by

conca tena t ion , but thi s may lead to confus ion w ith

pr oduct i n nu merica l alge br a . Let' s use ⊗. Then we

have m ul ti p r o duct of v e ctor s :





Thus,

• the Paul i a lg ebr a of quantum theor y i s la beled : C (3, 0) for

thr ee po si ti ve space -coor dina ted ter ms , zer o ne ga ti ve

ti me- coor d ina ted ter ms, i n it s me tric , w i th the ba si s : [1, δ1, δ2, δ3]

• the Dir ac alge br a , f or r e la ti vi s ti c theor y of the electr on i sla be lled : C (3, 1): thr ee pos it iv e space-coor din ates , one

ne ga t iv e t ime -coor dina te ; with the ba si s : [1 , d 1, δ2, δ 3, δ4].

In 1994 , ph ys ic is t, K. R. Gr eider , sho wed th at all r e la t iv istic

fie lds w ith spin s of 0, ½. 1 can be de veloped , v i a

mul ti pr oduct, wi thin the single Dir ac for ma li sm (a bov e),

w her eas the s tandar d tr ea tment i s a p atc hw or k of fo r mal is m .

This i s one mor e reason for comp lain ing agains t s tandar dma thema t ics , which a voids m u lt iv ec tor theor y , making

lear ning dif ficu lt for st udents .

MULTIP ROD UCT IS T HE G REAT UNIFIER

In a sen se , mul ti vector theor y i s bu il t ar ound a s ing le

mul ti vector pr oduct equa t ion , (9) , and it s e xtens ion

in to 3 -D , etc . . In mul ti vector calcu lus , mu lt ipr oduct

tak es the ind iv idua l oper ator s of g rad ( ∇A) , d iv (∇ •A), and c u r l (∇ ∧ A) of standar d vector calcu lus and

unif ies the m i nto a s ing le mul ti vector oper ator :∇ A =

∇ • A + ∇ ∧ A.

Wher eas, i n standar d vector calcu lus , we can onl y

deri ve t he gr ad ient of a s ca lar , the mul ti pr oduct

y ie lds the g rad ient of any mu lt iv ector .

Ther e is mor e unif ica t ion fr om d ir ected i nte gra t ion

(not the undir ected inte gration of s tandar d ca lculu s).

Given a vector -v alued function , f, of a vector , x, we

wr ite (noting separ ab i li ty , with mul ti p r oduct as

juxta po si tion ):



fx = (f•dx + f∧dx) =

fdx + f∧dx

The Ame rican phy sicist, David Hestenes -- author ity

on C l if f or d A lge br a - - find s t hat the i nver se of

g radient invok es the Cauc hy inte g ra l f or mu la, andhas sketc hed a theor y of the d ir ected i nte gr al , both

in R iemann ian and Le be sque for m .

He stene s ar gu es t hat "man y notion s of homo log y

theor y can mor e r ead il y be expr essed b y dir ected

in tegrals than by the sc alar -v al ued inte gral s of

cohomolog y theor y ... . It rema ins to be seen to what

de gree homo log y theor y can be regarded a s a theor yof di rected inte gral s" .

A ma thema t ical con sequence of th is unif ica t ion i s

tha t real and comple x ana lysis ar e unif ied, and

comple x anal ys is can be e xtended in w ays not

pos si bl e i n the s tandar d f or m .

A consequence of thi s un ific ation in ph ys ics is that

Maxw el l' s electr oma gnetic fie ld equa tion s -- usual lyexp ressed in e ight or f our equa tion s -- unif y as a

s ingle equa t ion (f or a s ing le pr oce ss ), v ia

mul ti pr oduct .

MULTIP RODUC T AS MULTIFACTOR

Please note tha t we can wri te: x1

2 + x22 + ... + xn

2 = (x1w + x2w + ... + xnw)2 (10).

for any un it vector , w s uch tha t w i • w i = 1 , for i = j,

zer o otherw ise.

Then , con ver sel y , we can factor :





APPEX DIX A: M ATHEMA TICS AND MOI

Some reader s mi ght th ink , fr om the intention of th is B ook,

tha t I a m a math ner d t rying to t ur n other s i nto mth ner ds .

Exactl y oppos ite is the case ! As my h is tor y un fold s, you 'l l

lear n tha t I onl y s hi fted t o a Mathe matics m ajor at N ew Yor k

Un iver si ty to esca pe fr om the Ph ysics ma jor forced upon me

at Co lumb ia U niver si ty. In f act, as y ou'l l lear n, a mur der m ade

me a m athem atic ian . A nd be ing a par ent m ade me a teac her .

I s pent app roximate ly five years, during American

par ti cip ation in W WII, as a Weather Obser ver and Weather

Forecaster in the Ar my Air Corps. After dischar ge, I sett led i n

NYC and met the great love of my l ife, Es ther C ar ol ine Ode ll

(1920-2000). After suf fering a l ifet ime from i nfant pol io,leaving her l eft leg par al yzed and ri ght foot de for med, E sther

developed breast cancer (w hich had killed her o lder sister) ,

had two m astectomi es , and d ied of meta sta tic br eas t cancer ,

On Augus t 28 , 1948 , Es ther and I w er e mar ried b y her fathe r,

The Rever end Ed ward A. Odel l, at Upper Mon tclai r

Pr esby terian Chur ch. In Septembe r, I be ga n st udie s at

Co lumb ia U niver sit y -- but onl y after a has sl e whic h ensur ed 8year s of un iver si ty mis educa tion - - 8 year s of fear and

loathing !

The tr ouble be ga n tha t su mmer of 1948 , w hen I'd r ece ived a

new co py of my los t d ischar ge pa per s and w as j udged e li gib le

under " The G. I . Bi ll" . Her e I was in the town w ith the

greatest jour nal ism schoo l i n the w or ld -- Co lumb ia. And ,

since I f ir st star ted reading H. G. Wel ls at 14, I had dreamedabout become a science wr i ter -- which w ould accommod ate

al l my other i nter ests . B ut the man who must sign my pa pe rs

at the VA, sa id, at fir st , I 'd h ave to m ajor in m e teor olog y ,

since I'd been a w eather ob ser ver and f or ecaster in the

Ser vice . I ob jected th at the me teor olog ical ma jor had been

dr opped at al l New Yor k City col le ges and un iver si tie s wi th





school , renamed " The Schoo l of Gener al S tudie s" , and the

matter of pr er equis it es for man y maj or s was st ill up f or

grabs.

But SGS was a se gregated co lle ge wi thin Co lumb ia

Un iver si ty. It had on ly a l im ited facul ty, and m ost of t hem had

no of fice s. You consu lted them on ca mpus par k benc hes . ThePr es ident of Colu mbia U., Dwight E isenho wer, had tha t ver y

September depar ted f or E ur ope to become C ommander of the

newl y or ganiz ed NATO -- without r esigning hi s Pr esidenc y. (In

the S er vice , the y cal l thi s "go ing AWOL"!) But , bef or e go ing,

Pr es . Eis enho wer appo inted h is tori an Lou is Hacker to be Dean

of SGS. Ha cker retur ned fr om O xfor d U ., accepted the pos t,

and went bac k to Oxfor d for the next four year s! (We f ir st saw

him t wo month s after gradua tion .)

The As sistan t D ean, Jack Arbo li no , w as and is a wonderfu l

man -- God bless him wher ever hs is! -- but his youth and t ime

at Co lumb ia l eft him l ittle clout. (Arbol ino la ter went to

Pr inceton to cr eate the "Ad vanced C redit Pr ogram", w hich

figur es i n tha t de lightfu l fi lm , Stand and De li ver ! , st ar ri ng

Ed ward J ame s O lmo s. )

So why did I go ahead, under s uch cond ition s? Because the

onl y other cho ice was no co lle ge a t al l . Es ther was the s ole

member of her famil y and, seem ing ly, the s ole per son i n her

home to wn, who thought I should be go ing to colle ge. 28-year-

old mar ried men don't go to col le ge! Never m ind tha t I

couldn' t go bef or e. Face it ! It was too late! T he on ly choice I

had was to star t Cou mbia under the se ri dicu lous cond ition s --

and bull my way thr ough . A nd list en, for the next four years,to " You'r e st il l in co lle ge?" "W hen ar e y ou go ing to fin ish? o r

gi ve up ?"

The Registr ar's O ffice of Co lumb ia U niver si ty tw ice lost all of

my reco rds after I m atricu la ted ther e.





don't sho w Frenc h I and Histor y I. It 's Colle ge Algebr a and

Ca lculu s I. Ar en 't t hose you'r e cr edit s. Isn 't thi s your Ar my

Ser ial Number --" , ratt li ng i t of f.

"T hat's my Ser ial Number" , I y elled. " And those ar e m y

cr edit s. I took those cour se s thi s Fal l!"

"Ok. Ok. Take 'em . But wher e ar e m y cr ed its ?"

Bac k at our ta ble , my fel low w or ker shook his head. "Took me

two weeks to find my credit s!"

The second ha ppe ning occur red at the be ginn ing of my last

and eighth S emes ter at Co lumb ia. Just to feel s afe, I c he cked

at the Regista r Of fice t o s ee if I would be r ead y for

Gr adua tion at the end of th is pr esent Seme ster .

Gr adua tion was impos sible becau se t he Regis tr ar Of fice had

no r ecor d that I had e ver atttended Co lumb ia. 114 Cr ed it

Hour s m issi ng! I was , again, a Non student .

Whil e I sat wi th face in hand s, mutter ing, a lovely young

woman cle rk s cur ried ar ound -- AND FOUND SOME OF MY

CR EDIT S - - con vinc ing other s tha t I w as no phantom . Bef or ethe da y was out, a ll my cr ed its wer e r esur rected . F or the time

being , I cea sed to be a Nonstudent.

I ne ver real ly relax ed a bout thi s matter unti l I w as able to

swit ch the tas se l on m y mo rtarboar d, s ign ify ing tha t I, along

wi th m y fello ws, had gradua ted.

I g radua ted fr om Columb ia U ni ver si ty in J une, 1952, with B. S .in Ph ys ics, but w ith an unr ecog nized equ iva lent maj or i n

mathema tics and a m inor in c hemi str y. With m y GI-B ill

suppor t d im ini sh ing fas ter t han or igina ll y es timated , I hoped

to w or k by da y and go to s chool at night a t C olu mbia in

phy si cs - - the ma jor I d id not c hoose but was for ced into b y

the V eter ans Ad min is tr ation (or for fei t G I- Bi ll suppor t). To get

going on a Mas ter' s Degree and to "te st the water s", I





ready when he made h is r ound s. T hi s invoked one of many

fight s I'v e had , o ver the years, wi th H ead Nur ses , on beha lf of

Esther . W hen I appea red w ith the shoe s tha t morning and

insi sted on putt ing the m i nto Es ther' s hands , the Head Nur se

sa id I w as v iolating v isiting hour s and de manded I tu rn them

over to her . I picked up the phone and thr eatened to phone Dr.

Cle veland , to s ay tha t the Head Nur se w oul dn't al low m e tocompl y wi th h is or der s.

The Head Nur se yie lded and I comp leted my m ission, mak ing

me -- me lodr ama tt ical ly! - - l ate f or m y mor ning clas s.

When I s eated mysel f, l ate , in the cla ss room , Pr of. Ha ll was

at the blac kboar d, seem ing qu ite ner vous , ta lking in a hal ting

voice, and he was not continu ing the e xplana tion about"pr oba bil it y in quantu m theor y" , as promised the day bef or e.

Ab ruptl y, he dismissed u s and depar ted . In the ha ll , s ome guy

queried , "Hey, did you hear a bout the mur der ?"

Briefl y, w hile I was at the hosp ita l, some poor m aniac had

enter ed the office of T he Ph ysi cs R ev iew jou rnal on the 9th

Floor of Pup in (the Phys ics bu ild ing) . W hen an 18-y ear-o ld

reception is t a sked h is bus ines s, he s hot her thr ough thehear t and ran away.

The man ran down the hal l. Pr of. Hal l look ed out fr om a

clas sroom and the a ssai lant shot at him, str iki ng the

doorfr ame be side him. T hen the assa ilant ran down the stair s,

esca ping.

The pol ice were puz zled . The gi r l had no boy friend s. The

family had no kn own enemie s. The "nut" fi le of T he Ph ysi cal

Review was consu lted, conta ining sever al le tter s fr om a man

who sa id he could pr ove th at the e lectr on d id not e xist and

they mu st s top sa yi ng o therwi se . Using an address in one of

the l etter s, the pol ice queried hi s mothe r, w ho sa id he w as

staying in a loca l hotel .



When encounter ed, the man adm itted his acti on. He had

chos en thr ee names fr om the Co lumb ia C ata log and planned

to k il l one or m or e of t hese per son s to m ake the w or ld listen

to h is theor y! Ir onica ll y, one of the chosen name s was tha t of

Pr of. H al l and another th at of a f or mer pr ofessor of m ine ,

Prof. Lu cy Hayner . But th is poor demented fel low didn' t kno w

how to find them . H e recogni zed the name of T he Ph ysi cal

Review (to which he 'd sent hi s le tter s) on the door of an

of fice. When the young woman s pok e to him , he became upset

and star ted shooting . Subsequent evidence r esul ted i n thi s

man being sent to a m ental ins titution .

Pr of. H al l was s o upse t b y thi s tha t he never an swer ed my

quest ion . F our year s at C olu mbia , and the fir st time a Phys ic s

Pr of agreed to an swer my ques tion , a demented man scar edhim out of i t! I dec ided to leave Co lumb ia f or N ew York

Un iver si ty. And to change fr om Ph ysi cs to Mathema tics .

MURDER MADE ME A MATHEMATICIA N!

A VA of fic ial said I had the max imum al lowance under th r G. I.

Bi ll. He said i t shou ld car ry fr om col le ge to gradua te schoo l,

up thr ough Mas ter' s Defr ee, and D octor ate and beyomd . Buttui tion at Co lumb ia and NYC incr eased s o fast, I didn 't have

enough for the fir st y ear a t NYC. So I took a job w ith an

insur ance br okerage , j us t off Wal l Str eet . Ou r s on, T imothy ,

was bor n, Ma y 12, 1954, and I was no w the sole ear ner.

When he w as a year old, I t ook my family to Inter A mer ican

Un iver si ty, San Ge rmán, Puer to R ico, to teac h math and

phy si cs , becau se I could get a ma id to he lp E sther w ith Tim.

In the sp ring of 1967 , I recei ved $50 ,000 for a Nati onal

Sci ence Founda tion Wor kshop for high school math t eac her s.

Ou r s on, C hri stopher was bor n tha t summer .

Two s ignif icant thi ngs ha ppene d to c hange my life as a

teac her . I di sco ver ed tha t st andar d teac htng of math w as

unneces sari ly rest ricted m a s noted in C ha p. One. And , in



pa pe rs fr om a pr eviou s Wor kshop , I lea rn about the "vector

for mat" of William Rowan H amil ton, w hich is the bas is of thi s

Book . I recei ved an excori ating le tter fr om a NS F of fic ial for

my teac hing about l imita tion s of st andar d teac hing. I became

per son s non gr ata to NSF. Not onl y request s wer e ignor ed , but

even simpl e l etter s w er e ignor ed , o ver the year s.

Then another e vent changed m y life f or ever. In J une, 1857,

Esther fel l over the s ide of a h il l, br eaking the f emur of her

par al yzed le g. She was in tr act ion at the l oca l ho spi tal f or

fi ve and a ha lf mon ths , then f lown to NYC for t wo and a half

mor e m onths . I w as a s ing le par ent of a smal l boy and a ba by

for seven month s. T his was i n the t ime of c loth d ia per s, I 've

washed out tw ent y at a t ime, wr ung them out , and hung the m

on the line .

I w as tea ching a he avy schedu le. Twelve to fifteen hour s was

nor mal, but I w as t eac hing twenty -t wo hour s a w eek ,

inc luding two Physics La bs. Somet ime s I had to t ake my

childr en to clas s, for one, two, thr ee , or f our hour s. The y

wer e so good .

Weak ened b y thi s br eak , E sther br oke one leg or theother tenmor e t imes , and I was a sinp le par ent and nur se for shor ter

period s. I mana ged to get my M.S. in M athem atic s in 1960 ,

and star ted wor k on my Doctor ate. But the load of teac hing,

going to school , and car ing f or m y family was too much, and,

wi th one -hundr ed-thr ee hou rs i n Mathe matics , I ended ABD

(al l but d is ser ta tion) , w hich limited my car eer . I taught at

Fair le gh-D ickin son in Ne w J er se y and at T he U ni ver sit y of

Maine at O rono , w her e E sther r ece ived a M .A . in C ompar ativeLi ter atur e.

Late r, I was mathema tici cian and compute r pr og rammer f or

The Naval Resear ch La bor ator y in W as hi ngton, D. C., unt il my

retir ement.

No t the st or y of a ner d.





Student s, you can l ear n onl ine about the P eano A xio ms f o r

Inte ger s . The se ar e im ple mented f or the st andar d mode l of

Inte geR s , But you can al so lear n on li ne th at t hese Ax iom s fit

a non standar d mode l of inte ger s , any ony of wh ic h i s g reater

than an y inte ger in the standar d m odel . You' ve hear d of ca se s

wher ein t he identi ty of some one was st olen along with the

per son' s Soci al Secur it y Nu mber , r esu lti ng i n great financ iallosses . T hi s is as if the N ons tandar d Inte ge rs ha ve s to len the

ident it y of the Standar d Mode l .

On li ne y ou also l ear about The Banac h-Tarski Pr ado x.

Accor ding to T he A xio ms of Euc li dean Geometr y (even as

revised and co mpleted by H ilbe rt) s ay you can cut the moon

into five piece s, put the piece s together a gain, and pu the

moon i n your poc ket. No one kno ws how to do it, butEuc lidean A xio ms say i t' s poss ib le because of fuzz ine ss

a bout piec ing together .

These t wo extr emes -- and ther e ar e o ther s -- cas t doubt on

use of A xio ms in S cimath .

Since extens ion is a kind of oppo site to in tens ion , you r ea li ze

it pr eclude s these pr oble ms .

You previou sl y under stood Pr otoT ype of inten sion , so can no w

under stand i ts Pr oto Type as f or mu la ted in ter ms of " set of al l

x s uch tha t pr opo si tion P(x) i s tr ue of it " and under stand

extensi on a s se t of in stance s s o p redica ted .

You have a P rototype in s ome one co llecting name s on a

peti tion for a ne ighborhood tr af fiC ligh t. Here, the in tens ion

inc ludes ae extens ion .

Student s, Standa rd Ar ithmet ic , w ith its Axi oms , allo ws

nonconst r ucti v e pr oof s . A pr oof of a theor em is not gi ven,

const ructng w hat the theor em descr ibe s, but cla ims to pr ove

"by contr ad iction ". The negati ve of the thes is is assu med;

thi s reasoning is sho wn to lead to a contr adict ion ; s o double



ne gative i s i nter pr eted as po si ti ve, and cla im is made tha t

the the si s is proven.

The model was the Euc lidean nonconst ructi ve pr oof th at the

rati o of the s ide of a squar e to it s d ia gonal i s not a fr act ion .

This m atter appar entl y ar ose becau se of "T he Pytha gor en

Theor em" is a bout the rat io of a s ide of an ob long t o i tsdia gonal . The side and width for m a rght tr iang le. Given an

oblong of le ngth f our un its , of w idth thr ee un it s. Then 32 + 4 2

= 5 2, wher e the l atter refer s to t he ob long's squar e.

But thi s doe sn't work wi th a squar e.

In Euc lid 's E lem ent s of G eom etr y appea rs a pr oof (a ppa rentl y

due to Hipp ias ) th at t he dia gonal of a un it su qar e is not a

fr action . The cr it ical not ion, in t he proof , i s tha t ever y

fraction can be reduced so tha t both numer ator and

denomin ator ar e not even number s, otherw is e t he com mon

factor of tw o can be div ided out . (Remember ! An even na tur al

number ha s the f or m, 2n , for some n atur al number n, and its

squar e has the for m, (2n) 2 = 4n 2= 2(2n 2). Simi la rly, an od d

na tur al number has the for m, 2n + 1, and its squar e has the

form, (2n + 1)2

= 4n2

+ 4n + 1 = 2(n2

+ n) + 1 .)

The inco mmensur ab le pr oof pr oceed s a s fol lows :

• Con side r a/b = √2 .

• Then a = √2b

• Squari ng both side s: a22b 2.

• The right-hand side has the for m of an even number

(twi ce some number) , mean ing t hat the l eft -hand numbe r,

a, is an even number .

• To denote i t as an even nu mber , y ou w rite a ≡ c, for some

natur al number c.

• Then you h ave (2c) 2 = 4c 2 = 2b 2.

• Dividing out the common factor of tw o , you have: 2c 2 = b2.





• A cut s epar a te s the r a tiona l nu mber s into two c las se s,

"lo wer" and " upper", s uc h tha t e ver y number of the l ower

c las s i s l es s t han ever y nu mber i n the upper c las s .

• Now, i f a repr esent a ti ve of the l ower c la ss can be

for mu la ted in a fr actiona l r e la tion to a pr esenta t iv e of

the opper cla ss , then the cut its elf is ra tional .

• Ot herwi se , the cut is i r rat iona l .

Dedek ind ther eby fr eed thi s d is ti nction fr om geometr y .

For our present case, w e can as sign to the upper clas s a ll

number s w hose sq uar es exceed tw o, and to the lo wer c la ss

al l number s whose sq uar e ar e l es s t han two . So, the cut i s

the s quar e r oot of t wo.

You can sho w th is b y cons ider ing, to seven digi ts , the

appr oxima tion of the squar e root oft wo:

• (1) 2 = 1 < 2 < 22 = 4;

• (1.4) 2 = 1.96 < 2 < (1.5) 2 = 2 .25 ;

• (1.41) 2 = 1.9881 < 2 < (1.42) 2 = 2.0264 ;

• (1.414) 2 = 1,999396 < 2 < (1.415) 2 = 2.002225 ;

• (1.4142) 2 = 1.9999616 4 < 2 < (1 .4143) 2 = 2 ,00024449 ;

• (1.41421) 2 = 1.9999899 924 < 2 < (1 .41422) 2 =

2,0000182084 ;

• (1.414213) 2 = 1. 9999984 09369 < 2 < (1 .414214 )2 =

2,000001237796 ;

• etc.

You notice tha t

• as you augment the a ppr oxim ation b y one digi t,• it s squar e a ppr oac hes clo ser to two,

• whi le i ts exceeder dim ini she s (antiton ical ly!) down

towar d t wo,

• and you (Ant iton ical ly!) appr oac h the s quar e root of t wo

as the cut.



You lear n tha t the Eudo xian theor y o f p r opor t ions moti vated

ancient Gr eek mathema tic ians to abaudon the d iscontinuou s

or d iscr ete s tr uctur es of a rith metic for the continuou s

st ructur es of geometr y to descri be rela tions betw een

se gment s and such. And, s ince t ime was con sider ed

continuou s , it was also separ ated fr om arithmet ic. You r ea li ze

thi s meant tha t concept s of dyna mica l mec han ics , su ch a sspeed , vel ocit y, acceler at ion, f or ce , etc ., cou ld not be defined

in ter ms of ar ith metic .

You see tha t even T he Fundamental T heor em of A ri thmet ic i s

pr oven by tw o pr oof-b y- contr adict ion ar gument s .

This book onl y pr oceed s con str uct iv el y (cr oss ing no

Boundar ies) .

You lear n how to expli ca te re lation s, function s, oper at ions :

• defin iti on of Relation : number of refer ents of a refer ence .

• gi ven R elation refer ence R, Relation s ar e c la ssi fied by

number of ref er ents coor din ated w ith R. Thus:

o Ru referents a unar y Rel ation o r a ttr ibute, as in "red" ;

o Ruv (uRv) r efer ent s a b inar y Rel ation, a s "ne xt to ";o Ruvw refer ents a ter nar y R ela tion , as in mar ria ge

cer emony with min iste r R br ide, g room

o Relation s ar e man y⇒ R⇒man y , as in many ob ser v e

many

o Relation s ar e man y⇒ R⇒ one , as in many v o ter s

elect ing one of ficia l

o Relation s are one⇒

R⇒

man y, as in one per son t aking

censu s of man y people

o Relation s are one ⇒ R⇒one , a s in s pous e i n

m onogam ous s oci ety

o The scope (r ange ) of a Relation input is its Doma in

(an In side)

o The scope of a R e la tion output is its Codoma in (an

Ins ide)



o Doma in of a R elation can d if fer i n type fr om

Codoma in ( Boundar y cr ossed betw een them) .

• Def init ion of Function : a R ela t ion tha t is man y-one or one -

one .

• Doma in of a Funct ion can dif fer in t ype fr om i ts Codoma in

(Boundar y cr ossed) , as in the Inventor y Funct ion w hose

Doma in compri se s number s, but C odomai n compr iseswar eho use i tems

• An Oper ation is a one-one Function w hose Doma in is

sa me t ype as i ts Codoma in (no Boundar y cr oss ing)

You kn ow your const ructi ve tool s in this book are funct ions

(wi th perha ps onl y domai n B oundar y cr os si ng) and oper ation s

(no Boundar y cr os si ng) , whic h ar e e xten siona l (no cr ossing of

rele vant B oundarie s), contr ar y to the (Boundar y cr oss ing)

in tens ional Ax iom s of S tanda rd Ma them ati cs . ( Topolog y i n

Relation s! )

Student s, toda y we wi ll tak e up the sub ject of par ti al or der or

par or der . You kno w tha t in ter ms of " inc luded in" ,

"subor dina te to" , and "bu si nes s or mil it ar y hier ar chy" .

Please note tha t, in a simp le or tota l or dering , of any tw o

member s, one i s subor dina te to the other . But . a par oder , you

can ha ve t wo member s of equal rank .

Student s, the Pr oto Type of a tota l or der ing i s the s et of

counting number s. The Pr otot ype of the par oder is, as noted

pr evi ous ly, the bu si nes s or militar y hier archy. But it cou ld

also be the factor s of nu mber s.

This l as t st ructur e sho ws ho w the par or der can be fur therdeveloped . F or, a par or der s uc h tha t an y tw o menber s ha s a

MINIM AX and a MA XIMIN i s a LAT TICE . The factor s of

number s compr is es a la tt ice: i ts MINIM AX is LE AST CO MM ON

MU LTIPLE (L CM); it s M AXIMI N i s GR EAT EST COMM ON DIVIS OR

(G CD) .



Yes. The tota l or der ed Ar ithmet ic has a SubAr ith metic w hi ch

is onl y par tia ll y or der ed . And thi s has ama zing con sequences!

Ther e orig ina ted for Logic an e xplic a ti o n of O r deri ng , knowm

as "tr uth ta ble s" . Given two i ndependent l ogica l st a tement s ,

A, B. (They are i ndeoent i f the condi tion of one has nothing to

so with the o ther ,) S ince the f ir st ha s two pos sibil it ie s (Tr ueor F alse) and s o has the second , the tr uth ta ble for thi s

st atement pa ir has four ro ws . Yes, the table for thr ee

independent st atement s has eight r ows; th at f or f our

indepemdent st atements ha s sixten rows; e tc, ; doubl ing w ith

eac h new independent st ate ment.

These tr uth ta bles have been re for mu la ted as I NDI CATOR

T A BLE S b y replac ing T R UE by ONE , FALS E b y ZE R O > T hey ar ethus u sed i n se t theor y and i n pr ob ab il it y theor y . Bu t onl y

her e do you f ind I NDI CATOR TABLE S f or La ttice s. How do we

kn ow tha t? Becau se the MINIMA X of a lattice is la be ll ed a s

its ZERO and i t MAXIMIN as its ONE . So the confli cting or

ambi guous use of the se ter ms would cause great confu sion .

But we can appl y INDICA TOR TABLES to t hat SubAr ith metic of

FACTORS, wi th great con sequences.

Arithme tic has pr ime number s, eac h wi th no factor ex cept

one and it se lf . Con sider the co mplemented dist ribu ti ve

la tt ices on pri mes , 2, 3, 5. Now GCD(2, 3, 5) = 1 , LCM(2, 3) =

6, LCM(2,5) = 10 , LCM(3,5)=15, LCN(6 , 10, 15)=30 .

When you appl y INDICA TOR TABL ES to the comp lemented

dist ribut iv e latt ice on f actor s of thi ty, you obtain the FREE

LATICE O N FA CTO RS OF THIR TY.The oper ation s of ar ithmet ic ar e un iva lent , otherwi se i ts

appli ca t ion w ould pr ovok e wides pr ead di sa gr eement. Bu t,

w i thin ar ithmet ic , i s a s ubar ithmet ic -- pecul iar l y kno wn a s

"the alg ebr a of factor s" -- w hich is m ul ti va lent.



The pri mar y oper at ions of factor theor y ar e l eas t common

mul ti ple (LC M ) and g rea te st common di vi sor (GC D ). B ut both

ar e mu lt iv alen t.

T hus , LCM(3 ,2) = LC M(3,6) , but 2 ≠ 6 .

A l so , G CD (6,10) = G CD (6,22) = G CD (6,14) = 2 . Th is imp li es , for

e xam ple , th a t LC M(7, 2) = LC M(7, x) ha s an i nfin ite number of

so lut ions , name ly , a l l mu lti ple s of 2 .

Th is i mp li es th at par tia l or dering s s uc h a s factor and

inc lus ion are not wel l- defined i n the w ay ad diti on and

mul ti pl ica t ion ar e, tha t is , a + b = a + c i f, and onl y if , b = c .

S imi lar l y , for p > 0 , p · q = p · r i f, and onl y if , q = r . R epe ating ,

the L CM and G C D oper a tor s ar e not we l l- defined .

As counter e xam p les : LC M(2, 3) = 6 and LC M(2, 6) = 6 , hence,

LC M(2, 3) = LC M(2, 6) , but , obviou sl y, 3 ≠ 6 . Si mi la r l y , G CD

(30, 60) = 30 and GC D(30 , 90) = 30 , hence, GC D(30 , 60) =

GC D(30 , 90) , but 60 ≠ 90 .

R esu lt : T h is a llo ws a " fr ee" f o rmal i sm of a rith metic

cons is ti ng of "fr ee na tur als or fr eena ts" and " fr ee in te ge rs or

frin teger s .

This may be over look ed as a resu lt of overlook ing the

va lua ble resour ce of ind ica to r ta bles (number ver sion s of

tr uth ta ble s in s ta tement log ic) . In the ca se of factor s of 30 =

2 · 3 · 6, wi th po ss ib le occur rences for eac h pri me (a bsent ,

pr esent : 0, 1 ), t hi s yields 23 = 8 independent pos sibi lit ies , s o

the i ndi cator s f or occur rence of it s pr ime s can be as si gned

as the binar y nota tion fr om z er o t o s even, r especti vel y:

0 0 00 0 10 1 01 0 01 0 11 0 0



1 0 11 1 01 1 1

The ind ica tor f or a compos ite number is one when all its

factor s ar e pr esent on that r ow, otherw ise zero.

This r esul ts i n the f ol lowing ind ica tor ta ble. Plea se note tha t,

if i(x) denotes tabula r i ndic ator of a rela tor , then i(LCM(a , b))= M AX( i(a), i(b)) and i(GCD(a , b)) = MI N(i (a), i(b)) . Then, you

find :

TABLE 1

1 2 3 5 6 10 15 30

0 0 0 0 0 0 0 00 0 0 1 0 1 1 10 0 1 0 1 0 1 10 0 1 1 1 1 1 10 1 0 0 1 1 0 10 1 0 1 1 1 1 10 1 1 0 1 1 1 11 1 1 1 1 1 1 1(1) (4) (4) (4) (6) (6) (6) (7)

A bal lot can be assigned t o count the nu mber s of

ones in eac h co lumn. T hi s is shown at ba se of eac h co lumn .

(You

notice no ba llot for 2, 3, 5

counts .) You note also dis tri bution of the bal lot over the

ranks

of th is Table:

1, 3, 3 , 1 - - a fami li ar bino mia l pa tter n

Since the sub system of ar ithme tic i s mu lt iv alen t , this

implies tha t the ind ica tor ta ble can be extended to

compr ehend

al l bal lot s, one to seven , by acqur ing factor s w ith ba llot s two,



thr ee, and f ive. (T his i s a non standar d comple tion

compar able to the standar d comple tion tha t shif ts fr om

rational to real number s.)

I f ind a "r ationa le" for thi s in another type of comple tion .

Emu la ting a B ASIS pl oy in s et -theor et ic topo log y, I star t wi ththe A TOMI C (P RIME) BASIS as {2 , 3 , 5} ,

to w hich I ad join MIN (1) for the

BASIS {1, 2, 3, 5} . I then find

tha t LCM. appl ied to

thi s extended bas is yie lds :

LCM{1, 2, 3, 5} = {1, 2, 3, 5, 6, 10, 15, 30}.

You find al l the f ac tor s of 30 ar e

obtained mer el y from oper ator LCM . But

what about oper ator GCD ?

Comp let ion i nvokes th is oper ator .

For ca lcul ating LCM, GCD of factor s,

I a ss ign alter native la be ls ,

respect ively, MAX(),MIN() appl ied to the ir assoc iated ind ica tor s. B ut,

for con venient labeling of an INDICA TOR (TABL E 2 , belo w), I

use such for ms as 2V3 for

thei r LCM and

2 ^ 3, for the ir

GC D; etc.

From T able 1 , we find MIN(i(2) , i(3)) =

0, 0, 0 , 0, 0, 0 , 1, 1,that is, 2^3

ha s bal lot 2. Similarly,

2^5 has ba llo t 2

: MIN(i(2) , i(5)) =

0, 0, 0 , 0, 0, 1 , 0, 1. Also 3^5:

MIN( i(3), i(5)) = 0 , 0, 0, 1 , 0, 0, 0 , 1.



What a bout LCD's of thes e new ly

obtained ele ments ? We find tha t MAX(MIN(2 , 3) ,

MIN(2 , 5)) = i ((2^ 3) V(2^5 )) = 0, 0 , 0 , 0, 0, 1 , 1, 1,

for B=3 . And MAX

(MIN(2 , 3) , MIN(3 , 5)) = i ((2^3)V(2^5)) = 0 , 0, 0, 1 , 0, 0, 1 , 1,

for B = 3 . AndMA X(MIN(2 , 5) , MIN(3 , 5)) = i ((2^5)V(3^5)) =

0, 0, 0 , 1, 0, 1 , 0, 1 for B = 3.

We no w h ave bal lot s 1, 2, 3 , 4, 6, 7 .

What of B = 5 ? We r ecoup th is, and

also a pr eviou sl y unl is ted ter m of B = 4.

We a ppl y dua li ty to resul ts in t he pr evious par agraph-but- one

to

find MIN(M AX(2 , 3) , MAX(2, 5)) =

i((2 v3)^(2v5)) = i(6^10) = 0 , 0, 0, 1 , 1, 1, 1 , 1, for

B = 5 .

MIN(M AX(2 , 3) , MA X(3, 5)) = i((2 v3)^(3v5)) =

i(6^15) = 0, 0 , 1, 1, 0 , 1, 1, 1 , forB = 5 .

MIN(M AX(2 , 5) , MAX(3, 5)) = i((2 v5)^(3V5))

= i (10^15)= 0, 1 , 0, 1, 0 , 1, 1, 1 , for

B = 5 .

We f ind one mor e r esult: MIN(MA X(2, 3),

MAX(2, 5), MAX(3 , 5)) = i((2V3)^(2V5)^(3v5))= i (6^(10V15)) = 0, 0, 0 , 1, 0, 1 , 1, 1, for

B = 4 , distinct from the other s

of the same ba ll ot.

For compr ession , we denote (2^3)v(2^5)

≡ X;

(2^3)v(3^5) &eqi v; Y;



(2 ^5) v(3^5) &eqi v; Z . Also, we label

the oper ands of the comp leted oper ator s as

"subdomi nants ". We then f ind the ta bled resu lts.

TABLE 2: SUBD OMIN AN TS OF 30 (ADJ OIN ED WIT H 6

INDIC ATOR)

2^3 2^5 3^5 X Y Z 6^10 6^15 10^15 6^10^15 60 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 1 0 00 0 0 0 0 0 0 1 0 0 10 0 1 0 1 1 1 1 1 1 10 0 0 0 0 0 1 0 0 0 10 1 0 1 0 1 1 1 1 1 11 0 0 1 1 0 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1(2) (2) (2) (3) (3) (3) (5) (5) (5) (4) (6)

(BALLOT)10 ^15 is

the e xcept ion) of Table 2 ar e

factor s or subdo minant s of 6.

You note Table 2 has r ank s 1, 2, 3, 4, 5, 6, 7

, so tha t rank va lue of an ele ment i s it s bal lot value ,

whic h is not the case in Table 1 . Please note alsotha t the distr ibut ion of ele ments over the se r ank ings is

1, 3, 3 , 4, 3, 3 , 1, clea rly not

a b inom ial pa tter n. I find no a lgori thm in the liter atur e to

deter mine the number of subdomi nants of the factor s of

n.

Having been a ler ted , I found , i n a book on in for mation

retrie val

(Ger ar d Sa lton , Autom atic Inf or ma t ion O rgani za t ion and

Re trie val , McGr aw- Hil l, 1968), the Hass e d iagram of

st andar d factor s of 30 cited as "descr ipt ion set " and the

Hasse

dia gram of the ta ble of subdom inants c ited as

"r eques t s pace" . [W ikiped ia has an ar tic le on him.]



Please note resemb lance of these resul ts to what is labe led,

in the liter atur e, a "fr ee " s tr uctur e, su ch as a f ree

alg ebr a . I then la be l the Natur al N umber s thu s extend ed as

"fr eena ts" ("fr ee n atur al s") and the Inte gers as "fri nte gers"

("fr ee inte ge rs") , i n the s ens e th at the m athem atic ian is fr ee

to u se al l oper ator s of a syste m to comp lete it .

"T he Fundamenta l T heor em of Ar ithmet ic" states tha t " a

number can be f actor ed in to a pr oduct of pri me number s i n

onl y

one way (e xcept for order of list ing) ." Do the a bove resu lt s

violate it?.

We ha ve an Ar ith metic with a Fundamenta l Theor em

espou si ng

Un iv alenc y, But th is A ri thmet ic has a Subari thmeti c th at is

Mul ti va lent. A nd a key wor d in FT is "factor" , i nvok ing

the s ubar ithmet ic , w hich can be extend ed bey ond FT,

Below please find the Ha sse Dia g ram of the comple mented

di st ribute la tt ice on factor s of thi r ty and the H as se D ia gram

of the deri ved fr ee la tice .

30



/\/ \/ | \/ | \/ | \

6 10 15|\ / \ /|

| \ / \ / || \/ \/ || /\ /\ || / \ / \ ||/ \ / \|2 3 5\ | /\ | /\ | /\ | /

\ | /1

30

/\/ |\/ | \/ | \/ | \/ | \6 10 15/ /|\ \/ / | \ \/ / | \ \

| / | \ \| / | \ \| / | \ || / | \ |6^10 6^15 10^15/ \ |\ \

\__________ / \ |

\________\_________ |



/ \______|___________\_______ | |

/ | \\| | 2 3 56^10^15

/ \ /|\ /\

/ \ / | \ / \/ \ / | \ /\

/ \ / | \ /\

/ \ / | \ /\

/ \ / | \ /\

/ \/ | \/

\ (2^3)V(2^5) (2^3) V (3^5)(2^5)v(3^5)

| \ \ / | \/ / |

| \ \ / | \/ / |

| \ \ / | \ // |

| \ \ / | \ // |

| \ \/ | \// |

| \ /\ | /\/ |

| \/ \ | / \/|

| /\ \ | / /\|

| / \ \ | / /\ || / \ \ | / /

\ || ___/______\___\_______|_______/_ _/

\ ||| / \ \ | /

\ ||| / \___\_____|

_____/______________\__ |



||/ \ | /\ | |

2 ^ 3 2 ^ 53 ^ 5

| ||

| |

| | ||

| ||

| ||

| ||

-------------------2 ^ 3 ^

5---------------------- ||

1

The above condots with find ings of the g reat Br it ish-

Ame rucan m athem atic ian -log ici an-ph il osophe r, A. N.

Whitehead (1861 -1947) in h is " Un iver sal A lg ebr a", 1898.

Whitehead i nd ica ted nonidempoten cy in nu merica l alge br a xx

≠ x , x ≠ 0, but ide m potenc y i n Bool ean A lge br a xx=x . And

ask ed if a " Un iver sal A lg ebr a" cou ld compr ehend both

syste ms . It alead y doe s, s ince factor theor y can be

for mu la ted as a Boo lean A lge br a . The fol lowi ng expl ica tessu ch a st ructur e, renamed since "Un iver sal Alge br a" has

acquir ed a difer ent meaning .

Def . 1. Given s tr uctur e A = <o 1, o2, o3, U, L> of ty pe <2 , 2 , 2, 0,

0> with (for all x in A) x o1 L = L, x o1 U = U , x o3 x ≠ x Then A

is a panalge br a if f x o 2 y = x o3 y w hene ver x o 3 y = L.

Def . 2. D = <2, 2> i s a supple mented l a t tice if f D i s a

di st ribut iv e la tt ice ; @ i s the commut ati ve, a ss oci ati ve ,



wel ldefined , non idempotent oper ation gener ati ng c ha in s of

pr oper (noma tomi c) j oin -i r r educib le s in D.

Theor em . A factor alge br a, Fn, is a pana lge br a . Pr oof : F n =

<gcd, lcm, •, n, 1>.



APPE NDI X C

TABLE OF MI NIM AX-MA XIMIN MATHEMA TIC AL S YST EMS

("B AR BIE DOLL" : M ATH C HAN GE MI MICS "C OSTUME CHANGE"

OF BAR BIE DOLL)

The pr ototype of the se is factor alge br a wi thi n ar ithme tic . FA

has two pr imar y oper a tion s . One is LCM (Least Co mmon

Mul tip le ) is a Minimax Oper a ti on , since the"Mu lt iple "maxi mize s it over i t s factor s , while the "Lea st"

mi nimi zes a l l mu lt iple s of the factor s . The other pr imar y

oper at ion , GCD (Gr eates t C ommon D iv iso r ), is a Maxi min

Oper at ion , since Di visor s m in im iz e o ver the Div idends , while

the " Gr eates t" maxi mize s al l d iv i sor s of t he d iv idend factor s .

(T hese l abel s ar e better than the s tandar d " Supr emum" and

"Infi mum" -- la be ls whic h obscur e thi s oper at ional contr ast .)

TABLE OF MINIMAX-MAXIMIN MATHEMATICAL SYSTEMS

SYSTEM MINIMAX MAXIMIN OPERANDS

ARITHMETIC

FACTORINGLCM GCD

ARITHMETIC

FACTORS

PECKING ORDER PECKING PECKED CHICKENS

BUSINESS

HIERARCHYBOSSING BOSSED PERSONNEL

MILITARYHIERARCHY

ORDERING ORDERED MILITANDS

ADDITIVE COLOR

OPTICSMIXED REMOVED

RED.BLUE,GREEN

LIGHT

SUBTRACTTIVE

COLOR OPTICSREFLECTED ABSORBED

YELLOW,

MAGENTA, CYAN

INKS. PAINTS

CHROMODYNAMICS

PARTICLE

THEORY (COLOR AS

CHARGE)

CREATED ANNIHILATED QUARKS

SET THEORY UNION INTERSECTION SETS

SET-THEORY

TOPOLOGYUNION INTERSECTION BASIS SETS

BOOLEAN RING

ADDITION

(SYMMETRIC

DIFFERENCE)

MULTIPLICATION POWERSET OF SET



STATEMENT LOGIC DISJUNCTION CONJUNCTION STATEMENTS

Documents

You Can Construct Arithmetic