A Theology of Mathematics

8/9/2019 A Theology of Mathematics

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A Theology of Mathematics

A theology of … what?!.........................................................................................2What is Mathematics?............................................................................................3Where are we now?................................................................................................5A note on Mathematics and Post Modernity..........................................................8Bea ty in Mathematics......................................................................................... "Morals in Mathematics.........................................................................................#he Mathematical mind $ strengths and wea%nesses........................................... 2Why do so many &eo&le disli%e Mathematics?.................................................... 3'oncl sion............................................................................................................ (Bi)liogra&hy......................................................................................................... *

“Numbers were beautiful things, numbers were funnythings, they were without a doubt ‘God stu ’” – Mister

God, This is Anna by Fynn

+oger M. ,rr -o em)er 2""3 'MW &ro/ect


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A Theology of Mathematics 2

A theology of … what?!The rst issue raised by this sub e!t is the sur"rise usuallygenerated by "utting together the two words “theology” and“mathemati!s”# $n his o"ening le!ture for the %hristian in the&odern 'orld !ourse, entitled the (a!red)(e!ular *i+ide, &arGreene as s the -uestion ‘'ho has a theology of mathemati!s.’There are +ery few "eo"le "re"ared to try and answer the -uestion/des"ite the fa!t, as he goes on to say, that we ha+e ea!h s"ent"erha"s an hour a day, +e days a wee 0during term time12 forele+en years or more in maths lessons during our s!hooling# 'hydo we ha+e no !oherent way of relating this a!ti+ity to our beliefs inGod. 'e seem to ha+e lost the sense that God has anything to dowith mathemati!s# $n!identally this is true of other things too –&ar is using mathemati!s "rin!i"ally as an e3am"le of the

di!hotomy between ‘religious’ and ‘se!ular’ a!ti+ity that is "resentin !ontem"orary 'estern !ulture#

This se"aration of mathemati!s from theology is histori!ally odd/many of the famous names from mathemati!s in the "ast were also

nown for their theologi!al writing#4laise 5as!al 06789:782 is "robably best nown today either for his‘5ensees’ whi!h are a !olle!tion of ‘thoughts’ mostly on the sub e!tof human su ering and faith or for ‘5as!al’s wager’ whi!h says “ IfGod does not exist, one will lose nothing by believing in him, whileif he does exist, one will lose everything by not believing 1 .”

;owe+er he was also an im"ortant mathemati!ian, who laid thefoundations for "robability and also ga+e his name to the ‘5as!altriangle’ 0whi!h was a!tually nown years before 5as!al himselfstudied it2# This is the table whi!h starts li e this<

11 1

1 2 11 3 3 1

1 4 6 4 1

where ea!h number is the sum of the two numbers in the rowabo+e it#This triangle was the basis for $saa! Newton’s 067=9–6>8>2 wor onbinomial e3"ansions – and he was another theologi!al writer< “God!reated e+erything by number, weight and measure# 8”# ?f !oursehe is best nown for wat!hing a""les falling#

@ohannes Ae"ler 06B>6:679C2 is another famous astronomer,theologian and mathemati!ian who des!ribed his wor in

0Blaise Pascal1 at www ga&.dcs.st and.ac. % 4history Mathematicians Pascal.html2 Bar lan 6ni ersity Physics 7e&t at htt& www.&h.)i .ac.il 9+P.&h&


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understanding "lanetary orbits, whi!h in+ol+ed mathemati!al woron elli"ses, as “thin ing God’s thoughts after him 9”#

4efore him Giordano 4runo 06B=D:67CC2 was a free thin er whowas e+entually illed by the $n-uisition – "art of his !rime wase3"lorations he had done on in nite whi!h was deemed hereti!alsin!e the %atholi! do!trine of the time held that only God !ould bein nite# ;ere was a dar er !onne!tion between theology andmathemati!s# 4efore this, for both Erabi! and !lassi!al Gree"hiloso"hers, mathemati!s was seen as !losely !ou"led to religiousthin ing#

;owe+er for most of us today this !onne!tion is lost, so $ intend toshow +arious ways that a %hristian mind !an intera!t with thestudy of mathemati!s#

What is Mathematics?This de!e"ti+ely sim"le -uestion is a!tually +ery hard to answer#$’ll try to gi+e a brief o+er+iew without assuming too mu!hmathemati!s – some of the slightly more te!hni!al bits are inse"arate bo3es and !an be s i""ed without "roblem#

The ancient world

The gy"tians new about right:angled triangles and used thisnowledge to build things li e the "yramids, whi!h are still with us

today#They new that a "ie!e of ro"e notted into twel+e e-ual "ortions!ould be used to measure a s-uare !orner : a bit li e this<

;owe+er, as far as we now, they didn’t seem to ha+e been able towor out why this was true/ "erha"s they weren’t interested in the‘why’ sin!e they mainly used mathemati!s to get things done#'e now that it wor s be!ause 9 8 =8 H B 8#

3 nstit te for 'reation +esearch at htt& www.icr.org & )s )tg a )tg "3:a.htm

Optional information: ythagoras!s theorem.

For any right:angled triangle the s-uare onthe hy"otenuse e-uals the sum of thes-uares on the other two sides#


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A Theology of Mathematics :

Es any s!hool !hild nows this is an e3am"le of ‘5ythagoras’stheorem’ whi!h was rst proved by the Gree "hiloso"her5ythagoras 0IB7J:=>B 4%2# $n fa!t the 4abylonians had wor ed outthe rule a millennium before him although they seem to ha+e beenunable to "ro+e it#

(o in this !ase mathemati!al nowledge "rogressed from<an observation that the 9:=:B triangle has a right angle toa r"le that a triangle with side a,b,! is right angled if a 8 b 8 H ! 8 toa theorem that this rule is true for any a, b and !

The Gree s were fas!inated by ‘"ure thought’ and many of thefoundations of mathemati!s were laid by them# ?ne of the greatestof these Gree mathemati!ians was u!lid 0I98B:87B 4%2# ;is‘ lements’ was one of thema or attem"ts to "ro+ide asystemati! summary of theresults they dis!o+ered, andis "ossibly the se!ond mosttranslated, "ublished andstudied boo e+er written0after the 4ible2#

u!lid wanted to "ro+ide a rm foundation for mathemati!s and sohe ga+e a great deal of em"hasis to rigorously "ro+ing theoremsfrom a3ioms# For e3am"le, in geometry his ho"e was that we !ould

!hoose a relati+ely small number of ey statements whi!h e+eryreasonable "erson !an see are self:e+idently true 0these usuallynown as a3ioms or "ostulates2 and then dedu!e the whole of

geometry as logi#al #onse$"en#es of these a3ioms#;e was able to get his starting set down to only +e "ostulates#

;e was unha""y with the fth "ostulate, mostly be!ause it wasinelegant, and wanted to be able to remo+e it# ;owe+er he wasunable to "ro+e it true from the rst four and neither !ould he nda sim"ler e-ui+alent# ?+er the ne3t two millennia manymathemati!ians tried, and failed, to do the same#

Optional Information % &"#lid!s post"lates:

62 E straight line segment !an be drawn oiningany two "oints#

82 Eny straight line segment !an be e3tendedinde nitely in a straight line#

92 Gi+en any straight line segment, a !ir!le !anbe drawn ha+ing the segment as radius andone end "oint as !entre#

=2 Ell right angles are !ongruent#

End the ‘odd man out’<

B2 $f two lines are drawn whi!h interse!t a thirdin su!h a way that the sum of the inner angleson one side is less than two right angles, thenthe two lines ine+itably must interse!t ea!hother on that side if e3tended far enough#


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u!lid ob+iously belie+ed the lines, !ir!les, et!# he des!ribed werethose of the real world and so his mathemati!s was a des!ri"tion or

!odi !ation of the nature of the uni+erse# ;is +iews led years laterto the statement “God is a geometer” whi!h seems to ha+e been!oined by Ae"ler# $t seems fair to say that most mathemati!ians –at least those from 'estern traditions – belie+ed that they weredis#overing truth about the uni+erse and that intuition, s!ien!e andmathemati!s were all di erent +iews of the same thing – reality#

Shaking the foundations

$n 6D89 this world +iew !hangedfore+er – although most "eo"ledidn’t realise it at rst#

@ohann 4olyai and Ni olay Koba!he+s y inde"endently dis!o+eredgeometries whi!h did not assume u!lid’s fth "ostulate# This"rodu!ed three main La+ours of geometry – u!lidean, ;y"erboli!and lli"ti!al#

Optional information ' Geometries

u!lid’s fth "ostulate is e-ui+alent tothe idea that there is a "ni$"e "arallelto any line through a "oint not on theline#

There are two main La+ours of non:eu!lidean geometry, whi!h !orres"ondto two di erent answers to the-uestion “;ow many "arallels arethere.”# $n hy"erboli! geometry thereis more than one "arallel line and inelli"ti!al geometry there is no su!hline#

Note that they all share some theorems– any theorem whi!h you !an "ro+eusing only the rst four of u!lid’sa3ioms is true in all of thesegeometries#


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A Theology of Mathematics (

The M7=,CCC -uestion is< whi#h one of these geometries mat#hes

the real world( The -uestion !annot really be answered by

e3"eriment sin!e an in nite line would ta e some time to!onstru!t and the -uestion !annot be answered from withinmathemati!s sin!e all three geometries are internally !onsistent#

Meanwhile, elsewhere in the forest…

Things were getting !onfusing in the world of arithmeti! too#Georg %antor 06D=B:6J6D2was "laying around within nity#

?ne of the biggest "roblems with in nity is trying to !om"are itwith itself – many "arado3es lie waiting to tra" the unwary#

%antor realised that there were two ways to !om"are the siOes ofsets of things – ta e ni+es and for s for e3am"le# ?ne way is to!ount the ni+es, !ount the for s and !he! the totals are the same#

Enother way is to "air u" the ni+es and for s until you run out of"airs# $f you ha+e nothing left then you ha+e the same number of

ni+es and for s/ if you ha+e something left o+er then this tells youwhi!h you’+e got more or – ni+es or for s# This se!ond methoddoesn’t in+ol+e remembering totals or e+en being able to !ount#

;e found that using this se!ond method of !om"aring siOes dealtwith the "arado3es and enabled a rigorous treatment of in nity# ;ewent on to "ro+e that there were many di erent in nities withdi erent siOes#$n "arti!ular he "ro+ed that there are more de!imal numbersbetween C and 6 0i#e# ‘"oint:somethings’2 than all the whole

numbers 0i#e# the numbers 6,8,9, et!# 2# ;e was able todemonstrate this by showing that howe+er you "air u" the de!imal

Optional Information % A paradox of in)nite

Ta e the natural numbers<6, 8, 9, =, B, 7,

Now double them<

8, =, 7, D, 6C, 68, There is an in nite number in both rows –whi!h row is bigger. ?b+iously the se!ond rowis bigger be!ause it is double the rst row#

(tart again but this time throw away e+eryother number<

8, =, 7, D, 6C, 68, Now whi!h row is bigger. ?b+iously the rstrow be!ause we too things out to !reate these!ond row#

4ut wait – the resultant row in both !ases is the

same – how !an it be both bigger and smallerthan the starting row.


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A Theology of Mathematics *

numbers with the integers there are always de!imals left o+er#Nowadays almost e+ery mathemati!ian a!!e"ts this result,although it seems e3tremely unli ely when it is rst e3"lained#

;e did not re!ei+e the same !onse-uen!es as Giordano 4runoabo+e, or e+en those of another earlier mathemati!ian 4ernhard4olOana 0a %Oe!h theologian)mathemati!ian who lost his tea!hing"ost at 5rague in 6D6J following his in+estigations into in nity2 buthe did ha+e mu!h o""osition to his ideas from othermathemati!ians and "hiloso"hers#$t is li ely that this o""osition !ontributed to his e+entual madness,although the sub e!t matter of his thoughts "robably had as mu!hto do with it#

%antor’s wor on in nity threw u" a hy"othesis– the so:!alled‘!ontinuum hy"othesis’# &u!h e ort was s"ent trying to "ro+e ordis"ro+e this hy"othesis# Aurt GPdel 06JC7:>D2 "ro+ed that thehy"othesis was !onsistent with the rest of arithmeti! and then 5aul%ohen 06J9=:2 "ro+ed that assuming this hy"othesis to be false wasalso !onsistent with the rest of arithmeti!# (o you !ould ta e it orlea+e it – both ways "rodu!ed a !om"letely !onsistent system#

?n!e again, li e u!lid’s fth "ostulate, how #o"ld yo" *now whi#hwas +tr"e! . $ still remember the sur"rise $ re!ei+ed while listeningto a maths le!turer who was "ro+ing a theorem when he said, “&y"roof of this theorem will use the !ontinuum hy"othesis# $f you

don’t belie+e this hy"othesis 0and $ don’t, but most mathemati!iansdo2 then the "roof is still "ossible but a lot harder”# $ had ne+erthought of belief before in the !onte3t of mathemati!s#

You just can’t prove everything

GPdel’s most im"ortant result howe+er is arguably the‘in!om"leteness theorem’# The full result is fairly !om"le3 and the"roof is -uite hard to gras"# ;owe+er the theorem states that,gi+en the rules of arithmeti!, there are statements of arithmeti!whi!h are tr"e but #annot be proved # This sho! ing !on!lusion wasthe death nell for the attem"t to build u" a !om"lete "i!ture of allmathemati!al truth from basi! a3ioms sin!e e+en arithmeti!, whi!hseemed de!e"ti+ely sim"le, was in!om"lete in this sense#

Where are we now?'ithout a doubt the relationshi" between mathemati!s and realityis less ob+ious than it was# This has a big im"a!t on how we thinabout mathemati!s theologi!ally#

Es far as $ !an tell there are three strands in the a""roa!hes being

ta en today#


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62 &athemati!s is an a priori truth about the real world< 8 8does e-ual =

82 &athemati!s is an empiri#al !onstru!tion from obser+ing theworld

92 &athemati!s is a "urely h"man !onstru!tion – it ought to bean arts sub e!t

These three basi! a""roa!hes are in "ra!ti!e often !ombined in +arious ways, but $’ll treat them se"arately as $ loo at what a%hristian mind !an a irm and #hallenge in these di erent +iews#

A Priori

The rst a""roa!h is highlighted when mathemati!ians tal about‘dis!o+ering’ a result# $t is a +ery !ommon +iew, both inside andoutside the world of mathemati!s, and is e+en reLe!ted in law sin!emathemati!al theorems !annot be "atented# This +iew is most li ethe traditional +iew that has been held almost uni+ersally, untilre!ently, sin!e the Gree s# &any mathemati!ians “really ha+e the

feeling of mo+ing in an abstra!t lands!a"e of numbers or guresthat e3ists inde"endently of their own attem"ts at e3"loring it =”

Es %hristians we !an a irm the sense of reality in this a""roa!h#'e would want to assert that absolute truth !an e3ist : and does soin God# $f "art of this truth is mathemati!s then when we domathemati!s we are in some sense ‘thin ing God’s thoughts after

himB

’# $n "re+ious times "eo"le ha+e used the e3isten!e ofmathemati!s as "art of “natural theology” attem"ts to "ro+e thee3isten!e of God# $mmanuel Aant 06>8=:6DC=2 was one su!h"hiloso"her and although many "eo"le today would re e!t the

+alidity of this sort of "roof it still seems to be hard to belie+e in theob e!ti+e truth of mathemati!s without somehow "utting some sortof god into your world +iew#

;owe+er we would want as %hristians to !hallenge attem"ts toe3alt mathemati!s as the ultimate truth : God may be a geometerbut he is mu!h more than ust a geometer# GPdel’s in!om"letenesstheorem stands as a reminder that mathemati!s !annot "ro+ee+erything that is true#

!pirical

The se!ond, em"iri!al, a""roa!h 0whi!h seems to ha+e similarity tothe gy"tians’ +iew2 is often asso!iated with the "hiloso"her @ohn(tuart &ill 06DC7:>92 who wrote that geometry “is built onhy"othesis/ that it owes to this alone the "e!uliar !ertaintysu""osed to distinguish it/ and that in any s!ien!e whate+er, by

: 7ehaene; &2:25 9ee note 3


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A Theology of Mathematics <

reasoning from a set of hy"otheses, we may obtain a body of!on!lusions as !ertain as those of geometry, that is, as stri!tly ina!!ordan!e with the hy"otheses, and as irresistibly !om"ellingassent, on !ondition that those hy"otheses are true 7”# Thoseholding this +iew often see mathemati!s ust as a tool, or alanguage, for doing other things – whether s!ien!e or e!onomi!s#&athemati!s in this a""roa!h has no ob e!ti+e reality – two "lustwo is four by e3"eriment and hen!e, "resumably, !ould be "ro+edfalse#

E %hristian !riti-ue would want to a irm that the reason that thisem"iri!al a""roa!h wor s at all is be!ause God !reated andsustains both the uni+erse and oursel+es# ;en!e it is not sur"risingthat we !an, at least in "art, understand the uni+erse# The boo of

@ob, for e3am"le, “shows that &an was intended to argue withGod >” – the for!e of the !on!lusion to the boo is that it is a re"ly to

@ob’s -uestions< @ob 9D 6 “Then the Kord answered @ob out of thestorm”# $t has often been noti!ed that !ontem"orary s!ien!e aroseout of a world +iew strongly inLuen!ed by %hristian belief in theorder of the uni+erse and the God:gi+en rationality of man enablingus to !om"rehend it#

E %hristian would also li e to "oint out that, without a belief inGod, the main "roblem with the em"iri!al a""roa!h is why doesmathemati!s wor li e this – instein said something li e “the mostin!om"rehensible thing about the world is that it is

!om"rehensibleD

”#

Artistic

The third a""roa!h at its most e3treme argues that mathemati!shas no e3ternal reality at all# $t is sim"ly an attem"t to im"ose"attern onto a !haoti! uni+erse# Few "eo"le go that far/ rathermore would be ha""y to e!ho G#;#;ardy 06D>>:6J=>2 who in hisboo ‘E &athemati!ian’s E"ology’ 0whi!h he wrote towards the endof his life2 says “Qeal mathemati!s”, as he referred to it, “must be

usti ed as art if it !an be usti ed at all# J ”$n this a""roa!h “if geometry is not God:gi+en, then it is "eo"le!reated, and how do we thin about human !reati+ity. 6C”# 'e !ana irm the in!redible ri!hness of the !reati+e "ro e!t whi!h ismathemati!s# ;owe+er, many %hristians might be unha""y at the‘"ri+atisation’ of mathemati!s whi!h is an e e!t of denying it anye3ternal reality# The near uni+ersal agreement o+er mathemati!altruth and the way e+en +ery abstra!t mathemati!s ee"s "o""ing

( 0=ogic1; . >. ; oted at htt& www. tm.ed research ie& m mill/s.htm* @ e 'ame 7own from ea en; 'harles Williams8

0'ollected otes from Al)ert Cinstein1 htt& rescom&.stanford.ed 4cheshire Cinstein otes.html< from @6nderstanding AnalysisD at htt& comm nity.middle) ry.ed 4a))ott 6A 6A &oint .html" htt& www.stthomas.ed cathst dies


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A Theology of Mathematics "

u" years later in s!ien!e are more !onsistent, to a %hristian mind atleast, with the rst two a""roa!hes#

The "i#le and $u!#ers

There are a few thoughts and "rin!i"les whi!h !an be drawn fromthe 4ible about arithmeti! and rationality#

Qeason and understanding are seen as gifts of God – soNebu!hadneOOar tem"orarily loses his reason as a "unishment in*aniel =# God e3"e!ts man to reason with him 0$s 6 6D2 and he is thesour!e of understanding although his thoughts are seen as higherthan the thoughts of man# There is little sign of a di!hotomybetween ‘faith’ and ‘reason’ in the 4ible – %hristians are en oinedto gi+e a ‘reason for the ho"e that you ha+e’ 065et9 6B2#;owe+er understanding in the 4ible does not arise in a +a!uum –5ro+erbs in "arti!ular stresses that man’s understanding andwisdom !omes out of a relationshi" with God# “$ belie+e therefore $am” is "robably !loser to this than the 'estern “$ thin therefore $am” of *es!artes#

God !reates order out of formlessness in Genesis 6/ whi!h isreLe!ted in the stru!tured way it is written# For e3am"le there aretwo tri"les of days ending with the s"e!ial > th day# God is seen as!reating and sustaining the order of !reation – sunrise and sunset,summer and winter 0Gen D 88 2# The 4ible sees the order and rhythm

of the world as a sign of God’s hand in it and $ thin this woulde3tend to arithmeti! also#

Then too the ;ebrews were +ery interested in numbers and theirsymbolism# &ost of the smaller numbers were asso!iated withs"e!i ! ideas, and so their use of ‘round numbers’ asa""ro3imations was more !om"li!ated than that of 'estern writers#?ne of the most im"ortant numbers was se+en, whi!h s"o e of!om"letion and "erfe!tion : as for e3am"le “the se+enfold s"irit ofGod” 0Qe+ 9 62#

The 4ible sees being able to number things as gi+ing some sort of"ower o+er them – the im"ortan!e is shown by the way all sorts ofthings get !ounted from "eo"le to drin ing +essels# *a+id gets intotrouble when he !ounts the $sraelites in 6 %hroni!les 86 whi!hseems to be be!ause he is abrogating God’s right alone to nowthis# $n the New Testament we hear from @esus that the ‘hairs of

your head are all numbered’ 0&att 6C 9C2#

Things that !ould not be !ounted – su!h as the stars, the !louds orthe grains of sand – are used to e3"ress the restri!ted nature of

man’s mind and !ontrast it with God’s omnis!ien!e# 'hen God is"romising Ebraham lots of des!endants he !om"ares them to the


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A Theology of Mathematics

stars, whi!h Ebraham !annot !ount# ;owe+er in 5salm 6=> Godnows the number of the stars and his understanding has no limit

0literally ‘no number’2# (o too in @ob 66 > Ro"har as s @ob therhetori!al -uestion “%an you "robe the limits of the Elmighty.”

The interest in numbers !an howe+er be!ome a wish to nd hiddensigni !an!e in them# This is usually nown as numerology and !anbe found in the -abbala of later @ewish thought – hidden numbersin 4ibli!al 0and other2 te3ts whi!h re+eal to the initiate the se!ret!ode un nown to others# E similar +iew seems to be behind someof the e3treme inter"retations of the boo of Qe+elation and thefamous number of the beast ‘777’# Elthough it is undoubtedly"ossible that God !ould embed "ro"heti! numbers in the 4ibli!alte3t it is un!lear why he would ha+e done so – the !odes andnumbers whi!h "eo"le nd are generally only understandable whenloo ing ba! at e+ents on!e they are !om"leted and the e+ents sodes!ribed seem arbitrary# $t !an be all too easy to nd a""arent!onne!tions between histori!al e+ents and numbers that are in fa!tsim"ly a !oin!iden!e – there are so many e+ents and so many waysto !ombine letters and numbers that the range of "ossibilitiesbe!omes immense# E sim"le e3am"le 0using the Aing @ames

+ersion2 is the ‘"roof’ that (ha es"eare wrote the 4ible : “5salms isthe most "oeti! boo of the 4ible# 5salm =7# Ta e the =7th# wordfrom the beginning# Edd the =>th# word from the end# Sou ha+ere+ealed the true author of the 4ible# 66 ”

Edditionally the bible itself does not !laim to be a boo of

numerology and @esus made no referen!es to this +iew of the;ebrew ?ld Testament des"ite ma ing many statements about itand its role#

A note on Mathematics and Post Modernity

&athemati!s does not sit +ery well with "ost:modernity# There area number of reasons for this, the most im"ortant being the!om"lete inability of mathemati!s to sur+i+e any sort ofin!onsisten!y#

Elan Turing 0best nown for his !ode brea ing in 'orld 'ar $$2a""arently attended a series of le!tures by the "hiloso"her'ittgenstein who argued that !ontradi!tions should not be re e!ted– Turing attem"ted to !ounter this +iew and e+entually ga+e u"attending the le!tures 68 # The in!lusion of ust one !ontradi!tion ina mathemati!al system allows you to "ro+e any result you want#;owe+er, the "ostmodern world +iew has gi+en mathemati!iansin!reased Le3ible about whi!h a3ioms are ta en as the starting"oint, e+en though they still use the same logi! to "rodu!e resultsfrom them#

Erom htt& www.geocities.com Athens #roy :"8 9ha%es&eare.html2 Barrow2""" &3""


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5erha"s %hristian thin ers !an learn something from mathemati!sabout ma ing a !lear se"aration between the !hoi!e of initiala3ioms from the !on!lusions su!h starting "oints lead to# Fore3am"le we might re!ognise that although in many !ases startingassum"tions !annot be "ro+ed the way in whi!h these are used torea!h to a !on!lusion !an be !hallenged# Es Tom 'right wrote“The !hur!h must re!o+er its faith in God:gi+en reason, not as aninde"endent sour!e of authority but as the tool for thin ing!learly 69 ”#

Kogi! and reason is +ery mu!h a "rodu!t of the !ons!ious mind –the un!ons!ious mind whi!h is so mu!h a "art of "sy!hology doesnot seem to follow the same rules of logi!# ?ne of the "roblems ofan o+er:rational a""roa!h to life is that it dis!ounts many of thethings whi!h are more asso!iated with the un!ons!ious – su!h assymbols, myths and "oetry# These are often stressed in a"ostmodern a""roa!h but sometimes it seems that the "endulumswings too far in the dire!tion of abandoning reason# $n the sameway that "sy!hology loo s to restore a healthy balan!e between the!ons!ious and the un!ons!ious there is a "ro"er balan!e betweenrationality and non:rationality#

?ne s"e!i ! area where $ belie+e there is a !onne!tion between"ost modernity and mathemati!s is in the relati+ely re!ente3"loration of so:!alled ‘%haos theory’ 0whi!h is "erha"s betterdes!ribed as “order without regularity”2 as seen in things li e

weather "atterns and the way ta"s dri"# The mathemati!s at thefoundations of this theory is not hard but the nlightenmentmindset didn’t nd it# This seems to be be!ause "eo"le were unableto a!!e"t that !om"li!ated beha+iour !ould arise from sim"le!auses and so loo ed for in!reasingly !om"li!ated !auses – theolder world +iew understood ‘order’ in a relati+ely deterministi!and "redi!table way whi!h did not t with the un"redi!table natureof !haos theory#

For e3am"le, !onsider the +ery sim"le formula f(x) = 2x 2 – 1 # $f youa""ly this formula to its own out"ut again and again you mighte3"e!t to get some sort of "attern# ;owe+er, if you try it with a"o! et !al!ulator or a !om"uter you’ll "robably be sur"rised – itloo s almost random# ;ere for e3am"le are the last BC numbersgenerated when $ started with an initial +alue for 3 of C#>D andre"eated the !al!ulation 6CCC times on a !om"uter#

3 0#hat s&ecial relationshi&1; #he F ardian; 8 ,ct 2""3


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-1.5

-1

-0.5

0

0.5

1

1.5

1 51

t might loo% random $ ) t it isnDt!


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A Theology of Mathematics :

Beauty in Mathematics

4eauty also has an im"ortant "la!e in mathemati!s –mathemati!ians and theoreti!al "hysi!ists ha+e an instin!ti+e"referen!e for elegant theorems/ 5aul *ira! is one of the best

nown of many who a""arently "referred a beautiful theory to onewhi!h sim"ly !om"lied with obser+ations# This ts well with ourunderstanding of the nature of the uni+erse from Genesis that itwas initially !reated ‘+ery good’#;ere are a !ou"le of e3am"les of beauty in mathemati!s#

%& The Mandel#rot set'

This set was highly "o"ular during the 6JJCs and e3tra!ts from it

a""eared e+erywhere#

This is an image of the entire set#

The set arises out of !haos theory and is generated by using ade!e"ti+ely sim"le formula f(z) = z 2 + c whi!h is a""lied again andagain to its own out"ut for di erent starting +alues of ‘!’# Thedetail that is re+ealed when magnifying "arti!ular areas of the setis astonishing and seems out of all "ro"ortion to the sim"li!ity ofhow it is de ned#

82 An incredi#le relationship #

Fi+e fundamental mathemati!al -uantities<e 0the base of natural logarithms : it also a""ears in

"robability2 0the ratio of the !ir!umferen!e of a !ir!le to its diameter2i 0the s-uare root of minus one, mu!h used in ele!tri!al

theory2oneOero

are related in this e-uation


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e π i 6 H C$ "ersonally nd this result 0and some other unli ely relationshi"s2almost unbelie+ably amaOing sin!e there seems no good reasonwhy -uantities whi!h were inde"endently dis!o+ered should ha+esu!h an intimate relationshi"#

For many "eo"le the beauty of mathemati!s is a sign of the beautywhi!h God "la!ed in the uni+erse# $ nd it hard to see why ‘beauty’should ha+e any relation to ‘truth’ if the uni+erse were in fa!tmeaningless#

Morals in MathematicsThe wor of "ure mathemati!s seems to ha+e little !onne!tion with

morality, other than the basi! integrity of +erifying, to the best of your ability, that you ha+e a!tually "rodu!ed a !om"lete "roof/whi!h sometimes re-uires mu!h more wor than a "artial one#

?ne ob+ious area where morals matter enormously in a""liedmathemati!s is statisti!s# 9>#= of statisti!s are made u" on thes"ot 6=# (eriously though, statisti!s ha+e a great deal of "ower intoday’s world and are used to ustify de!isions of all sorts# ;ereare a !ou"le of ways in whi!h morality is rele+ant#

The rst !all is to honesty in resear!hing and "resenting statisti!s#“There are lies, damned lies, and statisti!s 6B”, but statisti!s do notneed to be untruthful : there has been a substantial amount of wor done on what must be done to ensure that statisti!s are as truthfulas "ossible and an ethi!al a""roa!h to statisti!s must be informedby this wor # $f we are !reating fresh statisti!s it is im"ortant thatwe ta e !are, as best we !an, that what we "resent has beena!!urately resear!hed and is "resented without de!eit# Fore3am"le, what is the di eren!e between the two statements below.

0a2 “$ as ed, and 4ill reads the 4ible although Fred doesn’t”0b2 “E re!ent sur+ey found that BC of "eo"le read the 4ible”

The same fa!ts are "resented in both !ases, but the se!ond timethey are "resented in a way that im"lies far too mu!h# There are alot of te!hni!al ways to "re+ent this sort of falsehood – a goodbeginning is to ensure the number of "eo"le as ed, and how theywere sele!ted, is always in!luded#

There are more subtle ways – both deliberate and a!!idental – thatma e statisti!s misleading# Es another e3am"le, ta e the "hrase<“$n a re!ent sur+ey D out of 6C "eo"le "referred brand U !o ee”#

: Go donDt really eH&ect a footnote for this one; do yo ?5 Mar% #wain and Ben/amin 7israeli are )oth credited with saying this


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A Theology of Mathematics (

There are many dishonest ways to get a gure li e this# ?ne way isto ee" as ing grou"s of 6C "eo"le until, e+entually, you will nd by!han!e a grou" of ten "eo"le where eight of them ha""en to "referbrand U# The other grou"s that were sur+eyed are -uietlyforgotten# &orality here ob+iously means ee"ing all the resultsand not ust sele!ting the ones we ha""en to want#

Vnfortunately all those who !reate statisti!s are sub e!t to humansins of dishonesty, bias and laOiness and so we must ta e !are whenreading statisti!s and be ready to -uestion gures – whi!h mayre-uire some wor and !ourage on our "art# 'hene+er we read astatisti! we should be "re"ared, as *arrell ;u says, to ‘tal ba! ’to it< “Sou !an "rod the stu with +e sim"le -uestions, and by

nding the answers a+oid learning a remar able lot that isn’t so# 67 ”#(ometimes the bias may be ob+ious, or the gure gi+en with solittle su""ort that it means almost nothing, but often we ha+e to dosome wor to try and !he! whether what is !laimed is true#

$t is all too easy when "resented with statisti!s to sele!ti+elyremember and re:use the ones that !on rm our own beliefs and to"ass o+er the ones that !hallenge us# $ntegrity in this area meansfa!ing u" to statisti!s that we don’t li e rather than ignoring them/without forgetting that they !ould be wrong# $t is also tem"ting to

nd a !orrelation between two gures and then to um" too -ui! lyto de!iding one is the !ause and the other is the e e!t# $n "ra!ti!eit !an be +ery hard to identify what !auses what#

'hen using statisti!s from other sour!es it should be "art of ourintegrity to use them fairly, to in!lude any estimates of a!!ura!y,and to "ro+ide referen!es to the original sour!e of the gures#(omehow "eo"le seem to be eener to do this when -uoting anauthor, or using +erses from the 4ible, than when -uoting statisti!s#

The Mathematical mind – strengths and weaknesses

“The marble inde3 of a mind for e+er Woyaging through strange seas of thought, alone” :

illiam ordsworth

'ordsworth’s "i!ture !ould be des!ribing the ar!hety"almathemati!ian, someone la! ing in the normal human emotionsand wor ing "rin!i"ally alone# Elthough this is only "art of the"i!ture $ thin it does show some of the "otential dangers in ane3!essi+ely mathemati!al +iew of things# &athemati!s is a +eryabstra!t a!ti+ity, and abstra!tion has been de ned as ‘sele!ti+eforgetting’# Fi+e a""les and +e stones do not ha+e mu!h in!ommon, a"art from their “ +eness”, and to do mathemati!s with

( ow to =ie With 9tatistics & "


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A Theology of Mathematics *

them is to deliberately forget almost e+erything about the ob e!tsthemsel+es#

(ome mathemati!ians be!ome almost !om"ulsi+ely rational, tryingto use numbers to sol+e "roblems of human relationshi"s and dailyli+ing# $n the boo “E 4eautiful &ind” 0whi!h ins"ired thee"onymous lm2 we read the true story of a @ohn Nash, amathemati!ian who lost his mind to s!hiOo"hrenia – and thenagainst the odds re!o+ered it# ?ne of his most im"ortant results,and the one for whi!h he re!ei+ed the Nobel "riOe, was in gametheory whi!h is an attem"t to "ro+ide a systemati! theory ofrational human beha+iour by fo!using on the "laying of games#;owe+er, after his re!o+ery, he !omes to a life in whi!h “thoughtand emotion are more !losely entwined relationshi"s moresymmetri!al he has be!ome a great deal more than he e+erwas 6>”#

Qationality belie+es that there are reasons for the world and one ofthe strengths of a mathemati!al mind is the -uest for ndingmeaning in the world, e+en where it is not intuiti+e# ;owe+er this!an de!ay to be!ome the +iew that everything has a se!retmeaning# 0This also seems to be related to the numerologi!al use of the 4ible mentioned earlier#2 'hen @ohn Nash su ered froms!hiOo"hrenia he had to wrestle against beliefs that Qussians, oraliens, were !ommuni!ating with him through "atterns of numbersin news"a"ers, et!# that others !ould not nd# The tem"tation to

belie+e that there are hidden meanings nown only to the initiated– of whi!h we are of !ourse a "art : is a hard one to resist for many"eo"le#

The fth "ersonality ty"e in the nneagram model of "ersonality,the ‘?bser+er’, has a natural attra!tion to systems su!h asmathemati!s that “e3"lain uni+ersal "rin!i"les of intera!tion 6D”sin!e this ty"e of "erson tends to nd the outside world andemotions threatening and systems of thought !an "ro+ide !ontrolo+er this fear# The ‘$NT5’ tem"erament in the &yers:4riggss!heme des!ribes a similar ty"e of "erson#The strengths of this a""roa!h in!lude being able to remainob e!ti+e e+en when "ersonally in+ol+ed in a situation and beingable to dedu!e "rin!i"les from e3"erien!es# The wea nesses arety"i!ally some relu!tan!e to engage in relationshi"s and shynessabout !ontributing thoughts unless the "erson is !ertain they now‘almost e+erything’ about the sub e!t#

E %hristian +iew"oint of the wea nesses of this "ersonality ty"ewould want to stress that the 4ible re+eals both a God inrelationshi" – "erha"s most !learly in @esus’ "rayer in @ohn 6> : and

* A Bea tif l Mind &3888 0#he Cnneagram1; elen Palmer; &2"(


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also a God "re"ared to engage in this world’s troubles e+en to thee3tent of be!oming in!arnate as a human being# (o themathemati!al mind must be reminded that relationshi"s and

+ulnerability are +ery im"ortant !hara!teristi!s and to bede+elo"ed as "art of ea!h "erson’s !all to be!ome fully human#G#A# %hesterton’s adage “$f a thing’s worth doing it’s worth doingbadly 6J ” !an hel" "ro+ide !ourage to !ontribute e+en when the"erson feels inade-uate#

;ermann ;esse’s great no+el “The Glass 4ead Game” is a modern"arable of this – it des!ribes a grou" of "eo"le who see to e3"laine+erything through mathemati!al symbols and re"udiate emotionand strong relationshi"s# The hero of the story, @ose"h Ane!ht,defe!ts from this order, whi!h he leads, and e+entually gi+es u" hislife in sa!ri !e for a fellow human being#

?f !ourse, fortunately, not all mathemati!ians ha+e this "ersonalitybut it does seem that some of the same strengths and wea nesseswill a""ly to many of them# 'hether it is an e3"lanation for ta ingu" mathemati!s or an e e!t of studying it is hard to tell#

Why do so many people dislike Mathematics?

The !on+erse of a mathemati!al mind is innumera!y< “a failure todeal !omfortably with the fundamental notion of number and!han!e 8C”# This is -uite !ommon in our so!iety, and "eo"le e+en

ta e a "er+erse "ride in being bad at maths# This is both strange,sin!e few "eo"le ta e "ride in illitera!y or bad s"elling, andunfortunate, sin!e so mu!h of our so!iety relies on familiarity withnumbers#

'#'#(awyer wrote “The two main !onditions of su!!ess in any sortof wor are interest and !on den!e 86 ”# (adly in many !ases "eo"lela! not ust one but both of these !onditions when it !omes tomathemati!s# This is most ob+ious in s!hool where maths is often a"roblemati! sub e!t but in many !ases this la! is still "resent,albeit hidden, on!e formal lessons are o+er sin!e !onta!t withmathemati!s !an sim"ly be a+oided# &any "eo"le, highly!om"etent in many areas, fear mathemati!s#

E %hristian would want to begin with addressing this fear# “5erfe!tlo+e !asts out fear” 0@ohn = 6D2 a""lies to many things, and thisshould also in!lude mathemati!s# E lo+e of God and of his !reation!an stimulate interest in history, geogra"hy, s!ien!e and so on –in!luding mathemati!s# $t !an !ertainly gi+e us a theologi!alba! ing for loo ing at it#

<

#he 'ol m)ia World of otations; htt& www.)artle)y.com (( :< 22:


19/24

A Theology of Mathematics <

Qeason, whi!h "erha"s rea!hes its "urest form in mathemati!s, is agift of God and we !an !elebrate it# “&r God, This is Enna” is awonderful boo whi!h tells a story about Enna, a small girl withinsatiable !uriosity and able to see things from une3"e!ted "ointsof +iew# &aths for her was fun, a "uOOle, a bran!h of theology anda game for sharing# ;er enthusiasm and "layfulness is e3!iting butshe is unusual – few !hildren would be able to !o"e with so manyabstra!t !on!e"ts so young#

The use of abstra!t ideas is one of the "la!es where la! of interestand !on den!e arises# ?ne of the "roblems with mathemati!s isthat it does be!ome in!reasingly abstra!t throughout s!hooling# E!lassi!al model of !hild de+elo"ment designates the "re:teen "eriodas a ey "la!e for the rise of abstra!t thought# ;owe+er !hildren’sde+elo"ment is +ery indi+idual and so some will be earlier andsome later in learning to deal with abstra!tion# ?n to" of thisdi erent "eo"le ha+e more or less lo+e of abstra!tion e+en when!a"able of it# (o it !an be hel"ful to gi+e !on!rete e3am"les inmathemati!s to hel" "eo"le retain interest/ some tea!hing is betterthan others at "ro+iding this# $n addition, a wide range of su!he3am"les is hel"ful sin!e ea!h may demonstrate di erent fa!ets ofthe sub e!t# (omeone who has only seen e3am"les of !ounting withbeads or blo! s of wood may nd negati+e numbers really hard tounderstand – has anybody seen “–9” beads. The !on!e"t ofnegati+e numbers should be a lot easier to understand for someone

who has "layed with another e3am"le, su!h as distan!e, wherenegati+e numbers ha+e a more ob+ious use – if “=” means fourmiles north then “–=” has an ob+ious inter"retation as four milessouth#

5i! ing good e3am"les !an hel" ma e it fun as well – but it isim"ortant to a+oid !ontri+ed e3am"les# The traditional "roblem ofhow long it will ta e someone to ll a bath with both ta"s runningand the "lug out was dealt with swiftly in Enna’s world by “somemothers do ha+e ‘em – and they li+e 88 ”# This is an e3am"le of therigidity that sometimes dogs the tea!hing of mathemati!s – the!hild is e3"e!ted to sol+e the maths "roblem by using oftenuns"o en rules# These rules are needed to nd the right "roblemin mathemati!s within the story – there may be more than one wayof ‘sol+ing’ the story but only one will be mar ed as !orre!t# Therules may also a""ly inside mathemati!s itself – for e3am"le a "oortea!her might insist on their "referred way of doing long di+isione+en when the !hild already nows another way that also wor s#

+en something as sim"le as the “times tables” has more than oneway of doing it – the %hinese tea!h !hildren how to swa" thenumbers round when multi"lying so the smaller number always

!omes rst# This means they don’t need to learn the !ases where22 0Mr Fod #his is Anna1 & 23


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A Theology of Mathematics 2"

the rst number is bigger than the se!ond, so their tables are intotal ust o+er half the siOe of the 'estern ones#

Qigidity li e that abo+e !an lead to !hildren doing maths ‘by rote’and sim"ly mani"ulating symbols without really thin ing aboutwhat they mean# Es an e3am"le of this sort of error !onsider<

6)8 6)9 H 8)Bwhere the "erson doing the sum has added the ‘to"s’ and ‘bottoms’of the fra!tions together by misa""lying the rule for m"ltiplyingfra!tions to adding them#

E wrong answer li e this would not be a!!e"ted by someone with amental image of adding together half a "ie and a third of a "ie#

&aths !an be seen as a tool, a tool whi!h is useful for sol+ingthings# E small number of "eo"le ha+e the re+erse of this–e+erything is mathemati!s and so it is the tool# This a""roa!h !anseem e e!ti+e as in the "ro+erb “To a man with a hammer e+ery"roblem is a nail”# For someone li e this it is more im"ortant tostress other as"e!ts of life, things whi!h fall outside the area of"ure reason su!h as lo+e and intuition#

Vnli e some other sub e!ts mathemati!s is !ontinually building onwhat is already nown# This means that if something !auses"roblems the underlying reason might be an unlearnt lesson fromearlier on – re+isiting this sub e!t !an both hel" to regain!on den!e 0sin!e the "roblems here !an be sol+ed2 and then mean

the newer sub e!t matter suddenly seems to ‘!li! ’#

&an’s !reati+ity !omes out all o+er mathemati!s if it is allowed to#%hildren !ount on their ngers and often !ome u" with ways to addand subtra!t small numbers on their ngers# Kater on "i!tures andgra"hs !an hel" ma e numbers !ome to life# The “number line”

...___________________________________________... | | | | | | | | | | | -5 -4 -3 -2 -1 0 1 2 3 4 5

is a good way to see some of the se!rets of small numbers, andadding and subtra!ting them# E !al!ulator !an be a bit li e a ma"to e3"lore the bigger numbers – it!an "ro+ide games and "uOOles withnumbers# These do not ne!essarilyneed e3"laining – li e @esus telling agood "arable the ‘it!h’ of theunsol+ed "roblem !an get under thes in and lead the "erson to as theright -uestions for themsel+es#

A simple calculator game

$ 3 H 3 J 2 $ 33 H 33 J 22

$ 333 H 333 J 222


21/24


Numbers !an be seen as fun, as nown inhabitants of the mentalworld# (ome "eo"le ma e a few friends, some ma e a lot/ it seemsthe same with numbers# There is a well: nown story of the $ndianmathemati!ian Qamanu an whi!h shows his enormous lo+e ofnumbers# ;e was in hos"ital in ?3ford, and the nglishmathemati!ian ;ardy was +isiting him but found it hard to ndenough to"i!s of !on+ersation# ?n arri+al one day ;ardy startedthe !on+ersation by remar ing on the boring number of the !abthat he had !ome in – 6>8J# Qamanu an instantly re"lied that itwas not a boring number – it was the smallest number whi!h !anbe formed by adding two !ubes in two di erent ways# 0The twoways are 6 9 68 9 and J 9 6C 92#

$gnoran!e in mathemati!s is not treated in the same way as inother areas “Few edu!ated "eo"le admit to being !om"letelyuna!-uainted with the names (ha es"eare, *ante, Goethe, yetmost willingly !onfess their ignoran!e of Gauss, uler orKa"la!e 89 ”# This is not a good state of a airs and $ thin that thereare many things whi!h !an be done, some more e3"li!itly %hristianthan others, to en!ourage "eo"le of all ages but "erha"s"arti!ularly during s!hool years that mathemati!s need not beboring but !an e+en be enthralling#

23 nn meracy &**


22/24


onclusion$ ha+e tried to show that there are +arious lin s between theologyand mathemati!s but that, li e many su!h lin s, the "ri+atisation ofreligion in the 'est has greatly redu!ed the interest in and"er!e"tion of these lin s#

There are "hiloso"hi!al lin s between belief in God and thee3isten!e and "la!e of mathemati!s both ‘"ure’ and ‘a""lied’, and $ha+e tried to "ro+ide some theologi!al +iews on three main theoriesof what mathemati!s is# $ myself do not belie+e that any single oneof the three theories is a !om"lete e3"lanation of mathemati!s – itseems to me to !ontain elements of a "riori truth, em"iri!aldis!o+ery and !reati+ity# ;owe+er $ do belie+e that there areim"ortant things to be said from a theologi!al +iew"oint about

e+ery one of these a""roa!hes#There are some moral lin s, su!h as those in statisti!s wheretem"tations to laOiness or bias !an lead "eo"le into error#

Finally $ ho"e $ ha+e shown how some theologi!al basis to thesub e!t !an hel" "eo"le to lose their fear of mathemati!s andho"efully e+en !ome to en oy it#


23/24


Bi liography(ead trees <

Edams, *ouglas X %arwardine, &ar 06JJ62, /ast -han#e to 0ee ,5an, Kondon

4arrow, @ohn *# X Ti"ler, Fran @# 06JD72, The Anthropi#-osmologi#al rin#iple , ?V5, ?3ford

4arrow, @ohn *# 08CCC2, The oo* of 2othing , @onathan %a"e, Kondon4omford, Qodney 06JJJ2, The 0ymmetry of God , Free Esso!iation

4oo s, Kondon4rown, %olin 06J>D2, hilosophy and the -hristian 3aith , $W5, $llinois%arroll, Kewis 06JBD2, illow roblems 4 A Tangled Tale , *o+er

5ubli!ations, New Sor %rossley, @#N# X others 06J>82, hat is Mathemati#al /ogi#( , ?V5,

?3ford*a+ies 06JD92, The &dge of In)nity , ?V5, ?3ford

*ehaene, (tanislas 06JJD2, The 2"mber 0ense , 5enguin, Kondonastaway, Qob X 'yndham, @eremy 06JJD2, hy do "ses -ome inThrees( , Qobson, Kondon

Fynn 06J>=2, Mr God, This is Anna , Fount, KondonGardner, &artin 06JJC2, Mathemati#al -arnival , 5enguin, KondonGardner, &artin 06J7B2, Mathemati#al "55les and 6iversions ,

5eli!an, KondonGlei! , @ames 06JDD2, -haos , %ardinal, KondonGod 0 d2, The 7oly ible 82I9 , ;odder and (toughton, Kondon;esse, ;ermann 06JJC2, The Glass ead Game , ?wl 4oo s, New Sor ;ofstadter, *ouglas Q# 06JDC2, G;del, &s#her, a#h: An &ternal

Golden raid , 5enguin, Kondon;u , *arrell 06J>92, 7ow to /ie with 0tatisti#s , 5eli!an, KondonAiersey, *a+id X 4ates, &arilyn 06JD=2, lease rd &dition , Qoutledge,

Kondon

+idgway; Athelstan $ Cd K


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A Theology of Mathematics 2:

;errmann, Qobert E# “&odern &athemati!s : $ts Qelation to5hysi!al (!ien!e and Theology”,www#ser+e#!om)herrmann)math#htm

&!Kean, @e ery T# “&athemati!s and Theology< E %on+ersation”,www#stthomas#edu)!athstudies)6JJB)m!lean#htm

Warious authors at (t# Endrew’s Vni+ersity, “The &a!Tutor ;istoryof &athemati!s ar!hi+e”, www:ga"#d!s#st:and#a!#u )Ihistory)&athemati!ians)
http://www.serve.com/herrmann/math.htmhttp://www.stthomas.edu/cathstudies/1995/mclean.htmhttp://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/http://www.serve.com/herrmann/math.htmhttp://www.stthomas.edu/cathstudies/1995/mclean.htmhttp://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/

Documents

A Theology of Mathematics