Foundations of Mathematics or Mathematical Practice

8/10/2019 Foundations of Mathematics or Mathematical Practice

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198 JEAN PAUL V N BENDEGEM

ers seem to agree tha t models and theor ies u s e d b y t y p e Iphilosophers of mathematics are not in teres t ing . M t e r all, the i rapproach is a highly normative one, ignoring all aspects of realmathematical life. Either these models are cri t ic ized, o r t hey are

j u s t simply ignored. My point is tha t , al though acknowledgingtha t type I and type II researchers are in really different fields,the i r theories and models a r e (to use a fashionable term) to ala rge extent , commensurable. The basis allowing for t he poss ibil i ty of commensurabil i ty, i s cons t i tu t ed by t h e notion of a nartificial mathematician. In Type I research , the re a r e plenty ofart i f icial mathematicians around. The two most famous ones areHilbert s ideal mathematician and Brouwer s creat ive subjec t . InType II research, we obviously are talking about real mathematicians. t is therefore a natural quest ion to ask whether rea l and

art i f icial mathematicians are re la ted . And if so, can these p o ssible relations form t he background on which t compare t ype Iand type I I theor ies . As will be shown, t h e r e i s a gradua lt r ans i t ion from ext reme Type I theor ies to ext reme Type I Itheor ies .

The God l ike mathematician.

No doub t most working mathematicians assume s e t theory - i.e.

ZFC, Zermelo-Fraenkel s e t theory with Axiom of Choice - as thebes t ( type I) foundations around for mathematics a t the presen tmoment. The s tandard formulation consists of i) some version ofclassical f i r s t -o rder logic and (ii) the typica l se t - theore t i ca laxioms. In s u c h a foundational theory no mention i s made of amathematician. The s e t - t h eo r i s t will ( r ight ly) claim t h a t thelogical axioms and ru les mention only the logical s igns and these t axioms mention only se ts and operat ions on sets . However thefact tha t no propert ies of a mathematician are l is ted implicitly orexplicitly in the theory, does not imply tha t therefore the theorydeals only with mathematics a n d not with mathematicians. As t ra ight forward way to associate a mathematician with a mathematical t heory i s qui te simply to ask the following quest ion .Suppose the re is a being t h a t has the p r o p e r t y tha t i t knowseverything that the mathematical theory claims. What proper t iesdoes tha t being have? Note tha t t he quest ion i s a t r ivial one i fasked in a type I I approach. In t h a t case one s t r t s with t h emathematician (or t he mathematical community) and then s tudieshow the mathematician does mathematics . The ques t ion i s l e s st r ivial when asked in a t y p e I context . In order to clarify th iss t ra tegy, le t me p re sen t a f i r s t example.


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FOUNDATIONS AND PRACTICE 199

Most mathematicians would agree on t he following statements:(i) t h e r e i s someth ing l ike a mathemat ica l u n i v e r s e , (ii) t h i sun ive r se i s un ique and (iii) in i t al l mathematical problems a r ese t t led . The mathematicians ' t a s k is to d i scove r a n d c h a r t t h i s

universe , with t he knowledge t ha t a complete map is impossible .But suppose tha t t he re is a being with t he proper ty tha t i t hasfu l l knowledge of t he mathematical un ive r se . What ep i s t emicproper t i e s d o e s t h i s be ing possess? Two i m p o r t a n t p r o p e r t i e sfollow s t r a igh t away. Fi r s t , i t s knowledge i s s t rong ly complete .By (iii), all mathematical problems a re se t t led , t he re fo re g iven amathematical problem o r s ta tement A e i the r A is the case in t h emathematical un ive r se o r not-A i s t he case i n t h e mathemat ica lun ive r se . Second ly, b y ( i) a n d (ii), i t s knowledge i s weak lycomplete a s well. In mode l - theore t i c t e rms , (i) g u a r a n t e e s t h eexistence of a model, whereas (ii) guaran tees t he un iqueness o fth is modeL f t h i s hypothe t ica l mathematician has full knowledgeof t h i s model, th is obviously implies t he weak completeness . Fromt hese two proper t i e s , a th i rd , cruc ia l one i s der ived: t h i s be ingmust have t r u l y god-l ike powers The reason is quite simple. Foran epistemic s u b j e c t , to know a s t r o n g l y a n d weakly comple tef i r s t o r d e r t h e o r y, implies i t mus t h a v e a n a c t u a l i n f in i t e c a -pac i ty t s t o r e knowledge . I f t he capac i ty were r e s t r i c t e d topotent ial inf ini ty, t h e n undecidabil i ty resul ts become unavoidablea n d ful l knowledge o f t h e mathematical u n i v e r s e i s no l o n g e rposs ib le . E r r e t t Bishop summar ized h i s c r i t i q u e o n c las s i ca lmathematics when he wrote in h is [1976]2: .. c lass ica l mathematics c o n c e r n s i t s e l f with opera t ions t h a t c a n b e c a r r i e d o u t b yGod" and "I f God has mathematics of h is own t h a t needs to b edone , l e t him do i t himself" . In t e rms o f t h e above ana lys i s , a neven s t ronger s ta tement c a n be made: he i s the only one who c a ndo i t , he h a s to do i t himself .

t i s p e r h a p s i n t e r e s t i n g to p r e s e n t a n example o f t h eepistemic s t r e n g t h of t h i s God-mathematician (GM). Bishop h im

se l f i n t roduced in his (1985)3 t h e following example. Let (An) b ea b ina ry sequence . T h e n t h e GM will accep t t h e following p r i n -ciple, the so-cal led imited Principle o f mniscience (LPO): Ei the rt h e r e is a n n s u c h t h a t An 1, o r e lse An 0 fo r al l n. I f weassume t h a t GM c a n i ndeed dec ide t h i s prob lem, t h e n h e c a nso lve the fol lowing prob lem. Take a n unso lved mathemat ica lproblem, e.g. Fermat ' s Las t Theorem (FLT) o r Goldbach 's Conject u r e . Now c o n s i d e r t h e fol lowing sequence : i n t h e s e q u e n c e(An), An = 1 i f FLT i s provab le and An = 0 i f no t -FLT i sprovable . Obviously t h e s e q u e n c e (An) will cons i s t e i the r o f al l

ones, o r of al l zeros. However i t i s not obvious a t all which oneis t he case (at least for us , human mortals). Yet, if LPO holds, GM


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200 JEAN PA UL V N BENDEGEM

can decide the matter. Thus M can decide whethe r FLT o rnot-FLT holds. Note tha t for M mathematics ceases to be anin teres t ing enterpr ise , for the simple reason tha t eve ry th ing i sal ready known.

tis interest ing to note the close similarity between

GMandthe Demon of Laplace. In the v e r y same sense t h a t Laplace 's

Demon corresponds to the ideal physicis t , M corresponds to t heideal mathematician. For the Demon too, the universe ceases to bean in teres t ing place, a s i t holds no secrets . For the Demon too,time ceases to be real , j u s t as M l ives in a timeless realm. Onemight well wonder whether the paral lel breakdown of the Demonand M i s re la ted o r not.

he c o n s t r u c t i v i s t mathematician

f God-like mathematicians have li t t le o r nothing to do with us ,are we not best advised to scale this hypothet ical being down toour size? Basically, the re a re two opt ions : i) assume t h eexistence of a unique , mathematical universe , but deny one canhave a ful l knowledge of i t , a n d (ii) deny the exis tence of aunique mathematical universe al together. The second opt ioncorresponds roughly to the route taken by Brouwer, whereas thef irs t option is cur ren t ly explored in epistemic mathematics . The

crucia l d i ffe rence between these two approaches i s d i rec t lyl inked to the d i scovery-cons t ruc t ion dist inction. Are we w o nder ing around in a mathematical un ive r se wherein we discovermathematical theorems, or are we j u s t exploring a creat ion of o u rown making? I wil l not go into th is discussion - th is i s a qui teseparate topic - for i t is sufficient to note tha t in both cases theanswer will be t h e same to the following ques t ion: what i s theepistemic content of a hypothet ical mathematician whose capaci tyis limited to potential infinity? The answer is: what i s accessibleto t he mathematician on the bas i s of cons t ruc t ion and proof .Although p e r h a p s a t f i r s t s ight . th i s answer may seem a clearone, i t i s n e v e r t h e l e s s highly ambiguous . The his to ry of ( thephi losophy of) mathematics has shown us t ha t t he r e a r e manydifferent ways to sharpen this answer. In other words, there aremany cons t ruc t iv i s t mathematics CM) imaginable. However, as Iwill a rgue , they all share a se t of non-human proper t ies . Or, toput i t differently, M still has some dist inct ly type I proper t iesthat dist inguish i t clearly from a type II mathematician. Thus thedifferences do not appear to be essent ia l for the argumenta t ionpresen ted in th i s paper. Never theless , l e t me br ie f ly p re se n tth ree examples to i l lus t ra te the r i chness of the cons t ruc t iv i s t


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approach .For the intuitionist, CM I) knows A i f there is in principle, a

proof o r c o n s t r u c t i o n of A avai lab le , i .e. CM I) i s capab le o fproducing a proof of A or a construct ion for A. Obviously, CM I)

will re jec t LPO. But CM I) will a lso re jec t Markov 's pr inc ip leMP): I f An) is a b ina ry sequence such t ha t i t i s not t he case

for all n, tha t An = 0, t hen the re i s a n such t h a t An = 1. Thereason is tha t for the intuitionist not-A means tha t given a proofo r const ruc t ion of A t h i s proof o r construct ion can be ex tendedinto a proof of something absurd o r into an impossible c o n s t r u ct ion. Thus not -A s t a n d s for I f A, t hen absurd i ty . In the caseof MP, i f CM I) has shown tha t i t i s not the case t h a t for al l n ,An = 0, t hen he has only shown tha t the assumption t ha t all An =o leads to an absurdi ty. This given him no clue as to how t h e n,s u c h t h a t An = 1, can be found o r cons t ruc ted .

The Russ ian const ruc t iv is t , CM R), however accep t s MP. Thereason here is t h a t t he notion of construct ion is replaced by t henotion of algorithm in an extended sense. Cases such tha t , on t heone hand, one knows t h t t h e a lgor i thm will end on a c e r t a i ninput , but , on the o the r hand, no f ini te bound can be spec i f iedbefo rehand , r e accep ted . On t he o t h e r hand, CM R) will r e j e c tsome in tu i t ion i s t pr incip les , s u c h a s t he Fan Theorem (FT).

A th i rd vers ion is Bishop's cons t ruc t iv is t , CM B). This is t heweakes t ve r s ion , a s n e i t h e r MP nor e.g. FT a r e accepted . Themain advan tage of Bishop's construct ivism is t ha t i t is consis tentwith classical analys is (assuming t he consistency of the la t ter, ofcourse) . Both in tu i t ionis t and Russian cons t ruc t iv ism a r e e x t e ns ions of Bishop's const ruc t iv ism b u t both a re inconsis tent wi thclassical analys is . Fur thermore , intui t ionism is i ncons i s t en t wi thRussian construct ivism. Note too, t h a t these three approaches donot exhaus t t h e whole r a n g e of const ruct iv is t theor ies . I r e f e rthe r e a d e r to Beeson ' s [1985]5 fo r a n overview.

Let me now r e t u r n to t h e main l ine of the a rgument . What

proper t i e s of CM{x) - where x i s your favour i te b r a n d of c o ns t ruc t iv ism - ~ st i l l c learly of t y p e 1. Basically, t h e r e a r e twoa s p e c t s of pr ime impor tance . Actual ly, these two problems willa p p e a r only too famil iar to anyone acqua in ted wi th ep i s t emiclogic.

The f i r s t problem h a s to do with t r u e knowledge . I f CM(x)knows A, t h e n A m u s t b e t h e case . In o the r words , t h e c a s ewhere in CM x) knows A, b u t not -A i s a mathematical t heo rem,does not occur. I w l l r e fe r to th is principle as IE, the principleo f Immunity o Error I t is hardly necessary to a rgue tha t IE is

a typica l t ype I proper ty. Real mathematicians do bel ieve impossible t h ings from t ime to time. They did e.g. bel ieve s u c h n o n -


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sens ica l s ta tements a s ( . J - 1 r ' ~= -1. Moreover t h e y knew t h a tthese statements were nonsensical. The fact tha t these mathemat icians were fully aware of the absurdi ty involved, shows tha t anargument of t he following t ype does not apply. One might p r opose to weaken t he IE-pr inc ip le . Ins tead one could adop t theprinciple wIE (the leak Immunity Principle : I f one knows tha tone knows A t h e n A must be the case. In o t h e r words , j u s tknowing A does not guarantee the correctness of A. But, as said,tha t does not work. And i t is ra ther useless, to weaken wIE evenfu r the r, for what meaning could be given to t he s ta tement tha t'One knows t h a t one knows t h a t one knows t h t A without i tbeing the fact tha t one knows t ha t one knows tha t A ? Fur the rmore, they managed to deal with these absurdi t ies and to derivein te res t ing , impor tant and , above all , c o r r e c t mathematical

conclusions from them. To quote ano the r famous historical example, Berkeley did show convincingly tha t Newton's t r ea tmen tof infinitesimals was inconsistent , but most his tor ians will agreet h a t i t was a good t h ing for the development of mathematics ,analysis in part icular, tha t Newton largely ignored th is criticismand continued to develop this inconsistent theory. Actually, witht he adven t of n o n - s t a n d a r d analys is , one could a rg u e t h a tcons i s t en t ta lk abou t inf ini tesimals is , in fac t , poss ib le .

The second problem has to do with the pr incip le IC, t heprinciple o f Immediate Consequences. Suppose tha t CM(x) knows

A and t ha t B is a logical consequence of A. Then CM(x) must alsoknow B. IC is sure ly acceptable, for i t says noth ing but: i youknow A and there is a proof - according to your favouri te brandx - of B from A, then su re ly you must know B. But i f th i s i sacceptable, then i t has t he immediate, start l ing conclusion tha t iCM(x) knows A, t hen CM(x) must know l l logical consequencesfrom A. And th i s seems l e s s o r not t all accep tab le whendiscuss ing real mathematicians. Obviously no r ea l mathematicianhas such ins ight . I mentioned t the beg inn ing of th i s paper,ZFC as the foundations used today by most working mathematicians . Every mathematician who knows t hese axioms, t he re fo reknows all the logical consequences of t hese axioms, i .e. (s )heknows al l the theorems of s e t theory. One might ob jec t tha t forCM(x) to know t h a t B is a logical consequence of A, means t h a tCM(x) has a proof in principle o f B from A. Thus , to know alogical consequence, means quite simply to be able to p r e s e n t aproof when asked to do so. But th i s only increases the mystery:what kind of knowledge is th i s knowledge of proofs in p r i n -ciple ? One way o r another, th is must reduce to hav ing d i rec taccess to the mathematical un ive r se , where one c a n seewhether B is a logical consequence of A o r not. True , CM(x) can


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only see p a r t of t he un ive r se , never the les s , i t is somewhats tar t l ing to corne to the conclusion tha t GM and CM(x) a re closerr e l a t ives t h a n one might have imagined.

The f in i t i s t mathematician

How should we modify CM(x) such t h a t t he IE pr inc ip le and the1M principle no longer hold? In o rde r to r e j ec t the 1M pr inc ip le ,i t i s suff icient to replace the notion of proof in principle b y t henotion of rea l proof . A r ea l proof i s charac te r i zed b y the f ac ttha t i t should be recognizable as a proof b y a mathematician t h a tis bounded i n t ime and in space. Real proofs a r e sequences ofs igns wr i t ten in some l anguage o r other. t seems appropr i a t e tocall a mathematician t h u s l imited, a f in i te mathemat ic ian (FM).Actually, in t h i s case too, i t would be b e t t e r to s p e a k of FM(x)for, a s E r n s t Welti has shown in his excel lent , his tor ica l s t u d y,t he re a re many types of f ini t is t mathematics, s t r i c t o r otherwise ,a round . However, j u s t a s in the cons t ruc t iv i s t s c a s e , i t i s no tn e c e s s a r y to go into detai ls . I t is easy enough to see t h a t t h epresence of f ini te bounds must r e su l t in t h e violat ion of t h e 1Mpr inc ip le . F o r suppose , to keep matters simple, t h a t a n overa l lbound , say L, is def ined on t he l eng th of proofs . FM c a n o n l y

check and t h u s accep t o r r e j e c t proofs below a c e r t a i n u p p e rbound . S u p p o s e f u r t h e r t h a t FM has a c c e p t e d A a s a t heo remaf te r i n spec t ing the proof o f A, h a v i n g a l e n g t h l e s s t h a n L.Finally, suppose tha t FM h,as also accepted a proof of i f A, t h e nB , t h i s p roof equal ly having a l eng th l e s s t h e n L. t does no tfollow t ha t t he re fo re FM has to accept t h e proof o f B, s ince t heproof of B may have a l eng th l a rge r t h a n L. For t h e proof o f Bwill b e t he resu l t of t he concatenat ion of t h e proof of A a n d ofthe proof of if A then B . Thus i t is ra the r easy to re jec t t h e 1Mpr inc ip le . The IE pr inc ip le , however, i s a qui te d i f f e r e n t

problem.On the one hand, i t i s obvious t h a t the poss ib i l i ty of e r r o r

s h o u l d be allowed. The h i s to ry of mathemat ics p r e s e n t s a ni n t e r e s t i n g s t o r y of, what one could cal l , crea t ive mis takes .Precisely because mistakes were made, t h e mathematical commun i ty was ab le to see the nex t s t e p to t ake . But , o n t h e o t h e rhand, it i s no t clear a t all how one should proceed . What pr inc iples c a n b e formula ted about an ar t i f ic ia l mathemat ic ian t h a tallow t h i s be ing to make mistakes nd to l e a r n from them? Twoa l t e rna t ives p r e s e n t themselves. The f i r s t one i s to rep lace t he

u n d e r y ing con t rad ic t ion- f ree logic of mathemat ics b y a p a r a -cons i s t en t o r a dialectical logic. s The poss ib i l i ty i s t h e n allowed


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for to accep t tha t if FM knows A, then A is t h e case , tha t FMknows A , ye t tha t not-A is the case . However, this f i r s t a l t e rnative will sure ly have to be supplemented by some methods forrepa i r ing t he er ror. But, a s mus t be obvious , these methods

canno t be algori thms. f t hey were , i t would be suff ic ient toapply them each time a contradict ion ar i ses t h u s establishing amodified form of the IE-principle. f a n e r ro r occurs , i t can becalculated away . Thus heur is t ics have to be in t roduced. Inno

cen t though th i s conclusion may seem, i t i s of fundamenta limportance. So far, we always assumed t h a t whatever the ar t i f icial mathematician learns about the mathematical universe , i t i slearned t ru thful ly. At th is point , t h e poss ib i l i ty i s in t roducedtha t the artificial mathematician may be misled b y what he or shet h inks to be t he case in the mathematical universe . In o t h e r

words, th is universe i tself can no longer be used as a just if icato ry device. FM can no longer say, I believe o r I know A,because A is a mathematical fact , a n d , the re fo re , t r ue in t h emathematical universe. FM will have to look for o the r cri teria toconvince himself or herself , tha t he o r she knows A t ru thful ly.

The real individual mathematician.

The t ype I ph i losopher might r e m a r k a t th i s po in t t h a t i t is

clearly impossible to have mathematicians making er rors . f weres t r i c t ourse lves to FM-like mathemat ic ians , t h e n a n y proofprese nted will be a surveyab le proof because of the l imitsimposed on time and place resources. But a surveyable proof candecidably be found out to be er ro r- f ree o r not. f not, the er ro rcan be located and repaired. Why, then, do we need t he heur i st ics? The answer, in l l i t s simplicity, i s this : what most mathematicians write and r ead most of the t ime are not proofs in t heformal sense of the word. They are , what I have called elsewhereproof ou tlines. 9 That is, what are presented , are the major s teps

in the proof. The mathematician who wri tes the proof, t h e r e b yassumes t ha t a t ra ined mathematician with suff icient ly knowledgeof the part icular mathematical field t h e proof is about , is capableto fill in t he missing steps. The problem i s no t tha t mathematicians should be accused of laziness o r sloppiness, t he mat ter i squi te simply t ha t the demand of, formally speak ing , c o r r e c tproofs, is a n impossible one. I f something has been made clearby Principia Mathematica then sure ly, i t i s the fact tha t tha t i snot the way to do mathematics . f e r r o r s occur in a proofoutline, they do because the mathematician assumed wrongly tha ta par t icular s t ep could be filled in. Therefore , e r ro r s are l ikely


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FOUNDATIONS ND PRACTICE 2 5

to occur - the history of mathematics tells u s BOlO - and heur ist ics are needed to r e p a i r t h e s e er ro rs .

To i l lustrate this thesis , let me present th ree heurist ics tha thave been f requen t ly employed in mathematics to s e a r c h for

r r o r s and to repa i r the damage i f e r ro r occur red .The f i rs t example is well-known from Lakatos' brill iant s tudy

Proofs and Refutations on Euler 's conjecture, V-E+F = 2, i.e. thes ta tement t h a t , given a polyhedron, the number of ver t i cesminus t h e number o f e d g e s p lus the number of faces a lwaysequals two. The t h r ee -pa r t heur is t ic Lakatos a r r i v e s at , i s thefollowing:

Rule 1. I f you have a conjecture, se t out to prove i t and tore fu te it. I n s p e c t t h e proof carefu l ly to p r e p a r e a l is t of nont r ivial lemmas (proof-analysis) ; find counterexamples both to thecon jec tu re (global counterexamples) and to the suspec t lemmas(local counterexamples) .

Rule 2. I f you have a global counterexample disca rd yourconjecture, add to your proof-analysis a suitable lemma tha t w l lbe r e f u t e d b y the counterexample , and replace the d i sca rdedconjecture by a n improved one t ha t incorporates tha t lemma a s acondition. Do not allow a refutat ion to be dismissed a s a monster.Try to make al l h idden lemmas explicit .

Rule 3. I f you have a local counterexample , check to see

whetheri t

is not also a global counterexample.f

i t is , you caneas i ly apply Rule 2. 11The second example concerns a heurist ic tha t I have labelled

confining inconsistencies 12 Especially, in the pre-Newtonian andpre-Leibniz ian per iod in the development of analys is , manymathematicians - Giles Persone de Roberval, John Wallis, Fran


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The real social mathematician.

Indiv idual heur is t ics , however, do not te l l t he whole s to ry. Ofcourse , i t would be an easy way out to claim t ha t the mathematician, a s p a r t of t he mathematical community, on ly ex i s t s s amathematician in v i r tue of his or he r membership of tha t communi ty. But, t h a t does no t explain why, i f one is i n t e r e s t e d inunders tand ing the dynamics of mathematical change (as t y p e I Iphilosophers are) , social elements should be t aken into accoun t .As in the case of t he use of heurist ics , I believe t he re t be a tl eas t two major a rguments in s u p p o r t of t h i s thes i s .

The f i r s t a rgument r e l a t e s to a point , a rg u e d for in t h epreced ing pa ragraph . Mathematicians do not wri te proofs , b u tproof-out l ines . Proof-outlines do not have a s t anda rd form in thesense tha t formal proofs do. They a re not a sequence of fo rmulas , where each formula i s e i the r a n axiom or the resu l t of theapplication of a derivat ion ru le on formulas a l ready occur r ing inthe l is t . Ins tead they can t ake many d i f f e r e n t forms. As a nexample, compare these two proofs of t he same theorem, namelythe fundamenta l theorem of ar i thmet ic . (Two minor no ta t iona lchanges have been in t roduced. Ins tead of us ing subsc r ip t s , t henota t ion a b i s u s e d to i nd ica te a with s u b s c r i p t b. a to t h epower b is wri t t en [a,b])18

Version 1. To prove t he r e su l t , note f i r s t t h a ti

a prime pdivides a product mn of natura l numbers then e i ther p divides mo r p d iv ides n. Indeed i p does not d iv ide m t h e n (p,m) = 1whence there exist in tegers x, y such tha t px my = 1; t h u s wehave pnx: mny = n a n d hence p divides n. More genera l ly weconclude t h a t i p divides n '1n '2 .. n 'k then p divides n 1 forsome 1 Now suppose tha t , a p a r t from the fac tor iza t ion N =[p '1,J 1] . . . [p k, j 'k] d e r i v e d above , t h e r e i s a n o t h e r decomposition and tha t p ' is one of t he primes occurr ing there in . Fromt he preceding conclusion we obtain p ' = p 1 for some L Hence wededuce tha t , i f t he s t anda r d f ac to r i za t ion fo r NIp ' i s un ique ,t h e n so also i s tha t for N The fundamenta l theorem follows b yinduct ion .

Version 2. First, N must have a t least one representa t ion , N =[p 1,a 1][p 2,a '2] ... [p n ,a n] (1). Let a be the smallest divisorof N which is > 1. t must be prime, since i not, a would have adivisor > 1 a n d < a. This divisor, < a , would divide N a n d th i sc o n t r a d i c t s the def in i t ion of a. Write a now a s p 1, a n d t h equotient N/p ' l , as N '1. Repeat the process with N 1. The processmust terminate, s ince N > N l > N 2 > > 1. This genera tes Eq.(1). Now i t he re were a second representa t ion , by the corol laryof Theorem 6, each p 'i must equal some q i , since p iIN. Likewise


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208 JEAN PAUL VAN BENDEGEM

each q 'i must equal some p L Therefore p 'i q 'i and m n. fb 'i > aAi, divide [p 'i,a 'i] into Eqs. 1) and (2). (Note: Eq. (2) i sthe second representat ion: [q l,b '1][q 2,b '2] . [q n,b n]). Thenp i would divide the quotient in Eq. (2) but not in Eq. 1). Thiscontradic t ion shows

tha t a i = bi.

These two proofs a r e suff ic ient ly di ffe ren t to w a r r a n t thein t roduct ion of the notion of s ty le in mathematics. t is not anexaggera t ion to claim tha t a mathematician develops a cer ta int ype of style and tha t one can identify him or her by it. t alsoimplies - and here the social element en te r s the pic ture - tha tmathematicians shar ing the s a m e ' s ty le will u n d e r s t a n d eacho t h e r be t t e r. After all, they do speak t h e same language, or,should one ' say, the same mathematical dialect . Seen from th i sperspect ive , the Bourbaki projec t , apa r t from i t s mathematical

content , was an equally important pro jec t in i t s proposal for anew mathematical style. The Bourbaki volumes asp i red to be anew foundat ions of mathematics , b u t a t the same time, t h e ycons t i tu ted a manual of s ty le for i t .

The second a rgument has to do with a recent , in ter twined,two-fold development or, better, change in mathematical pract ice.

Long proofs a r e not uncommon in mathematics , a s is wellknown. However, i t is a qui te r e cen t phenomenon t h a t someproofs t u r n out to be so long tha t an individual mathematician isincapable of survey ing i t . The exemplar in th i s case i s the

classification theorem of finite groups , est imated a t about 15.000pages . 19 Ins t ead one can only claim t h a t t h e proof i s social lysu rveyab le , no t indiv idual ly surveyab le . Mathematician A haschecked pa r t X and mathematician B p a r t Y, and put t ing the i reffo r t s t oge the r, t h e y come to t h e conclus ion t h a t t h e wholeproof i s cor rec t . Neither A nor B individual ly c a n make th i sclaim, but together they can. Or, in other words, the proof as amathematically accepted proof ex i s t s only on the social level .Hence, the basic uni t to consider is not the individual mathemat ic ian, bu t t he mathematical community.

The re la ted pa r t has to do with computer-proofs . Since the'drama' of the four-colour theorem, i t has become apparen t t h a tthe presence of the computer a s a symbol manipulat ing device,mus t have i t s effec t on mathematical pract ice . f p a r t of t h eproof has been carr ied out by computer, and the calculations aretha t cumbersome and in t r ica te tha t nei ther a human mathematician, nor the mathematical community, i s l ike ly to check i t indetail , a r e we t h e n in a posi t ion to accep t the proof? f one i stempted to answer ' yes ' to th is ques t ion , t h e n one must accep tthe conclusion tha t 'prCXJf', as classical ly unders tood, i s not t h eon ly way to es tab l i sh new mathematical resu l t s . This i s rea l ly


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go ing beyond heur i s t i c s , fo r heur i s t i c s point the way to aclass ica l proof , whereas , in t he computer case , th i s compute rcalculat ions a r e the b e s t available. The problem is not a r e c e n tone. Mark Steiner in his Mathematical Knowledge 20 already made a

case for o the r methods , bes ides mathematical proof (in t hec lass ica l s e n s e , i.e. u p to the rea l individual mathematician(RIM, to es tab l i sh the t r u t h of a mathematical proposi t ion .Pe rhaps one is not inclined to follow along such a route, bu t , i funders tand ing mathematical pract ice is t he goal, t hese a s p e c t swill have to b e t aken in to account .

Although the rea l social mathematician RSM) is not t he e n dof the continuum - sure ly we should go fur ther and consider therea l social mathematician in society a t la rge RSLM) - I hope tohave made a conv inc ing case for t he idea t h a t GM and RSM,a l though wor lds apar t , a r e re la ted .

t en ta t ive conclus ion

The sub jec t of th is paper, basically, was to answer th is question:f X is any t y p e of mathematician, then for X to know A where A

is a mathematical s ta tement , ~ e n sexact ly what? We have p r o -g r e s s e d from the God-l ike mathematician, GM fo r whom t h eanswer was qu i t e s t r a igh t fo rward . For GM to know A, is simplyequiva lent to A being t r u e in t he unique mathematical universe .Along come t h e cons t ruc t iv i s t s who want to sca le down GM tosome kind of ideally real mathematician CM x). CM x) knows A i fCM x) has a proof o r cons t ruc t ion avai lable of A, in pr inc ip le .Replacing t h e in pr incip le p r t b y ac tua l ly , CM x) is t r n s -formed into some kind of f ini t is t mathematician, FM. But - re ly ingon some wel l -known arguments abou t epistemic logic - s i tt u r n s out , e v e n FM is still a highly idealized being. Establishinga l ink with t h e rea l - indiv idual o r socialized - mathematician

must force u s in to in t roduc ing elements in t h e s to ry t h t onewould perhaps not expeCt in the mathematical context: heur is t ics ,fai lure and e r r o r (and therefore revision), s ty le (and the re fo reaes the t i c s ) a n d t he social coherence of the mathemat ica l community.

Although i t is clear tha t the t ransi t ion from GM to RSM, i s agradua l one, t h e differences between the extremes of t he c o nt inuum a r e enormous . But t h a t has been known all along. T h emore i n t e res t ing p a r t is tha t i t s gradual . Seen from t h e v i e wpoint of the RIM or the RSM, FM, CM x) and GM are to be seen as

increas ing, t h e r e b y, simplifying and helpful abs t rac t ions . Notetoo t h a t all mathematicians ment ioned , art if icial and o the rwise ,


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a r e only snapsho t s from an immense gal lery of possibi l i t ies .Perhaps the reader wonders why I am so ins is tent on th i s

point . Basically, the re a re two reasons. The f i rs t reason has todo with our unders t and ing of RIM and RSM. As said, FM, CM x)and the like, may t u r n out to be v e r y helpful fictions, in muchthe same way, t ha t propositional logic is a quite interest ing, ye thighly fictional logic. In th i s paper, I hope to have made thepoint that , e.g., epistemic logic is really worth while to look into.Thereby, I am also claiming tha t the p r o j e c t to formulate atheory of mathematical pract ice , will benef i t from t h e use offormal tools such as epistemic logic. In t h e bes t of cases , i tshould be possible to formulate theorems abou t the n a t u r e ofmathematical practice. As must be obvious, th is position is o tsimilar to Wittgenstein s at t i tude. Without going into details, one

example may suff ice to make the dis t inc t ion c lear. For Witt-genste in , the social coherence of the mathematical community,does not need to be explained for. t j u s t happens to be tha tway, and i t would not make sense to ask a mathematician why heo r she is willing to accept the verdict of his o r her colleague asfinal. 21 In th is case, I want to f ind a rguments t h a t explain t he(necessi ty of the) coherence. One argument mentioned sure ly istha t , i f A wants to check the proof of B, i t increases efficiency,i f A and B share the same mathematical s ty le . But t h e l a t t e rfea tu re i s prec ise ly a n impor tant element t h t con t r ibu tes to

social coherence . t is , t the same t ime, a r e f u s a l to l e t thehis tory, psychology, sociology and economy of mathematics d e -genera te into a loose collection of in teres t ing , anecdotal , t h e r e -fore accidental, bi ts and pieces. The second reason, re la ted t othe f i r s t one, is tha t i t is still possible t o maintain t he existenceof a unique mathematical universe while holding t h e view t ha tthe way mathematics i s done, is b e s t desc r ibed us ing a RIM o rRSM type of model. What is said here, will sound only too familiarto any philosopher. The only thing t ha t I am claiming, is tha t theminimal real is t position holds for mathematics as well. t does notfollow - and I emphasize this point most s t rongly - tha t taking asociological, psychological o r whatever point of view, implies theimpossibility of the existence of something l ike, th mathematics.True, one might argue, tha t a separate ent i ty such as the uniquemathematical universe, is not called for, but, a s must e clear, a nappeal to the pract ice of mathematics to deny i t s existence, doesnot c a r r y the force many au thors expect o r want i t to do.

t would be wishful thinking to believe t ha t t he above pleawil l br ing together type I and type II philosophers of mathemat-ics. Perhaps they do not need to be brought together physically.

f a common language is available - and a modest proposal for a


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cand ida te is ske tched ou t in th i s paper - i t will be the re forwhoever wants to use it. Now all too often, a false dichotomy isd rawn.

Bevoegdverklaard Navorser NFWO - Ri jksun ive rs i t e i t GentVrije Univers i te i t Brusse l

A f i r s t d r a f t of th is p a p e r was presen ted a t the Center fo rPhilosophy of Science, Universi ty of Pi t tsburgh, October 1988 oninvi ta t ion of e r r y Massey. This vers ion has benef i ted f romcri t icisms bo th from J e r r y Massey, Ken Manders and the o the rfellows of t h e Center p re s e n t a t t h e moment. Especial ly KenManders cr i t icisms were important but , taken seriously (as theyshould), they cons t i tu ted a new r e s e a r c h program.

NOTES

1 An excellent overview of the l i t e r a tu r e i s p rov ided inThomas Tymoczko (ed), New Direct ions in the PhilosopllY o fMathematics Birkhauser, Stu t tga r t , Boston, 1986. A few a d -dit ional impor tan t works not ment ioned are: David Bloor,Knowledge and Social I m a g e r y RKP London, 1976; SalRestivo, The Social Relat ions o f Phys ics Mysticism a n dMathematics Reidel Dordrecht, 1983 and Eric Livingston, TheEthnomethodologica l Foundations o M a t h e m a t i c s RKP,London, 1986 (for a cri t ical review of th is work, see DavidBloor, The Living Foundations of Mathematics , Social Studieso f Science vol. 17 2 1987, pp.337-358. Also: Phil ip Kitcher(ed.), Philosophie des Mathemat iques-Phi losophy of Mathematics Revue In ternat ional de Philosophie, 4/1988, 167.

2 Er re t t Bishop, Foundat ions o Cons t ruc t ive A n a l y s i s

McGraw-Hill, New York, 1967, p.2.3 Erre t t Bishop, Schizophrenia in Contem porary Mathematics ,in: Murray Rosenblatt (ed.): Erre t t Bishop: Reflections on im

and His Research AMS Prov idence , Rhode Is land, 1985,pp.1-32. See also Douglas Bridges a n d Fred Richman, Variet ies o Construct ive Mathematics Cambridge UP, Cambridge ,1987 for a n excel lent and in t roduc tory discussion and p r e -senta t ion o f pr incip les s u c h a s LPO.

4 For the epistemic mathemat ics approach , see Stewar tShapiro (ed.) , In tens ional M a t h e m a t i c s North-Hol land,

Amsterdam, 1985. Related ar t ic les are : Nicolas D Goodman,The Knowing Mathematician , Bynthese 60 1, 1984 pp.21-38;


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Stewart Shapiro, On the Notion of Effectiveness , History andPhilosophy o Logic 1, 1980, pp.209-230. A historical ly interes t ing con t r ibu t ion i s Kur t Gadel, An In te rpre ta t ion ofthe Intuitionistic Sentential Logic , in: Jaakko Hintikka (ed.),The Philosophy o a tllema t ics , Oxford UP, Oxford, 1969,pp.128-129.

5 Michael J. Beeson, Foundations o Const ruc t ive MathematicsSpr inger, Heidelberg, 1985, esp. chap te r III , pp.47-57.

6 Classics in the field of epistemic logic are: Jaakko Hintikka,Knowledge and the Known: Historical Perspect ives in Epis te-mology Reidel, Dordrecht , 1974: b y the same au thor, TheIn tent ions o In tent ional i ty and Other New Models for Mo-dalities Reidel, Dordrecht, 1975 and Karel Lambert (ed.), TheLogical Way o Doing Things Yale UP, New Haven, 1969.

7 Erns t Welti, ie Philosophie des s t r i k t e n Finit ismus. E n t -wicklungstheore t i sche und mathematische Unte r suchungeni iber Unendlichkeitsbegriffe in Ideengeschichte und heut igerMathematik Peter Lang, Bern, 1987. Also: Erns t Welti, ThePhilosophy of Str ic t Finitism , Theoria II, 5-6, 1987, pp.575-582.

8 For an overview of paracons i s t en t and dialectical logic, seeRichard Routley and Graham Pr ies t (eds.), Essays on Para-consis tent Logic Philosophia Verlag, Miinchen, to appear.Also: Ayda 1. Arruda, A Survey of Paraconsistent Logic , in:

A.I. Arruda, R. Chuaqui and N.C.A. da Costa, MathematicalLogic in Latin America North-Holland, Amsterdam, 1980,pp.1-41.

9 Jean Paul Van Bendegem, Non-Formal Proper t i e s of RealMathematical. Proofs , in: Arthur Fine and Jar re t t Leplin, PSA1988, Volume One, PSA, East Lansing, pp.249-254.

10 Besides the example of Euler s con jec tu re mentioned in thetext , e r r o r s have occur red in Fermat s Las t Theorem - seemy Fermat s Last Theorem seen as an Exerc ise in Evolut ionary Epistemology, in: Werner Callebaut and Rik Pinxten

(eds.), Evolutionary Epistemology Reidel, Dordrecht , pp.337-363 - in The Four-Colour Theorem - Ian Stewart in his TheProblems o Mathematics Oxford UP, Oxford, 1987, p.111speaks of a comedy of er ro rs - and n the Goldbach conjectu re , in the Riemann hypothesis , in the Bieberbach c o njecture .

11 Imre Lakatos, Proofs and Refutat ions. The Logic o Mathe-matical Discovery ed i t ed b y John Worrall and Elie Zahar,Cambridge UP, Cambridge, 1976, p.50.

12 Jean Paul Van Bendegem, Dialogue Logic and Complexity ,Prepr in ts 16, Ghent, 1985.


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13 See Carl B. Boyer, The History o the Calculus and i t sConceptual Development, Dover Books, New York, 1959 forhistorical detai ls . Wallis proof dates from 1656-57 and i s tobe found in his Opera M8 thematica (see Carl B. Boyer, Opecit . , pp.168-174).

14 I t would be a mistake to believe tha t the pract ice of conf ini n g inconsis tencies i s typical for the mathematical per iodpreced ing the age of r igour. Dirac s de l t a - func t ion i s asimilar, qui te r e cen t case .

15 See J . Vers luys , 96 bewijzen voor he t theorema vanPythagoras , A. Vers luys , Amsterdam, 1914. (96 Proofs oPythagoras Theorem).

16 Fermat s Last Theorem seen as an Exercise in EvolutionaryEpis temology , see note 10.

17 For a n excellent overview, see Matthew L. Ginsberg (ed . ) ,Readings in Nonmonotonic Reasoning, Morgan Kaufmann, LosAltos, 1987.

18 The i r s t vers ion is to be found in Alan Baker, A ConciseIn t roduc t ion to the Theory o f Numbers , Cambridge UP,Cambridge , 1984, p.4.. The second vers ion i s t aken f romDaniel Shanks , Solved and Unsolved Problems in NumberTheory, Chelsea Publ . eYe New York, 1978 (second edit ion),pp.6-7. The corol lary of Theorem 6 s ta tes tha t I f a prime p

divides a produc t of numbers, i t must divide a t leas t one ofthem.19 See Daniel Gorenstein, Classifying the Finite Simple Groups ,

Bulletin o the AMS, 14, 1986, pp.1-98 .20 Mark Ste iner, Mathematical Knowledge, Cornell UP, I thaca ,

London, 1975.21 This sho r t excurs ion in to the Wittgens te in ian f ield i s no t

meant to t ake in to account all the in t r i ca te deta i l s a n dcomplexities one f inds in Crispin Wright, Wittgenstein on theFoundat ions o Mathematics, Duckworth, London, 1980 a n d

Stuar t G. Shanker, Wittgenstein and the Turning-Point in thePhi losophy o Mathematics , Croom Helm, London, 1987.

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