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QA 9.[SchizophreS c h i z o p h r e n i ai nc o n t e m p o r a r yE r r e t tm a t h e m a t i c sA .B i s h o p' iUniversity of California, San DiegoLa J o l l a , C a l i f o r n i aDistributed in conjunction w i t hthe Colloquium Lectures given a t theUniversity of M o n t a n a , Missoula, M o n t a n aAugust 2 1 — 2 4 , 1973,seventy-eighth summer meeting of theAmerican M a t h e m a t i c a l Society.UNIVERSITY OF TOLEDO LIBRARIESparLstlc

UNIVERSITY OF TOLEDO LIBRARIESInformally distributed manuscripts and articles should be treatedas a personal communication and are not for library use. Referenceto the contents of this publication should have the prior approvalof the author.Copyright 1973 by th« American Mathematical SocietyPrinted in the United States of America,***j«s.,:- - - - - Jauaa&fr.i.

QA 9.1Schizopttre3 28;SCHIZOPHRENIA IN CONTEMPORARY MATHEMATICSbyErrett BishopDuring the past ten years I have given a number of lectures on the subjectof constructive mathematics.My general impression is that I have failed tocommunicate a real feeling for the philosophical issues involved.here today, I still have hopes of being able to do so.Since I amPart of the difficultyis the fear of seeming to be too negativistic and generating too much hostility.Constructivism is a reaction to certain alleged abuses of classical mathematics.Unpalatable as it may be to have those abuses examined, there is no other way tounderstand the motivations of the constructivists.Brouwer's criticisms of classical mathematics were concerned with what Ishall refer, to as "the debasement of meaning".His incisive criticisms wereone of his two main contributions to constructivism.(His other was to establisha new terminology, involving a re-interpretation of the usual connectives andquantifiers, which permits the expression of certain important distinctions ofmeaning which the classical terminology does not.)The debasement of meaning is just one of the trouble spots in contemporarymathematics.Taken all together, these trouble spots indicate that something islacking, that there is a philosophical deficit of major proportions.What it isthat is lacking is perhaps not clear, but the lack, in all of its aspects constitutes a syndrome I shall tentatively describe as "schizophrenia".sarl.stlc4 260

-2-One could probably make a long list of schizophrenic attributes of contemporary mathematics, but I think the following short list covers most of theground:rejection of common sense in favor of formalism; debasement of meaningby wilful refusal to accomodate certain aspects of reality; inappropriatenessof means to ends; the esoteric quality of the communication; and fragmentation.Common sense is a quality that is constantly under attack.supplanted by methodology, shading into dogma.It tends to beThe codification of insight iscommendable only to the extent that the resulting methodology is not elevatedto dogma and thereby allowed to impede the formation of new insight.Contemporarymathematics has witnessed the triumph of formalist dogma, which had its inceptionin the important insight that most arguments of modern mathematics can be brokendown and presented as successive applications of a few basic schemes.The ex-perts now routinely equate the panorama of mathematics with the productions ofthis or that formal system.of symbols.Proofs are thought of as manipulations of stringsMathematical philosophy consists of the creation, comparison, andinvestigation of formal systems.Consistency is the goal.In consequencemeaning is debased, and even ceases to exist at a primary level.The debasement of meaning has yet another source, the wilful refusal of thecontemporary mathematician to examine the content of certain of his terms, suchas the phrase "there exists".He refuses to distinguish among the differentmeanings that might be ascribed to this phrase.meaning it has for him.Moreover he is vague about whatWhen pressed he is apt to take refuge in formalistics,declaring that the meaning of the phrase and the statements of which it forms apart can only be understood in the context of the entire set of assumptions andtechniques at his command.Thus he inverts the natural order, which would beto develop meaning first, and then to base his assumptions and techniques on the

-3-rock of meaning.Concern about this debasement of meaning is a principal forcebehind constructivism.Since meaning is debased and common sense is rejected, it is not surprisingto find that the means are inappropriate to the ends.Applied mathematics makesmuch of the concept of a model, as a tool for dealing with reality by mathematicalmeans.When the model is not an adequate representation of reality, as happensonly too often, the means are inappropriate.One gets the impression that someof the model-builders are no longer interested in reality.become autonomous.Their models haveThis has clearly happened in mathematical philosophy:themodels (formal systems) are accepted as the preferred tools for investigating thenature of mathematics, and even as the font of meaning.Everyone who has taught undergraduate mathematics must have been impressedby the esoteric quality of the communication.It is not natural for "A impliesB" to mean "not A or B", and students will tell you so if you give them the chance.Of course, this is not a fatal objection.The question is, whether the standarddefinition of implication is useful, not whether it is natural.The constructivist,following Brouwer, contends that a more natural definition of implication would bemore useful.This point will be developed later.One of the hardest concepts to, communicate to the undergraduate is the concept of a proof.concept is esoteric.Most mathematicians, when pressed to say what they mean bya proof, will have recourse to formal criteria.by contrast is \/eryThe constructive notion of proofsimple, as we shall see in due course., perhaps more troublesome, is the concept of existence.tWith good reason—theEqually esoteric, andSome of the problemsassociated with this concept have already been mentioned, and we shall return to, the subject again.Finally, I wish to point out the esoteric nature of the

-4-classical concept of truth.As we shall see later, truth is not a source oftrouble to the constructivist, because of his emphasis on meaning.The fragmentation of mathematics is due in part to the vastness of the subjectbut it is aggravated by our educational system.A graduate student in pure mathe-matics may or may not be required to broaden himself by passing examinations invarious branches of pure mathematics, but he will almost certainly not be requiredor even encouraged to acquaint himself with the philosophy of mathematics, itshistory, or its applications.We have geared ourselves to producing researchmathematicians who will begin to write papers as soon as possible.This anti-social and anit-intellectual process defeats even its own narrow ends.Thesituation is not likely to change until we modify our conception of what mathematics is.Before important changes will come about in our methods of educationand our professional values, we shall have to discover the significance of theoremand proof.If we continue to focus attention on the process of producing theorems,and continue to devalue their content, fragmentation is inevitable.By devaluation of content I mean the following.To some pure mathematiciansthe only reason for attaching any interpretation whatever to theorem and proof isthat the process of producing theorems and proofs is thereby facilitated.content is a means rather than the end.For therOthers feel that it is important to havesome content, but don't especially care to find out what it is.Still others, forwhom Godel (see for example [16]) seems to be a leading spokesman, do their bestto develop content within the accepted framework of platonic idealism.Onesuspects that the majority of pure mathematicians, who belong .to the union of thefirst two groups, ignore as much content as they possibly can.If this suspicionseems unjust, pause to consider the modern theory of probability.of an event is commonly taken to-be a real number between 0 and 1.The probabilityOne might

-5-n a i v e l y expect t h a t t h e p r o b a b i l i s t s would concern themselves w i t h t h e computationof such r e a l numbers.I f s o , a q u i c k l o o k a t any one o f a number o f modern t e x t s ,f o r i n s t a n c e the e x c e l l e n t book o f Doob [ 1 4 ] , should s u f f i c e t o disabuse him o fthat expectation.F r a g m e n t a t i o n ensues, because much i f n o t most o f t h e t h e o r yi s useless t o someone who i s concerned w i t h a c t u a l l y f i n d i n g p r o b a b i l i t i e s .Hew i l l e i t h e r develop h i s own semi-independent t h e o r i e s , o r e l s e work w i t h ad hoctechniques and r u l e s o f thumb.I do n o t c l a i m t h a t r e i n v o l v e m e n t o f t h e p r o b -a b i l i s t s w i t h the b a s i c q u e s t i o n s o f meaning would o f i t s e l f r e v e r s e the processof fragmentation o f t h e i r d i s c i p l i n e , only t h a t i tI n r e c e n t y e a r s a small number o f c o n s t r u c t i v i s t si s a necessary f i r s t(see [ 3 ] ,[9 ] ,step.[ 1 0 ] , [11 ] ,[ 1 2 ] , [ 2 3 ] , and [ 2 4 ] ) have been t r y i n g t o h e l p the p r o b a b i l i s t s t a k e t h a tWhether t h e i r e f f o r t s w i l lstep.u l t i m a t e l y be a p p r e c i a t e d remains t o be seen.When I a t t e m p t t o express i n p o s i t i v e terms t h a t q u a l i t y i n which contemporarymathematics i s d e f i c i e n t , t h e absence o f which I have c h a r a c t e r i z e d as" s c h i z o p h r e n i a " , I keep coming back t o t h e term " i n t e g r i t y " .Not t h eintegrityo f an i s o l a t e d f o r m a l i s m t h a t p r i d e s i t s e l f on t h e m a i n t a i n a n c e o f i t s ownstandards o f e x c e l l e n c e , b u t an i n t e g r i t y t h a t seeks common ground i n there-searches o f pure m a t h e m a t i c s , a p p l i e d m a t h e m a t i c s , and such m a t h e m a t i c a l l yo r i e n t e d d i s c i p l i n e s as p h y s i c s ; t h a t seeks t o e x t r a c t t h e maximum meaning fromeach new development; t h a t i s guided p r i m a r i l y by c o n s i d e r a t i o n s o f c o n t e n tr a t h e r than elegance and formal a t t r a c t i v e n e s s ; t h a t sees t o i t t h a t t h e matheI m a t i c a l r e p r e s e n t a t i o n o f r e a l i t y does n o t degenerate i n t o a game; t h a t seeksj t o understand t h e place o f mathematics i n contemporary s o c i e t y .may n o t be p o s s i b l e o f r e a l i z a t i o n - , b u t t h a t i s n o t i m p o r t a n t .ThisintegrityI l i k e too f c o n s t r u c t i v i s m as.one a t t e m p t t o r e a l i z e a t l e a s t c e r t a i n aspects o fthinkthis

-6-idealized integrity.This presumption at least has the possible merit of pre-venting constructivism from becoming another game, as some constructivisms havetended to do in the past.In discussing the principles of constructivism, I shall try to separate thoseaspects of constructivism that are basic to the philosophy from those that aremerely convenient (or inconvenient, as the case may be).Four principles standout as basic:(A)Mathematics is common sense.(B)Do not ask whether a statement is true until you know what it means.(C)A proof is any completely convincing argument.(D)Meaningful distinctions deserve to be maintained.Surprisingly many brilliant people refuse to apply common sense to mathematicsA frequent attitude is that the formalization of mathematics has been of greatvalue, because the formalism constitutes a court of last resort to settle anydisputes that might arise concerning the correctness of a proof.Common sensetells us, on the contrary, that if a proof is so involved that we are unable todetermine its correctness by informal methods, then we shall not be able to testit by formal means either.Moreover the formalism can not be used to settlephilosophical disputes, because the formalism merely reflects the basic philosophyand consequently philosophical disagreements are bound to result in disagreementsabout the validity of the formalism.Principle (B) resolves the problem of constructive truth.For that matter,it would resolve the problem of classical truth if the classical mathematicianswould accept it.We might say that truth is a matter of convention." This simplymeans that all arguments concerning the truth or falsity of any given statement**ii'

-7-about which both parties possess the same relevant facts occur because they havenot reached a clear agreement as to what the statement means.For instance inresponse to the inquiry "Is it true the constructivists believe that noteverybounded monotone sequence of real numbers converges?", if I am tired I answer"yes".Otherwise I tell the questioner that my answer will depend on what mean- 'ing he wishes to assign to the statement (*), that everyor real numbers converges.bounded monotone sequenceMoreover I tell him that once he has assigned a precisemeaning to statement (*), then my answer to his question will probably be clear tohim before I give it.The two meanings commonly assigned to (*) are the classicaland the constructive.It seems to me that the classical mathematician is not asprecise as he might be about the meaning he assigns to such a statement.I shallshow you later one simple and attractive approach to the problem of meaning inclassical mathematics.clear.However in the case before us the intuition at least isWe represent the terms of the sequence by vertical marks marching to theright, but remaining to the left of the boundB.'The classical intuition is that the sequence gets cramped, because there areiinfinitely many terms, but only a finite amount of space available to the leftofB.Thus it has to pile up somewhere.That somewhere is its limitL.BjThe constructivist grants that some sequences behave in precisely this way.call those sequences stupid.Let me tell you what a smart sequence will do.IIt

-8-wiill pretend to be stupid, piling up at a limit (in reality a false limit)Then when you have been convinced that it really is piling up atLf.L f , it willtake a jump and land somewhere to the right 1BjumpL e t us postpone a s e r i o u s d i s c u s s i o n o f t h i s example u n t i l we have d i s c u s s e d thec o n s t r u c t i v e r e a l number system.interpretation willThe p o i n t I w i s h t o make now i s t h a t under neithit h e r e be any disagreement as t o t h e t r u t h o f ( * ) , once t h a ti n t e r p r e t a t i o n has been f i x e d and made p r e c i s e .Whenever a s t u d e n t asks me whether a p r o o f he has g i v e n i s c o r r e c t ,beforeanswering h i s q u e s t i o n I t r y t o d i s c o v e r h i s concept o f what c o n s t i t u t e s a p r o o f .Then I t e l lhim my own c o n c e p t , (C) above, and ask him whether he f i n d shisargument c o m p l e t e l y c o n v i n c i n g , and whether he t h i n k s he has expressed h i m s e l fc l e a r l y enough so t h a t o t h e r i n f o r m e d and i n t e l l e g e n t people w i l lpletelya l s o be com-convinced.C l e a r l y i t i s i m p o s s i b l e t o accept (C) w i t h o u t a c c e p t i n g ( B ) , becauseitd o e s n ' t make sense t o be convinced t h a t something i s t r u e unless you know whati t means.The q u e s t i o n o f t e n a r i s e s , whether a c o n s t r u c t i v i s t would accept a nonc o n s t r u c t i v e p r o o f o f a numerical r e s u l t i n v o l v i n g no e x i s t e n t i a lsuch as Goldbach's c o n j e c t u r e o r Fermat's l a s t theorem.by ( C ) :quantifiers,My answer i ssuppliedI would want t o examine t h e p r o o f t o see whether I found i t ' c o m p l e t e l yconvincing.Perhaps one should keep an open m i n d , b u t I f i n d i t hard t o b e l i e v e

-9-t h a t I would f i n d any p r o o f t h a t r e l i e d on the p r i n c i p l e o f t h e excluded m i d d l e'f o r instance completely convincing.F o r t u n a t e l y the problem i si because such proofs do n o t seem t o a r i s e .hypothetical,I t does r a i s e t h e i n t e r e s t i n gpointi t h a t a c l a s s i c a l l y a c c e p t a b l e p r o o f o f Goldbach's c o n j e c t u r e m i g h t n o t be conI s t r u c t i v e l y a c c e p t a b l e , and t h e r e f o r e t h e c l a s s i c a li n t e r p r e t a t i o n s o f Goldbach's c o n j e c t u r e must d i f f e rand t h econstructivei n some fundamentalrespect.; We s h a l l see l a t e r t h a t t h i s i s indeed t h e case.C l a s s i c a l mathematics f a i l s t o observe meaningful d i s t i n c t i o n s having t o dowith integers.This b a s i c f a i l u r e r e f l e c t s i t s e l f a t a l l development o f mathematics.Consider t h e number* Riemann h y p o t h e s i s i s t r u e and1ifitl e v e l s o f the c l a s s i c a ln Q , d e f i n e d t o beis false.0iftheThe c o n s t r u c t i v i s t does notwish t o prevent t h e c l a s s i c i s t from w o r k i n g w i t h such numbers ( a l t h o u g h he maypersonally b e l i e v e t h a t t h e i r i n t e r e s t i s l i m i t e d ) .1He does want thecalssicistto d i s t i n g u i s h such numbers from numbers which can be "computed", such as t h e10 1 0numbern-, of primes less than10.Classical mathematicians do concern! themselves sporadically with whether numbers can be "computed", but only on anad hoc basis.The distinction is not observed in the systematic development ofclassical mathematics, nor would the tools available to the classicist permithim to observe the distinction systematically even if he were so inclined.The constructivists are frequently accused of displaying the same insensitivityto shades of meaning of which they accuse the classicist, because they do not.distinguish between numbers that can be computed in principle, such as thenumbern1defined above, and numbers that can be computed in fact."violate their own principle (D).not easy to refute.Thus theyThis is a serious accusation, and one that isRather than attempting to refute it, I shall give you my

-10-personal point of view.First, it may be demanding too much of the constructivistto ask them to lead the way in the development of usable and systematic methodsfor distinguishing computability in principle from computability in fact.If andwhen such methods are found, the construct!"vists will gratefully incorporate theminto their mathematics.going to be found.Second, it is by no means clear that such methods areThere is no fast distinction between computability in prin-ciple and in fact, because of the constant progress of the state of the art amongother reasons.The most we can hope for is some good systematic measure of theefficiency of a computation.Until such is found, the problem will continue tobe treated on an ad hoc basis.I was careful not to call the numbern«defined above an integer.Whetherwe do call it an integer is of no real importance, as long as we distinguish itin some way from numbers such asand calln-,n-,.a constructive integer.For instance we might callnQan integerThe constructivists have not accepted thisterminology, in part because of Brouwer's influence, but also because it does notaccord with their estimate of the relative importance of the two concepts.I shalreserve the term "integer" for what a classicist might call a constructive integerand put aside, at least for now, the problem of what would be an appropriate termfor what is classically called an integer (assuming that the classical notion ofan integer is indeed viable).Thus we come to the crucial question, "What is an integer?"ready seen, the question is badly phrased.As we have al-We are really looking for a definitionof an integer that will be an efficient tool for developing the full content ofmathematics.Since it is clear that we always work with representations ofintegers, rather than integers themselves (whatever those may be), we are reallytrying to define what we mean by a representation of an integer.Again, an

-11-integer is represented only when some intelligent agent constructs the representation, or establishes the convention that some artifact constitutes a representation.an integer?"Thus in its final version the question is, "How does one representIn practice we shall not be so meticulous as all this in our useof language.- We shall simply speak of integers, with the understanding that-weare really speaking of their representations.This causes no harm, because.theoriginal concept of an integer, as something invariant standing behind all ofits representations, has just been seen to be superfluous.Moreover we shalli not constantly trouble to point out that (representations of) integers exist' only by virtue of conventions established by groups of intelligent beings.Afterthis preliminary chatter, which may seem to have been unnecessary, we present our definition of an integer, dignified by the title of theFundamental Constructivist ThesisEvery integer can be converted in principle to- decimal form by a finite,purely routine, process.Note the phrase "in principle".It means that although we should be ableto program a computer to produce the decimal form of any given integer, thereare cases in which it would be naive to run the program and wait around for theresult.Everything else about integers follows from the above thesis plus the rulesof decimal arithmetic that we learned in elementary school.Two integers are equalif their decimal representations are equal in the usual sense.The order relationsand the arithmetic of integers are defined in terms of their decimal representations

-12-W i t h t h e c o n s t r u c t i v e d e f i n i t i o n o f t h e i n t e g e r s , we have begun o u r s t u d yo f the t e c h n i c a limplementation of the c o n s t r u c t i v i s t philosophy.Our p o i n t o fview i s t o d e s c r i b e the mathematical o p e r a t i o n s t h a t can be c a r r i e d o u t byf i n i t e b e i n g s , man's mathematics f o r s h o r t .In c o n t r a s t , c l a s s i c a lconcerns i t s e l f w i t h o p e r a t i o n s t h a t can be c a r r i e d o u t by God.t h e above numbernQFor i n s t a n c e ,i s c l a s s i c a l l y a w e l l - d e f i n e d i n t e g e r because God can p e r -form t h e i n f i n i t e search t h a t w i l ltrue.mathematicsdetermine whether t h e Riemann h y p o t h e s i sisAs another example, t h e smart sequences p r e v i o u s l y d i s c u s s e d may be a b l et o o u t w i t you and me ( o r any o t h e r f i n i t e b e i n g ) , b u t they w i l lo u t w i t God.n o t be a b l e t oThat i s why s t a t e m e n t ( * ) i s t r u e c l a s s i c a l l y b u t n o tconstructively.You may t h i n k t h a t I am making a j o k e , o r a t t e m p t i n g t o put down c l a s s i c a lm a t h e m a t i c s , by b r i n g i n g God i n t o t h e d i s c u s s i o n .my b e s t t o develop a secure p h i l o s o p h i c a lThis i s n o t t r u e .I am doingf o u n d a t i o n , based on meaning r a t h e rthan f o r m a l i s t i c s , f o r c u r r e n t c l a s s i c a l p r a c t i c e .The most s o l i dfoundationa v a i l a b l e a t p r e s e n t seems t o me t o i n v o l v e the c o n s i d e r a t i o n o f a being w i t hn o n - f i n i t e p o w e r s — c a l l him God o r whatever you w i l l — i n a d d i t i o n t o t h e powerspossessed by f i n i t ebeings.What powers s h o u l d we a s c r i b e t o God?A t t h e veryl e a s t , we s h o u l d c r e d i thim w i t h l i m i t e d o m n i s c i e n c e , as d e s c r i b e d i n t h e f o l l o w i n g l i m i t e d p r i n c i p l eo f omniscience (LPO f o r s h o r t ) :eitherLPOn. 0for allkIf{n }\i s any sequence o f i n t e g e r s , t h e nor there e x i s t s akwithn k t 0.By a c c e p t i n gjas v a l i d , we a r e s a y i n g t h a t t h e being whose c a p a b i l i t i e s our mathematicsd e s c r i b e s i s a b l e t o search t h r o u g h a sequence o f i n t e g e r s t o determine whetherithey a l lvanish o r n o t .L e t us r e t u r n t o the t e c h n i c a l'*developemnt o f c o n s t r u c t i v e m a t h e m a t i c s ,s i n c e i t i s s i m p l e r , and postpone t h e f u r t h e r c o n s i d e r a t i o n o fclassical

-13-mathematics until later.Our first task is to develop an appropriate languageto describe the mathematics of finite beings.Brouwer.For this we are indebted to(See references [1 ], [ 6 ] , [15], [20], and [21]exposition than we are able to give here.)for a more completeBrouwer remarked that the meaningscustomarily assigned to the terms "and", "or", "not", "implies", "there exists",and "for all" are not entirely appropriate to the constructive point of view, andhe introduced more appropriate meanings as necessary.The connective "and" causes no trouble.Aand also proveTo prove "A and B", we must proveB, as in classical mathematics.To prove"AorB" we mustgive a finite, purely routine method which after a finite number of steps eitherleads to a proof ofclassical use ofAor to a proof ofThis is veryB.different from the"or"; for example LPO is true classically, but we are notentitled to assert it constructively because of the constructive meaning of "or".The connective "implies" is defined classically by takingto mean "notAorB".This definition would not be of much value constructively, Brouwer therefore defined"AimpliesB"to mean that there exists an argument'. which shows how to convert an arbitrary proof ofan example, it is clear that(AimpliesC)"1ofCC""{(Aimplies"Aimpliesis the following:B"and"BWe define(BimpliesB.C)}To takeimpliesimpliesBC"into a proof of"AA, convert it into a proofand then converting that proofC."notA"to mean thatA.that it is inconceivable that a proof of0 1"into a proof ofandgiven any proof ofby first converting it into a proof of' into a proof of"notB)Ais always true constructively; the argument that convertsarbitrary proofs ofimplies"A implies B"is a true statement.is contradictory.ABy this we meanwill ever be given.The statement"0 1"For example,means that when the

-14-numbers"0"and"1"are expressed in decimal form, a mechanical comparisonof the usual sort checks that they are the same.Since they are already indecimal form, and the comparison in question shows they are not the same, it isimpossible by correct methods to prove that they are the same.would be defective, either technically or conceptually.(AandnotA)"impossible to proveAny such proofAs another example, "notis always a true statement, because if we prove notAit isA—therefore, it is impossible to prove both.Having changed the meaning of the connectives, we should not be surprisedto find that certain classically accepted modes of inference are no longercorrect.or notThe most important of these is the principle of the excluded middle—"AA".Constructively, this principle would mean that we had a method whichin finitely many, purely routine, steps would lead to a proof of disproof of anarbitrary mathematical assertionA.Of course we have no such method, and no-body has the least hope that we ever shall.It is the principle of the excludedmiddle that accounts for almost all of the important unconstructivities ofclassical mathematics.A".Another incorrect principle is"(not notA)impliesIn other words, a demonstration of the impossibility of the impossibilityof a certain construction, for instance, does not constitute a method forcarrying out that construction.I could proceed to list a more or less complete set of constructively validrules of inference involving the connectives just introduced.superfluous.sense.This would beNow that their meanings have been established, the rest is commonAs an exercise, show that the statment"(A - 0 1) - notis constructively valid.A"

-15-The classical concept of a set as a collection of objects from some preexistent universe is clearly inappropriate constructively.Constructive mathe-matics does not postulate a pre-existent universe, with objects lying aroundwaiting to be collected and grouped into sets, like shells on a beach.Theentities of constructive mathematics are called into being by the constructingintelligence.suspect.set?".From this point of view, the veryquestion "What is a set?" isRather we should ask the question, "What must one do to construct aWhen the question is posed this way, the answer is not hard to find.Definition.To construct a set, one must specify what must be done toconstruct an arbitrary element of the set, and what must be done to prove twoarbitrary elements of the set are equal.Equality so defined must be shown tobe an equivalence relation.As an example, let us construct the set of rational numbers.a rational number, define integersthat the rational numbersp/qpandandqpi/q-.q t 0.and prove thatare equal, proveTo constructpq, p,q.While we are on the subject, we might as well define a functionIt is a rule that to each elementequal elements ofBofAassociates an elementbeing associated to equal elements ofThe notion of a subsetan element ofxAQof a setAf : A - B.f(x)ofB,A.is also of interest.A Q , one must first construct an element ofTo proveTo constructA, and then provethat the element so constructed satisfies certain additional conditions,characteristic of the particular subsetif they are equal as elements ofAQ.Two elements ofAQare equalA.Contrary to classical usage, the scope of the equality relation never extends{beyond a particular set.Thus it does not make sense to speak of elements of

-16-d i f f e r e n t s e t s as being e q u a l , unless p o s s i b l y those d i f f e r e n t s e t s are bothsubsets o f t h e same s e t .T h i s i s because f o r t h e c o n s t r u c t i v i s t e q u a l i t y i s ac o n v e n t i o n , whose scope i s always a g i v e n s e t ; a l ld i s t i n c t from t h e c

mathematics has witnessed the triumph of formalist dogma, which had its inception in the important insight that most arguments of modern mathematics can be broken down and presented as successive applications of a few basic schemes. The ex perts now routinely equate the panorama of mathematics

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