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# Introduction to Mathematical Thought MATH 124

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This 11 page Class Notes was uploaded by Breanne Schaden PhD on Saturday October 3, 2015. The Class Notes belongs to MATH 124 at Boise State University taught by Staff in Fall. Since its upload, it has received 31 views. For similar materials see /class/218011/math-124-boise-state-university in Mathematics (M) at Boise State University.

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A217 The Three Crises in Mathematics Logicism Intuitionism and Formalism A217 1 Introduction httpwwwmembersshawcakschindlermathphilhtm Note there are a couple of special symbols used in the original which will not translate well in HTML so I ve substituted endnotes in square brackets which refer to a lay description of what the symbols look like Those who know any set theory will not need these visual aids 7 MS source Ernst Snapper The Three Crises in Mathematics Logicism Intuitionism and Formalism in Mathematics Magazine 52 Sept 1979 20716 reprinted in Douglas M Campbell John C Higgins eds Mathematics People Problems Results vol 2 Belmont CA Wadsworth 1984 183193 Ernst Snapper was born in the Netherlands in 1913 He received his PhD from Princeton in 1941 and was immediately offered a position at Princeton which he held from 1941 to 1945 Since then he has held special chairs at the University of Southern California lVIiami University Indiana University and Dartmouth College This essay is as cogent and readable an introduction to the philosophy of modern mathematics as exists in the English language While the main thrust of Snapper s essay is to explain the three approaches to mathematics noted in the title it is clear that he also has another message Brie y it is that arguments about the philosophy of mathematics are arguments about philosophy Mathematicians often feel that their special talents place them outside that arena in which other mortals contend This conceit almost surely extends to matters of philosophy as well as politics finance and so on Although no such survey exists or is contemplated a poll of current mathematicians on their opinions of logicism intuitionism and formalism would be fascinating It would be astonishing ifas many as 1 0 percent of the sample were to connect these three di erent views of mathematics with their philosophical counterparts This is due not so much to an ignorance of philosophy as to a refusal to admit that mathematics in any way partakes of the vague and arbitrary nature of philosophy What Snapper clearly reveals is that these three approaches to mathematics can be rephrased as views about the nature of abstract entities These views in one guise or another date back to the ancient Greeks debate as to which is the quotcorrectquot approach has been an ongoing philosophical entertainment for well over 2 000 years Whether the mathematical aspect of this debate will add any new insights to those already o ered by the philosophers remains to be seen What is certain is that failure on the part of the mathematicians to admit the philosophical dimensions of their dispute will result in the mathematicians simply reinventing in another language a number of very old philosophical wheels This seems at best a dubious use of mental resources and Snapper s article probably ought to be required reading for anyone even mildly interested in the foundations of mathematics THE THREE SCHOOLS mentioned in the title all tried to give a rm foundation to mathematics The three crises are the failures of these schools to complete their tasks This article looks at these crises quotthrough modern eyesquot using whatever mathematics is available today and not just the mathematics which was available to the pioneers who created these schools Hence this article does not approach the three crises in a strictly historical way This article also does not discuss the large volume of current technical mathematics which has arisen out of the techniques introduced by the three schools in question One reason is that such a discussion would take a book and not a short article Another one is that all this technical mathematics has very little to do with the philosophy of mathematics and in this article I want to stress those aspects of logicism intuitionism and formalism which show clearly that these schools are founded in philosophy A2172 Logicism This school was started in about 1884 by the German philosopher logician and mathematician Gottlob Frege 18481925 The school was rediscovered about eighteen years later by Bertrand Russell Other early logicists were Peano and Russell s coauthor of Principia M athematz39ca A N Whitehead The purpose of logicism was to show that classical mathematics is part of logic If the logicists had been able to carry out their program successfully such questions as quotWhy is classical mathematics free of contradictionsquot would have become quotWhy is logic free of contradictionsquot This latter question is one on which philosophers have at least a thorough handle and one may say in general that the successful completion of the logicists program would have given classical mathematics a firm foundation in terms of logic Clearly in order to carry out this program of the logicists one must first somehow define what quotclassical mathematicsquot is and what quotlogicquot is Otherwise what are we supposed to show is part of what It is precisely at these two definitions that we want to look through modern eyes imagining that the pioneers of logicism had all of presentday mathematics available to them We begin with classical mathematics In order to carry out their program Russell and Whitehead created Principia M athematz39ca which was published in 1910 The first volume ofthis classic can be bought for 345 Thank heaven only modern books and not the classics have become too expensive for the average readerPrincz39pia as we will refer to Principia M athematz39ca may be considered as a formal set theory Although the formalization was not entirely complete Russell and Whitehead thought that it was and planned to use it to show that mathematics can be reduced to logic They showed that all classical mathematics known in their time can be derived from set theory and hence from the axioms of Principia Consequently what remained to be done was to show that all the axioms of Principia belong to logic Of course instead of Principia one can use any other formal set theory just as well Since today the formal set theory developed by Zermelo and Fraenkel ZF is so much better known than Principia we shall from now on refer to ZF instead of Principia ZF has only nine axioms and although several of them are actually axiom schemas we shall refer to all of them as quotaxiomsquot The formulation of the logicists program now becomes Show that all nine axioms of ZF belong to logic This formulation of logicism is based on the thesis that classical mathematics can be defined as the set of theorems which can be proved within ZF This definition of classical mathematics is far from perfect as is discussed in However the above formulation of logicism is satisfactory for the purpose of showing that this school was not able to carry out its program We now turn to the definition of logic In order to understand logicism it is very important to see clearly what the logicists meant by quotlogicquot The reason is that whatever they meant they certainly meant more than classical logic Nowadays one can define classical logic as consisting of all those theorems which can be proven in first order languages discussed below in the section on formalism without the use of nonlogical axioms We are hence restricting ourselves to first order logic and use the deduction rules and logical axioms of that logic An example of such a theorem is the law of the excluded middle which says that if p is a proposition then either p or its negation p is true in other words the proposition p p is always true where is the usual symbol for the inclusive quotor If this definition of classical logic had also been the logicists definition of logic it would be a folly to think for even one second that all of ZF can be reduced to logic However the logicists definition was more extensive They had a general concept as to when a proposition belongs to logic that is when a proposition should be called a quotlogical propositionquot They said A logical proposition is a proposition which has complete generality and is true in virtue of its form rather than its content Here the word quotpropositionquot is used as synonymous with quottheoremquot le or physles or what have you on the eontrary thrs law holds wrth eornplete generalrty thatrs for any proposmonp whatsoeyer Why then does rt hold7 The logelsts answer e Beeause oflts form H theyrne by form syntaetreal forrn L tn ln t or dthe negatron not denoted by 1 and 1 respeehyely On the one hand logle as d Flu d h r l l 1 On the h rh wd l l de nrtron t H v I eoneept of alogleal proposmon famllrarnmh 114quot wt d F L ww h andthe axlom ofeholee eannotpossrbly be eonsldered as logleal proposmons For example the axlom of lnflmty says that there exlst lnflmte sets Why do we aeeept ths axlom as belng txue7 The reason ls that the set ofpolnts m Euelldean 3rspaee Henee we aeeeptthrs axlom on grounds of our everyday expenenee wrth sets and th5 Tu w r 1 when eornrnon eyeryday rl ttmh h l l TWP V quantr ers the for all quanh er n 11 way he wo 1 and the there exlsts quanh er 1were rntrodueedrnto logle by Frege and l l l hl V N 1 rexample They have seope The philosophy oflogicism is sometimes said to be based on the philosophical school called quotrealismquot In medieval philosophy quotrealismquot stood for the Platonic doctrine that abstract entities have an existence independent of the human mind Mathematics is of course full of abstract entities such as numbers functions sets etc and according to Plato all such entities exist outside our mind The mind can discover them but does not create them This doctrine has the advantage that one can accept such a concept as quotse quot without worrying about how the mind can construct a set According to realism sets are there for us to discover not to be constructed and the same holds for all other abstract entities In short realism allows us to accept many more abstract entities in mathematics than a philosophy which had limited us to accepting only those entities the human mind can construct Russell was a realist and accepted the abstract entities which occur in classical mathematics without questioning whether our own minds can construct them This is the fundamental difference between logicism and intuitionism since in intuitionism abstract entities are admitted only if they are man made Excellent expositions of logicism can be found in Russell s writing for example and A2173 Intuitionism This school was begun about 1908 by the Dutch mathematician L E J Brouwer 1881 1966 The intuitionists went about the foundations of mathematics in a radically different way from the logicists The logicists never thought that there was anything wrong with classical mathematics they simply wanted to show that classical mathematics is part of logic The intuitionists on the contrary felt that there was plenty wrong with classical mathematics By 1908 several paradoxes had arisen in Cantor s set theory Here the word quotparadoxquot is used as synonymous with quotcontradictionquot Georg Cantor created set theory starting around 1870 and he did his work quotnaivelyquot meaning nonaxiomatically Consequently he formed sets with such abandon that he himself Russell and others found several paradoxes within his theory The logicists considered these paradoxes as common errors caused by erring mathematicians and not by a faulty mathematics The intuitionists on the other hand considered these paradoxes as clear indications that classical mathematics itself is far from perfect They felt that mathematics had to be rebuilt from the bottom on up The quotbottomquot that is the beginning of mathematics for the intuitionists is their explanation of what the natural numbers 1 2 3 are Observe that we do not include the number zero among the natural numbers According to intuitionistic philosophy all human beings have a primordial intuition for the natural numbers within them This means in the first place that we have an immediate certainty as to what is meant by the number I and secondly that the mental process which goes into the formation of the number I can be repeated When we do repeat it we obtain the concept of the number 2 when we repeat it again the concept of the number 3 in this way human beings can construct any finite initial segment 12 n for any natural number n This mental construction of one natural number after the other would never have been possible if we did not have an awareness of time within us quotAfterquot refers to time and Brouwer agrees with the philosopher Immanuel Kant 17241804 that human beings have an immediate awareness of time Kant used the word quotintuitionquot Anschauung in German which doesn t have the same connotation as intuition in English it has more the connotation of perception at a given point in time illLS for quotimmediate awarenessquot and this is where the name quotintuitionismquot comes from It is important to observe that the intuitionistic construction of natural numbers allows one to construct only arbitrarily long finite initial segments 12 n It does not allow us to construct that whole closed set of all the natural numbers which is so familiar from classical mathematics It is equally important to observe that this construction is both quotinductivequot and quoteffectivequot It is inductive in the sense that if one wants to construct say the number 3 one has to go through all the mental steps of first constructing the I then the 2 and finally the 3 one cannot just grab the number 3 out of the sky It is effective in the sense that once the construction of a natural number has been finished that natural number has been constructed in its entirety It stands before us as a completely finished mental construct ready for our study of it When someone says quotI have finished the mental construction of the number 3quot it is like a bricklayer saying quotI have nished that wallquot which he can say only after he has laid every stone in place We now turn to the intuitionistic de nition of mathematics According to intuitionistic philosophy mathematics should be de ned as a mental activity and not as a set of theorems as was done above in the section on logicism It is the activity which consists in carrying out one after the other those menial constructions which are inductive and effective in the sense in which the intuitionistic construction of the natural numbers is inductive and effective Intuitionism maintains that human beings are able to recognize whether a given mental construction has these two properties We shall refer to a mental construction which has these two properties as a construct and hence the intuitionistic de nition of mathematics says Mathematics is the mental activity which consists in carrying out constructs one after the other A major consequence of this de nition is that all of intuitionistic mathematics is effective or quotconstructivequot as one usually says We shall use the adjective quotconstructivequot as synonymous with quoteffectivequot from now on Namely every construct is constructive and intuitionistic mathematics is nothing but carrying out constructs over and over For instance if a real number r occurs in an intuitionistic proof or theorem it never occurs there merely on grounds of an existence proof It occurs there because it has been constructed from top to bottom This implies for example that each decimal place in the decimal expansion of r can in principle be computed In short all intuitionistic proofs theorems de nitions etc are entirely constructive Another major consequence of the intuitionistic de nition of mathematics is that mathematics cannot be reduced to any other science such as for instance logic This de nition comprises too many mental processes for such a reduction And here then we see a radical difference between logicism and intuitionism In fact the intuitionistic attitude toward logic is precisely the opposite from the logicists attitude According to the intuitionists whatever valid logical processes there are they are all constructs hence the valid part of classical logic is part of mathematics Any law of classical logic which is not composed of constructs is for the intuitionist a meaningless combination of words It was of course shocking that the classical law of the excluded middle turned out to be such a meaningless combination of words This implies that this law cannot be used indiscriminately in intuitionistic mathematics it can often be used but not always Once the intuitionistic de nition of mathematics has been understood and accepted all there remains to be done is to do mathematics the intuitionistic way Indeed the intuitionists have developed intuitionistic arithmetic algebra analysis set theory etc However in each of these branches of mathematics there occur classical theorems which are not composed of constructs and hence are meaningless combinations of words for the intuitionists Consequently one cannot say that the intuitionists have reconstructed all of classical mathematics This does not bother the intuitionists since whatever parts of classical mathematics they cannot obtain are meaningless for them anyway Intuitionism does not have as its purpose the justi cation of classical mathematics Its purpose is to give a valid de nition of mathematics and then to quotwait and seequot what mathematics comes out of it Whatever classical mathematics cannot be done intuitionistically simply is not mathematics for the intuitionist We observe here another fundamental difference between logicism and intuitionism The logicists wanted to justify all of classical mathematics An excellent introduction to the actual techniques of intuitionism is Let us now ask how successful the intuitionistic school has been in giving us a good foundation for mathematics acceptable to the majority of mathematicians Again there is a sharp difference between the way this question has to be answered in the present case and in the case of logicism Even hardnosed logicists have to admit that their school so far has failed to give mathematics a rm foundation by about 20 However a hardnosed intuitionist has every right in the world to claim that intuitionism has given mathematics an entirely satisfactory foundation There is the meaningful de nition of intuitionistic mathematics discussed above there is the intuitionistic philosophy which tells us why constructs can never give rise to contradictions and hence that intuitionistic mathematics is free of contradictions In fact not only this problem of freedom from contradiction but all other problems of a foundational nature as well receive perfectly satisfactory solutions in intuitionism Yet if one looks al intuitionism from the outside namely from the viewpoint of the classical mathematician one has to say that intuitionism has failed to give mathematics an adequate foundation In fact the mathematical community has the mathematical community done this in spite of many very attractive features of intuitionism some of which have just been mentioned One reason is that classical mathematicians atly refuse to do away with the many beautiful theorems that are meaningless combinations of words for the intuitionists An example is the Brouwer fixed point theorem of topology which the intuitionists reject because the fixed point cannot be constructed but can only be shown to exist on grounds of an existence proof This by the way is the same Brouwer who created intuitionism he is equally famous for his work in nonintuitionistic topology A second reason comes from theorems which can be proven both classically and intuitionistically It often happens that the classical proof of such a theorem is short elegant and devilishly clever but not constructive The intuitionists will of course reject such a proof and replace it by their own constructive proof of the same theorem However this constructive proof frequently turns out to be about ten times as long as the classical proof and often seems at least to the classical mathematician to have lost all of its elegance An example is the fundamental theorem of algebra which in classical mathematics is proved in about half a page but takes about ten pages of proof in intuitionistic mathematics Again classical mathematicians refuse to believe that their clever proofs are meaningless whenever such proofs are not constructive Finally there are the theorems which hold in intuitionism but are false in classical mathematics An example is the intuitionistic theorem which says that every realvalued function which is defined for all real numbers is continuous This theorem is not as strange as it sounds since it depends on the intuitionistic concept of a function A realvalued function f is defined in intuitionism for all real numbers only if for every real number r whose intuitionistic construction has been completed the real number r can be constructed Any obviously discontinuous function a classical mathematician may mention does not satisfy this constructive criterion Even so theorems such as this one seem so far out to classical mathematicians that they reject any mathematics which accepts them These three reasons for the rejection of intuitionism by classical mathematicians are neither rational nor scientific Nor are they pragmatic reasons based on a conviction that classical mathematics is better for applications to physics or other sciences than is intuitionism They are all emotional reasons grounded in a deep sense as to what mathematics is all about If one of the readers knows of a truly scientific rejection of intuitionism the author would be grateful to hear about it We now have the second crisis in mathematics in front of us It consists in the failure of the intuitionistic school to make intuitionism acceptable to at least the majority of mathematicians It is important to realize that like logicism intuitionism is rooted in philosophy When for instance the intuitionists state their definition of mathematics given earlier they use strictly philosophical and not mathematical language It would in fact be quite impossible for them to use mathematics for such a definition The mental activity which is mathematics can be defined in philosophical terms but this definition must by necessity use some terms which do not belong to the activity it is trying to define Just as logicism is related to realism intuitionism is related to the philosophy called quotconceptualismquot This is the philosophy which maintains that abstract entities exist only insofar as they are constructed by the human mind This is very much the attitude of intuitionism which holds that the abstract entities which occur in mathematics whether sequences or orderrelations or what have you are all mental constructions This is precisely why one does not find in intuitionism the staggering collection of abstract entities which occur in classical mathematics and hence in logicism The contrast between logicism and intuitionism is very similar to the contrast between realism and conceptualism A very good way to get into intuitionism is by studying Chapter IV of and in this order A2174 Formalism l A True one 1893 and the seeonol one ml 0 N rm l V us or rnturtrom sm g Fu ld n t m ete The next questron ls How do we formallze aglven axlomanzedtheorg t t n to formallze 739 offlve rtems These fountems Wm vanables7 2 ngbols for the eonneetayes of everyday speeeh say 1 for not 1 for and 1 for the rnelusrye or 1 for lthen and 1 for lf anol onlylf iwho ean talk about anythrng at all wrthoutusrng eonne tr 7 3 n 4 The two quantr ers the for all quanta er 1 andthe there emst quantlfler 1 the rst one ls usedto say sueh thrngs as 4 Instead we turn to the lth ltem Srnee Tls an amomatazeol theory rt has sorcalled unde nedterms one has to ehoose an appropnate symbol l r ofT 1 rm t n terms ofplane Euelrolean geometry oeeur polnt hue and lnmdenme and for eaeh one ofthem an of arrthmetre oeeur zero addluon and multrplreatron and the symbols one ehooses for them are ofcourse 0 wd r 0 term namely the membershlp relataon one ehooses ofcourse the usual symbol 1 forthatrelatlon These symbols one for eaeh unde nedterm othe amomatazeol theory T are olten ealleol the parameters of the rst oroler language andhenee the parameters make up the fl h Since the parameters are the only symbols in the vocabulary of a first order language which depend on the given axiomatized theory T one formalizes T simply by choosing these parameters Once this choice has been made the whole theory T has been completely formalized One can now express in the resulting first order language L not only all axioms de nitions and theorems of T but more One can also express inL all axioms of classical logic and consequently also all proofs one uses to prove theorems of T In short one can now proceed entirely with L that is entirely quotformallyquot But now a third question presents itself quotWhy in the world would anyone want to formalize a given axiomatized theoryquot After all Euclid never saw a need to formalize his axiomatized geometry It is important to ask this question since even the great Peano had mistaken ideas about the real purpose of formalization He published one of his most important discoveries in differential equations in a formalized language very similar to a first order language with the result that nobody read it until some charitable soul translated the article into common German Let us now try to answer the third question If mathematicians do technical research in a certain branch of mathematics say plane Euclidean geometry they are interested in discovering and proving the important theorems of the branch of mathematics For that kind of technical work formalization is usually not only no help but a definite hindrance If however one asks such foundational questions as for instance quotWhy is this branch of mathematics free of contradictionsquot then formalization is not just a help but an absolute necessity It was really Hilbert s stroke of genius to understand that formalization is the proper technique to tackle such foundational questions What he taught us can be put roughly as follows Suppose that Tis an axiomatized theory which has been formalized in terms of the first order language L This language has such a precise syntax that it itself can be studied as a mathematical object One can ask for instance quotCan one possibly run into contradictions if one proceeds entirely formally withinL using only the axioms of Tand those of classical logic all of which have been expressed in Lquot Ifone can prove mathematically that the answer to this question is quotnoquot one has there a mathematical proof that the theory Tis free of contradictions This is basically what the famous quotHilbert programquot was all about The idea was to formalize the various branches of mathematics and then to prove mathematically that each one of them is free of contradictions In fact if by means of this technique the formalists could have just shown that ZF is free of contradictions they would thereby already have shown that all of classical mathematics is free of contradictions since classical mathematics can be done axiomatically in terms of the nine axioms of ZF In short the formalists tried to create a mathematical technique by means of which one could prove that mathematics is free of contradictions This was the original purpose of formalism It is interesting to observe that both logicists and formalists formalized the various branches of mathematics but for entirely different reasons The logicists wanted to use such a formalization to show that the branch of mathematics in question belongs to logic the formalists wanted to use it to prove mathematically that that branch is free of contradictions Since both schools quotformalizedquot they are sometimes confused Did the formalists complete their program successfully No In 1931 Kurt Godel showed that formalization cannot be considered as a mathematical technique by means of which one can prove that mathematics is free of contradictions The theorem in that paper which rang the death bell for the Hilbert program concerns axiomatized theories which are free of contradictions and whose axioms are strong enough so that arithmetic can be done in terms of them Examples of theories whose axioms are that strong are of course Peano arithmetic and ZF Suppose now that Tis such a theory and that Thas been formalized by means of the first order language L Then Godel s theorem says in nontechnical language quotNo sentence of L which can be interpreted as asserting that T is free of contradictions can be proven formally within the language Lquot Although the interpretation of this theorem is somewhat controversial most mathematicians have concluded from it that the Hilbert program cannot be carried out Mathematics is not able to prove its own freedom of contradictions Here then is the third crisis in quot quot Of course the tremendous importance of the formalist school for presentday mathematics is well known It was in this school that modern mathematical logic and its various offshoots such as model theory recursive function theory etc really came into bloom Formalism as logicism and intuitionism is founded in philosophy but the philosophical roots of formalism are somewhat more hidden than those of the other two schools One can find them though by re ecting a little on the Hilbert program Let again T be an aXiomatized theory which has been formalized in terms of the first order language L In carrying out Hilbert s program one has to talk about the languageL as one object and while doing this one is not talking within that safe languageL itself On the contrary one is talking about L in ordinary everyday language be it English or French or what have you While using our natural language and not the formal language L there is of course every danger that contradictions in fact any kind of error may slip in Hilbert said that the way to avoid this danger is by making absolutely certain that while one is talking in one s natural language aboutL one uses only reasonings which are absolutely safe and beyond any kind of suspicion He called such reasonings quotfinitary reasoningsquot but had of course to give a definition of them The most explicit definition of finitary reasoning known to the author was given by the French formalist Herbrand It says if we replace quotintuitionisticquot by quotfinitaryquot By a nitary argument we understand an argument satisfying the following conditions In it we never consider anything but a given nite number of objects and of functions these functions are well de ned their definition allowing the computation of their values in an unequivocal way we never state that an object exists without giving the means of constructing it we never consider the totality of all the objects X of an infmite collection and when we say that an argument or a theorem is true for all these X we mean that for each X taken by itself it is possible to repeat the general argument in question which should be considered to be merely the prototype of these particular arguments Observe that this definition uses philosophical and not mathematical language Even so no one can claim to understand the Hilbert program without an understanding of what finitary reasoning amounts to The philosophical roots of formalism come out into the open when the formalists define what they mean by finitary reasoning We have already compared logicism with realism and intuitionism with conceptualism The philosophy which is closest to formalism is quotnominalismquot This is the philosophy which claims that abstract entities have no eXistence of any kind neither outside the human mind as maintained by realism nor as mental constructions within the human mind as maintained by conceptualism For nominalism abstract entities are mere vocal utterances or written lines mere names This is where the word quotnominalismquot comes from since in Latin nominalis means quotbelonging to a namequot Similarly when formalists try to prove that a certain aXiomatized theory T is free of contradictions they do not study the abstract entities which occur in T but instead study that first order languageL which was used to formalize T That is they study how one can form sentences in L by the proper use of the vocabulary of L how certain of these sentences can be proven by the proper use of those special sentences of L which were singled out as aXioms and in particular they try to show that no sentence of L can be proven and disproven at the same time since they would thereby have established that the original theory T is free of contradictions The important point is that this whole study of L is a strictly syntactical study since no meanings or abstract entities are associated with the sentences of L This language is investigated by considering the sentences of L as meaningless eXpressions which are manipulated according to eXplicit syntactical rules just as the pieces of a chess game are meaningless figures which are pushed around according to the rules of the game For the strict formalist quotto do mathematicsquot is quotto manipulate the meaningless symbols of a first order language according to eXplicit syntactical rulesquot Hence the strict formalist does not work with abstract entities such as in nite series or cardinals but only with their meaningless names which are the appropriate eXpressions in a first order language Both formalists and nominalists avoid the direct use of abstract entities and this is why formalism should be compared with nominalism The fact that logicism intuitionism and formalism correspond to realism conceptualism and nominalism respectively was brought to light in Quine s article quotOn What There Isquot pages 183196 Formalism can be learned from any modern book on mathematical logic A2175 Epilogue Where do the three crises in mathematics leave us They leave us without a rm foundation for mathematics After Godel s paper appeared in 1931 mathematicians on the whole threw up their hands in frustration and turned away from the philosophy of mathematics Nevertheless the in uence of the three schools discussed in this article has remained strong since they have given us much new and beautiful mathematics This mathematics concerns mainly set theory intuitionism and its various constructivist modifications and mathematical logic with its many offshoots However although this kind of mathematics is often referred to as quotfoundations of mathematicsquot one cannot claim to be advancing the philosophy of mathematics just because one is working in one of these areas Modern mathematical logic set theory and intuitionism with its modifications are nowadays technical branches of mathematics just as algebra or analysis and unless we return directly to the philosophy of mathematics we cannot expect to find a firm foundation for our science It is evident that such a foundation is not necessary for technical mathematical research but there are still those among us who yearn for it The author believes that the key to the foundations of mathematics lies hidden somewhere among the philosophical roots of logicism intuitionism and formalism and this is why he has uncovered these roots three times over References E Snapper What is mathematicsAmer Math Monthly no 7 86 1979 551 557 Various symbols are used for negation from a macron a line above the element or term a tilde in front of it or in Snapper s case a symbol which looks like an L rotated 180 degrees somewhat lik 7 This is the union symbol again various forms can be used but Snapper uses one that looks like a sansserif V This is also the logical operator or As per 3 This reads p or notp As per 5 As per 3 As per 2 This looks like an upsidedown sansserif A This looks like a backwards sansserif E B Russell Principles ofMathematics 1st ed New York W W Norton 1903 Available in paperback B Russell and A N Whitehead Principia Mathematica 1st ed Cambridge Cambridge University Press 1910 Available in paperback B Russell Introduction to Mathematical Philosophy New York Simon and Schuster 1920 Available in paperback A Heyting Intuitionism An Introduction Amsterdam 1966 I bid A A Fraenkel Y BarHillel and A Levy Foundations of Set Theory Amsterdam 1973 M Dummett Elements of Intuitionz39sm Oxford Clarendon Press 1977 A S Troelstra Choice Sequences Oxford Oxford University Press 1977 As per 2 An upsidedown sansserifV 1800 rotation of the symbol described in 3 As per 3 A right arrow like 9 A doubleheaded arrow like 69 As per 9 As per 10 Like a sansserif lowercase Greek epsilon 8 J van Heijenoort From F rege to Godel Cambridge MA Harvard University Press Available in paperback P Benacerraf and H PutnamPhilosophy ofMathemalics New YorkPrenticeHall 1964

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