L.E.J. Brouwer And A. Heyting On Foundational Labels .
L.E.J. Brouwer and A.Heyting on foundationallabels: their creation anduse
by Miriam FranchellaDepartment ofHistorical StudiesUniversità degli StudiMilano
The use of the three labels (logicism, formalism,intuitionism) to denote the three foundational schoolsof the early twentieth century is now firmly part of theliterature. Still, they were not introduced by thefounding fathers each for his own school.Furthermore, neither their number nor their adoptionhas been stable over the twentieth century. In this talkwe will see the role and attitude that intuitionists ofthe first era (Brouwer and Heyting) have had in theproduction and use of foundational labels.
I will advance the thesis that not only thecreation but also the use of labels, far frombeing a mere gesture of academic referenceto literature, can be a sign of the culturaloperation each scholar wants to do.
Brouwer’s thesis (1907)In his doctoral thesis Brouwer used the labels at hand at themoment on the foundational “market” (i.e. those mentioned inthe mathematical/philosophical journals where the papers byCantor, Poincaré, Hilbert, Couturat, Russell had appeared).Therefore, we find: “axiomatics”, “cantorianism”and“logistics” (as proposed by Couturat). We do not find either“logicism”, because such foundational label appeared only in1928, or “intuitionism”, because Brouwer had not yet chosenanything specific for his own viewpoint.
Brouwer analyzed the various foundationalapproaches starting from his own: his own wasthe right one and the differences with itexplained the failure of other approaches.On this basis, he began to analyse : 1) thefoundations on axioms, 2) Cantor’s transfinitenumbers, 3) the Peano-Russell logistic, 4) thelogical foundation after Hilbert.
The first field embraced the “recent investigations ofPasch, Schur, Hilbert, Peano, Pieri”, that had pointed outsome “holes” in Euclidean treatment of geometry: theyhave “convincingly demonstrated that [Euclideangeometry] as a logical linguistic structure is imperfect,namely that here and there tacitly axioms areintroduced”. However, Brouwer stigmatized that theirvery target were not Euclidean imperfections, butpathological geometries. “They constructed linguisticstructures and then they required a proof of consistency,but nobody proved that consistency was a conditionsufficient for existing mathematically.“
The second field that Brouwer consideredwas Cantorianism, that he criticizedbecause “Cantor loses contact with thefirm ground of mathematics” (CW I, 81) inhis definition of the second number class.Namely, the concept of “den Inbegriff aller’mentioned something which cannot bethought of, for a totality constructed bymeans of ‘and so on’ could only be thoughtof if ‘and so on’ referred to an ordertype of equal objects, and this was not thecase.
Then he passed to Peano and Russell’s treatment of logic.Brouwer saw it as an attempt to put a remedy to the fact thatclassical logic was inadequate for mathematics. “Thelogisticians consider the propositional functions as the freeorigin of logic and mathematics [ ] But in the intellect onecannot give a linguistic system of statements andpropositional functions priority over mathematics, for noassertions about the external world can be intelligentlymade besides those that presuppose a mathematicalsystem that has been projected on the external world” (CWI, p. 88). Therefore it was not surprising that they, like theCantorians, came up against contradictions.
The conclusion that he drew about logistics was that itwas not suitable as a foundation of mathematicsbecause it was separated from mathematics and the bestto which it could aim was being a faithful stenographiccopy of the language of mathematics (“which itself is notmathematics but no more than a defective expedient formen to communicate mathematics to each other and toaid their memory for mathematics” - CW I, p. 92).
Thereafter he turned back to Hilbert, that gave“the most uncompromising conclusion of themethods we attack, which illustrates mostlucidly their inadequacy” (CW I, p. 92).He stressed that Hilbert was even morecriticizable than “the logicians” because in hisworks it was possible to find a list of stages thatwere confused:
1. Construction of intuitive mathematical systems2. Mathematical speaking or writing (the expression of 1)3. The mathematical study of language: “we notice logicallinguistic structures, raised according to principles fromordinary logic or through the logic of relations”4. Forgetting the sense of the elements of the logical figures in3. and imitating the construction of these figures by a newmathematical system of second order.5. The language that may accompany 4.6. The mathematical study of language 5.7. Forgetting the sense of the elements of the logical figures in6. and imitating the construction of these figures by a newmathematical system of third order.8. The language that may accompany 5.
1911 It was only in the 1911 review of the Mannouryvolume "Methodologisches Philosophisches undzur Elementar-Mathematik" that Brouwer suddenlyintroduced the contraposition intuitionism vsformalism (that was not present in Mannoury’sbook ): “[ ] the author defends the ‘formalist’conception, which has also been advocated byDedekind, Peano, Russell, Hilbert and Zermelo,against ‘intuitionists’ like for instance Poincaré andBorel.
This formalist conception recognizes no other mathematicsthan the mathematical language and it considers itessential to draw up definitions and axioms and to deducefrom these other propositions by means of logicalprinciples which are also explicitly formulated beforehand”(CW I, 121).
After defining formalism, Brouwer wondered whatcould be the reason for accepting those axioms andrecalled that Russell’s answer was to verify thelogical existence of mathematical entities, whileHilbert’s answer had been the project to verify thatthe logical figure of ‘contradiction’ could not bederived by the axioms. He ended by stressing thatboth Russell and Hilbert could not do without “theintuitive application of complete induction” andtherefore “they have invigorated by their reasoningrather intuitionism than formalism” (CW I, 121).
Furthermore, he added that Mannoury couldcriticise intuitionism because he had in mindonly Poincaré’s version of it, that presented twoweak points: the rejection of every infinitenumber, including the denumerable, and theidentification of mathematical existence withnon-contradictoriness. Brouwer was sure that“it is only after these mistakes have beenredressed, and after the basic intuition of twoity has been accepted, that intuitionismbecomes invulnerable” (p. 122).
In Mannoury’s book were described two oppositeviewpoints about mathematics:Kantianism andSymbolism. Thus, the question arises how Brouwercame to employ the labels ositionKantianism/Symbolism in the index of the book (p.262), inside the third group of subjects belonging to thefirst chapter (of the second part of the book) devoted tomathematical logic: “Kritik des Symbolischen Logik; dieBeurteilung der Widerspruchslosigkeit der logischenFormeln; Kantianismus und Symbolismus (139-149)”.In the inner pages of the book we find “Kantianismus”as attached mainly to Poincaré, defined as “thetalented representant of Kantianism in mathematics (p.144).
The authors quoted on the opposite side are Peano,Couturat and Hilbert. Mannoury specifies that among other(unnamed) Kantianists there is Aurel Voss, that, in his 1908lecture “Über das Wesen der Mathematik”, defended “thehigher meaning of mathematics, in opposition toformalism”. Therefore, the word “Formalism” for theenemies of “Kantianism” is present in a footnote.
We can imagine that Brouwer decided to changethe label “Kantianism” for his foundational schoolin order to point out the novelty of his ownposition (even if he admitted his ‘debt’ to Kant).Since the intuition of two-ity had been indicatedby himself as the key for granting theunvulnerability of his position, it seemscomprehensible that he used the label“intuitionism”. Once he decided to change themain of the two labels, it could have come naturalto him to change also the other, that was more“unstable” inside the book and that was alreadysubstituted by “formalism” in a footnote.
It should be considered also that Felix Klein, in his firstEvanston lecture (1893) had distinguished three maincategories among mathematicians - logicians, formalistsand intuitionists: “the word logician is here used, of course,without reference to the mathematical logic of Boole, Peirce,etc; it is only intended to indicate that the main strength ofthe men belonging to this class lies in their logical andcritical power, in their ability to give strict definitions, and toderive rigid deductions therefrom” (p. 2). As an example hequoted Weierstrass.
“The formalists excel mainly in the skillful formaltreatment of a given question, in devising for it an“algorithm”. “To the intuitionists, finally, belongthose who lay particular stress on geometricalintuition (Anschauung), not in pure geometry only,but in all branches of mathematics. What BenjaminPeirce has called ‘geometrizing a mathematicalquestion’ seems to express the same idea”.Examples: Lord Kelvin and von Staudt.We see that the content of the categories isdifferent from Brouwer’s meaning, but the labels“formalism” and “Intuitionism” had been coined.Brouwer could have been inspired by them
1912In 1912, in his introductory lecture “Intuitionism andformalism”, he stated that there were two mainpoints of view on what grounded the exactness ofmathematics: “The question where mathematicalexactness does exist, is answered differently by thetwo sides; the intuitionist says: in the humanintellect; the formalist says: on paper” (CW I, p. 125).
He traced back an old form of intuitionism in Kant,but added that Kant’s intuitionism was weak. Itbecame more credible when he abandoned apriorityof space and built mathematics on the only intuitionof time. “In the construction of these sets neither theordinary language nor any symbolic language canhave any other role than that of serving as a nonmathematical auxiliary, to assist the mathematicalmemory or to enable different individual to build upthe same set.” (CW I, p. 128).
Brouwer passed to describe the formalists as scholarsthat started from the belief that human reason did nothave at its disposal exact images either of straight linesor of large numbers (numbers larger than three, forexample). Then, they concluded that such entities “donot have existence in our conception of nature any morethan in nature itself”, but they grounded their nonmathematical conviction of legitimacy of their systems onthe efficacy of their projection into nature.
“For the formalist, therefore, mathematical exactnessconsists merely in the method of developing the seriesof relations [ ] And for the consistent formalist thesemeaningless series of relations to which mathematicsare reduced have mathematical existence only whenthey have been represented in spoken or writtenlanguage together with the mathematical-logical lawsupon which their development depends, thus formingwhat is called symbolic logic” (CW I, p.125). In order tobe sure of the consistency of the language that theyused, formalists avoided daily language and introducednew ones. Here we find Peano labelled as formalist.
According to Brouwer, intuitionists and formalists agreedas for finite sets: in that field the two tendencies differedsolely in their method, not in their results. On the contrary,when infinite sets are considered, “the formalist introducesvarious concepts, entirely meaningless to the intuitionist,such as ‘the set whose elements are the points of space’,‘the set whose elements are the continuous functions of avariable’, etc.” (CW I, p. 130) Brouwer ended his lectureby stating that he saw no hope that an agreement in afinite period could be reached. He quoted Poincaré: “Leshommes ne s’entendent pas, parce qu’ils ne parlent pasle mȇme langue et qu’il y a des langues qui nes’apprennent pas” [The men disagree, because they donot speak the same language and there are languagesthat cannot be learned] (CW I, p. 138).
1928In his 1928 “Intuitionistische Betrachtungen über denFormalismus” Brouwer went on with his „duel“ withformalism and listed four viewpoints that the intuitionistsasserted and that he was sure that, soon or later, alsoformalists would share: there would be the end of the“Grundlagenstreit” and from that time on, the choicebetween formalism and intuitionism would be only aquestion of taste.The four viewpoints mentioned were:1)The formalists aim to build a formal image ofmathematics while they also have an intuitive theory ofthe laws of such a construction. They admit thatintuitionistic mathematics of the integers is indispensablefor their intuitive theory.2)The refusal of a blind application of the principleof excluded middle, that has a sure validity only for finitedomains.
3)The identification of the principle of excludedmiddle with the principle of the solvability of everymathematical problem.4)The knowledge that justifying formalisticmathematics through the proof of its noncontradictoriness contains a vicious circle: it isgrounded on the law that let to pass from thedouble negation to the affirmation; but this law isgrounded, at its turn, on the principle of excludedmiddle.
The first viewpoint, “prepared by Poincaré”, wasexpressed by Brouwer in 1907 [CW I, 94] and shared informalistic literature as testified by the use of the word“metamathematics”. The second viewpoint, thatappeared in Brouwer 1908 for the first time, hadsomehow been accepted by the formalists insofarHilbert admits “die beschränkte inhaltliche Gültigkeit“ ofthe principle of excluded middle. Still, Brouwer noted,he himself extended his doubt to all of Aristotelianlogical laws.The third and the fourth viewpoints had not yet beenaccepted by the formalists.
1928 is also the year of Brouwer’s “BerlinerGastvorlesungen”. Van Dalen published in 1991Brouwer’s manuscripts of the lessons [Brouwer1991]. We see the same structure that we will readin the postwar years. In the first chapter devoted tothe historical position of intuitionism, hedistinguished three periods:1) The first lasted till the 19th century, wascharacterized by the belief in the existence ofproperties of time and space, independent both oflanguage and logic, and was called by Brouwer”Kantian viewpoint”. The period finished as aconsequence of the non-euclidean geometry and ofthe theory of relativity that interrupted the belief inthe Kantian theory of space and let mathematics bebased on the theory of numbers.
2) Therefore, the second period was characterized by thearithmetization of geometry [old-formalist school(Dedekind, Peano, Russell, Couturat, Hilbert, Zermelo) ].In the meanwhile, the pre-intuitionistic school (Kronecker,Poincaré, Borel, Lebesgue) was “completely different”.Still, pre-intuitionism went on with using logic (including theprinciple of excluding middle) in mathematical inferences(Herleitungen). To the same period belongs the newformalistic school, represented by Hilbert, Bernays,Ackermann and von Neumann.3) The third period was characterized by the two acts(Handlungen) of Intuitionism.
Through the first act, it separated mathematicsand logic, and grounded mathematics on thelanguageless activity of mind having its origin inthe perception of time. Through the second act itrecognized the possibility of producing: 1) theinfinitely proceeding sequences of mathematicalentities (previously acquired); 2) hematical entities previously acquired .
We see that Brouwer kept the two labels that he hadintroduced (formalism vs intuitionism), by adding aKantian viewpoint that in the 1912 lecture simplybelonged to “intuitionism” and that in his final workswill disappear at all. Furthermore, he distinguishedinside each label two standpoints: an old (or pre-)one and a new one. It is clear that in this wayBrouwer could stress that his intuitionism was theend point of a finalized path, and not a variant of analready present school.
In “Mathematik, Wissenschaft und Sprache” (talk given inVienna in 1928), Brouwer mentioned the formalisticschool (CW I p. 422) and stated that its fault resided in itsbelief in classical logic. The origin of such beliefconsisted in the fact that their laws were trustable whenthey were referred to finite domains. But thistrustworthiness led man to a superstitious faith in themiraculous power of language. Brouwer opposed againthe intuitionists to the formalists: the intuitionistsdestroyed the confidence in logical laws when applied toinfinite domains by giving counterexamples to the validityof the principle of excluded middle in those domains andcriticized the formalists for building a linguistic castleinstead of a mathematical building.
“Die Struktur des Kontinuums” (second Vienna talk) wasdevoted to explain the novelty of Brouwer’s treatment of thecontinuum through the exploitation of the duo-unity in terms oflawless sequences. In order to do this, Brouwer pointed out thedifferent approaches to the continuum in recent history ofmathematics. The partition among schools was expressed asfollows: his position was called “intuitionist”; then there were theformalists (stemming from Dedekind, Peano, Russell, Zermeloand Hilbert) which paid attention only to mathematical languageand took care of avoiding from the theorization of thecontinuum the production of contradictions; and the oldintuitionists (stemming from Poincaré and Borel) for whom onlythe denumerable part of the continuum had a content, i.e. couldbe built by constructive means starting from the intuition of theduo-unity, while for the more-then-denumerable continuum thereference to a merely linguistic source was necessary.
During the 1928 Brouwer had ahard conflict with Hilbert, describedin detail in van Dalen’s 1990 article“The battle of the frog and themice”. Hilbert re-founded the“Mathematische Annalen” to strikeBrouwer off the editorial committee.Although the other editors tried tobake the pill, Brouwer felt very hurt.Consequently, we find a break ofhis publications, with some smallexceptions, from 1930 to 1948.
Logicism at the horizonIn 1928, in his revised edition of «Einleitung in dieMengenlehre», Abraham Fraenkel distinguished threeBegründungsarten of Set Theory: logicism, intuitionismand formalism, by specifying that he preferred logicism.His preference for the logicism in the sense of “insertionof mathematics into the wider framework of logic” wasnot only due to the hope that the difficulties linked to thereducibility axiom would be solved, but even if it wouldnot happen, insofar as also the other schools were notsuccessful, he would count this as a proof of the limits ofhuman thought regarding such questions
Fraenkeldistinguishedbetween“logistics” and “logicism”. He stressed(1928, 263) that the first label was notunivocal, because it was used in theliterature both to indicate the formalaspects of the new logic that had beendeveloped in the 19th century (algebraof logic, symbolic logic, etc.) and forthe foundational point of view.Therefore, he believed that it wasbetter to indicate the foundationalschool through a specific label:“Logizismus”
In 1929, in his «Abriss der Logistik mit besonderemRücksichtigung des Relationstheorie und ihre Anwendungen»,Carnap presented “Logistik oder symbolische Logik” bystressing that the whole mathematics was a branch of it. In afootnote [p. 2] he listed Ziehen, Meinong and the three scholarsof the Geneva 1904 symposium (Couturat, Itelson, Lalande) asthose who had proposed the label. Then he specified that: “Aphilosophical tendency with a strong or, perha
intuition (Anschauung), not in pure geometry only, but in all branches of mathematics. What Benjamin Peirce has called ‘geometrizinga mathematical question’ seems to express the same idea”. Examples: Lord Kelvin and von Staudt. We see that the content of the categories is different from Brouwer’smeaning, but the labels
Figure 2: Henri Poincar e, left, and Luitzen Egbertus Jan Brouwer, right. Brouwer, in 1910, presented his xed point theorem: 1. Brouwer Fixed-Point Theorem in R: Given that set KˆRnis compact and convex, and that function f: K!Kis continuous, then there exists some c2Ksuch that f(c) c;
risk management in project finance by dr.philippe j.s. de brouwer lecturer at vlerick business school and university of warsaw; and head analytics development at royal bank of scotland group) www.de-brouwer.com * philippe@de-brouwer.com october 3, 2015 university of warsaw cluster: "finance" section: finance
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Brouwer’s Collected Works, on p. 568 in the footnote Heyting, as the editor, stated: ‘Brouwer seems not to have known Frege’s work. He has never mentioned it’. Kuiper (2004, p. 223) stated: That Brouwer was at least familiar with some of Frege’s work and can be concluded from the following:
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