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by Corry Shores
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[Mancosu & van Stigt, “Intuitionistic Logic,” entry directory]
[The following is summary. I am not a mathematician, so please consult the original text instead of trusting my summarizations. Bracketed comments are my own. Proofreading is incomplete, so please forgive my mistakes.]
Summary of
Paolo Mancosu & Walter P. van Stigt
“Intuitionistic Logic”
in
From Brouwer to Hilbert:
The Debate on the Foundations of Mathematics in the 1920’s
Part IV
“Intuitionistic Logic”
Part IV’s Introduction:
“Intuitionistic Logic”
4.1
Brouwer
Brief summary:
(4.1.1) Brouwer fought his “foundational battle” of intuitionistic mathematics initially on philosophical grounds: {1} he defends his claim that the true nature of mathematics is constructive thought construction against the “logicist-formalist confusion of mathematics with its symbolic representation;” {2} he takes the “first act of Intuitionism” to be “ ‘the separation of mathematics and language,’ exposing the true nature of logic as no more than a science, a mathematical analysis of the symbolic record of a mathematical thought-construction” (275) (see section 1.4.1.2); {3} he particularly “condemned the traditional practice of using logical principles – distilled from past mathematical records – as operators on words and sentences to generate new mathematical truths. The Principle of the Excluded Middle was singled out as obviously flawed when applied to the mathematics of the infinite, its use in his reconstruction of mathematics expressly avoided” (275) (See section 1.4.1.3). (4.1.2) But with his text “Intuitionist Splitting of the Fundamental Notions of Mathematics,” Brouwer “now embarks on an investigation in the field of logic proper” (275). Brouwer even came to soften his critique of the use of language in mathematics. He was part of the “Signific movement,” which called for “new words expressing spiritual values for the languages of western nations,” and which “was particularly relevant to Brouwer’s programme of reconstructing mathematics; new words were needed to represent his new notions and distinctions – and communicate his message to the mathematical world” (276). (4.1.3) In his “Intuitionist Splitting of the Fundamental Notions of Mathematics,” Brouwer fashions new words for some of his intuitionistic ideas (see section 4.1.2). He begins with mathematical truth and with absurdity, which is proven impossibility. From this we learn that the Principle of Excluded Middle and Double Negation (here called “The Principle of Reciprocity of Complementary Species”) are not applicable in intuitionistic mathematics. (While Brouwer will keep this rejection of double negation, he holds something similar, namely that) “Truth implies absurdity-of-absurdity, but absurdity-of-absurdity does not imply truth” (276). (It seems that this means, from an affirmed formula you can derive its doubled negation, but from a double negation you cannot derive its unnegated form. Maybe the idea is the following. Negation of negation is somehow like lacking a constructed proof, but that does not prove the affirmative form. For, it could still be shown to simply be absurd later. But, suppose you have constructed a valid proof of something. Nothing in the future could ever then show it to be absurd. So from a proven formula you can derive its double negation or the absurdity of it being absurd.) But, a triple negation, “Absurdity-of-absurdity-of-absurdity is equivalent with absurdity” (276). Brouwer does not want to use the symbolic notation for negation, because he does not want to portray logical operators as mathematical (and thus as intuitive). But we can eliminate enchained absurdities (reduce negations) in the expected way: “a finite sequence of absurdity predicates can be reduced to either absurdity or absurdity-of-absurdity ... by striking out pairs of absurdity predicates, provided that the last absurdity predicate of the sequence is never included in the cancellation” (276). (4.1.4) Thus in his paper “Intuitionist Splitting of the Fundamental Notions of Mathematics,” Brouwer for the first time makes explicit his rejection of double negation. (4.1.5) Rolin Wavre calls this “Brouwerian Logic” (Logique Brouwerienne), and many others joined the debate about or made contributions to intuitionist logic, including Kolmogorov, Lévy, Avstidisky, Barzin & Errera, Borel, Khinchin, Glivenko, and Heyting. (4.1.6) But Brouwer did not get too much involved in the debate over intuitionistic logic on account of his still negative view of formal logic (see section 1.4.1.2 and section 1.4.1.3). Heyting then takes over the cause of intuitionistic logic.
[Brouwer’s Foundational Battle]
[Brouwer’s Quest for New Words to Express Intuitionism]
[New Terminology for Multiple Negation]
[Brouwer’s First Explicit Rejection of Double Negation]
[The Response in the Logic Community]
[Heyting’s Leadership in Intuitionistic Logic]
Summary
[Brouwer’s Foundational Battle]
[Brouwer fought his “foundational battle” of intuitionistic mathematics initially on philosophical grounds: {1} he defends his claim that the true nature of mathematics is constructive thought construction against the “logicist-formalist confusion of mathematics with its symbolic representation;” {2} he takes the “first act of Intuitionism” to be “ ‘the separation of mathematics and language,’ exposing the true nature of logic as no more than a science, a mathematical analysis of the symbolic record of a mathematical thought-construction” (275) (see section 1.4.1.2); {3} he particularly “condemned the traditional practice of using logical principles – distilled from past mathematical records – as operators on words and sentences to generate new mathematical truths. The Principle of the Excluded Middle was singled out as obviously flawed when applied to the mathematics of the infinite, its use in his reconstruction of mathematics expressly avoided” (275) (See section 1.4.1.3).]
[ditto]
“Intuitionist Splitting of the Fundamental Notions of Mathematics” (B1923C1) is an important landmark in Brouwer’s Intuitionist campaign and in the history of Intuitionism. Brouwer had fought his “foundational battle” so far mainly on philosophical grounds: a defense of the true nature of mathematics as constructive thought construction against the logicist-formalist confusion of mathematics with its symbolic representation. His “first act of Intuitionism” was “the separation of mathematics and language,” exposing the true nature of logic as no more than a science, a mathematical analysis of the symbolic record of a mathematical thought-construction. His paper “The Unreliability of the Logical Principles” (B1908C) in particular condemned the traditional practice of using logical principles – distilled from past mathematical records – as operators on words and sentences to generate new mathematical truths. The Principle of the Excluded Middle was singled out as obviously flawed when applied to the mathematics of the infinite, its use in his reconstruction of mathematics expressly avoided.
(275)
[Brouwer’s Quest for New Words to Express Intuitionism]
[But with his text “Intuitionist Splitting of the Fundamental Notions of Mathematics,” Brouwer “now embarks on an investigation in the field of logic proper” (275). Brouwer even came to soften his critique of the use of language in mathematics. He was part of the “Signific movement,” which called for “new words expressing spiritual values for the languages of western nations,” and which “was particularly relevant to Brouwer’s programme of reconstructing mathematics; new words were needed to represent his new notions and distinctions – and communicate his message to the mathematical world” (276).]
[ditto]
With the “Intuitionist Splitting of the Fundamental Notions of Mathematics,” Brouwer’s approach to logic enters a new phase. While maintaining his fundamental stand on the separate identities of mathematics and logic and the nonproductive role of logic in mathematics, he now embarks on an investigation in the field of logic proper. During the past few years there had been a marked softening of his | negative appraisal of language. He had become actively involved in the “Signific movement” of linguistic reform. Despair of the possibility of human communication had (temporarily) made place for a more optimistic conviction that the instrument of language is capable of improvement. The Manifesto of his newly created “Signific Circle” expressed as its aim “the coining of new words expressing spiritual values for the languages of western nations.” The creation of new words was particularly relevant to Brouwer’s programme of reconstructing mathematics; new words were needed to represent his new notions and distinctions – and communicate his message to the mathematical world.
(275-276)
[New Terminology for Multiple Negation]
[In his “Intuitionist Splitting of the Fundamental Notions of Mathematics,” Brouwer fashions new words for some of his intuitionistic ideas (see section 4.1.2). He begins with mathematical truth and with absurdity, which is proven impossibility. From this we learn that the Principle of Excluded Middle and Double Negation (here called “The Principle of Reciprocity of Complementary Species”) are not applicable in intuitionistic mathematics. (While Brouwer will keep this rejection of double negation, he holds something similar, namely that) “Truth implies absurdity-of-absurdity, but absurdity-of-absurdity does not imply truth” (276). (It seems that this means, from an affirmed formula you can derive its doubled negation, but from a double negation you cannot derive its unnegated form. Maybe the idea is the following. Negation of negation is somehow like lacking a constructed proof, but that does not prove the affirmative form. For, it could still be shown to simply be absurd later. But, suppose you have constructed a valid proof of something. Nothing in the future could ever then show it to be absurd. So from a proven formula you can derive its double negation or the absurdity of it being absurd.) But, a triple negation, “Absurdity-of-absurdity-of-absurdity is equivalent with absurdity” (276). Brouwer does not want to use the symbolic notation for negation, because he does not want to portray logical operators as mathematical (and thus as intuitive). But we can eliminate enchained absurdities (reduce negations) in the expected way: “a finite sequence of absurdity predicates can be reduced to either absurdity or absurdity-of-absurdity ... by striking out pairs of absurdity predicates, provided that the last absurdity predicate of the sequence is never included in the cancellation” (276).]
[ditto]
“Intuitionist Splitting” is in fact such an exercise of creating new words, in this case words expressing the various relations between points and between points and species of points. In line with his own rules of correct logical practice, Brouwer starts from his concepts of mathematical truth and absurdity (i.e., proven impossibility), resulting immediately in the inapplicability of the Principle of the Excluded Middle and of what he calls “The Principle of Reciprocity of Complementary Species,” which asserts the equivalence of truth and double negation. He replaces the latter principle by a restricted form of complementarity: “Truth implies absurdity-of-absurdity, but absurdity-of-absurdity does not imply truth.” He then proceeds to analyze the effects of these distinct “correctness predicates,” considered as operators on “assertions of property,” that is, propositions. He states and proves one theorem: “Absurdity-of-absurdity-of-absurdity is equivalent with absurdity.” (Brouwer seems reluctant to use symbols for logical operations as if to emphasize their distinct nonmathematical nature.) The corollary of this theorem and his restricted form of complementarity is “that a finite sequence of absurdity predicates can be reduced to either absurdity or absurdity-of-absurdity ... by striking out pairs of absurdity predicates, provided that the last absurdity predicate of the sequence is never included in the cancellation.”
(276)
[Brouwer’s First Explicit Rejection of Double Negation]
[Thus in his paper “Intuitionist Splitting of the Fundamental Notions of Mathematics,” Brouwer for the first time makes explicit his rejection of double negation.]
[ditto]
The logical analysis of absurdities was intended as an introduction to the main part of the paper, sections 2, 3, and 4, where the absurdity and absurdity-of-absurdity operators are applied to terms expressing relations between points and point species, so generating a variety of new relations. Although it takes up less than one page of the paper, this introductory paragraph is the most important part of “The Intuitionist Splitting.” The inapplicability of the traditional principle of complementarity and double negation was implied in Brouwer’s earlier rejection of the Principle of the Excluded Middle. In this paper, however, Brouwer spelled it out in public and drew the attention of mathematical logicians to its implications for traditional logic.
(276)
[The Response in the Logic Community]
[Rolin Wavre calls this “Brouwerian Logic” (Logique Brouwerienne), and many others joined the debate about or made contributions to intuitionist logic, including Kolmogorov, Lévy, Avstidisky, Barzin & Errera, Borel, Khinchin, Glivenko, and Heyting.]
[ditto]
There was a quick response from Rolin Wavre, who coined the term “Logique Brouwerienne,” and from Kolmogorov. Others joined the debate or made their contributions to the development of Brouwer’s alternative, Intuitionist logic: Lévy, Avstidisky, Barzin and Errera, Borel, Khinchin, Glivenko, and Heyting (see Thiel 1988).
(276)
Thiel, C., 1 988, Die Kontroverse um die intuitionistische Logik vor ihrer Axiomatisierung durch Heyting im Jahre 1930, History and Philosophy of Logic 9, pp. 67-75.
(285)
[Heyting’s Leadership in Intuitionistic Logic]
[But Brouwer did not get too much involved in the debate over intuitionistic logic on account of his still negative view of formal logic (see section 1.4.1.2 and section 1.4.1.3). Heyting then takes over the cause of intuitionistic logic.]
[ditto]
Brouwer himself stayed aloof from the debate, remaining true to his conviction that engagement in logic “is an interesting but irrelevant and sterile exercise.” He approved of the contributions of his loyal student Arend Heyting, “Die formalen Regeln der intuitionistischen Logik” (1930b) and “Die formalen Regeln der intuitionistischen Mathematik” (1930c). When asked by the editor of the Bulletin de | l’Academie Royale de Belgique in 1930 to round off the debate, Brouwer referred him to Heyting’s papers as the authoritative and “masterly” account of Intuitionist Logic. By that time he had already retired from the foundational battlefield and effectively passed the Intuitionist leadership to Heyting, who kept the cause of Intuitionism alive and changed its course. Let us now tum to the debate on intuitionistic logic.
(276-277)
From:
Mancosu, Paolo & Stigt, Walter P. van. (1989). “Intuitionistic Logic” In: From Brouwer to Hilbert: The Debate on the Foundations of Mathematics in the 1920’s, edited by Paolo Mancosu. Oxford: Oxford University.
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