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by Corry Shores
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[Walter P. van Stigt, entry directory]
[Stigt, “Brouwer’s Intuitionist Programme,” 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
Walter P. van Stigt
“Brouwer’s Intuitionist Programme”
in
From Brouwer to Hilbert:
The Debate on the Foundations of Mathematics in the 1920’s
Part I.
L.E.J. Brouwer
Ch1.:
“Brouwer’s Intuitionist Programme”
1.1
The Intuitionist-Formalist Controversy
Brief summary:
(1.1.1) Brouwer’s text “Foundations of Set Theory Independent of the Principle of Excluded Middle” diagnosed a crisis in the foundations of mathematics. This was noted by and had strong influence on Hermann Weyl. In his “Intuitionist Set Theory,” “Brouwer set out the consequences for established mathematics of his Intuitionist theses, in particular, his rejection of the logical Principle of the Excluded Middle: ‘the use of the Principle of the Excluded Middle is not permissible as part of a mathematical proof ... [it] has only scholastic and heuristic value, so that theorems which in their proof cannot avoid the use of this principle lack all mathematical content.’” (1.1.2) Hilbert saw Brouwer’s program as posing a threat to Cantorian set theory and to his own program, so he launched a counterattack in 1922. This begins the Intuitionist-Formalist debate of the 1920s. (1.1.3) The Intuitionist-Formalist debate revolved around two issues: {1} “The nature of mathematics: either human thought-construction or theory of formal structures” and {2} “The role of the Principle of the Excluded Middle in mathematics and Brouwer’s restrictive alternative logic” (2). (1.1.4) “Brouwer’s main concern was the nature of mathematics as pure, ‘languageless’ thought-construction” (2). His program aimed to convince the mathematical world of this view. His “Intuitionist Splitting of the Fundamental Notions of Mathematics” opened debate among logicians regarding an alternative “Brouwer Logic,” (including contributions by Kolmogorov, Borel, Wavre, Glivenko, and Heyting). But Brouwer was not concerned with logic, because he believed that “logic and formalization were ‘an unproductive, sterile exercise’ with no direct relevance to mathematics and its foundations” (2). (1.1.5) Hilbert’s program retains classical mathematics and bases “its validity on a proof of the consistency of its formalization” (3). Brouwer’s program had a constructive interpretation of mathematics. Both held some interest in the mathematical community, but Brouwer’s program failed to gain traction. (1.1.6) “The Brouwer-Hilbert debate grew increasingly bitter and turned into a personal feud” (3). Hilbert expels Brouwer from the board of the Mathematische Annalen. (1.1.7) Brouwer’s professional rejection led him to stop publicizing his program, even at the same time that Hilbert’s formalist program was shown to be fundamentally flawed.
[Brouwer’s Rejection of the Principle of Excluded Middle and Its Crisis for Mathematical Foundations]
[Weyl and Brouwer versus Hilbert in the Intuitionist-Formalist Debate]
[The Two Issues of the Debate]
[Brouwer’s Immediate Influence in Mathematics and Logic]
[The Immediate Reception of Hilbert’s and Brouwer’s programs]
[The Personal Side of the Brouwer-Hilbert Debate]
[The End of Brouwer’s Program]
Summary
[Brouwer’s Rejection of the Principle of Excluded Middle and Its Crisis for Mathematical Foundations]
[Brouwer’s text “Foundations of Set Theory Independent of the Principle of Excluded Middle” diagnosed a crisis in the foundations of mathematics. This was noted by and had strong influence on Hermann Weyl. In his “Intuitionist Set Theory,” “Brouwer set out the consequences for established mathematics of his Intuitionist theses, in particular, his rejection of the logical Principle of the Excluded Middle: ‘the use of the Principle of the Excluded Middle is not permissible as part of a mathematical proof ... [it] has only scholastic and heuristic value, so that theorems which in their proof cannot avoid the use of this principle lack all mathematical content.’” ]
[ditto]
In 1920 Hermann Weyl diagnosed “a new crisis in the foundations of mathematics” (Weyl 1921), sparked off by the publication of Brouwer’s “Foundations of Set Theory Independent of the Principle of Excluded Middle” (B1918B and B1919A). In a series of lectures at the Mathematical Colloquium of Zürich, he dramatically renounced his own Das Kontinuum and hailed Brouwer’s set theory and interpretation of the continuum as “the revolution”: “… und Brouwer – das ist die Revolution!” (Weyl 1921, p. 99), the one mathematician who at last had solved the problem of the continuum, which since ancient times had defeated even the greatest | minds. At the same time, in “Intuitionist Set Theory” (B 1919D) Brouwer set out the consequences for established mathematics of his Intuitionist theses, in particular, his rejection of the logical Principle of the Excluded Middle: “the use of the Principle of the Excluded Middle is not permissible as part of a mathematical proof ... [it] has only scholastic and heuristic value, so that theorems which in their proof cannot avoid the use of this principle lack all mathematical content.” (p. 23)
(1-2)
[Weyl and Brouwer versus Hilbert in the Intuitionist-Formalist Debate]
[Hilbert saw Brouwer’s program as posing a threat to Cantorian set theory and to his own program, so he launched a counterattack in 1922. This begins the Intuitionist-Formalist debate of the 1920s.]
[ditto]
Both Brouwer’s challenge and Weyl’s support raised the alarm among the Cantorian and Formalist establishment of Gottingen. Hilbert, who had recognized Brouwer’s major contribution to topology and had welcomed him as a member of his inner circle, grew increasingly impatient with his old friend and alarmed by the implied threats to Cantorian set theory and his own programme. He launched a counterattack in 1922:
What Weyl and Brouwer do amounts in principle to following the erstwhile path of Kronecker: they seek to ground mathematics by throwing overboard all phenomena that make them uneasy ... if we follow such reformers, we run the danger of losing a large number of our most valuable treasures. (Hilbert 1922, p. 200)
The ensuing Intuitionist-Formalist “debate” dominated the foundational scene throughout the 1920s. Brouwer and Hilbert remained the main protagonists, each drawing support for his cause beyond national frontiers and an even greater audience of interested observers and commentators.
(2)
[The Two Issues of the Debate]
[The Intuitionist-Formalist debate revolved around two issues: {1} “The nature of mathematics: either human thought-construction or theory of formal structures” and {2} “The role of the Principle of the Excluded Middle in mathematics and Brouwer’s restrictive alternative logic” (2).]
[ditto]
The debate centered on two different, though related, issues:
1. The nature of mathematics: either human thought-construction or theory of formal structures;
2. The role of the Principle of the Excluded Middle in mathematics and Brouwer’s restrictive alternative logic.
(2)
[Brouwer’s Immediate Influence in Mathematics and Logic]
[“Brouwer’s main concern was the nature of mathematics as pure, ‘languageless’ thought-construction” (2). His program aimed to convince the mathematical world of this view. His “Intuitionist Splitting of the Fundamental Notions of Mathematics” opened debate among logicians regarding an alternative “Brouwer Logic,” (including contributions by Kolmogorov, Borel, Wavre, Glivenko, and Heyting). But Brouwer was not concerned with logic, because he believed that “logic and formalization were ‘an unproductive, sterile exercise’ with no direct relevance to mathematics and its foundations” (2).]
[ditto]
Brouwer’s main concern was the nature of mathematics as pure, “languageless” thought-construction. He had set himself the task of bringing the mathematical world around to his view, convincing them of the need for reform, and had started the programme of reconstructing mathematics on an Intuitionist basis. Most of his publications in the period 1918-1928 were part of this programme; only a few dealt directly with the “negative” aspects of his Intuitionist campaign: the misuse of logic, in particular the Principle of the Excluded Middle, and the flaws in the Formalist programme. Understandably these papers aroused greater interest and further controversy. His excursion into the field of logic (“Intuitionist Splitting of the Fundamental Notions of Mathematics,” Bl923C), in which he drew the immediate conclusions from his strict interpretation of negation and his rejection of the Principle of the Excluded Middle, created considerable excitement among logicians and started a debate about an alternative, “Brouwer Logic.” This debate was joined by Kolmogorov, Borel, Wavre, Glivenko, Heyting, and others (see Part IV). Brouwer himself did not take a further active part, remaining true to his conviction that logic and formalization were “an unproductive, sterile exercise” with no direct relevance to mathematics and its foundations.
(2)
[The Immediate Reception of Hilbert’s and Brouwer’s programs]
[Hilbert’s program retains classical mathematics and bases “its validity on a proof of the consistency of its formalization” (3). Brouwer’s program had a constructive interpretation of mathematics. Both held some interest in the mathematical community, but Brouwer’s program failed to gain traction.]
[ditto]
The main Intuitionist-Formalist “debate” was a contest between the leaders of two opposing philosophies of mathematics, each with its own programme and competing for the support of the mathematical world. Apart from the occasional direct | exchange, each camp concentrated on its own programme. Hilbert’s Programme, retaining the “whole treasure of classical mathematics” and basing its validity on a proof of the consistency of its formalization, attracted widening support and an able team of collaborators. Brouwer’s constructive interpretation of mathematics, much in line with the natural outlook of the working mathematician, was enthusiastically received and raised early hopes. However, his austere programme of reconstruction within the Intuitionist constraints failed to gather momentum. His increasing isolation was partly due to his inability to work with others, but more important, Brouwer’s and Weyl’s hopes that the “natural” Intuitionist approach would lead to a simplification of reformed mathematics did not materialize. Indeed, it proved “unbearably awkward” in comparison with traditional mathematics relying on the methods of classical logic. Even Weyl had to accept this with regrets:
Mathematics with Brouwer gains its highest intuitive clarity. He succeeds in developing the beginnings of analysis in a natural manner, all the time preserving the contact with Intuition much more closely than had been done before. It cannot be denied, however, that in advancing to higher and more general theories the inapplicability of the simple laws of classical logic eventually results in an almost un bearable awkwardness. And the mathematician watches with pain the larger part of his towering edifice, which he believed to be built of concrete blocks, dissolve into mist before his eyes. (Weyl 1949, p. 54)
(2-3)
[The Personal Side of the Brouwer-Hilbert Debate]
[“The Brouwer-Hilbert debate grew increasingly bitter and turned into a personal feud” (3). Hilbert expels Brouwer from the board of the Mathematische Annalen.]
[ditto]
The Brouwer-Hilbert debate grew increasingly bitter and turned into a personal feud. The last episode was the “Annalenstreit,” or, to use Einstein’s words, “the frog-and-mouse battle.” It followed the unjustified and illegal dismissal of Brouwer from the editorial board of the Mathematische Annalen by Hilbert in 1928 and led to the disbanding of the old Annalen company and the emergence of a new Annalen under Hilbert’s sole command but without the support of its former chief editors, Einstein and Carathéodory.
(3)
[The End of Brouwer’s Program]
[Brouwer’s professional rejection led him to stop publicizing his program, even at the same time that Hilbert’s formalist program was shown to be fundamentally flawed.]
[ditto]
For Brouwer it was the last straw. His failure to “simplify” Intuitionist methods and make Intuitionism the universally accepted mathematical practice had eroded his self-confidence. The conspiracy of his fellow Annalen editors and “lack of recognition” left him bitter and disillusioned. He abandoned his Intuitionist Programme and withdrew into silence just about the time when the Formalist Programme was shown to be fundamentally flawed. Some “books” were left uncompleted and unpublished. The 1928 “Vienna Lectures,” The Structure of the Continuum (B1930A) and “Mathematics, Science and Language” (B1929), and his paper “Intuitionist Reflections on Formalism” (B1928A2) mark the end of Brouwer’s creative life and his Intuitionist campaign. They reflect the stage his programme had reached and the mood of its founder at the time. The Structure of the Continuum summarizes his Intuitionist vision and analysis of the continuum. In “mathematics, Science and Language” he returns to the pessimism of his philosophy of science and language, which had inspired his Intuitionist rebellion. “Intuitionist Reflections on Formalism” is Brouwer’s final assessment of the state of play in the contest between Intuitionism and Formalism and an emotional outburst at the lack of recognition. It lists outstanding differences as well as “the Intuitionist Insights” adopted by Formalists “without proper mention of authorship,” such as the notion of meta-mathematics.
(3)
From:
Stigt, Walter P. van. (1989). “Brouwer’s Intuitionist Programme” 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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