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L. E. J. Brouwer: Toward intuitionistic logic. (English) Zbl 0833.01017

Although his name is connected to intuitionistic logic, L. E. J. Brouwer was no logician. Nevertheless, he had particular attitudes towards basic concepts of logic derived from his definition of mathematics. The paper under review gives a detailed and thoroughgoing account of the development of Brouwer’s logical opinions, starting with his mystical first pamphlet, Leven, kunst en mystiek (1905), up to his last published paper and some unpublished papers from the 1950s.
In Brouwer’s early views logical laws are of linguistic nature and thus a source of weaknesses. They are, e.g., connected to particular languages and cultures. Starting with the conviction that logic, being a collection of banalities, is an irrelevant part of mathematics, he only “expressed a cautious openness to logic during the last years of his life” (p. 307), not at least through the success of Arend Heyting’s axiomatization of intuitionistic logic (cf. ibid., n. 5).
As a guideline for her presentation the author follows Brouwer’s changing analyses of the nonvalidity of the law of excluded middle. She identifies three different interpretations of Brouwer’s during his career: “prior to 1928, it meant that one was unable to maintain that the law holds; later, it meant that the general law was contradictory; and finally it took on both meanings simultaneously” (p. 313).
In a last chapter, the author discusses Brouwer’s contributions to the description of the domain of validity of the law of excluded middle, especially his considerations concerning the “law of reciprocity of complementarity”, \(\bigwedge_P \bigwedge_x(\neg\neg Px\to Px)\), and the “law of testability”, \(\bigwedge_P \bigwedge_x (\neg Px\vee \neg\neg Px)\).

MSC:

01A60 History of mathematics in the 20th century
03-03 History of mathematical logic and foundations

Biographic References:

Brouwer, L. E. J.
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