Heyting’s Arguments

Part of the Synthese Library book series (SYLI, volume 279)


There is rather conflicting evidence on how Arend Heyting, probably Brouwer’s most faithful disciple, assessed his master’s philosophical arguments for the intuitionistic revision of mathematics. On the one hand, he faithfully quotes Brouwer’s remarks explaining the concept of the intuition of two-ity, or Brouwer’s views on the relation between mathematics and logic. On the other hand, however, the way Brouwer’s views are quoted or explained gives the impression that Heyting either holds them irrelevant for the intuitionistic program, or may even be inclined to maintain a certain distance from his teacher’s philosophical positions. As we shall see, Brouwer’s Kantian approach, which consists in speculation about the epistemic abilities of the knowing subject and consequently which mathematical objects are allowable, is discarded. In a similar vein, Brouwer’s concepts of consciousness, mind, causal attention, and the like play no role in Heyting’s argument for intuitionism. The notion of mathematical intuition receives a new reading. The concept of number is introduced in psychological terms and related to a faculty that serves to discern entities in somebody’s mental content. Furthermore, Brouwer’s overall negative appraisal of logic is replaced by a much more liberal stance that allows for investigations of the logic of intuitionistic mathematics. There is a considerable shift of attitude towards philosophy as well. Heyting claims that “no philosophy is needed to understand intuitionistic mathematics” and that intuitionistic mathematics is simpler than any philosophy. (1974, p. 79) He therefore does not attempt to justify the intuitionistic revision philosophically. The superiority of intuitionistic mathematics is argued on the grounds of its simplicity, higher subtlety or even its independence from controversial philosophical assumptions. As a result of these changes, much of the militant spirit that was associated with Brouwer’s attack on classical mathematics is gone. They also suggest that coming to Heyting’s arguments for intuitionism is like entering a new landscape. Let us study it in more detail.


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© Springer Science+Business Media Dordrecht 1999

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