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L. E. J. Brouwer, the founder of mathematical intuitionism, believed that mathematics and its objects must be humanly graspable. He initiated a program rebuilding modern mathematics according to that principle. This book introduces the reader to the mathematical core of intuitionism – from elementary number theory through to Brouwer's uniform continuity theorem – and to the two central topics of 'formalized intuitionism': formal intuitionistic logic, and formal systems for intuitionistic analysis. Building on that, the book proposes a systematic, philosophical foundation for intuitionism that weaves together doctrines about human grasp, mathematical objects and mathematical truth.
All cognition, that is, all presentations consciously referred to an object, are either intuitions or concepts. Intuition is a singular presentation (repraesentatio singularis), the concept is a general (repraesentatio per notas communes) or reflected presentation (repraesentatio discursiva).
(Jäsche Logic, §1)
In whatever manner and by whatever means a cognition may relate to objects, intuition is that through which it is in immediate relation to them, and to which all thought as a means is directed. But intuition takes place only in so far as the object is given to us.
(Critique of Pure Reason, A19/B33)
Space is not a discursive or, as we say, general concept of relations of things in general, but a pure intuition.
Charles Parsons has taught us that the Kantian conception of intuition is a multi-faceted notion and that this complexity affects Kant's philosophy of mathematics. In this essay, I focus on these two lessons, but also broaden them a bit. Specifically, I have three goals:
Parsons has taught us that the notion of immediacy – which he interprets phenomenologically – must be separately added to the traditional criterion of singularity that has been stressed by all commentators on Kant's definition of intuition. In this essay, however, I shall point out that Kant offers not two but three marks of human intuition: There is singularity; there is, as Parsons insists, immediacy; and there is also something I shall call “reference.” Kant calls it the “object givingness” of intuition. It is there quite clearly at A19.
L. E. J. Brouwer and David Hubert, two titans of twentieth-century mathematics, clashed dramatically in the 1920s. Though they were both Kantian constructivists, their notorious Grundlagenstreit centered on sharp differences about the foundations of mathematics: Brouwer was prepared to revise the content and methods of mathematics (his “Intuitionism” did just that radically), while Hilbert's Program was designed to preserve and constructively secure all of classical mathematics.
Hilbert's interests and polemics at the time led to at least three misconstruals of intuitionism, misconstruals which last to our own time: Current literature often portrays popular views of intuitionism as the product of Brouwer's idiosyncratic subjectivism; modern logicians view intuitionism as simply applying a non-standard formal logic to mathematics; and contemporary philosophers see that logic as based upon a pure assertabilist theory of meaning. These pictures stem from the way Hilbert structured the controversy.
Even though Brouwer's own work and behavior occasionally reinforce these pictures, they are nevertheless inaccurate accounts of his approach to mathematics. However, the framework provided by the Brouwer-Hilbert debate itself does not supply an adequate correction of these inaccuracies. For, even if we eliminate these mistakes within that framework, Brouwer's position would still appear fragmented and internally inconsistent. I propose a Kantian framework — not from Kant's philosophy of mathematics but from his general metaphysics — which does show the coherence and consistency of Brouwer's views. I also suggest that expanding the context of the controversy in this way will illuminate Hilbert's views as well and will even shed light upon Kant's philosophy.