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4. Hilbert’s formalism
Hilbert accepted the synthetic a priori character of (much of) arithmetic and geometry, but rejected Kant’s account of the supposed intuitions upon which they rest. Overall, Hilbert’s position was more complicated in its relationship to Kant’s epistemology than were those of the intuitionists and logicists. Like Russell, he rejected Kant’s specifically mathematical epistemology – in particular, his conception of the nature and origins of its a priori character. Like Russell, too, he rejected the common post-Kantian belief in the epistemological asymmetry of arithmetic and geometry. Hilbert was, however, unique among those mentioned here in endorsing the framework of Kant’s general critical epistemology and making it a central feature of his mathematical epistemology. Specifically, he adopted Kant’s distinction between the faculty of the understanding and the faculty of reason as the guide for his pivotal distinction between the so-called ‘real’ and ‘ideal’ portions of classical mathematics (Hilbert 1926: 376–7, 392).
Hilbert took ‘real’ mathematics to be ultimately concerned with the shapes or forms (Gestalten) of concrete signs or figures, given in intuition and comprising a type of ‘immediate experience prior to all thought’ (1926: 376–7;  1967: 464–5). Hilbert proposed this basic intuition of shape as a replacement for Kant’s two a priori intuitions of space and time. Like Kant’s a priori intuitions, however, Hilbert, too regarded his finitary intuition as an ‘irremissible pre-condition’ of all mathematical (indeed, all scientific) judgment and the ultimate source of all genuine a priori knowledge (1930: 383, 385).
The genuine judgments of real mathematics were the judgments of which our mathematical knowledge was constituted. The pseudo-judgments of ideal mathematics, on the other hand, functioned like Kant’s ideas of reason. They neither described things present in the world nor constituted a foundation for our judgments concerning such things. Rather, they played a purely regulative role of guiding the efficient and orderly development of our real knowledge.
Hilbert did not, therefore, affirm the necessity of either arithmetic or geometry in any simple, straightforward way. Rather, he distinguished two types of necessity operating within both. One, pertaining to the judgments of real mathematics, consisted in the (presumed) fact that the apprehension of certain elementary spatial and combinatorial features of simple concrete objects is a pre-condition of all scientific thought. The other, pertaining to the ideal parts of mathematics, had a kind of psychological necessity, a necessity borne of the manner in which our minds inevitably or best regulate the development of our real knowledge.
This conception of the necessity of mathematics was different from both Kant’s and the logicists’ and intuitionists’. So, too, was Hilbert’s view of the cognitive richness of mathematics, which he attributed both to the objective richness of the shapes and combinatorial features of concrete signs and to the richness of our imaginations in ‘creating’ complementary ideal objects.
In its overall structure, Hilbert’s mathematical epistemology thus resembled Kant’s general critical epistemology. This included his so-called ‘consistency’ requirement (that is, the requirement that ideal reasoning not prove anything contrary to that which may be established by real means), which resembled Kant’s demand that the faculty of reason not produce any judgment of the understanding that could not in principle be obtained solely from the understanding (1781/1787: A328/B385).
Detlefsen, Michael. Hilbert’s formalism. Mathematics, foundations of, 1998, doi:10.4324/9780415249126-Y089-1. Routledge Encyclopedia of Philosophy, Taylor and Francis, https://www.rep.routledge.com/articles/overview/mathematics-foundations-of/v-1/sections/hilberts-formalism.
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