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1. History 2: The Kantian Matrix, Which Grants Mathematics a Constitutive Intermediary Epistemological Position MAME 7113 Continuation....

2. Evolution of Kantian Ideas in Nineteenth-Century Philosophy The Intuitionist Response Synthetic A Priori and Formalism in the Early Twentieth Century Kant’s Exclusion of Complete Infinity The Rise of Logical Positivism and Contemporary Tensions

3. 1. Evolution of Kantian Ideas in Nineteenth-Century Philosophy Fries, among other philosophers, viewed certain elements of reason as figments of reason's creative imagination. He argued that just because certain ideas are analytic and a priori, it does not necessarily follow that they are real. This approach, however, was not satisfactory for the leading nineteenth-century philosophers. While the extravagant claims of German idealism eventually subsided, the trend of reconciling reason with reality continued, albeit in less ambitious forms. By the time of Gottlob Frege (1848–1925), and partly due to his own efforts, logic had been developed to a point where it could describe a wide variety of arithmetic structures and derivations.

4. "The thought we have expressed in the Pythagorean theorem is timelessly true, true independently of whether anyone takes it to be true. It needs no owner. It is not true only from the time when it is discovered; just as a planet, even before anyone saw it, was in interaction with other planets" (Frege, 1984, p. 363).

5. 2.The Intuitionist Response This refers to Kant’s notion of "pure intuition," the experience of space and time that precedes empirical observation. Mathematicians such as Henri Poincaré and L.E.J. Brouwer sought to retain mathematics' special status as an "in-between"-neither fully empirical nor purely abstract.

6. 3. Synthetic A Priori and Formalism in the Early Twentieth Century Around the same period, Hermann Cohen and Ernst Cassirer took a different approach, preferring to follow Kant by adhering to something like the synthetic a priori. However, instead of rejecting higher infinitary analysis, they chose to enrich it. Cohen (1883) considered the infinitesimal as the distinguished synthesizing foundation of mathematics, while Cassirer (1910) adopted modern notions of function and relation, rendering the focus on mathematical "objects" obsolete.

7. 3. Synthetic A Priori and Formalism in the Early Twentieth Century Hilbert's formalism, on the other hand, seems more aligned with Fries’s articulation of arithmetic versus syntax. Hilbert's meta-language constituted a minimal mathematics that aimed to respect the demands of intuitionists, logicists, and nominalists who sought to reduce mathematics to a formal abstraction based on empirical counting. Unlike Cohen and Cassirer, Hilbert was less concerned with Kant’s synthetic a priori and more focused on a foundational approach that avoided philosophical disputes.

8. 4. Kant’s Exclusion of Complete Infinity Due to the Kantian exclusion of complete infinity from intuition, operations involving complete infinities and infinitesimals were to be considered strictly syntactic (Fries, 1822, pp. 280, 294). Mathematical operations with infinities, as used by mathematicians such as Wallis, Euler, and Fontenelle (Boyer, 1959), had no grounding in empirical experience and were therefore relegated to what Fries saw as games of reason.

9. 5. The Rise of Logical Positivism and Contemporary Tensions Logical positivism, which emerged in the early twentieth century, sought to dismantle mathematics' intermediary position. It allowed only empirical synthetic a posteriori assertions and logical analytic a priori ones. This division split mathematical claims into two aspects: an empirical descriptive aspect and a formal syntactic one . Building on this tradition, contemporary realism and nominalism push for a clear choice between these two frameworks, rejecting the dualism.