Eric Schmid – Transcendental Mathematics Book

In Transcendental Mathematics, Eric Schmid confronts the fundamental epistemological problem that has haunted philosophy since Kant’s fourfold division of knowledge into analytic a priori, synthetic a priori, synthetic a posteriori, and the notoriously empty fourth cell of analytic a posteriori, a schema whose boundaries have been successively contested throughout analytic philosophy, from Carnap’s logical empiricism (which sought to collapse the synthetic a priori into the analytic a priori through linguistic convention) through Quine’s holistic dissolution of the analytic-synthetic distinction altogether, to Per Martin-Löf’s constructive type theory, which, as Neil Tennant documents in The Taming of the True, reconstitutes the synthetic a priori through a meaning-theoretic framework where mathematical truth emerges through judgment acts rather than abstract correspondence. Schmid argues that homotopy type theory provides the technical apparatus to reconcile the perennial tension between Platonist mathematical realism (mathematics as the discourse on being qua being, independent of mind) and Husserlian phenomenology (mathematical objects as constituted through intentional acts of categorial intuition), positions that seemed irreconcilable yet both partially correct. The work’s significance is precisely this: where Kant bequeathed philosophy the problem of how mathematical knowledge could be both necessary and ampliative, and where twentieth-century philosophy either eliminated the synthetic a priori (Quine) or absorbed it into the analytic (Carnap), Schmid’s “transcendental mathematics” shows that identity types interpreted as paths, and the univalence axiom equating equivalence with identity, ground the synthetic a priori in homotopy-invariant structure, so that mathematical objects are simultaneously constructed through constitutive activity (the phenomenological moment of proof and judgment) and objectively constrained by the geometry of type space (the realist moment of path structure), making the conditions of mathematical thought transcendentally necessary in virtue of the intrinsic homotopical character of identity itself.

Eric Schmid earned an MS in Applied Mathematics with Distinction from DePaul University (2025) after briefly attending the University of Chicago’s MS in Computer Science program. He holds a BA from New York University, where he studied Continental Philosophy and Visual Art with a Minor in Mathematics. Schmid is a Member of Technical Staff at a stealth AI startup, where he works on domain-specific languages, categorical semantics, and applied category theory. In 2007, he co-authored a paper on the economics of targeted drugs in Targeted Oncology as part of an internship at Northwestern University Feinberg School of Medicine. He has co-organized two conferences at the Emily Harvey Foundation on the philosophy of science and logic, and in 2024 published a conversation between logician Prof. Colin McLarty and philosopher of mathematics Dr. Andrei Rodin on his publishing imprint. His 2022 book Prolegomenon to a Treatise on Mathematical Structuralism, Constructive Computationalism, De-ontologized Metaphysics of Hermeneutics, and the Synthetic A Priori was published by Bauer Verlag. Preprints include “Applied Sheaf Theory For Multi-agent Artificial Intelligence (Reinforcement Learning) Systems: A Prospectus” and, with F. Tohmé and W. Chin, “Sheaves on Quivers via a Grothendieck Topology on the Path Category.” As an artist, Schmid has exhibited at Vilma Gold (London), Neue Alte Brücke (Frankfurt), Croy Nielsen (Berlin/Vienna), Svetlana (New York), Triest (New York), and Galleria Federico Vavassori (Milan). He has released music on Mille Plateaux, Recital, Regional Bears, Bánh Mì Verlag, Careful Catalog, Robert & Leopold, Reserve Matinee, Index Clean, Pentiments, fqw, and Psychic Liberation. He has been published in a monograph of Yngve Holen’s work (Distanz Verlag), previously hosted a monthly podcast on Anòmia Radio, and sometimes appears on Montez Press Radio.