This Prolegomenon to a Treatise will examine the possibility for a foundational system which conceptually unifies ordinary mathematics (category theory, topos theory, or homotopy type theory are the options). For that matter, this treatise will discuss structure in the sense of philosophical structuralism (not to be confused with French structuralism) and if category theory provides a structuralism as Awodey suggests. Recent research after the univalence axiom has demonstrated a new contender; does Homotopy Type Theory provide a foundation for mathematics? We will borrow Negarestani’s notion of ‘localization’ and try to provide a more rigorous context for a ‘site’ directly within topos theory, which has ramifications for any ontological framework. Moreover, we will discuss the necessary background ontology to any foundational ‘topos’ which will depend on the use of the Tarski axiom.

We will address the issue of the “synthetic a priori” and whether we can, in the Kantian manner, treat geometry as the “a priori”, either at the macro-level, as Einstein has shown, where the Minkowski spacetime develops a “spacetime” where the only invariant is the speed of light; or on the meso-level, “the relativity of simultaneity” where “distant simultaneity — whether two spatially separated events occur at the same time — is not absolute, but depends on the observer’s reference frame”; or on the micro-level (of neurons) where perception uses a “cut loci” which is a geometric form (from differential geometry) in so that we can develop a “neurogeometry of vision” as Jean Petitot has done.

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Foto Slyvia John