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‚Äôs notion of ‚Äėlocalization‚Äô and try to provide a more rigorous context for a ‚Äėsite‚Äô directly within topos theory, which has ramifications for any ontological framework. Moreover, we will discuss the necessary background ontology to any foundational ‚Äėtopos‚Äô which will depend on the use of the Tarski axiom.
We will address the issue of the ‚Äúsynthetic a priori‚ÄĚ and whether we can, in the Kantian manner, treat geometry as the ‚Äúa priori‚ÄĚ, either at the macro-level, as Einstein has shown, where the Minkowski spacetime develops a ‚Äúspacetime‚ÄĚ where the only invariant is the speed of light; or on the meso-level, ‚Äúthe relativity of simultaneity‚ÄĚ where ‚Äúdistant simultaneity ‚ÄĒ whether two spatially separated events occur at the same time ‚ÄĒ is not absolute, but depends on the observer‚Äôs reference frame‚ÄĚ; or on the micro-level (of neurons) where perception uses a ‚Äúcut loci‚ÄĚ which is a geometric form (from differential geometry) in so that we can develop a ‚Äúneurogeometry of vision‚ÄĚ as Jean Petitot has done.
Foto Slyvia John
By Eric Schmid