Categorical Foundations
About
What if mathematics could be built on structure rather than sets?
Book I launches the Panta Rhei series by constructing Category τ from nine axioms on a small, explicit signature. The result is a foundational framework that is categorical (unique up to isomorphism), rigid (no nontrivial automorphisms), and designed to be structurally decidable in its core predicates—aiming for mathematics that is not only elegant, but computationally and conceptually tractable.
Across thirteen parts, the book develops a complete internal world:
- Core theory: τ is determined by relations rather than labels, with label-independence and rigidity as central theorems.
- Internal set theory: “membership” emerges from divisibility; bounded powersets arise from the successor structure.
- Metric and order: a Cayley word metric yields computable distances and decision procedures.
- Geometry from algebra: the Tarski axioms for Euclidean geometry are verified—including the parallel postulate as a theorem, not an axiom.
- Topos and arithmetic: τ forms a well-pointed elementary topos with an NNO; internal number systems grow naturally from ℕτ through ℂτ, supporting constructive analysis.
- Calibration and computation: a canonical invariant ιτ = 2/(π + e) anchors a calibration dictionary, alongside a native computation model (the τ-Tower Machine) and a tractability principle based on bounded interfaces.
Book I is offered as a research program—rigorous, testable, and intended for critical engagement—closing with a bridge to Book II: Categorical Holomorphy, where the “holomorphic side” of the theory takes center stage.
"Nine axioms for a foundation of mathematics — the rest is theorem.”
Free reader downloads: To get a fast, high-signal overview of this volume, we provide two PDFs extracted from the original published pages of the book: the Table of Contents (see the full structure at a glance) and the Q&A Appendix (a reader’s guide to key ideas and common questions). Both are free to view and share for review/academic reference.
New to the series? Start with the Q&A Appendix; use the TOC to choose your entry points.