Both metamathematics and physics are posited to emerge from samplings by observers of the unique ruliad structure that corresponds to the entangled limit of all possible computations. The possibility of higher-level mathematics accessible to humans is posited to be the analog for mathematical observers of the perception of physical space for physical observers. A physicalized analysis is given of the bulk limit of traditional axiomatic approaches to the foundations of mathematics, together with explicit empirical metamathematics of some examples of formalized mathematics. General physicalized laws of mathematics are discussed, associated with concepts such as metamathematical motion, inevitable dualities, proof topology and metamathematical singularities. It is argued that mathematics as currently practiced can be viewed as derived from the ruliad in a direct Platonic fashion analogous to our experience of the physical world, and that axiomatic formulation, while often convenient, does not capture the ultimate character of mathematics. Among the implications of this view is that only certain collections of axioms may be consistent with inevitable features of human mathematical observers. A discussion is included of historical and philosophical connections, as well as of foundational implications for the future of mathematics.

- Abstract
- Mathematics and Physics Have the Same Foundations
- The Underlying Structure of Mathematics and Physics
- The Metamodeling of Axiomatic Mathematics
- Some Simple Examples with Mathematical Interpretations
- Metamathematical Space
- The Issue of Generated Variables
- Rules Applied to Rules
- Accumulative Evolution
- Accumulative String Systems
- The Case of Hypergraphs
- Proofs in Accumulative Systems
- Beyond Substitution: Cosubstitution and Bisubstitution
- Some First Metamathematical Phenomenology
- Relations to Automated Theorem Proving
- Axiom Systems of Present-Day Mathematics
- The Model-Theoretic Perspective
- Axiom Systems in the Wild
- The Topology of Proof Space
- Time, Timelessness and Entailment Fabrics
- The Notion of Truth
- What Can Human Mathematics Be Like?
- Going below Axiomatic Mathematics
- The Physicalized Laws of Mathematics
- Uniformity and Motion in Metamathematical Space
- Gravitational and Relativistic Effects in Metamathematics
- Empirical Metamathematics
- Invented or Discovered? How Mathematics Relates to Humans
- What Axioms Can There Be for Human Mathematics?
- Counting the Emes of Mathematics and Physics
- Some Historical (and Philosophical) Background
- Implications for the Future of Mathematics
- Some Personal History: The Evolution of These Ideas
- Notes & Thanks
- Graphical Key
- Glossary
- Bibliography
- Index

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