"Metamathematics is the study of what mathematics is made of, and how it’s put together. It’s about the foundations of mathematics, and the structure that lies below that most remarkable intellectual enterprise that we call mathematics. If we think of mathematics as an abstraction, then metamathematics is an abstraction of that abstraction—and something that probes deep into foundational questions.

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I view this book as much as anything as a beginning, and in particular a beginning for a new metamathematics. I’ve tried to give at least an indication of the most obvious directions in which to go. But there is much to be done, and many new results—both theoretical and empirical—to be found. There was great activity around metamathematics roughly a century ago. And it’s my hope that the ideas in this book can finally rekindle the excitement of those times. And that with the methods I describe it’ll now be possible to make more progress on some of the deepest questions about the character and foundations of mathematics, and to finally be able to convincingly answer questions like 'What is mathematics?'"

Preface

The Physicalization of Metamathematics and Its Implications for the Foundations of Mathematics

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 · Annotated Bibliography

The Concept of the Ruliad

The Entangled Limit of Everything · Experiencing the Ruliad · Observers Like Us · Living in Rulial Space · The View from Mathematics · The View from Computation Theory · What's beyond the Ruliad? · Communicating across Rulial Space · So Is There a Fundamental Theory of Physics? · Alien Views of the Ruliad · Conceptual Implications of the Ruliad · Appendix: The Case of the "Multiplicad" · Thanks & Note

The Empirical Metamathematics of Euclid and Beyond

Towards a Science of Metamathematics · The Most Famous Math Book in History · Basic Statistics of Euclid · The Interdependence of Theorems · The Graph of All Theorems · The Causal Graph Analogy · The Most Difficult Theorem in Euclid · The Most Popular Theorems in Euclid · What Really Depends on What? · The Machine Code of Euclid: All the Way Down to Axioms · Superaxioms, or What Are the Most Powerful Theorems? · Formalizing Euclid · All Possible Theorems · Math beyond Euclid · The Future of Empirical Metamathematics · Thanks · Note Added

Implications for Mathematics and Its Foundations, Section 12.9 from A New Kind of Science (2002)

Notes

Index