Metamathematics: Foundations & Physicalization

“What is mathematics?” is a question that has been debated since antiquity. This book presents a groundbreaking and surprising answer to the question—showing through the concept of the physicalization of metamathematics how both mathematics and physics as experienced by humans can be seen to emerge from the unique underlying computational structure of the recently formulated ruliad. Written with Stephen Wolfram's characteristic expositional flair and richly illustrated with remarkable algorithmic diagrams, the book takes the reader on an unprecedented intellectual journey to the center of some of the deepest questions about mathematics and its nature—and points the way to a new understanding of the foundations and future of mathematics, taking a major step beyond ideas from Plato, Kant, Hilbert, Gödel and others.
Contents
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)
Index