Metamathematics: Foundations & Physicalization

Book Synopsis: “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

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