Metamathematics: Foundations & Physicalization

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“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

From the Publisher

"An exceptionally straightforward and elegant approach to what is 'really going on' behind logic, math and by extension computer science."

— Verified Purchaser

From the Preface

"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.

... ... ...

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?'"

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)

Notes

Index

About the Author

Stephen Wolfram is an award-winning scientist and bestselling author, and the creator of some of the world's most respected software systems—Mathematica, Wolfram|Alpha and Wolfram Language. For more than 35 years, he has been CEO of the global technology company Wolfram Research, as well as responsible for a series of groundbreaking advances in basic science, including the recent Wolfram Physics Project.

A few of Stephen Wolfram's other books:

What Is ChatGPT Doing ... and Why Does It Work?A New Kind of ScienceThe Second Law: Resolving the Mystery of the Second Law of ThermodynamicsA Project to Find the Fundamental Theory of PhysicsIdea Makers: Personal Perspectives on the Lives & Ideas of Some Notable PeopleAn Elementary Introduction to the Wolfram Language, Third Edition

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Page Count
456 1280 584 778 362 384

Publication Year
2022 2002 2023 2020 2021 2022

Publisher ‏ : ‎ Wolfram Media, Inc.
Publication date ‏ : ‎ December 16, 2022
Language ‏ : ‎ English
Print length ‏ : ‎ 456 pages
ISBN-10 ‏ : ‎ 1579550762
ISBN-13 ‏ : ‎ 978-1579550769
Item Weight ‏ : ‎ 2.5 pounds
Dimensions ‏ : ‎ 7.5 x 1.25 x 9.5 inches
Best Sellers Rank: #220,371 in Books (See Top 100 in Books) #62 in Mathematical Physics (Books) #80 in Mathematical Logic #106 in Mathematics History
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Price: $44.95 - $29.31
(as of Oct 16, 2025 14:03:26 UTC - Details)