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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 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 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 · 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 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 ·
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 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 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 · 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 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 ·
- GenresScienceMathematics
456 pages, Kindle Edition
Published December 16, 2022
11 people are currently reading
136 people want to read
About the author
Stephen Wolfram
47 books464 followersStephen Wolfram is the founder & CEO of Wolfram Research, creator of Mathematica, Wolfram|Alpha & Wolfram Language, author of A New Kind of Science and other books, and the originator of Wolfram Physics Project.
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Displaying 1 - 2 of 2 reviews
May 6, 2023
Wolfram is and apparently always has been obsessed with the phrase 'all possible ...' and in particular all possible rules, or computations, or patterns of inference. The claim is that if we assume that this is what fundamentally is then we can see how physics - and now in this book how mathematics as we know it - can be derived, or emerges. In the raw it is unimaginably complex and saturated with undecidability but simpler patterns emerge at the human level. An analogy to the relationship between statistical mechanics and thermodynamics is used often.
The book is not terribly well written - in the sense of engaging and easy enough to follow. The author plunges straight into complex symbols and arcane diagrams without taking the time to explain the background and build some comprehension. Later discussions of the 'ruliad' get tedious and repetitive and in places not terribly clear. The Ruliad !! Going to have to come up with a better name. It's off-putting. Sounds crank-ish.
But that doesn't matter too much. The big question is 'is there anything to all of this?' Wolfram and his gang have certainly put a lot of work into it. So it's worth a look. .... And maybe there is. And if there isn't - well, nice try. In the case of physics we look for rules so the assumption that at the base level it's all built out of all possible 'rules' or computations in the abstract is worth considering. I guess. Computations crunching away on something? Raw state - undefined. Something like that. .... We assume the physical universe is computable - and describable by rules. And historically so far so good. But is it so at its most fundamental level? And why make the vastly simplifying assumption that it is discrete ( finite or at most countably infinite). The concept of a continuum ( the next level of infinity up ( ...maybe ) ) has worked pretty well for a few centuries. Differential equations are vastly more subtle and more powerful than difference equations ( over the Integers ).
The case of mathematics is more subtle again. If we said something like mathematics was the study of all possible ( there's that phrase again ) structures and functions we are immediately ( more or less ) confronted with a dizzying array of higher and higher levels of infinity - all the way up to the Very Large Cardinals and beyond. The Computable functions ( or rules, or computations, or patterns of inference) constitute a tiny slither of a subset of a set with the very smallest (non finite ) cardinality. And these are supposed to be the basis of all that is possible!! Not likely.
They may be the basis of all that we can say and reason and calculate. If so then so much the worse for us. But maybe these too can be extended in interesting ways. This was the original meaning of 'metamathematics' as used by the pioneers including Godel. Using the higher infinities to find ways around the recently discovered intrinsic incompleteness and undecidability of mathematical theories. It has been shown for example that an undecidable statement in a given theory can be made decidable in a bigger theory which includes an additional sufficiently large Large Cardinal Axiom. I don't know about you. Takes my breathe away. When you use the phrase 'all possible' you have to consider carefully what kind of set you think you're 'quantifying' over. It makes a big difference.
The book is not terribly well written - in the sense of engaging and easy enough to follow. The author plunges straight into complex symbols and arcane diagrams without taking the time to explain the background and build some comprehension. Later discussions of the 'ruliad' get tedious and repetitive and in places not terribly clear. The Ruliad !! Going to have to come up with a better name. It's off-putting. Sounds crank-ish.
But that doesn't matter too much. The big question is 'is there anything to all of this?' Wolfram and his gang have certainly put a lot of work into it. So it's worth a look. .... And maybe there is. And if there isn't - well, nice try. In the case of physics we look for rules so the assumption that at the base level it's all built out of all possible 'rules' or computations in the abstract is worth considering. I guess. Computations crunching away on something? Raw state - undefined. Something like that. .... We assume the physical universe is computable - and describable by rules. And historically so far so good. But is it so at its most fundamental level? And why make the vastly simplifying assumption that it is discrete ( finite or at most countably infinite). The concept of a continuum ( the next level of infinity up ( ...maybe ) ) has worked pretty well for a few centuries. Differential equations are vastly more subtle and more powerful than difference equations ( over the Integers ).
The case of mathematics is more subtle again. If we said something like mathematics was the study of all possible ( there's that phrase again ) structures and functions we are immediately ( more or less ) confronted with a dizzying array of higher and higher levels of infinity - all the way up to the Very Large Cardinals and beyond. The Computable functions ( or rules, or computations, or patterns of inference) constitute a tiny slither of a subset of a set with the very smallest (non finite ) cardinality. And these are supposed to be the basis of all that is possible!! Not likely.
They may be the basis of all that we can say and reason and calculate. If so then so much the worse for us. But maybe these too can be extended in interesting ways. This was the original meaning of 'metamathematics' as used by the pioneers including Godel. Using the higher infinities to find ways around the recently discovered intrinsic incompleteness and undecidability of mathematical theories. It has been shown for example that an undecidable statement in a given theory can be made decidable in a bigger theory which includes an additional sufficiently large Large Cardinal Axiom. I don't know about you. Takes my breathe away. When you use the phrase 'all possible' you have to consider carefully what kind of set you think you're 'quantifying' over. It makes a big difference.
December 29, 2022
Very interesting
Displaying 1 - 2 of 2 reviews