The Empirical Metamathematics of Euclid and Beyond

Stephen Wolfram

The Empirical Metamathematics of Euclid and Beyond

Towards a Science of Metamathematics

One of the many surprising things about our Wolfram Physics Project is that it seems to have implications even beyond physics. In our effort to develop a fundamental theory of physics it seems as if the tower of ideas and formalism that we’ve ended up inventing are actually quite general, and potentially applicable to all sorts of areas.

One area about which I’ve been particularly excited of late is metamathematics—where it’s looking as if it may be possible to use our formalism to make what might be thought of as a “bulk theory of metamathematics”.

Mathematics itself is about what we establish about mathematical systems. Metamathematics is about the infrastructure of how we get there—the structure of proofs, the network of theorems, and so on. And what I’m hoping is that we’re going to be able to make an overall theory of how that has to work: a formal theory of the large-scale structure of metamathematics—that, among other things, can make statements about the general properties of “metamathematical space”.

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A Short Note on the Double-Slit Experiment and Other Quantum Interference Effects in the Wolfram Model

Jonathan Gorard (University of Cambridge/Wolfram Research)

A Short Note on the Double-Slit Experiment and Other Quantum Interference Effects in the Wolfram Model

This bulletin is a short note detailing how single-slit, double-slit and multi-slit photon diffraction and interference patterns can be successfully reproduced using the author’s own formulation of quantum mechanics in the Wolfram model. The author has benefited greatly from many fruitful conversations with Stephen Wolfram, as well as from the encouragement (and infectious enthusiasm) of Hatem Elshatlawy. Continue reading

A Candidate Geometrical Formalism for the Foundations of Mathematics and Physics

Jonathan Gorard (University of Cambridge/Wolfram Research)

Formal Correspondences between Homotopy Type Theory and the Wolfram Model

This bulletin is a writeup of work done in collaboration with Xerxes Arsiwalla and Stephen Wolfram, as publicly presented and discussed in livestreams here, here and here. This bulletin is intended to be a high-level survey of the effort so far; a more formal article, intended to give rigorous formulations and proofs of the various ideas discussed here, is currently in preparation for submission to an appropriate journal. Continue reading

A Burst of Physics Progress at the 2020 Wolfram Summer School

Stephen Wolfram

A Burst of Physics Progress at the 2020 Wolfram Summer School

And We’re Off and Running…

We recently wrapped up the four weeks of our first-ever “Physics track” Wolfram Summer School—and the results were spectacular! More than 30 projects potentially destined to turn into academic papers—reporting all kinds of progress on the Wolfram Physics Project.

When we launched the Wolfram Physics Project just three months ago one of the things I was looking forward to was seeing other people begin to seriously contribute to the project. Well, it turns out I didn’t have to wait long! Because—despite the pandemic and everything—things are already very much off and running!

Six weeks ago we made a list of questions we thought we were ready to explore in the Wolfram Physics Project. And in the past five weeks I’m excited to say that through projects at the Summer School lots of these are already well on their way to being answered. If we ever wondered whether there was a way for physicists (and physics students) to get involved in the project, we can now give a resounding answer, “yes”.

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Exploring Rulial Space: The Case of Turing Machines

Stephen Wolfram

Generalized Physics and the Theory of Computation

Let’s say we find a rule that reproduces physics. A big question would then be: “Why this rule, and not another?” I think there’s a very elegant potential answer to this question, that uses what we’re calling rule space relativity—and that essentially says that there isn’t just one rule: actually all possible rules are being used, but we’re basically picking a reference frame that makes us attribute what we see to some particular rule. In other words, our description of the universe is a sense of our making, and there can be many other—potentially utterly incoherent—descriptions, etc.

But so how does this work at a more formal level? This bulletin is going to explore one very simple case. And in doing so we’ll discover that what we’re exploring is potentially relevant not only for questions of “generalized physics”, but also for fundamental questions in the theory of computation. In essence, what we’ll be doing is to study the structure of spaces created by applying all possible rules, potentially, for example, allowing us to “geometrize” spaces of possible algorithms and their applications. Continue reading

Event Horizons, Singularities and Other Exotic Spacetime Phenomena

Stephen Wolfram

The Structure and Pathologies of Spacetime

In our models, space emerges as the large-scale limit of our spatial hypergraph, while spacetime effectively emerges as the large-scale limit of the causal graph that represents causal relationships between updating events in the spatial hypergraph. An important result is that (subject to various assumptions) there is a continuum limit in which the emergent spacetime follows Einstein’s equations from general relativity.

And given this, it is natural to ask what happens in our models with some of the notable phenomena from general relativity, such as black holes, event horizons and spacetime singularities. I already discussed this to some extent in my technical introduction to our models. My purpose here is to go further, both in more completely understanding the correspondence with general relativity, and in seeing what additional or different phenomena arise in our models. Continue reading

The Wolfram Physics Project:
The First Two Weeks

Stephen Wolfram

First, Thank You!

We launched the Wolfram Physics Project two weeks ago, on April 14. And, in a word, wow! People might think that interest in fundamental science has waned. But the thousands of messages we’ve received tell a very different story. People really care! They’re excited. They’re enjoying understanding what we’ve figured out. They’re appreciating the elegance of it. They want to support the project. They want to get involved.

It’s tremendously encouraging—and motivating. I wanted this project to be something for the world—and something lots of people could participate in. And it’s working. Our livestreams—even very technical ones—have been exceptionally popular. We’ve had lots of physicists, mathematicians, computer scientists and others asking questions, making suggestions and offering help. We’ve had lots of students and others who tell us how eager they are to get into doing research on the project. And we’ve had lots of people who just want to tell us they appreciate what we’re doing. So, thank you!

The Wolfram Physics Project: The First Two Weeks

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