the physicalization of metamathematics

[Stephen Wolfram]

I just posted something I think may be of interest to FOM subscribers: 
https://writings.stephenwolfram.com/2022/03/the-physicalization-of-metamathematics-and-its-implications-for-the-foundations-of-mathematics

It’s a (rather unexpected, at least to me) outgrowth of our recent (and very active) Physics Project https://www.wolframphysics.org/

Here’s an abstract:
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Both metamathematics and physics are posited to emerge from samplings by observers of the unique ruliad structure that corresponds to the entangled limit of all possible computations. The possibility of higher-level mathematics accessible to humans is posited to be the analog for mathematical observers of the perception of physical space for physical observers. A physicalized analysis is given of the bulk limit of traditional axiomatic approaches to the foundations of mathematics, together with explicit empirical metamathematics of some examples of formalized mathematics. General physicalized laws of mathematics are discussed, associated with concepts such as metamathematical motion, inevitable dualities, proof topology and metamathematical singularities. It is argued that mathematics as currently practiced can be viewed as derived from the ruliad in a direct Platonic fashion analogous to our experience of the physical world, and that axiomatic formulation, while often convenient, does not capture the ultimate character of mathematics. Among the implications of this view is that only certain collections of axioms may be consistent with inevitable features of human mathematical observers. A discussion is included of historical and philosophical connections, as well as of foundational implications for the future of mathematics.
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I’m not sure if others will care about it ... but I at least am very excited about what we’ve figured out ... not least because it changes my mind about things I’ve long assumed about the nature and foundations of mathematics.

Even though what I’ve written is quite long (~200 pages) I consider it just a very beginning, with a great many loose ends, missing technical details, inadequate levels of precision, etc.  I’m looking forward to others improving and developing it. 

This is an open science project.  All images in the writeup are clickable and give Wolfram Language code immediately runnable on the desktop or the cloud.  The working notebooks from the project (including all their missteps) are available at https://www.wolframphysics.org/archives/index/.  I’ve been livestreaming working sessions about the project https://livestreams.stephenwolfram.com/ (as well as posting a great many hours of video work logs).  

--- Stephen Wolfram
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