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John Baez (Feb 28 2026 at 18:22):Here is an example of a yes-or-not question about The Stacks Project:
Is it possible to express all of our knowledge of physics within the framework of The Stacks Project?
I said "I recommend that new mathematicians learn the art of asking well-posed yes-or-no mathematical questions". Your question is not a mathematical question. I could try to define a yes-or-no mathematical question - for starters, such questions are answered by proving theorems - but instead let me illustrate the concept with some examples:
"is 2+2 = 5?"
"does the fraction 7/12 appear as a truncation of the continued fraction expansion of ?"
"is every Noetherian ring finitely presented?"
is there a functor from sheaves over a topological space X to topological stacks over X?
Until one gets good at asking and answering such questions, one isn't truly engaging with mathematics.
José said:
Thanks. I’m not looking to contradict you; rather, I want to emphasize the crucial distinction between emergent physics and fundamental physics.
From the perspective of the Wolfram Physics Project, the continuous structures that define modern physics, such as Hilbert spaces and partial differential equations, are essentially "low-resolution" models. While they are useful for calculation, they are not fundamental. Instead, the "high-resolution" reality is a finite, discrete structure governed by integral bilinear forms on Z-modules (a structure mentioned in The Stacks Project).
A big difference between what you're calling "low-resolution models" and "high-resolution models" of physics is that the former are extremely successful in making predictions that get confirmed by experiment, while there's absolutely no empirical evidence for the latter: at present they are purely speculations. In fact I don't know a single testable prediction made by these self-proclaimed "high-resolution models". That's why I'm mainly ignoring them. If this ever changes, of course my attitude will change. For now I pay just enough attention so that I'll notice if something substantial emerges from these speculations.
Morgan Rogers (he/him) (Feb 28 2026 at 19:57):Another warning @José to please stop posting text produced by LLMs in this forum. Thanks.
José M. R. Caballero (Feb 28 2026 at 23:59):Ok, I deleted the table.
José M. R. Caballero (Mar 01 2026 at 00:29):John Baez said:
A big difference between what you're calling "low-resolution models" and "high-resolution models" of physics is that the former are extremely successful in making predictions that get confirmed by experiment, while there's absolutely no empirical evidence for the latter: at present they are purely speculations. In fact I don't know a single testable prediction made by these self-proclaimed "high-resolution models". That's why I'm mainly ignoring them. If this ever changes, of course my attitude will change. For now I pay just enough attention so that I'll notice if something substantial emerges from these speculations.
I completely agree with that, but given the immense complexity of the internet, especially in the era of LLMs, we are dealing with a physical system whose "high-resolution model" is well understood (as a configuration of logic gates), yet whose "low-resolution model" remains a mystery. The work of Leslie Lamport [1] on distributed systems can be interpreted as suggesting that the "low-resolution model" of the internet exhibits properties akin to special relativity. If branching were introduced to the internet (perhaps motivated by a new cryptographic protocol) it could give rise to dynamics resembling quantum phenomena [2]. In this way, large computational systems like the internet display behaviors that closely parallel those found in fundamental physics. Therefore, even if "high-resolution models" currently fall outside the realm of empirical measurement, studying the relationship between them and "low-resolution models" is fully justified by its applications in the theory of complex systems.
[1] Lamport, Leslie. "Time, clocks, and the ordering of events in a distributed system." Concurrency: the Works of Leslie Lamport. 2019. 179-196.
[2] Video explanation: When the universe branches, what happens to me? with Stephen Wolfram.
John Baez (Mar 01 2026 at 00:39):In this way, large computational systems like the internet display behaviors that closely parallel those found in fundamental physics.
Closely? Let's say "very roughly". But anyway, if you want to work on this parallel please go ahead.
José M. R. Caballero (Mar 01 2026 at 03:30):I guess the concept of sheaf is crucial to understanding the "low-resolution model" of the Internet. That could be a starting point to read The Stacks Project from a network theoretic point of view. Indeed, hyperbolic geometry is related to complex networks: Hyperbolic Geometry of Complex Networks
José M. R. Caballero (Mar 01 2026 at 04:59):In the framework of the Weil Conjectures, the Lefschetz trace formula relates the number of points on a variety over a finite field to the traces of the Frobenius endomorphism acting on its ℓ-adic étale cohomology groups. In the study of complex networks, this role of "counting" is performed by the adjacency matrix, where the traces of its powers correspond to the number of closed walks of a given length. Consequently, in the ideal case scenario, by constructing an algebraic stack 𝒳_G associated with a network G, the characteristic polynomial of the network's adjacency matrix should be identified with the characteristic polynomial of the Frobenius operator acting on the first cohomology group H¹(𝒳_G, ℚ_ℓ) (or a related weighted sum across Hⁱ).
José M. R. Caballero (Mar 01 2026 at 05:02):A yes-or-no question could be: Is there a construction associating an algebraic stack 𝒳_G with a network G satisfying the above mentioned property?