Using a set of rules for hypergraph evolution they construct a directed graph. Then they decide to embed it into a lattice that they equip with the Minkowski metric. This embedding is completely ad hoc. It establishes as much connection between their formalism and relativity, as writing the two formalisms next to each other on the same page would. Their “proof” of Lorentz covariance consists of observing that they can apply a Lorentz transformation (but there is nothing non-trivial it preserves). At some point they mention the concept of “discrete Lorentzian metric” without giving the definition. As far as I know it is a completely non-standard notion and I have no idea what it means. Later they talk about discrete analogues of concepts in Riemannian geometry and completely ignore the Lorentzian signature. Then they claim to derive Einstein’s equation by assuming that the “dimensionality” of their causal graph converges, which is supposed to imply that something they call “global dimension anomaly” goes to zero. They claim that this global dimension anomaly corresponds to the Einstein-Hilbert action in the continuum limit. Only, instead of concluding the action converges to zero, they inexplicably conclude the variation of the action converges to zero, which is equivalent to the Einstein equation.
Thanks for writing this. I hesitated before commenting, because I am not an expert on physics, but something just felt wrong. It took some time to pinpoint the source of wrongness, but now it seems to me that the author is (I assume unknowingly) playing the following game:
1) Find something that is Turing-complete
The important thing is that it should be something simple, where the Turing-completeness comes as a surprise. A programming language would be bad. Turing machine would be great a few decades ago, but is bad now. A system for replacing structures in a directed graph… yeah, this type of thing. Until people get used to it; then you would need a fresh example.
2) Argue that you could build a universe using this thing
Yes, technically true. If something is Turing-complete, you can use it to implement anything that can be implemented on a computer. Therefore, assuming that a universe could be simulated on a hypothetical computer, it could also be simulated using that thing.
But the fact that many different things can be Turing-complete, means that their technical details are irrelevant (beyond the fact they they cause the Turing-completeness) for the simulated object. Just because a Turing machine with 1-dimensional tape could simulate the universe, it doesn’t mean that the universe is “truly 1-dimensional”, “truly consists of discrete cells containing symbols”, or anything like that. It just means it could be simulated on such system, because such system is Turing-complete; nothing more, nothing less.
So it’s simultaneously “yes, you could simulate A using B” and “A and B are quite different”. However, you can...
3) Find some shallow analogies between the thing and the laws of physics
Remember, these are analogies, so you can point out any similarity you find, and change the topic whenever the similarity is exhausted. For example, the Turing machine has a tape consisting of discrete cells… and if you squint, that is kinda like the quantum physics, because the quantum physics also has some discrete values, and… uhm, end of paragraph. But also the Turing machine moves at a limited speed, at most 1 step per turn… and that is kinda like theory of relativity, you know, with the limited speed of light. And perhaps the tape looks like a string, which would point towards string theory, who knows. You would notice different similarities with the graph replacement rules, but you could find some if you try hard enough.
This gives the fake impression that we have found a similarity deeper than “Turing-complete, therefore universal, therefore allows to simulate any universe”. (If you can simulate any universe, it means you cannot conclude anything useful about the universe from the fact that it can be simulated.) It gives the impression that this specific system is somehow more relevant to our universe than any other Turing-complete system. That it provides insights into the true nature of our universe. Which in fact it does not, because all we have is the Turing-completeness and a few shallow analogies. The fact that X can simulate a universe, and also has a few shallow analogies with the known laws of physics, means less than it may seem.
(So, thanks for confirmation that the analogies with laws of physics are indeed gibberish.)
Love this description. All of the results I’ve skimmed look an awful lot like showing that a thing which can correspond to space and time (your “thing that is Turing-complete”) allows you to rederive the things about our space and time.
That being said, I still think exploring various translations and embeddings of mathematical, physical, and computational paradigms into each other is a very valuable exercise and may shed light on very important properties of abstract systems in the future. Also, cool compressed explanation of how some concepts in physics fit together, even if somewhat shallowly.
The property such universes have in common is that they are computable on a hypothetical classical computer with unlimited capacity. (Potentially very inefficiently, like maybe computing one Planck unit of time in a tiny part of the simulated universe would require greater computing capacity than our universe could provide during its entire existence. These are mathematical abstractions unrelated to the real world.)
That implies a few things, for example that in none of these universe you could solve a halting problem. (That there would be certain potentially infinite calculations, whose output you would not be able to predict in limited time.)
But Wolfram’s theory doesn’t provide any benefit here, other than circular reasoning. “If the universe is Turing-complete, then it can be simulated using Wolfram’s latest favorite system, which would imply that it is Turing-complete.” Why not just skip that part?
Using a set of rules for hypergraph evolution they construct a directed graph. Then they decide to embed it into a lattice that they equip with the Minkowski metric. This embedding is completely ad hoc. It establishes as much connection between their formalism and relativity, as writing the two formalisms next to each other on the same page would.
Some discussion by Gorard here, which makes it sound like the Minkowski embedding was meant as an illustration, and not meant to do any heavy lifting. Given that, it’s not surprising that it might seem a bit ad hoc.
The problem is, there is no heavy lifting. “We made a causal network, therefore, special relativity”. Sorry, but no, you need to actually explain why the vacuum seems Lorentz invariant on the macroscopic level, something that’s highly non-obvious given that you start from something discrete. A discrete object cannot be Lorentz invariant, the best you can hope for is something like a probability measure over discrete objects that is Lorentz invariant, but there is nothing like that in the paper. Moreover, if the embedding is just an illustration, then where do they even get the Riemannian metric that is supposed to satisfy the Einstein equation?
Thanks for this reply. It’s interesting to see such wildly different answers to this question. I guess that’s what you get when most people are giving hot takes built after a skim. You seem to have looked at this more closely than anyone else so far, so I’m updating heavily in your direction.
I think Wolfram’s “theory” is complete gibberish. Reading through “some relativistic and gravitational properties of the Wolfram model” I haven’t encountered a single claim that was simultaneously novel, correct and non-trivial.
Using a set of rules for hypergraph evolution they construct a directed graph. Then they decide to embed it into a lattice that they equip with the Minkowski metric. This embedding is completely ad hoc. It establishes as much connection between their formalism and relativity, as writing the two formalisms next to each other on the same page would. Their “proof” of Lorentz covariance consists of observing that they can apply a Lorentz transformation (but there is nothing non-trivial it preserves). At some point they mention the concept of “discrete Lorentzian metric” without giving the definition. As far as I know it is a completely non-standard notion and I have no idea what it means. Later they talk about discrete analogues of concepts in Riemannian geometry and completely ignore the Lorentzian signature. Then they claim to derive Einstein’s equation by assuming that the “dimensionality” of their causal graph converges, which is supposed to imply that something they call “global dimension anomaly” goes to zero. They claim that this global dimension anomaly corresponds to the Einstein-Hilbert action in the continuum limit. Only, instead of concluding the action converges to zero, they inexplicably conclude the variation of the action converges to zero, which is equivalent to the Einstein equation.
Alas, no theory of everything there.
Thanks for writing this. I hesitated before commenting, because I am not an expert on physics, but something just felt wrong. It took some time to pinpoint the source of wrongness, but now it seems to me that the author is (I assume unknowingly) playing the following game:
1) Find something that is Turing-complete
The important thing is that it should be something simple, where the Turing-completeness comes as a surprise. A programming language would be bad. Turing machine would be great a few decades ago, but is bad now. A system for replacing structures in a directed graph… yeah, this type of thing. Until people get used to it; then you would need a fresh example.
2) Argue that you could build a universe using this thing
Yes, technically true. If something is Turing-complete, you can use it to implement anything that can be implemented on a computer. Therefore, assuming that a universe could be simulated on a hypothetical computer, it could also be simulated using that thing.
But the fact that many different things can be Turing-complete, means that their technical details are irrelevant (beyond the fact they they cause the Turing-completeness) for the simulated object. Just because a Turing machine with 1-dimensional tape could simulate the universe, it doesn’t mean that the universe is “truly 1-dimensional”, “truly consists of discrete cells containing symbols”, or anything like that. It just means it could be simulated on such system, because such system is Turing-complete; nothing more, nothing less.
So it’s simultaneously “yes, you could simulate A using B” and “A and B are quite different”. However, you can...
3) Find some shallow analogies between the thing and the laws of physics
Remember, these are analogies, so you can point out any similarity you find, and change the topic whenever the similarity is exhausted. For example, the Turing machine has a tape consisting of discrete cells… and if you squint, that is kinda like the quantum physics, because the quantum physics also has some discrete values, and… uhm, end of paragraph. But also the Turing machine moves at a limited speed, at most 1 step per turn… and that is kinda like theory of relativity, you know, with the limited speed of light. And perhaps the tape looks like a string, which would point towards string theory, who knows. You would notice different similarities with the graph replacement rules, but you could find some if you try hard enough.
This gives the fake impression that we have found a similarity deeper than “Turing-complete, therefore universal, therefore allows to simulate any universe”. (If you can simulate any universe, it means you cannot conclude anything useful about the universe from the fact that it can be simulated.) It gives the impression that this specific system is somehow more relevant to our universe than any other Turing-complete system. That it provides insights into the true nature of our universe. Which in fact it does not, because all we have is the Turing-completeness and a few shallow analogies. The fact that X can simulate a universe, and also has a few shallow analogies with the known laws of physics, means less than it may seem.
(So, thanks for confirmation that the analogies with laws of physics are indeed gibberish.)
Love this description. All of the results I’ve skimmed look an awful lot like showing that a thing which can correspond to space and time (your “thing that is Turing-complete”) allows you to rederive the things about our space and time.
That being said, I still think exploring various translations and embeddings of mathematical, physical, and computational paradigms into each other is a very valuable exercise and may shed light on very important properties of abstract systems in the future. Also, cool compressed explanation of how some concepts in physics fit together, even if somewhat shallowly.
It might lead to new insights by showing that some properties are shared by all Turing-simulateable universes.
The property such universes have in common is that they are computable on a hypothetical classical computer with unlimited capacity. (Potentially very inefficiently, like maybe computing one Planck unit of time in a tiny part of the simulated universe would require greater computing capacity than our universe could provide during its entire existence. These are mathematical abstractions unrelated to the real world.)
That implies a few things, for example that in none of these universe you could solve a halting problem. (That there would be certain potentially infinite calculations, whose output you would not be able to predict in limited time.)
But Wolfram’s theory doesn’t provide any benefit here, other than circular reasoning. “If the universe is Turing-complete, then it can be simulated using Wolfram’s latest favorite system, which would imply that it is Turing-complete.” Why not just skip that part?
Some discussion by Gorard here, which makes it sound like the Minkowski embedding was meant as an illustration, and not meant to do any heavy lifting. Given that, it’s not surprising that it might seem a bit ad hoc.
The problem is, there is no heavy lifting. “We made a causal network, therefore, special relativity”. Sorry, but no, you need to actually explain why the vacuum seems Lorentz invariant on the macroscopic level, something that’s highly non-obvious given that you start from something discrete. A discrete object cannot be Lorentz invariant, the best you can hope for is something like a probability measure over discrete objects that is Lorentz invariant, but there is nothing like that in the paper. Moreover, if the embedding is just an illustration, then where do they even get the Riemannian metric that is supposed to satisfy the Einstein equation?
Thanks for this reply. It’s interesting to see such wildly different answers to this question. I guess that’s what you get when most people are giving hot takes built after a skim. You seem to have looked at this more closely than anyone else so far, so I’m updating heavily in your direction.