Multicomputation as a General Paradigm for Theoretical Science
writings.stephenwolfram.com
writings.stephenwolfram.com
What's promising about TEGs (and their incidence hypergraph, the rewrite hypergraph) is that they offer a clean methodology to decompose the behavior of a non-deterministic automaton into its causally independent parts. We're still trying to understand how to think about them, but the most promising approach seems to use the lens of (modular) representation theory, which gives us a rich mathematical toolkit to work with.
If this methodology works, there will be possibility to represent many kinds of systems in disparate fields, ranging from distributed computation to physics to biology to machine learning, in the common language of TEGs and their representations. Of course it may turn out to be merely a recasting of older ideas. In particular the Krohn-Rhodes theorem [1], categorical Petri nets [2], and the GNS construction [3] seem like they might be describing the same or an analogous procedure.
I hope to soon be describing this approach in full detail using quiver geometry [4].
[0]: https://github.com/maxitg/SetReplace/blob/master/Research/Lo...
[1]: https://www.wikiwand.com/en/Krohn–Rhodes_theory
[2]: https://arxiv.org/abs/2101.04238
As for the stuff I'm actively working on: even if it is successful, and uniquely suited to solve some particular problem, I'd imagine it would take at least a few years to be actively applied in the right places -- and it might end up being me helping to apply it!
If you look back at for example graph theory, it was explored on the pure mathematics and computer science side for many decades before it spawned e.g. network science and got applied in sociology, economics, etc.
So, yeah, don't hold your breath! If you'd prefer not to read any blog posts heralding XYZ as the next big thing before XYZ has already led to a concrete breakthrough, I think that's a totally fair. I myself am quite happy to work quietly on things until there is a satisfying and complete application, but Stephen isn't like that. Both stances have pros and cons.
However, it's really hard to infer properties from them in a static manner using model checking or abstract interpretation.
How are we going to infer properties about petri nets or the models Wolfram is proposing? Geometry or differential equations have been so successful because they are amenable to algebraic and analytic approaches.
One application of these might be to resolve the meaning of probabilities under the many-worlds interpretation of quantum mechanics. When an interaction takes place, a number of worlds are created for each possible outcome, but it was never clear how to formally derive the probability of finding yourself in any given world (the Born rule).
Under the section titled "Observers, Reference Frames and Emergent Laws" in this article, you can see some branches merging again in that first graph, so perhaps the probabilities of the Born rule are due to parallel computations that merge in this way, ie. the probability you'd find yourself in the BBBB world rather than the AA world at step 3 of the computation is 2/3 vs. 1/3, respectively. If rules generate recurring patterns as shown there, these might show up as stable probabilities in aggregate.
Yes.
> One application of these might be to resolve the meaning of probabilities under the many-worlds interpretation...
There's some ways to go before we Hilbert spaces, operator algebras, and the like, but yes, the idea would be that some path counting procedure would be used to derive the Born rule. It would be great to bridge with Carroll's self-locating uncertainty paper [0]
> Under the section titled "Observers, Reference Frames and Emergent Laws" in this article, you can see some branches merging again in that first graph, so perhaps the probabilities of the Born rule are due to parallel computations that merge in this way...
Right!
The brilliant point of this paper is to point out that in the same way entropy describes our ignorance of the microscopic state of a system, the action (the time integral of the Lagrangian [itself equal to kinetic energy – potential energy]) seems to quantify our ignorance of the microscopic law that governs a system.
It’s not made explicit, but as a lattice gauge theorist it’s an easy analogy to make: the gauge configurations that contribute correspond to different Dirac operators (ie. PDEs) for matter—-gauge symmetry is in some sense a “rulial space” but… you know… discovered 100 years ago.
[0] http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.42....
You know, it's bizarre, I recently pursued a similar approach, attempting to derive quadratic kinetic energy of an automaton controlled by a computer program (it used a circular tape to decide whether to move left or right). I set up a coarse-groaning, identifying energy with information, well, negentropy of the state of the tape, and arrived at similar results: higher energy = fewer microstates, in other words, there are fewer programs that move at the speed of light than at some slower speed. Which makes sense! There is only one tape that tells the automaton to move right at every step. Toffoli says exactly this: "A low-energy state is 'cheap' because it is 'common', there are so many ways to achieve it". Conservation of energy is exactly microscopic reversibility! What could be cleaner?
That whole approach is on the shelf for now, because I first want to develop a discrete version of contact, and thence symplectic geometry, since that will provide the mathematical formalism to express a kind of "discrete classical physics", rather than just a single toy model.
Anyway, I'm beyond grateful for this reference!
No. Stephen Wolfram is wealthy. He doesn't need to operate or publish within the academic system.
I glance at his writings over the past few years and most often think that unless he uses his theory to solve a problem others care about, his line of research will be abandoned after he passes.
I did buy and read most of it, as I also have bought many of Penrose's popular books, so perhaps I'm just a disgruntled purchaser.
As for how non-local entanglement can occur, I'll just offer my current speculation, having spent about a year working on this kind of approach. You'll have to give up SR in its traditional sense. There will be a preferred reference frame, which is the rule application order that the automata uses in any particular history. The challenge becomes to explain why it is not observable and you have approximate Lorentz covariance. I think this will be less hard than one imagines -- even in Feynman's lectures you see explanations of SR that involve an aether (or a global frame if you prefer) and clocks defined by light bouncing between mirrors moving in this aether. Of course all physical laws have to be covariant, and explaining how this happens requires you to know how the physical laws are "microphysically". But the graph itself in graph-automata models is a good candidate for an ether, with particles being e.g. topological defects. Somehow covariance must reflect how a dynamical account of defect behavior changes as one foliates rule application order, mysterious but not inconceivable.
Now, entanglement: one imagines entanglement is implemented by long-range connections, which in graph-type models could take many forms. This is a kind of discrete version of the "ER = EPR" proposal. But they will have to be such a limited form of connection that they do not permit signaling, and I think they way they can do this is via some sort of knot-theoretic braiding. Only be comparing measurement outcomes classically will it be possible to deduce the way the braiding was effected by measurement and confirm you had e.g. a GHZ state.
Now, QM is more than just entanglement, but in the words of Jaynes: "QM is a peculiar mixture describing in part realities of Nature, in part incomplete human information about Nature - all scrambled up by Heisenberg and Bohr into an omelette that nobody has seen how to unscramble." When scrambled, all the ingredients look inextricably connected. I think the unscrambling will seem beyond hope until one has the exact recipe to recreate the omelette.
That doesn't mean what he's saying has any value. I'm not saying what he's written here doesn't, but the prodigiously smart are just as capable of being intellectually lazy as anyone else. Anyone making grandiose claims like "our Physics Project [... is] showing us something even bigger and deeper: a whole fundamentally new paradigm for making models and in general for doing theoretical science" doesn't get to rest on their laurels if they want to be taken seriously.
In this case we abstract over time in a system of computation. To be more precise the system is a non-deterministic automata, so we abstract over time and all the possible branches too.
Mapping out all state of a non-deterministic automata, you eventually find overlaps of different branches. To Wolfram this is a big revelation since instead of the system getting more complicated you found some simplification, aka. generality.
I respect his ability to integrate a huge body of scientific knowledge in a single brain and then articulate it in books like A New Kind of Science, but I have to wonder whether his approach is designed to maximize scientific progress, or to maximize his personal reputation. His eagerness to claim credit for ideas that could hardly be credited to a single person is a bit of a warning sign.
Exerpt: "But then—basically starting in the early 1980s—there was a burst of progress based on a new idea (of which, yes, I seem to have ultimately been the primary initiator): the idea of using simple programs, rather than mathematical equations, as the basis for models of things in nature and elsewhere."
He also draws out everything more than it needs to be in this long winded narrative and explicit self-congradulatory form that makes me usually wait until someone else reputable with a computer or computational science background reads it and summarizes to see if it's worth suffering through the time to read it myself. Not only is it questionable (Conway and many others come to mind, would have to check dates to see origins but honestly who cares), it's just off-putting.
Wolfram's 'new science' of physics improves in no way at all over existing models when it comes to making predictions about the physical world.
http://backreaction.blogspot.com/2017/11/how-popper-killed-p...
Wolfram's theory though is worse in that it doesn't appear to be even testable, doesn't build on things we know describe the world well,
I don’t think he is the next timecube guy, but I am more glad he is sharing than not.
I spent way too much time looking at the Physics Project when it came out, there is just nothing of substance there. And no, he is not doing experiments that back up his claims, he's just running some random simulations and waving his hands at the pictures that come out.
One key aspect of the scientific method is to acknowledge that it's very easy to fool yourself into thinking you have understood something "up to ironing out some details". Every crackpot is convinced of this. And that's why science operates on the foundation of convincing others of the understanding we have reached. Not any random others, but other people that have invested the time to master what is already known, and are making a good faith effort to understand. I am not talking about peer review, I am talking about the long conversation that unfolds over many many publication, conference talks, heated seminar discussions, etc.
Nothing Wolfram has done/contributed to in the last 30 years that has convinced anyone of note. Deep thinkers in all branches of science that he touches upon have unanimously found his output to be a vanity project of no scientific value. That is not to say that there are no interesting and valuable ideas in the texts he produces, it's simply that these ideas are already known, not his, and he does not acknowledge or tackle the problems that they have. In turn he has not made a good faith effort to engage with the work of others that is pertinent to his claims (usually because this would show that his claims are vastly exagerated and known to be problematic).
He good at one thing: Selling himself as a genius outsider to people not willing or qualified to come to their own conclusions on his work.
(I rather enjoyed this review of ANKS which goes into some detail for a number of the points I make here: http://www.bactra.org/reviews/wolfram/ I personally looked quite deeply into the "papers" produced at the time of his "Physics Project", by Gorard. Some of these are things I have studied very deeply in the past and feel qualified to judge. There simply is _nothing_ there.)
The 'applications' section at the end is just ramblings at best and borderline crankpotty at worst. Which is weird since he is undeniably smart and educated, maybe an overflow?