The Wolfram Model & Wolfram Physics Project

[Auto-Didact]

You actually were, in what I quoted in my post #23.

I maintain what I said in #23, if it is understood in the same sense as that calculus is 'part of physics', while in actuality it is part of analysis; this isn't something to get too caught up on because using language to communicate necessarily brings with it some unavoidable vagueness.

In fact, any results found by varying parameters of any iterative map in a computer algebra system is experimental mathematics. Wolfram is a pioneer in this field, because he of course has not merely created Mathematica but fully masters it.

As a side note, the key thing I learned in medicine is that if specific vagueness within some context can be made precise, then there is no problem whatsoever apart from the purely subjective feeling of being uncomfortable around vagueness in a setting without context.

Applied bifurcation theory and stability analysis may or may not involve experimental mathematics, but they are not part of it.

You are again strictly correct but missing the point, namely that using a computer algebra system, such as Mathematica, to actually carry out computational analyses in bifurcation theory and stability analysis in the context of dynamical systems research in practice is de facto doing experimental mathematics.

The original discovery of cellular automata by Wolfram, the original discovery of chaos by Lorenz and the discovery of the Mandelbrot set by Mandelbrot were all mathematical results found by computational experiments i.e. were all instances of experimental mathematics.

Almost all dynamical system theory research is done using computers at some point, instead of using pen and paper for carrying out computations; just recall the key work of Feigenbaum, Lorenz, Mandelbrot, Smale et al. In the universities that I work, the actual subject is taught as a branch of applied mathematics, while the actual experimental research is done using mostly Mathematica.

Of course there are also things done with only pen and paper or chalk and blackboard alone but that is usually more on the theory side (often even done by theoretical and/or mathematical physicists) and even then it is typically based on data gathered from computational experiments.

Dynamical systems theory is a proper branch of pure and applied mathematics. I do not know of any reputable mathematician (in dynamical systems or a different field) that would seriously argue otherwise. (The MSC by the AMS is the subject classification used throughout mathematics, e.g. in any journal of a pure or applied nature.)

The mathematicians I am speaking about are certainly reputable but they only tend to argue pejoratively about the field in private, never in public; of course, those that do look down at dynamical systems tend not to actually be very familiar with the field; they of course accept publications and are more critical of the form than of the content.

They are usually just instinctively criticizing much of the non-rigorous style of research as too foreign from what they themselves do or consider as proper mathematics; in fact, they usually 'insult' the field by saying things like "so you see, the fact that experiments play a key role in the work of those researchers proves that what they are doing is actually not really mathematics, but instead physics".

Good luck pursuing your interests.

Thanks. I never realized that you were a dynamical systems theorist, we seem to be a rare breed on the physicsforums. Judging from your name I'm assuming you are from the Netherlands, do you by any chance happen to have ever met either Ruelle or Takens?

Is the Wolfram approach anything like Category Theory but applied to physics? My understanding is that Category Theory uses digraphs to create abstractions of underlying mathematical proceses whereas Wolfram is using hypergraphs to create abstractions of underlying physical processes.

I get the same feeling, but I'm not fully comfortable answering because I am not a category theorist myself. Maybe ask John Baez? He is active on Twitter.

So can you show somewhere that he is using these methods, even if he doesn't call them by these names?

In the introductory pages of Wolfram's 450 page manuscript he explicitly doesn't mention what the nodes are in his model, but leaves it abstract as 'element': w.r.t. the communication to physicists this seems to be done on purpose to maintain some mystery. His evolution rules of his graphs are described in section 2.6 (page 10) and section 2.7; by actually stating all the axioms of his graphs he essentially has given away what this subject is about.

Moreover, see page 72 of his manuscript in which he shows that the graphs that he constructs purely computationally are statistically indistinguishable from the result of a search done on some actual phenomena in the real world using machine learning methods.

As a side note, I'm getting the feeling that most people commenting here and IRL don't seem to be making these connections. Maybe I should write a review instead of trying to address these things on here? Then again I'm already writing a corona management guideline for primary care at the moment... I don't think my wife will appreciate me taking on more work as it is.

I've studied geometric complexity theory. It is the proposed application of algebraic geometry to the separation of complexity classes (separation means, e.g., proving P is distinct from NP). I call it a proposed application because only a handful of separations have been proven this way so far, although there has been a lot of work aimed at eventually dealing with the more difficult cases.

The GCT method involves constructing algebraic-geometric objects that encode complexity classes, and showing that there are obstructions to embedding one such object into another, which shall in turn imply that the corresponding complexity class of the first object cannot be reduced to the complexity class of the second.

To offer my honest perspective as an outside researcher: the field of GCT is brand new; how many active researchers are there realistically speaking? The fact that there are any meaningful results at all seems to me itself quite amazing.

As a side note, seeing you are familiar with GCT: in how far do the obstructions in GCT map onto the cohomological obstructions in Abramsky's work on non-locality?

This method is not on display anywhere in the Wolfram literature. Indeed the only claim so far is a claim of reduction (equivalence), not of separation, namely the claim that P=BQP, and the methods hinted at, from what I can see, are not even geometric, let alone related to the use of representation theory etc as done in GCT per se.

As I said earlier, Wolfram likes to dance around the point instead of getting straight to the point; perhaps he expects others to flesh out his vaguer points through their own research? I mean, he has after all opened this Project to the universities for all others to participate.

Wolfram aside, I am wondering whether GCT can possibly count as a branch of information geometry. What I understand of information geometry is that involves geometrizing information spaces, e.g. putting a metric on them. I suppose GCT geometrizes and/or algebrizes complexity classes, but there's presently a huge gap between its specific aims and methods, and what anyone else does.

The implication I am making is that they use the same methodology for different purposes i.e. the methods have the same form but do not necessarily refer to the same (type of) content. Nevertheless the interesting question naturally arises whether or not parts of these matured methodologies in one field are directly applicable in another field, e.g. as in the case with the cohomological obstructions in Abramsky's work.

With respect to the geometry of information spaces, the applications are way more obvious because of the direct and central roles that information and entropy play in thermodynamics, mathematical statistics (Fisher metric), machine learning and biophysics.

 

[Auto-Didact]

That's what I was asking. If it doesn't then is it physics or metaphysics?

Excuse me, but since when is the mathematical study of (variants of) equations from physics in order to generalize them suddenly not physics anymore?

By that logic, when Dirac took the Schrodinger equation and generalized it purely mathematically into the equation which now bears his name, he wasn't doing physics either.

 

[martinbn]

Excuse me, but since when is the mathematical study of (variants of) equations from physics in order to generalize them suddenly not physics anymore?

By that logic, when Dirac took the Schrodinger equation and generalized it purely mathematically into the equation which now bears his name, he wasn't doing physics either.

Can you give me an example where the black box did this?

Ps writing down equations is easy, finding useful equations is harder and happens rarely, but deriving the consequences of the equations is the hardest part and the essence.

 

[Auto-Didact]

@mitchell porter & @HBrown if I recall correctly, Wolfram talks about the link to computational complexity, machine learning and so on here and here, among other places.

Can you give me an example where the black box did this?

For example, Wolfram claims on his blog that his black box reproduces the vacuum EFE and full EFE.

Ps writing down equations is easy, finding useful equations is harder and happens rarely, but deriving the consequences of the equations is the hardest part and the essence.

With this I agree to a certain extent, because the step of deriving the consequences can often be translated into a straightforward procedure, sometimes even capturable in a flowchart, i.e. inside some definite often pre-defined framework.

Any method that can be described in this manner isn't very impressive, since it only requires hard work of learning the content inside some given framework, instead of both the hard work of learning the content in some framework and the intuition to go beyond the known framework.

Also, by 'finding equations' I of course mean 'finding useful equations', but I would argue that even finding the wrong equations is useful in the broader theoretical context of gaining scientific understanding; for example, starting with the Schrodinger equation and arriving at the Klein-Gordon equation first when one is actually trying to arrive at the Dirac equation.

In fact, if I question a grad student that is attempting to carry out this derivation and he doesn't know about the route to the Klein-Gordon equation, nor does he even recognize the Klein-Gordon equation, I would take away points and doubt his understanding of what he is actually doing, especially if he wants to become a mathematical physicist.

 

[Auto-Didact]

Here is a visual summary of the main results of the Wolfram Physics Project:

 

[mitchell porter]

As a side note, seeing you are familiar with GCT: in how far do the obstructions in GCT map onto the cohomological obstructions in Abramsky's work on non-locality?

At the most concrete level, they don't look to be very similar. Abramsky's obstructions seem to be about preventing a kind of foliation, whereas GCT's obstructions prevent an embedding of one algebraic variety into another.

if I recall correctly, Wolfram talks about the link to computational complexity, machine learning and so on here and here, among other places.

In the first link he's only talking about complexity but not computational complexity; but in the second one he does mention computational complexity. He says first of all that in his systems, amount of fundamental computation is anchored to fundamental physics. (I will note in passing that this has its analogues in conventional physics, e.g. the Bekenstein bound or the Landauer limit.) Then he says

"there’ll be an analog of curvature and Einstein’s equations in rulial space too—and it probably corresponds to a geometrization of computational complexity theory and questions like P?=NP."

Also later he enthuses about how "it almost seems like everyone has been right all along" and his framework has "hints of" every major quantum gravity research program, and aligns naturally with numerous modern mathematical ideas - and here he mentions GCT again. But the previous comment is the most substantive.

Here, specifically with respect to computational complexity, I think he's simply being naive. This part does indeed sound like information geometry. But the whole difficulty of computational complexity theory, is not in quantifying properties of the algorithms, it lies in showing that one problem can not be mapped onto another from a putatively different complexity class. If you think of how difficult problems in geometry and topology can be (e.g. Poincare conjecture); that's also how problems like P?=NP look, when expressed geometrically. It seems like they'll need that full armoury of Fields-Medal-level techniques, and beyond. Simply expressing the problem in a particular context (like Wolfram's directed hypergraphs) will not itself be a silver bullet.

 

[Auto-Didact]

At the most concrete level, they don't look to be very similar. Abramsky's obstructions seem to be about preventing a kind of foliation, whereas GCT's obstructions prevent an embedding of one algebraic variety into another.

Thanks, that saves me a lot of reading up on the obstructions in GCT; I must say that the demonstration of prevention of embedding intuitively makes a lot of sense as an approach to finding a proof, which also incidentally makes the traditional (more set theoretic) viewpoint of having one class be a subclass of the other seem almost pedestrian in comparison.

Here, specifically with respect to computational complexity, I think he's simply being naive. This part does indeed sound like information geometry. But the whole difficulty of computational complexity theory, is not in quantifying properties of the algorithms, it lies in showing that one problem can not be mapped onto another from a putatively different complexity class.

I never truly got into computational complexity theory precisely because of its non-geometric flavour, but I do know both graph theory and series acceleration from numerical analysis, and also that both of these fields eventually tie into issues from computational complexity theory (at least in so far as the standard texts mention that they do). Moreover, both of these fields tie into CCT in a different fashion, where the series acceleration connection clearly is about quantifying properties of algorithms, while the graph theoretic connection not so much.

To be blunt, the beauty of graph theory is that it naturally contains certain methods to transform an entire topic into a different discipline, e.g. a particular subdiscipline of graph theory can be transformed into set theory and similarly certain particular kinds of graph theoretic methods are de facto really just algebraic topology in disguise. This, or something very similar, is the connection I think that Wolfram is making after having read his manuscript and blogs.

A possibility to do CCT in a geometric fashion seems to be therefore even more interesting, because it opens up the field to a lot of researchers who would otherwise most likely just ignore it, e.g. purely because of its traditional non-geometric form; such biases of only being receptive to certain forms of presentation may sound silly, but they seem to play an enormous role in the practice of mathematics, physics and science more generally.

If you think of how difficult problems in geometry and topology can be (e.g. Poincare conjecture); that's also how problems like P?=NP look, when expressed geometrically. It seems like they'll need that full armoury of Fields-Medal-level techniques, and beyond. Simply expressing the problem in a particular context (like Wolfram's directed hypergraphs) will not itself be a silver bullet.

I don't doubt that it requires Field-Medal-level techniques. What I do doubt is whether or not it has been sufficiently creatively approached from all possible mathematical angles that are already available today, instead of only being creatively approached from a filtered audience of mathematicians who don't mind working on non-geometric forms of mathematics; if the pre-filter group has a higher creativity e.g. because of their geometric intuition, then it's no wonder that the problem hasn't been solved yet.

 

[Andrea Panza]

I would like to know if any of the contributors to this post is aware of the recent developments posted by Wolfram as bulletins in the last couple of months about the emergence of general relativity and the recapitulation of QM phenomena

https://www.wolframphysics.org/bulletins/

My level of understanding is too low to completely apreciate whether his claims are truly revolutionary (and so my question might not be appropriate for a post that is labelled as Advanced), so I come here to the member of this community to hear what the wise men of the mountain :) have to say.

Regards