A The Wolfram Model & Wolfram Physics Project

[Auto-Didact]

TL;DR Summary

https://www.wolframphysics.org/

As seen in the video above, ten years ago Stephen Wolfram teased a computational physics program which he simply called 'Physics'; some of this work was outlined in his magnum opus "A New Kind of Science". A few days ago he has finally given a huge update of this in his blog: https://writings.stephenwolfram.com...damental-theory-of-physics-and-its-beautiful/

This is a project of at least 50 years in the making, starting from about the time that Wolfram left theoretical physics to work on what would become Mathematica and his other well-known products. The Wolfram Physics Project is essentially the result of a brute force search for a mathematical framework underlying physical theory; it is characterized by Feynman's guiding principle for theory construction, namely to recognize when 'more seems to come out than was put in'.

A brief summary of the methodology seems to be that he is modelling discrete mathematical structures - namely directed hypergraphs - and then evolving these structures using simple evolution rules; all of this is done purely computationally. In this evolutionary process, higher dimensional network or simplicial complex-like structures arise automatically; he then compares these arisen graph theoretic structures with the graph theoretic structure underlying known physical theory in order to 'find our own world'.

Wolfram has 3 technical papers with two physics students:

A Class of Models with the Potential to Represent Fundamental Physics

Some Relativistic and Gravitational Properties of the Wolfram Model

Some Quantum Mechanical Properties of the Wolfram Model

Moreover, Wolfram has also opened the Wolfram Physics Project up as a community based project (with universities being able to opt in) for the entire world to join him in trying to find the correct model that describes our world: https://www.wolframphysics.org/

 

[Auto-Didact]

 

[martinbn]

What problems can one solve with this approach?

 

[Auto-Didact]

Automated unification of the mathematical frameworks underlying two (or more) separate established/mature physical theories.

Right now, carrying out this process usually requires a lot of training as well as sufficient experience for the researcher in question; obviously, all of this is done by hand today.

In other words, if it turns out that Wolfram's project works generically, then a huge part of the job of mathematical physicists and applied mathematicians will be lost to automation.

 

[Devils]

His book seemed to go nowhere. I remember reading that great minds often come up with grandiose improbable ideas in their old age. Let's see how this goes.

(The only work of Wolfram I have read was when he was at school & he definitely is a prodigy).

 

[MathematicalPhysicist]

Automated unification of the mathematical frameworks underlying two (or more) separate established/mature physical theories.

Right now, carrying out this process usually requires a lot of training as well as sufficient experience for the researcher in question; obviously, all of this is done by hand today.

In other words, if it turns out that Wolfram's project works generically, then a huge part of the job of mathematical physicists and applied mathematicians will be lost to automation.

Yes, just like computer programmers will lose their jobs long before he accomplishes his goals.
For jobs lost, jobs will be gained... as I wrote before only pushing one button twice a day... on/off.

 

[martinbn]

Automated unification of the mathematical frameworks underlying two (or more) separate established/mature physical theories.

Right now, carrying out this process usually requires a lot of training as well as sufficient experience for the researcher in question; obviously, all of this is done by hand today.

In other words, if it turns out that Wolfram's project works generically, then a huge part of the job of mathematical physicists and applied mathematicians will be lost to automation.

Anything specific?

 

[Auto-Didact]

Anything specific?

Pick any completed unification of theory from the history of physics. Maxwellian Electrodynamics, Special Relativity, General Relativity, Analytical Mechanics, etc.

More specifically, e.g. put in the theory of Magnetism and the theory of Electrity into Wolfram's black box and out rolls, not merely Electrodynamics, but also Special Relativity in the vector calculus formalism, the tensor calculus formalism, the covariant formalism, the differential geometry formalism, the Clifford algebra formalism and so on.

You can basically use his program to unify the conceptual frameworks - i.e. the mathematical frameworks - underlying any physical theory whatsoever - or any sufficiently mathematicized economic, or biological, or sociological, or generally scientific theory whatsoever.

Even stronger, you can even use it to automatically find mathematical frameworks which naturally unify (parts of) different research programs, such as all possible unified mathematical frameworks that underly both string theory and loop quantum gravity; the possibilities seem to be practically endless.

 

[Auto-Didact]

Yes, just like computer programmers will lose their jobs long before he accomplishes his goals.
For jobs lost, jobs will be gained... as I wrote before only pushing one button twice a day... on/off.

Indeed, there are multiple possibilities: either automation or augmentation.
Just like in radiology and dermatology, it is possible this form of AI will merely be a form of augmentation instead of automation for the mathematical physicist. This will almost definitely be the case, certainly in the initial phase; judging across a few decades from now however is a whole other story.

 

[MathematicalPhysicist]

Indeed, there are multiple possibilities: either automation or augmentation.
Just like in radiology and dermatology, it is possible this form of AI will merely be a form of augmentation instead of automation for the mathematical physicist. This will almost definitely be the case, certainly in the initial phase; judging across a few decades from now however is a whole other story.

Well, everyone is using some sort of software for their calculations (also pure mathematicians, they have sage and other computer programmes).

BTW, there's a book that I planned eventually to purchase on Clifford analysis which has an old software for the lengthy calculations called REDUCE if you know this software. (quite old).
But now with the Covid-19 I hesitate to purchase it since who knows if I will even get my purchase.
Who knows if we will ever get to live until computer programmers will lost their jobs for automation...

 

[MathematicalPhysicist]

Pick any completed unification of theory from the history of physics. Maxwellian Electrodynamics, Special Relativity, General Relativity, Analytical Mechanics, etc.

More specifically, e.g. put in the theory of Magnetism and the theory of Electrity into Wolfram's black box and out rolls, not merely Electrodynamics, but also Special Relativity in the vector calculus formalism, the tensor calculus formalism, the covariant formalism, the differential geometry formalism, the Clifford algebra formalism and so on.

You can basically use his program to unify the conceptual frameworks - i.e. the mathematical frameworks - underlying any physical theory whatsoever - or any sufficiently mathematicized economic, or biological, or sociological, or generally scientific theory whatsoever.

Even stronger, you can even use it to automatically find mathematical frameworks which naturally unify (parts of) different research programs, such as all possible unified mathematical frameworks that underly both string theory and loop quantum gravity; the possibilities seem to be practically endless.

Seems too good to be true.

 

[mitchell porter]

automation ... this form of AI

I thought Wolfram was just proposing to base physics on a kind of cellular automaton. Is he using AI to search the possible rules? Where does he talk about this?

 

[HBrown]

I read through his initial post.
He's clearly excited about some ideas, but they are still vague, and to some extent he seems aware of this.
He seems to make the common mistake of being just vague enough about some important pieces that he can oscillate between different concepts as it suits his hand waving. For example how he wants to extract physical notions from an abstract graph, even simple things like distance change flavor in various hand wavy descriptions.

In the end, he has an idea that he enjoys, and he has resources to chase them.
It doesn't sound compelling to me, but arguing motivations and what sounds plausible is pointless, he's going to spend his time on it regardless. So I guess I wish him luck. Looks like a long shot.

 

[martinbn]

Pick any completed unification of theory from the history of physics. Maxwellian Electrodynamics, Special Relativity, General Relativity, Analytical Mechanics, etc.

More specifically, e.g. put in the theory of Magnetism and the theory of Electrity into Wolfram's black box and out rolls, not merely Electrodynamics, but also Special Relativity in the vector calculus formalism, the tensor calculus formalism, the covariant formalism, the differential geometry formalism, the Clifford algebra formalism and so on.

You can basically use his program to unify the conceptual frameworks - i.e. the mathematical frameworks - underlying any physical theory whatsoever - or any sufficiently mathematicized economic, or biological, or sociological, or generally scientific theory whatsoever.

Even stronger, you can even use it to automatically find mathematical frameworks which naturally unify (parts of) different research programs, such as all possible unified mathematical frameworks that underly both string theory and loop quantum gravity; the possibilities seem to be practically endless.

If this is an accurate description then it sounds like a meta-theory, not physics. Not so sure if this is useful. The input is theories that are well understood, the output is theories that are also well understood, but only if you can recognize them otherwise you get all kinds of possibilities without any hope of filtering a useful one.

May be I should have been more specific myself. By a problem I mean the type of problems physicists solve. Say heat conduction in a rod, a vibrating membrane, an orbiting planet, or anything like that. Any example at all. How does his black box deal with it?

 

[S.G. Janssens]

You can basically use his program to unify the conceptual frameworks - i.e. the mathematical frameworks - underlying any physical theory whatsoever - or any sufficiently mathematicized economic, or biological, or sociological, or generally scientific theory whatsoever.

If this is indeed what he believes his "program" is capable of, then it is not just comfortably vague, or "too good to be true" (post #11), but it is also blatantly arrogant. Based on his "New Kind of Science" I did not expect anything less.

 

[MathematicalPhysicist]

If this is indeed what he believes his "program" is capable of, then it is not just comfortably vague, or "too good to be true" (post #11), but it is also blatantly arrogant. Based on his "New Kind of Science" I did not expect anything less.

Sometimes it appears that history repeats itself...

 

[Auto-Didact]

I thought Wolfram was just proposing to base physics on a kind of cellular automaton. Is he using AI to search the possible rules? Where does he talk about this?

Read the three papers and his book. It also helps to take a moment of reflection and look at this, not from the perspective of a contemporarily practicing theoretical physicist, but instead from the perspective of a modern applied mathematician, computational scientist or data scientist that is knee-deep within the modern deep learning revolution (NB: I'm fairly ankle-deep in this as a consequence of supervising (under)graduate students doing such research).

I read through his initial post.
He's clearly excited about some ideas, but they are still vague, and to some extent he seems aware of this.
He seems to make the common mistake of being just vague enough about some important pieces that he can oscillate between different concepts as it suits his hand waving. For example how he wants to extract physical notions from an abstract graph, even simple things like distance change flavor in various hand wavy descriptions.

In the end, he has an idea that he enjoys, and he has resources to chase them.
It doesn't sound compelling to me, but arguing motivations and what sounds plausible is pointless, he's going to spend his time on it regardless. So I guess I wish him luck. Looks like a long shot.

If this is indeed what he believes his "program" is capable of, then it is not just comfortably vague, or "too good to be true" (post #11), but it is also blatantly arrogant. Based on his "New Kind of Science" I did not expect anything less.

I understand your skepticism and actually appreciate it in order to paint a contrast. What Wolfram in his enthusiasm doesn't emphasize enough is that his method doesn't only produce a "unique correct answer", but instead also a veritable multitude of answers; a selection procedure (from mathematical physics/condensed matter theory flavored statistical physics) is then used to trim this forest down to find the "fittest" answers (NB: as in survival of the fittest). Subsequently these fittest answers are then used axiomatically to rederive known physical theories, as a kind of second order selection method. All of this is quite typical hybrid methodology in modern machine learning.

The above hybrid methodology is a highly creative but very specific application of evolutionary algorithms applied to axiomatization applied to derivation from first principles applied to theory construction, which each are of course each tried and true methods. What isn't conventional knowledge to physicists is that this hybrid method isn't actually new; it is known as "experimental mathematics" and has actually been quite conventional since the 80s within applied mathematics and engineering under another moniker namely, "dynamical systems theory".

A presentation or communication problem which often arises is that most physicists (and many mathematicians and computer scientists as well) simply aren't accustomed to or familiar with this type of research and therefore are unable to recognize it when they come across it and so end up reliably misjudging its scientific value. What is truly novel is that Wolfram seems to have automated a large part of the experimental mathematics methodology, and so actually given it a rigorous grounding, when before it was seen as more of an art form, i.e. like skillful experimentation often is seen.

In any case, the importance of this discovery cannot be stressed enough: what Wolfram has discovered is a revolution not just for theoretical physics, but actually for pure mathematics as well; namely Wolfram has laid bare, in a purely axiomatic format, that there are as of yet unidentified existing links between the theory of analysis, computational complexity theory and algebraic geometry which is not generally recognized by either the physics literature or the physics community, apart from a handful of subfields within mathematical physics.

In fact, the last time that I can recall off the top of my head that such a huge mathematical discovery was actually done in mathematical physics proper on the basis of deepening the understanding of older theories was when Euler, Lagrange and Hamilton reviewed Newtonian Mechanics using extremely sophisticated mathematics for their time and ended up inventing Analytical Mechanics as well as emphasizing and/or literally inventing the associated mathematical disciplines.

If this is an accurate description then it sounds like a meta-theory, not physics. Not so sure if this is useful. The input is theories that are well understood, the output is theories that are also well understood, but only if you can recognize them otherwise you get all kinds of possibilities without any hope of filtering a useful one.

May be I should have been more specific myself. By a problem I mean the type of problems physicists solve. Say heat conduction in a rod, a vibrating membrane, an orbiting planet, or anything like that. Any example at all. How does his black box deal with it?

Those actually aren't "physics problems", but instead applied physics problems i.e. actually engineering problems and therefore, generally speaking, completely uninteresting to the discipline of theoretical physics directly; of course, this doesn't mean they are uninteresting to experimental physics, since such problems obviously can lead to better experiments. However that is clearly a matter of secondary concern for the enterprise of theoretical physics - especially when experimental physics has already reached the degree of sophistication as today - and it is even of tertiary concern for mathematical physics.

More directly, Wolfram's black box is a complete overkill for such applied physics problems since such problems - by their very hands-on nature - literally require to be defined within an already given mathematical framework; Wolfram's blackbox is instead for addressing problems with entire frameworks themselves, i.e. by actually removing what is given. Actually doing such a thing successfully literally requires a seasoned expert in theoretical physics and/or mathematical physics, e.g. a practicing physicist at the level of John Baez and often in that age category as well.

Suffice to say, Wolfram's earlier invented, pedestrian tools - i.e. Mathematica and WolframAlpha - tend in almost all cases to be sufficient for tackling most applied physics problems with a fair amount of effectiveness such that even most undergraduates are typically capable of quickly making some headway towards finding solutions to such applied physics problems; difficult outstanding applied problems from this perspective are merely remaining 'low hanging fruit' which require at most a few grad students and perhaps a good doctoral advisor.

 

[mitchell porter]

What Wolfram in his enthusiasm doesn't emphasize enough is that his method doesn't only produce a "unique correct answer", but instead also a veritable multitude of answers; a selection procedure (from mathematical physics/condensed matter theory flavored statistical physics) is then used to trim this forest down to find the "fittest" answers (NB: as in survival of the fittest). Subsequently these fittest answers are then used axiomatically to rederive known physical theories, as a kind of second order selection method. All of this is quite typical hybrid methodology in modern machine learning.

So far I have not found any statement anywhere from him, which talks about using machine learning, evolutionary algorithms, etc, in order to choose among models.

what Wolfram has discovered is a revolution not just for theoretical physics, but actually for pure mathematics as well; namely Wolfram has laid bare, in a purely axiomatic format, that there are as of yet unidentified existing links between the theory of analysis, computational complexity theory and algebraic geometry which is not generally recognized by either the physics literature or the physics community, apart from a handful of subfields within mathematical physics.

The only thing I have seen about computational complexity theory is a passing claim in the paper on "Some quantum mechanical properties...", that they can show P=BQP (P being polynomial-time algorithms, BQP being its quantum counterpart), a claim which, on account of its made-in-passing character, looks certain to be not only wrong, but dreadfully naive, by the standards of actual computational complexity theory. The computer scientists who study complexity of quantum algorithms have a professional interest in classical algorithms that can approximate quantum ones. The claim by Jonathan Gorard of Wolfram Inc seems to be made on the basis of a very simple way of imitating or approximating quantum mechanics.

As for "algebraic geometry"... It has occurred to me that one could try to imitate a field theory with tensor-valued fields, using cellular automata, by having a CA system where the grid discretely approximates space-time, and where the CA cell types discretely approximate the possible values of the vector/matrix/tensor-valued field. Is this anything like what they do?

 

[arivero]

I read Wolfram's PhysRev papers in my second year of undergraduate. At that time the Wilson-Kogut renormalisation and Kadanoff's block-spin were still hot topics, so the natural question I remember to discuss with some friends was "how do you renormalise a celular automata".

 

[Demystifier]

If this is an accurate description then it sounds like a meta-theory, not physics.

That's correct, but it doesn't mean that it doesn't have a value. I would compare it with logic and set theory, which are often viewed as meta-mathematics rather than mathematics. But it doesn't mean that logic and set theory are not important for mathematics.

Or more generally, almost any branch of science has its meta-science. For instance, chemistry is a meta-science for biology. It doesn't mean that chemistry is not needed in biology.

 

[martinbn]

That's correct, but it doesn't mean that it doesn't have a value. I would compare it with logic and set theory, which are often viewed as meta-mathematics rather than mathematics. But it doesn't mean that logic and set theory are not important for mathematics.

Or more generally, almost any branch of science has its meta-science. For instance, chemistry is a meta-science for biology. It doesn't mean that chemistry is not needed in biology.

That of course is true. They do have value. But no one would say that they had a break through in say elliptic differential equations if they couldn't prove a single statement about these equations. But it seems like that in the case of Wolfram.

 

[martinbn]

Those actually aren't "physics problems", but instead applied physics problems i.e. actually engineering problems and therefore, generally speaking, completely uninteresting to the discipline of theoretical physics directly; of course, this doesn't mean they are uninteresting to experimental physics, since such problems obviously can lead to better experiments. However that is clearly a matter of secondary concern for the enterprise of theoretical physics - especially when experimental physics has already reached the degree of sophistication as today - and it is even of tertiary concern for mathematical physics.

Well, I completely disagree with this. These are theoretical physics problems. In fact I would say that this is exactly what physics is about.

More directly, Wolfram's black box is a complete overkill for such applied physics problems since such problems - by their very hands-on nature - literally require to be defined within an already given mathematical framework; Wolfram's blackbox is instead for addressing problems with entire frameworks themselves, i.e. by actually removing what is given. Actually doing such a thing successfully literally requires a seasoned expert in theoretical physics and/or mathematical physics, e.g. a practicing physicist at the level of John Baez and often in that age category as well.

So this black box would not be able to solve any of these problems, not even those that are well understood and can be solved by physics students.

Suffice to say, Wolfram's earlier invented, pedestrian tools - i.e. Mathematica and WolframAlpha - tend in almost all cases to be sufficient for tackling most applied physics problems with a fair amount of effectiveness such that even most undergraduates are typically capable of quickly making some headway towards finding solutions to such applied physics problems; difficult outstanding applied problems from this perspective are merely remaining 'low hanging fruit' which require at most a few grad students and perhaps a good doctoral advisor.

Not sure if this is relevant.

 

[S.G. Janssens]

What isn't conventional knowledge to physicists is that this hybrid method isn't actually new; it is known as "experimental mathematics" and has actually been quite conventional since the 80s within applied mathematics and engineering under another moniker namely, "dynamical systems theory".

Sorry, but this is not correct. I am a mathematician working in dynamical systems theory, classified here (2010) and here (2020) by the AMS. Suggesting that "experimental mathematics" and "dynamical systems" are synonyms is wrong.

"Experimental mathematics" can suggest directions of research in dynamical systems theory (and so can many other fields in mathematics and the sciences), and dynamical systems theory can suggest new mathematical experiments (analogous remarks apply), but you cannot equate them.

 

[Auto-Didact]

So far I have not found any statement anywhere from him, which talks about using machine learning, evolutionary algorithms, etc, in order to choose among models.

This is precisely Wolfram's biggest problem: he doesn't say that he is using evolutionary algorithms and/or machine learning, while he is. It is likely that he just disregards the terminology of machine learning, because he is by training a theoretical physicist and - like most physicists, including myself - therefore just blatantly disregards the conventional terminology of machine learning, computer science and applied mathematics, quite simply because he can't be bothered to go through the trouble of translating mathematical terminology between different mathematical disciplines; he isn't interested in communication to other kinds of mathematical scientists but only in doing mathematical science himself (NB: and as everyone knows, the physics terminology is superior).

The only thing I have seen about computational complexity theory is a passing claim in the paper on "Some quantum mechanical properties...", that they can show P=BQP (P being polynomial-time algorithms, BQP being its quantum counterpart), a claim which, on account of its made-in-passing character, looks certain to be not only wrong, but dreadfully naive, by the standards of actual computational complexity theory. The computer scientists who study complexity of quantum algorithms have a professional interest in classical algorithms that can approximate quantum ones. The claim by Jonathan Gorard of Wolfram Inc seems to be made on the basis of a very simple way of imitating or approximating quantum mechanics.

Wolfram's latest work is directly related to a novel subdiscipline in computational complexity theory, namely (algebro-)geometric complexity theory, see the Wikipedia page and arxiv references. Wolfram's approach in this field seems to be completely novel; moreover, geometric complexity theory is related to the still semi-nascent interdisciplinary field called information geometry (i.e. algebro-geometric information theory), which is of course related to thermodynamics and so theoretical physics; there are even more specific problems that can be experimentally approached, e.g. in theoretical biophysics.

As for "algebraic geometry"... It has occurred to me that one could try to imitate a field theory with tensor-valued fields, using cellular automata, by having a CA system where the grid discretely approximates space-time, and where the CA cell types discretely approximate the possible values of the vector/matrix/tensor-valued field. Is this anything like what they do?

Not exactly. His latest work isn't really about CA, but instead about directed hypergraphs, in which these (hyper)graphs themselves are then evolved via an evolution rule a la CA. Alternatively, conventional graph theory methods, via topological considerations, allows a direct transformation of the entire framework into a purely algebro-geometric construction.

Well, I completely disagree with this. These are theoretical physics problems. In fact I would say that this is exactly what physics is about.

This is quite frankly a misunderstanding of what theoretical physics entails as a discipline. Theoretical physics is, as the name suggests, primarily concerned with the theoretical side of the science physics, i.e. with the invention of new physical theories and working out of mature physical theories. Solving applied physics problems is literally of secondary concern for the endeavor of theoretical physics; if solving such applied problems seems to be characteristic of theoretical physics today then that is an accident of history, and even worse a symptom of the crisis that theoretical physics is currently in.

More descriptively, theoretical physics as a scientific endeavor is primarily concerned with finding mathematical models of physical phenomenon, i.e. with finding what are called physical theories. A physical theory is generally speaking essentially a mathematical function with an input and output structure which directly serves as a mathematical model of the behavior of some specific physical phenomenon; this mathematical function, as well as its input and output space and all other mathematical properties (continuity, analyticity, boundary conditions, embedding space, etc), naturally tends to belong to a class of mathematical frameworks from pure and/or applied mathematics (NB: for simplicity, I am also referring to anything which is a direct generalization of a function - such as a functional - as a function).

Examples of physical theories are obvious: a mathematical model of gravity, of electromagnetism, of light, of atoms, of heat, of particles, etc. The finding and creation of such mathematical models of physical phenomena i.e. making such physical theories is the job of theoretical physics; the testing of such theories is one of the main jobs of experimental physics, and finally the application of the theories that have survived experimental testing is the job of applied physics; solving problems such as the ones you pose squarely falls in this final category.

So this black box would not be able to solve any of these problems, not even those that are well understood and can be solved by physics students.

They actually can be solved by his black box, because this black box is directly integrated with both WolframAlpha and Mathematica, i.e. WolframAlpha translates and integrates input/output from the black box into Mathematica code. Even better, the black box is written and used in the same Wolfram Language, i.e. will even automatically be integrated into the next version of Mathematica, exactly as what happened with WolframAlpha and Wolfram's own neural network programming suite.

But I want to make clear why wanting to use this particular black box for applied physics problems is almost a ridiculous request through the following analogies: it is like trying to let the paper boy do his newspaper delivery round by using a rocket ship to visit the houses in his neighborhood instead of just using a bicycle, or like consulting a surgeon for a small cut which doesn't even require a bandage, or like utilizing a nuclear weapon in order to take care of a rat infestation within an old building.

 

[HBrown]

I'm sorry, but Auto-Didact you are putting a lot of words in Wolfram's mouth. I feel what you are saying is unrecognizable to the actual content of what he wrote.

It would be easier to discuss these things if you clearly separated ideas this inspires for you (where you think this will grow as a field, what it may accomplish, and other speculations) from what Wolfram is actually saying. It is already hard enough to discuss because Wolfram is still fleshing out his ideas. Please do not mix in another layer of speculation on top of this.

 

[Auto-Didact]

Sorry, but this is not correct. I am a mathematician working in dynamical systems theory, classified here (2010) and here (2020) by the AMS. Suggesting that "experimental mathematics" and "dynamical systems" are synonyms is wrong.

"Experimental mathematics" can suggest directions of research in dynamical systems theory (and so can many other fields in mathematics and the sciences), and dynamical systems theory can suggest new mathematical experiments (analogous remarks apply), but you cannot equate them.

Pardon me, they aren't synonymous and I am not equating them, but instead relating them in exactly the way that you say: huge swats of experimental mathematics are literally completely part of the basic conventional methodology within dynamical systems theory (analogous to calculus being completely part of the basic conventional methodology in physics). In fact, much of experimental mathematics methodology used in dynamical systems theory tends to be seen as so pedestrian to dynamical systems researchers that they themselves don't even bother to give these methods a name, much less refer to them as 'experimental mathematics'.

It would of course be more correct to regard experimental mathematics as a subfield within mathematics itself, but that completely misses the main point I am trying to make, namely that large parts of experimental mathematics (e.g. doing computational experiments, applied bifurcation theory, stability analysis etc) are in fact conventional methodologies within dynamical systems research in actual practice, whether or not dynamical systems researchers use or recognize the terminology 'experimental mathematics'.

In other words, if a physicist wants to know how to use experimental mathematics in his own research it is productive (or at least, it was in my case) to talk to a dynamical systems theorist or to peruse the dynamical systems theory literature, instead of talking to a (pure) mathematician or perusing the mathematics literature which in my experience is quite counterproductive because many mathematicians don't even seem to recognize dynamical systems theory as proper mathematics.

 

[Auto-Didact]

I'm sorry, but Auto-Didact you are putting a lot of words in Wolfram's mouth. I feel what you are saying is unrecognizable to the actual content of what he wrote.

It would be easier to discuss these things if you clearly separated ideas this inspires for you (where you think this will grow as a field, what it may accomplish, and other speculations) from what Wolfram is actually saying. It is already hard enough to discuss because Wolfram is still fleshing out his ideas. Please do not mix in another layer of speculation on top of this.

I can give specific references if you want, where is the exact confusion?

 

[HBrown]

Mitchel Porter already brought up an explicit example.

Furthermore, you are expanding the hype beyond what even Wolfram is claiming, with your added layer of abstraction to a black box that can automatically unify any mathematical research we shove in the box

Even stronger, you can even use it to automatically find mathematical frameworks which naturally unify (parts of) different research programs, such as all possible unified mathematical frameworks that underly both string theory and loop quantum gravity; the possibilities seem to be practically endless.

First, it is incredibly difficult to figure out how to extract physical meaning from the hypergraphs. This is not something that just falls out of a black box, or occurs by iterating some machine learning program trying to maximize some fitness function. Extracting meaning from the hypergraphs is something that comes from interpretation, figured out by hard work of a human, and placed on there by a human.

Second, if anything, the "input" is an initial condition and a graph evolution rule. He then is looking at what happens and trying to extract meaning. With luck he can extract QM or something. This is the opposite of how you are trying to present it.

Third, let me remind you that Wolfram is claiming his series of ideas for how to interpret the graph evolution make SR and even GR come out inevitably. This means you cannot have Newtonian gravity with its perfect Kepler orbits (no GR corrections), or Newtonian mechanics with Galilean symmetry (because his ideas _require_ a finite information speed limit), or many other mathematically consistent possibilities. What you are hyping doesn't seem to line up with what he is suggesting at all.

Anyway, my point was to be careful to separate your speculations from Wolfram's.

 

[Auto-Didact]

Mitchel Porter already brought up an explicit example.

More specific, do you mean the connection with algebraic geometry or computational complexity theory (or something else)?

Furthermore, you are expanding the hype beyond what even Wolfram is claiming, with your added layer of abstraction to a black box that can automatically unify any mathematical research we shove in the box

That isn't exactly my intention; my intention is to make Wolfram's claims as bluntly as possible, instead of dancing around the claims as he does by cryptically hiding the main points in hundreds of pages, walls of blogtexts and hours of video (just look at his blogs, youtube posts, books and so on).

First, it is incredibly difficult to figure out how to extract physical meaning from the hypergraphs. This is not something that just falls out of a black box, or occurs by iterating some machine learning program trying to maximize some fitness function. Extracting meaning from the hypergraphs is something that comes from interpretation, figured out by hard work of a human, and placed on there by a human.

Agreed, doing this requires serious experience and familiarity with applied discrete mathematics as well as knowledge how it relates to fields such as dynamical systems theory and algebraic geometry; it just so happens that this issue was exactly one of the core issues of the main research topic of a research group that I am leading. If I am speaking in a too as-a-matter-of-fact sort of way that is a consequence of my intimate familiarity with the material in question.

Second, if anything, the "input" is an initial condition and a graph evolution rule. He then is looking at what happens and trying to extract meaning. With luck he can extract QM or something. This is the opposite of how you are trying to present it.

His input isn't merely an initial condition of some system like in calculus/physics, it is something of a completely different nature; this is de facto the key point that he is dancing around by shrouding it in unnecessary mystery.

Third, let me remind you that Wolfram is claiming his series of ideas for how to interpret the graph evolution make SR and even GR come out inevitably. This means you cannot have Newtonian gravity with its perfect Kepler orbits (no GR corrections), or Newtonian mechanics with Galilean symmetry (because his ideas _require_ a finite information limit), or many other mathematically consistent possibilities. What you are hyping doesn't seem to line up with what he is suggesting at all.

He is being severely careful in his wording, which leads to the length of the manuscript of 400pp what can be easily stated in about 20pp. The point is though that many different limiting models of deeper theories can be identified by a combination of statistical analysis over the limits by varying the input models and then identifying the unique correct derivation through the method of experimental mathematics e.g. using Mathematica; this strategy itself is a completely conventional method in machine learning.

Anyway, my point was to be careful to separate your speculations from Wolfram's.

Fair point.

 

[Demystifier]

That of course is true. They do have value. But no one would say that they had a break through in say elliptic differential equations if they couldn't prove a single statement about these equations. But it seems like that in the case of Wolfram.

Did Wolfram said that his approach solves a specific problem in physics? Maybe I missed something, but it seems to me that he only claims to have a general framework for a "theory of everything".

 

[S.G. Janssens]

Pardon me, they aren't synonymous and I am not equating them

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

namely that large parts of experimental mathematics (e.g. doing computational experiments, applied bifurcation theory, stability analysis etc) are in fact conventional methodologies within dynamical systems research in actual practice, whether or not dynamical systems researchers use or recognize the terminology 'experimental mathematics'.

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

In other words, if a physicist wants to know how to use experimental mathematics in his own research it is productive (or at least, it was in my case) to talk to a dynamical systems theorist or to peruse the dynamical systems theory literature, instead of talking to a (pure) mathematician or perusing the mathematics literature which in my experience is quite counterproductive because many mathematicians don't even seem to recognize dynamical systems theory as proper mathematics.

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.)

Good luck pursuing your interests.

 

[Devils]

Sorry, but this is not correct. I am a mathematician working in dynamical systems theory, classified here (2010) and here (2020) by the AMS. Suggesting that "experimental mathematics" and "dynamical systems" are synonyms is wrong.

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.

 

[mitchell porter]

This is precisely Wolfram's biggest problem: he doesn't say that he is using evolutionary algorithms and/or machine learning, while he is.

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

Wolfram's latest work is directly related to a novel subdiscipline in computational complexity theory, namely (algebro-)geometric complexity theory, see the Wikipedia page and arxiv references. Wolfram's approach in this field seems to be completely novel; moreover, geometric complexity theory is related to the still semi-nascent interdisciplinary field called information geometry (i.e. algebro-geometric information theory), which is of course related to thermodynamics and so theoretical physics; there are even more specific problems that can be experimentally approached, e.g. in theoretical biophysics.

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. 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.

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.

 

[Devils]

After listening to the videos, Wolfram makes claims that the photon may not be massless, and he can explain why anti-matter does not have anti-mass. It will be interesting to see whether his system can make predictions or explain phenomena.

 

[martinbn]

Did Wolfram said that his approach solves a specific problem in physics? Maybe I missed something, but it seems to me that he only claims to have a general framework for a "theory of everything".

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

 

[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