Unary, binary, ternary, k ary hyperedges in Wolfram Physics
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- เผยแพร่เมื่อ 28 ก.ย. 2022
- Here are answers to some fundamental questions about hypergraphs:
A hyperedge can connect any number of nodes: one, two, three, four, seventeen or any other number.
And a hypergraph can include any of these different kinds of hyperedge, or all of them.
Let’s take a look at what this means for Wolfram Physics... and at some of the beautiful hypergraphs it allows us to generate!
Prefer to listen to the audio? Search for The Last Theory in your podcast player, or listen at lasttheory.com/podcast/017-un...
The full article is at lasttheory.com/article/unary-...
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Hey, I just wanted to say that I absolutely love your channel! It's so professional, and I'm really impressed by the quality of your videos considering the low subscriber count.
I have a suggestion that I think could really enhance the viewing experience for your audience: have you considered adding some quiet, subtle music to your videos? I've seen this done really effectively on channels like 3blue1brown, and it really helps to create a more immersive and professional atmosphere. Plus, it could help keep people engaged and interested in your content.
Another idea is to consider using manim or a similar tool to add some visually stunning, animated math graphics to your videos. I think this could really punch up the production value and make your videos stand out even more.
Just some thoughts, and I hope you find them helpful! I'm a huge fan of your work and I can't wait to see what you do next.
Thanks Ben, I really appreciate that!
And thanks for the suggestions. Yes, for sure, background music and more animations could make these videos more compelling. (I hadn't come across manim before, thanks for the pointer.)
I'd really like to do more animations of hypergraphs, which requires more coding (I use my own simulation software rather than Stephen Wolfram's). Takes a lot of time, but definitely worth it.
I'll continue to work on making these videos more appealing. Thanks for the feedback, it really helps me focus on where I need to improve.
I appreciate the effort you put into this series, keep it up! I have a question on hyperedges though - why did Wolfram pick hyperedges over simple binary edges? In his introduction to his Wolfram Physics, Stephen Wolfram argues that 2-edges don't have a rich enough structure to produce a relevant physics model, but I never understood why this is the case
That's a great question, and I don't know the answer yet. I believe Stephen Wolfram might have backtracked on this, that he might now believe that binary edges are enough. I think the answer is related to Turing Completeness, but I need to look into it further. Thanks for the question, and I'll be sure to post a video on this as soon as I have a better answer!
Presumably, it's for the same reason that you can imagine a 2d world but it wouldn't be as rich as our 3d one. We don't seem to need more than 3d though. 3 is a magic number!
@@fig7047I don't think this is the correct answer. It has to do with the mathematical rules of graph theory. It comes out of necessity.
Thank you for this. This should have 913k views
Thanks! Maybe some day it'll get to 913k!
every time you use the example of "snake, bats and crabs" im reminded of a talk discussing human perception of reality, which stated that evolution does not lead toward perception of the truth but rather perception of fitness. its always good to keep in mind that we are probably not the center of the universe and that anything we see/know is only known because it helps our survival.
Yes. It gets _really_ interesting when you start to think about other consciousnesses that perceive reality in different ways according to _their_ position in rulial space. How do the aliens perceive the universe?
@@lasttheory i feel like this is the most important thing to keep in mind when trying to develop any theory in physics, if its really a theory of everything then it can and will explain the phenomenon of consciousness
@@FrostCraftedMC Yes. It's exciting that there seems a path, at least, to physics explaining, or at least describing, consciousness!
Yes, but survival fitness often requires those perceptions to be true. The fact that science has proved to be extremely useful to our survival demonstrates this clearly, IMO.
This reminds me of a Dialectical Chiasmic Cube Structure! Cool!
Ah, interesting. I'll look into that further!
@@lasttheory Glad to hear, These tho are less theoretical physics terms, and are more philosophical metaphysical terms, but still very interesting and worth checking out!
It seems to me these days theoretical physics is bumping up against Meta-physical Philosophy (Cough Hegelian/Spinozian Cough) more and more, so may be worth looking into!
Great work keep it up!
@@animefurry3508 Thanks! I know less about Hegel & Spinoza than I'd like, so I have some homework to do. You're right, the boundary between physics and philosophy is becoming ever more fascinating!
@@lasttheory My suggestion for when studying Philosophy is not to go Historically Chronologically from example from Antiquity with Plato and Aristotle, to Middle ages, then Enlightenment, and then Contemporary.
This approach tho having its merit really only produces the History of Philosophy and progression of ideas, rather then the generation of new ones.
So i suggest going in reverse, pick someone you admire contemporary, and go backward chronologically, learning about whom influenced them, and so on and so forth.
That way you can bring new insight into old thinkers, which then change how you read both the old thinkers and the new ones!
...
Have fun! And good luck!
My personal favorite contemporary philosopher is Slavoj Zizek!
What difference between k-ary hyperedges with one rule and applying set of rules simultaneously to chain of binary edges? Can we emulate high k-ary hyperedges by lower one?
Another excellent question. If we can emulate a hypergraph with a graph - in other words, if there are rules that can be applied to binary edges in a Turing Complete way - then maybe hyperedges aren't necessary. I'm looking into this, and will record an episode on it when I understand it better! Thanks, as ever, for the questions, Yarov: keep them coming!
@@lasttheory Each k-ary hyperedge can be thought of as a group of k binary edges between each entry/node in the algebraic representation {1,2,3...k} and a new node, N. [{1,N},{2,N},...{k,N}].
Unless im missing something obvious, this seems reasonable.
@@Kowzorz Right, thanks. So that gives us a new concept of an ordered group of binary edges, which obviates the need for k-ary edges. Is the concept of ordered groups of binary edges any simpler, though, than k-ary edges?
So what determines the evolution rule? do we have probabilistic approach like Feynman integrals?
This is the question!
I love it that you bring up the idea of a probabilistic approach, because that really appeals to me. I imagine something like this: simple rules are more probable, complex rules are less probable.
But the truth is, we don't know. The Wolfram Physics Project has moved away from the idea of _one_ rule. Jonathan Gorard is interested in the idea of a finite set of rules that meet certain criteria (though we don't yet know what these criteria might be). Stephen Wolfram is promoting the idea of the ruliad: _every possible_ rule.
This is the question that makes Wolfram Physics such an exciting framework: there's so much to explore in it! Thanks for asking.
I'm a little confused by the notation in the video... wouldn't a unary edge be {1,1}? {1} just looks like a single node unconnected to itself or any other node. A self-connection is different from an unconnected single node, right? {1} is 0-ary, {1,1} is unary, and {1,2} is binary? Maybe a more clear algebraic notation exists for this...
Right, thanks for asking. Maybe, to make it less confusing, I need to explain more clearly that the numbers in the brackets refer to individual nodes.
So {1, 2} is an edge between node 1 and node 2, and {34, 56789} is an edge between node 34 and node 56789. Both are binary nodes, since they're between two different nodes.
To make it even clearer, we could label the nodes with _names_ instead of numbers. so {A, B} is an edge between node A and node B, and {Al, Betty} is an edge between the node called Al the node called Betty. Again, both are binary nodes, since they're between two different nodes.
Unary edges involve only _one_ node. So {1}, {2}, {34} and {56789} are all unary nodes.
And ternary edges involve _three_ nodes, e.g. {1, 2, 3} or {Al, Betty, Charlie}.
Self-loops are just edges that involve the same node twice, e.g. {1, 1} or {Charlie, Charlie} or {Al, Betty, Al}. See my video _Loops and self loops in the hypergraph_ th-cam.com/video/Wk97OLlj020/w-d-xo.html for more on this.
Hope that helps!
Look, I’m really just a TH-cam user ending up here. Not an expert, just some enthausiast. Do have a question, just not sure if it’s a dumb one. In these simulations every step or compute cycle the hypergraph keeps growing. Is it proposed that this also happends in reality then? Or is there some cleaning mechanism as well at play? I really don’t get it yet, is this growth of information also supposed to be connected to the inflationary growth of the universe itself? Or shouldn’t I think about it like that at all? Great video’s in any case!! 😊
It's certainly not a dumb question!
Yes, absolutely, that's the way I look at it: rules that cause the hypergraph to grow are causing the universe to expand. So yes, your way of thinking about it is a good way of thinking about it: the growth of the graph corresponding to the inflationary growth of the universe.
And yes, there might be a "cleaning mechanism", too. Some rules cause the hypergraph to shrink, e.g. replace 2 edges with 1 edge. Obviously, if such a shrinking rule were the only rule, the universe would go on contracting. But if we allow for every possible rule, rather than just one rule, then growing and shrinking rules might balance each other out.
Thanks for watching, Nick, and thanks for the question!
@@lasttheory so cool of you to reply with so much detail! 🫡 if there is a relation, than that’s pretty wild. I mean, would that mean that the smallest planck scale is actually a node in this hypergraph model for instance? And is the node density the same everywhere? If not what would that mean for models like general relativity? Lots to think about, definetly going to watch more on this. For now wondering how this model can be empirically tested. On first sight it feels to me like there should be a lot of ways to do this, any toughts on that?
@@nickbroekman9360 More good questions, thanks Nick. Stephen Wolfram thinks the scale of the graph is much, much smaller than the Planck scale, with edges being around 10^-100 metres. That makes it difficult to test the theory directly, because it's so hard to probe beyond the Planck scale. But there may be indirect tests. For example, what if, as you suggest, the graph isn't the same everywhere? What if the dimensionality of the graph is slightly different from 3 in some regions of space? If the theory predicts such a divergence, it might allow a testable prediction.
@@lasttheory Super cool stuff, really. I’ve been following up some interviews on lex Friedman’s channel. So cool.
I have no idea how he calculated the scale of nodes and edges. But lets say he is 100% correct. Than that says something interesthing maybe? If I understood correct, Planck scale is the tiniest measurement of discrete information we think there is, right (at least in the current standard model)? I could be wrong here, but tought that was kind of the thing with planck scale.
Still it needs massive amounts of nodes and edges to construct these smallest discrete chuncks planck of information.
If you would use a computer as an analogy, is this kind of the relationship between machine code and some higher level language? If so, doesn’t this difference in size tell us a lot as well? And if so, couldn’t we derive a lot of information and perhaps testable stuff from this?
I’m just spitballing, but maybe a computer scientist might have something to say here 😂.
Again, I might just not understand Planck scale, in that case chuck this comment straight into the bin I guess 😅
@@nickbroekman9360 Yes, the relationship between machine code and a high-level language is a pretty good analogy!
The thing with the Planck scale is that it's difficult for us large-scale beings to probe beyond it. And that has led physicists to suggest that there's nothing beyond it (beware invisible things!) But that's obviously not the case: just because we can't see things smaller than the Planck scale, it _doesn't_ mean there's nothing there.
And yes, with any luck, what's there _will_ have an influence on things larger than the Planck scale, things we can test. Too early to say, but let's hope!
After thinking about this for a year im more certain that there's something to it
OK Mark, now that we have come to ‘anything goes’ and so, which rules to apply to the unary to k -ary edges in order to yield our Universe, I finally have a (stupid?)question. Does that mean _all_ possible rules would need to be applied?
This is _definitely_ not a stupid question. In fact, it's the crucial question.
Stephen Wolfram's answer would, I believe, be yes: we should apply all possible rules to the graph.
This generates a multiway graph of all possible universes generated by all possible rules. As you might imagine, this is impossibly complex, and we'll need causal invariance and maybe even consciousness to simplify it to something resembling the universe we perceive.
This is where it starts to get really fascinating: much more to come on this!
@@lasttheory So, something parallel to Feynman integrals??
@@value8035 Yes, maybe! As I mention in reply to one of your other comments, The Wolfram Physics Project has _not_ been going this way, but I'm very interested in these kinds of possibilities.
One more question: Why ignore the weights of edges? Why should all the weights edges be 1?
Right, yes, there's no weighting to the edges in the Wolfram model. They're all equally "strong". We _could_ introduce weights, but that would add complexity. Do you think there's something in physics that might be explained by weighted edges that can't be explained by equal-weight edges?
@lasttheory I do not, however, potentially time dilation, entanglement, and quantum tunneling.
I feel that this is wrong in explaining 3 dimensions in the fact that all your nodes and edges are represented in the same plain and all the same size. In a 3 dimensional situation all your nodes and edges could be the same size but visually ones closer to you would be larger than those away from you. Also there in 3 dimensions would be places where you could only see nodes and places where two edges would look like one or two nodes like one.
Yes, well spotted, Christopher: I've been rendering graphs in two dimensions. But I've written my software so that it can render graphs in three dimensions instead, and I'm going to do exactly as you say and render them in such a way that it _looks_ three-dimensional (I'll draw further nodes & edges smaller and fuzzier than closer nodes & edges, as if you're looking at the graph through thin mist). I'll be showing these in my videos soon! But really, at the scale of the graphs I'm showing, neither two- nor three-dimensional renderings are quite right, because when you measure the Hausdorff dimensionality, these graphs are of fractional dimensionality, e.g. 3.37-dimensional (see my dimensionality trilogy th-cam.com/video/dqnUpq2guX0/w-d-xo.html for more on this). Thanks for the comment, as ever, it's fun to hear your perspectives, and really helps me decide what to focus on next!
what we need is a rule that can take any single node and evolve into a hypergraph with any k-ary hyper edges.
Then it means that we can apply this rule to an already existing network of any size and we will get the same result. Conclusion - it is not necessary to start from one node.
Yes, that's really interesting. The way Wolfram's rules are specified, the -arity of the resulting hyperedges is defined. But if we consider the rulial space of all possible rules, including hyperedges of all possible -arities, then there are rules that do exactly what you say: generate hypergraphs with k-ary hyperedges where k can by anything.
@@YarUnderoaker i see your point! i meant only that the rule should work starting with a graph of a single node.
having read the replies to this comment, i wanna say i think there actually needs to be a way to describe a rule that could bring a universe into existence. there has to be a way to explain why the rule is the rule. in a constantly expanding universe, where energy cannot be created or destroyed(granted that is another convention we may need to rethink like previous videos explain), there has to be a reason all this energy we observe exists. the fact that we are having this convo proves we exist, in some sense in *the* Universe. assuming wolfram right(the entire point of these videos), how does the universe "know" the rule? how can this rule be determined without writing it down? what is going through all the nodes of the universe and applying the rule? and why? whats making the machinery of the universe tick. where did all the energy come from? was it always here? basically, how can a rule exist before the universe it evolves?
@@lasttheory i think wolfram physics gets most interesting in the discussion of rulial space. i have very little to base this prediction on but i believe if wolfram physics is 'true' (whatever that means haha) then we must exist in the universe of all possible rules. thats kinda already stated by the logic of wolfram physics, but more specifically i think we live in the 'newest' branch of rulial space. we arent on the tip of some random branch where only 1 rule gets applied but rather we are the stem all branches come from and lead to
what about -1ary
Ah! I can conceive of an edge involving 3 nodes, or 2 nodes, or 1 node. I fear I lack the imagination to conceive of an edge involving -1 nodes. If you know what that might look like, I'd love to hear ;-)
Ok, so why a graph or hypergraph in the first place? Is there something special about this mathematical model that makes it capable of modelling any system, in the same way that the lambda calculus or Turing machine do for for um... computation.
And in the end, how are they going to know they haven't just created a model that just looks a lot like the world we see, in the same way as classical physics simulations also look realistic, despite in reality having many simplifications and inaccuracies?
Any fact can be represented as a relationship between things, which is exactly what an edge is. The rule is applied on a certain fact, which is just a Horn Clause.
I made a couple comments here already, trying to explain why the Wolfram Model looks the way that it does...there's a lot of misconception about how it works, because there's just a lack of knowledge about what tools are being used to construct the model.
I would suggest looking up "What is a Complex System?" a complex systems 101 video which will give a good enough preamble as to why network graphs in complex systems is being used as the basis for the Wolfram Model.
Any kind of pre-complex systems newtonian modeling is massively outdated. If you go into any kind of nonlinear dynamical systems...which is practically any system that isn't a ball in a box, you will encounter the errors of Newtonian modeling (I'm sure you know about chaos, the three body problem etc...)
Complex Systems swoops in as a way to discuss systems that are nonlinear like this, and they are modeled as networks. If you get into this stuff, you realize that everything points like a huge finger in the sky to "the world is a computation guys" in big fluffy balloon letters. It just wasn't obvious about what the true mechanism behind complex systems was until people like Wolfram pointed it out with self-evident proofs like the CA classes being equivalent and Turing Universal. I would say that The Wolfram Physics Model, is the unified Formalization that Complex Systems Studies was looking for, for decades.
Here's the thing though...Networks is not the only way to model the things...it's just how the wolfram model chose to present it. More specifically, The Wolfram Physics is concerned more with metamathematics, which unifies all the mathematical theories into a single meta description for how mathematics is "created," as they are all forms of computation. So, the Wolfram Model is **more** fundamental then mathematics and really set itself up as a computational model of physics. You can model it in many ways using any of those mathematicise, but network theory/complex systems is in my view the right way one *should* do it because the tools are a lot simpler, more visual, and easy to understand. He has done different modeling's of course, using different mathematics, and there's a lot of neat stuff in his NKS book and some journals where he goes into that.
Cheers,
@@NightmareCourtPictures Thanks, I'll look into that. Yes I am aware of the issues with classical modelling - I've written real time simulations, which cheat by using inelastic collisions, simplified friction, and of course do not model relativistic effects. These simulations however fool millions of people on a daily basis, so my point was how do you really know you've got the rule, and not one that just looks right, especially given the similar constraints of only being able to run small scale simulations due to computational irreducibility?
Edit: Consider this, if we took our really not-real physics models used in games and VFX back 300 years to Issac Newton, and managed to convince him discrete integration was ok (😀), would he be able to tell using the instruments and computation resources available that these models are not really what underpins our universe?
@@lordlucan529 Fascinating discussion.
I have a video in mind to address the question of why a graph? why a hypergraph? It's to do with the fundamentalness (if that's a word!) of relationships. I need to hone the idea before I put it out there!
As to how to know we've got it right, if there _is_ only one rule (I'm not sure that's the case, but just supposing...) I wonder whether the way to show that it's right might be to show that it gives rise to elementary particles with the exact rest masses we see in reality. Something like that. Most rules are obviously wrong, i.e. not capable, on their own, of giving rise to our universe, so we might only have to focus on a few promising rules. Still, it might be (impossibly) difficult computationally to simulate an elementary particle.
Great questions, thanks Andy!
@@NightmareCourtPictures Thanks for this response. "Networks is not the only way to model the things...it's just how the wolfram model chose to present it" - yes, I think this is right.
I’m completely lost or am I stupid ?
If you're lost, that's _my_ fault, not yours!
Did you watch my previous video _What is a hypergraph?_ th-cam.com/video/AbPGsRdNhds/w-d-xo.html
That helps set the stage for this one. The basic idea is that in a graph, an edge joins two nodes, but in a hypergraph, an edge can join three nodes.
In _this_ video, I name these edges: an edge that joins two nodes is a _binary_ edges; an edge that joins three nodes is a _ternary_ edge. And I generalize further, introducing _unary_ edges that "join" only one node, for example.
I hope that helps! Let me know if there's anything more I can do to clarify, thanks!
@@lasttheory I’ve watched all your videos, I find them fascinating , I’ve been following Mr wolfram for decades I’ve never really understood his concepts , I don’t think it’s the way you are presenting his ideas ,
Thank you for the great presentations
You cannot fundamentally unify the universe into one theory or equation because of its natural diversity and the belief that just because we can manipulate nouns(person, places, and things) in useful and decietful ways does not mean we can form a calculation that explains the universe alone. If we could then we wouldn't have different calculations to show the speed of light or polarity or mass vs. area vs. density. Your calculations will use all we have today and all we compute till we either find the answer or die off. I believe that when if ever (in my eyes not probable) come to the equation that explains the universe to our minds it will be so large that it might as well be infinite in length or your np problem you never complete.
That would be the Wolfram model. it is a description for the kind of mechanism that feels "so large" that it seems incomprehensible...but the model does indeed unify everything in a way that is ridiculously simple that you could describe it in about a sentence : All systems are Turing Universal (equivalent to each other) and *The Universe is itself a Turing Machine*
Complex systems and Network Theory is the formal tools Wolfram and others use to describe how this works. It is a very visual set of tools and very easy to learn (look up Complex Systems 101 on You-tube).
It doesn't even require much math to understand...and really there is no "equation" at the bottom in the fundamentals of the Wolfram Physics Model. It's just a self-evident principle discovered through experiment, and then modeling the consequences of such a principle, which is what the wolfram model is...a modeling of the universe based on that principle. In fact there is an exact mechanism that describes what you said here called Computational Irreducibility : "does not mean we can form a calculation that explains the universe alone."
That phenomenon of Irreducibility is a fundamental component of the model, which is the statement that because all systems computational sophistication is equivalent, then systems can not out compute other systems...and therefor the behaviour or complexity of these systems is irreducible...and can never in principle be "reduced" to a description that can predict what the system does, and by proxy can not predict what the universe will do, unless you are the universe itself.
Final words: Watch the 12 part New Kind of Science Series by Wolfram on his channel. He reads the book, and in that book he derives that principle of computational equivalence. That book is the foundation of the Wolfram Model, and that's where Complex System/Network Theory comes in to model it.
Cheers,
You cannot fundamentally unify chemistry into one theory or equation. If we could then we wouldn't have different calculations to show density or colour or melting point or flamability or conductivity. Such a "unifying table" would have no use in calculating density from first principles because the calculation would take forever.
(The use of a unifying theory isn't in using it to re-derive the universe, that's computationally intractable and the reason why we have abstraction. The use of it is in filling in the holes between known and observable phenomena and in developing new hypotheses. We didn't strictly need to understand atomic theory to do chemistry, but it certainly helped!)
The universe is not a machine that created itself based on the calculations that is used to describe it that's like saying mutation of single beings in a group of beings in a species was calculated and imposed upon that individual in order for the species to continue on because those that didn't mutate died from some change in climate or ozone difeiciency or change in oxygen, carbon dioxide levels. The universe can be describe in various aspects of its existence and the happenings within it but it is an always changing and mutating entity that may just turn all your computations on it's heads. How about you all start small and unify a theory of human beings in a computation then the earth then the sun (which atomically and chemistry wise has been done and was probably easier than the previous two). How do you ever expect to predict or define the whole when you don't even understand the parts.
@@christopherevansanders3629 Look uhh...this argument your trying to make...you're only having it with yourself. The Wolfram Model (and complex systems) has literally the same understanding of the universe as what you are attempting to describe...which just means you don't seem to understand the Wolfram Model at all.
I assume that's why you are here on this channel...to learn about it and frankly you should try because everything you are saying...are aspects of how this model works bro.
If you want to know how the model works, you need just a smidge of background in Complex Systems. If you ACTUALLY studied biology or biological systems like myself...you have to use complex systems to research or model anything in the field, and by proxy are inevitably drawn like a trail of bread crumbs to computational theories of physics.
I guarantee, that if you look up the 101 video : "What is a Complex System?" you will realize exactly what I'm trying to tell you...and it should become obvious why the wolfram model looks and operates the way that it does...and the visuals given by Last Theory will make more sense to you.
Cheers,
These are hard discussions to have... so easy to talk at cross-purposes. Thanks for the interesting questions and for the thorough responses!
Who woke up to this?
So 'edges' are like surfaces formed by combining vectors?😊
Thanks Mitchell! The edges are simply lines between nodes. The way I draw the binary hyperedges is a little confusing, in that I draw a surface formed by the two lines of the hyperedge, but there's no reality to that surface, it's just an arbitrary way to indicate that the two lines belong to the hyperedge.