# Wolfram Physics Project: Working Session Aug 18, 2020 [Physicalization of Empirical Metamathematics]

May 23, 2024
Interesting, it is almost a complete graph, what it shows is the set of entanglement relationships between terms. What the hell is that mathematically? So do you understand what this does? This says that two nodes are joined if they have a common ancestor here. So in a sense, this is saying. these are two different results, there are two things that would be possible endpoints of a theorem, so the question is: what is this entanglement graph? I mean, this is throwing us into math. Mathematics is about which theorems are true stuff. mathematics is more about what the possible proofs are, it just makes sense, whatever, yes, yes, I think the paths on the left are proofs, the paths on the right, the left are proofs, these here are essentially part of proofs of higher order because they say that given If you have two proofs of the same theorem, you can ask how you know what the relationship is between those two proofs, but in reality this is not even a proof of the same theorem, they are proofs of nearby theorems.
He wonders what the map of nearby theorems is. correct theorems, so what we're saying is that if we were to cluster, you know, this is an interesting point in a quantum measurement, what we end up doing is clustering nearby quantum states. We don't have a direct analogue of that in mathematics in our course. grading in math is fine, so what's the analog? So that would be like saying we have a collection of possible theorems that are close theorems and what the heck are close theorems in mathematics uh mathematical subjects well, unless not axiom unless it is unless it is different systems of axioms, but okay , nearby theorems can be about similar things.

## wolfram physics project working session aug 18 2020 physicalization of empirical metamathematics...

I guess there's another way to think about near theorems, which are things that can be easily accessed in a lemma from a certain term, in other words, given a term, there are things that can be done. kind of ball around that term which are things that are easily accessible, yeah right, so it's, it's, I mean, there's a lot of lemmas, so some theorems can be proven shortly from others, they begin with a set of quotes of close terms, also known as things that are Do you know why the equivalence test is easy? As a consequence of the involuntary manslaughter theorem, it has been proven.
What do you mean there was a statement from Andrews Wild at the end of his test? I don't remember the name of the conjecture, but a consequence of it was the strong attack theorem, so what is it, how is that related to this? It's kind of close, it was a very powerful conjecture and Fermat's Last Theorem was a trivial consequence. I see what you're saying. I can see that. you're claiming that a powerful conjecture is something that's a big backbone, right, a powerful theorem, it better be a big backbone in this graph, of which there are corollaries, I mean, basically, yeah, this is, these could be called corollaries, the close ones, you know, close theorems. are basically corollaries corollaries brian is saying that the close theorems are those that use similar flames uh yes, that's true too, for which most of the proof is similar, but I claim that the corollary is perhaps the correct way to think about close theorems, so we have a collection of corollaries and one thing we ask is, again, is it so confusing between these paths and the full theorems that a collection of correlates are probably a gd6 package yes, in a sense, the question whether it is a positively or negatively curved space, for example, it is a question of whether the corollaries at the two ends are closer, you know whether in the middle gd6 converges or diverges, in other words whether these corollaries are, you know If in the middle of trying these different corollaries you end up with totally different paths to try them, even though in the end they're all, they're very similar, yeah, I mean, it's a little confusing, but, um. um, you know, trying to understand what hmm, well, the question can be seen as related to category theory, yeah, uh, but it's yeah, and I mean this is all like a week, I mean, you can think about this, I think if you wanted to.
You could think of the structure of this as a weak category with these unidirectional paths or this is like categories, but I haven't made it out that there is no transitivity here. I haven't done the transitive closure if I did the transitive closure which is what you would do in ordinary category theory then I would have basically corrected every lemma, in other words in category theory the version of this would be that in this graph you take the transitive closure of this graph, which means that I'm I'm letting it be one step to get to everything, but what's the relationship with category theory?
Once you know these paths are tests, but now you can see the homotopy between parts, which is the transformation from one test to another. -called higher order proofs and when you look at that, that's the knowledge analogous to the higher categories and the infinite limit of that is this infinite groupoid object, so um and jonathan has a lot of conjectures about how that works and how it's related with I mean growth actually studied a lot the limit of effectively infinitely high order testing or what it came down to infinitely high order testing um and uh um that's um uh yeah so that's kind of it. the relationship with category theory um let's see, I mean problem um yeah, there are people commenting here this goes right into the arms of set theory uh boy, okay, here's a conjecture that's still a little confusing oh boy , this is fine, there is this property of causal invariance that is really important in

### physics

because it is what leads to general relativity and it is what leads to objective reality in quantum mechanics, it is basically the statement that even though you have two paths Here they diverge, no divergence is definitive, you can always go back to where you were, you always can. you can always know like this path here goes you know you're going to abab you could say oh my god I've gotten lost here but actually you can always go back to abab okay so the question is in

### metamathematics

what is it?
So causality and variance matter for the physical interpretation of these things. What is the importance of causal variants for the mathematical interpretation of these things? In other words, just like the law of the excluded middle, I'm not sure, I don't believe it. I think it's like the axiom of extensionality, which is actually Euclid's first common notion, so the axiom of extensionality says that two things that are equivalent to one thing are equivalent to each other, see what I'm saying, if these two things here are equivalent to that thing so this is kind of saying that they are equivalent to each other, maybe that makes sense, I mean, this is what critical lemma pairs and fear enhancers use, it's that idea of extensionality, two things that are equal to a particular thing are equal. between us now, I don't know if I mean that the conjecture would be that somehow that assumption is what leads to good, so one of the great mysteries of human mathematics is that there are so many things that can be done without falling into undecidability, and so I've been wondering about this for decades, why human mathematics is so feasible because in the computational universe of possible programs you know that your average question is undecidable.
Does this stop? Doesn't this contain all this kind of stuff? It is normally undecidable in mathematics. The task of mathematicians has been to prove theorems and they have proven many of them and you don't hear, you know, you don't see mathematicians on a normal day tearing their hair out because they discovered what they were trying to discover. proving is undecidable is actually a very rare endpoint for mathematics that people say, "My God, this thing we were trying to do turns out to be undecidable" has happened only a limited number of times in the history of mathematics problem posed to groups tenth Hilbert problem i I don't know, maybe uh uh homeomorphisms of um uh varieties beyond four dimensions or four dimensions and beyond things like that, but there are comparatively few examples where the things that mathematicians have wanted to do deeply they turned out to be undecidable and that's surprising to me and so one possible hypothesis is that the fact that things don't end up being undecidable is some consequence of a fundamental assumption that is made in establishing the axiom systems for mathematics that if you choose a arbitrary system of axioms you would no longer have that characteristic, but if you choose a system of axioms that, for example, has this property of causal variance, which I think is equivalent to the axiom of extensionality, which is found in basically all systems of axioms, it's fundamental to the way equality works, but I think you already know. that might lead to the idea, I mean, that might be what leads you to, by the way, in set theory there is an axiom of extensionality which is a much more sophisticated axiom of extensionality that works for infinite sets and so on. , um and uh um, you know, so the hypothesis might be okay, my meta-hypothesis is that, in

### physics

, as computationally limited observers of the universe, we have to group certain close states together in some sense, that's the nature of our our in the extent to which we are not able to do enough calculations to separate all those pieces and the hypothesis would be that something similar happens in mathematics and the cause is because of the assumption of this extension that could have to do with, I just don't know it yet, I I mean how it relates to equality and things. so, but in other words, the construction of mathematics on set theory that has this axiom of extensionality could be building on something that is built on a you know, not too much undecidability, uh, which is the same thing, yeah, let me See that. that is the same reason we don't, that we are able to derive 20th century physics, which is what makes statements about the universe that are not mired in irreducibility and undecidability, that we can make global statements like the Einstein's equations and things like that without having to say oh, let's look at what happens on the elementary length scale, etc., so that's some uh um uh yeah, so, by the way, um um, sure, I mean, I think um um uh, yeah, right, so I mean the various ones. comments here on our live stream about um it should be possible to describe the multiway graph using lambda calculus uh yeah this multiway graph is closely related to combinators and lambda calculus.
I could talk about that another time. the centenary of Moses Sean Finkel's invention of combinators, which was the first example of a universal computing system, although it was at least 16 more years before anyone understood what he had done, but yes, December 19 from the 1920s were combinators, which is kind of the precursor to the lambda calculus and in fact, these multi-way graphs are the combinator evaluation graphs or lambda calculus um, so there's a relationship there and it's um um uh Brian He is commenting that perhaps what our mathematics uses is self-selective. provable axioms so you don't know from the axioms the axioms well, it's okay that the axioms can state that the axioms of extensionality are important yes, that may be true.
I've always thought that the theorems that the way we choose to explore mathematics is by holding to situations in which theorems can be proven, in other words, that the way we generalize things in mathematics is not just oh, we'll choose a system of arbitrary axioms or we'll choose an arbitrary theorem to prove that we're always moving along a path that maintains, I guess, I guess the Point there, yeah, seen in these terms, all it says is that we've already proven a bunch of theorems and we never get to a very long point here, that's what it basically says, yeah, that's the statement that in mathematics you're usually no, your average theorem doesn't go that many steps, so in other words, unless you already have a bunch of theorems in the bag that allow you to get to the next theorem without too many steps, you just won't go there, I think that's actually it. right, so the statement would be oh, this is a mess of a notebook, um, but, um, you know, the basic statement would be, when doing mathematics, you don't follow a long path, you use theorems that you already have, and you follow a fairly long path. short and probably yes, so I mean hmm well that's basically saying that you only get to things that can be reached by fairly short paths, there might be things where there could be an arbitrary long path, it might be undecidable , but I'll never see that because you only get the things that you can get to with uh um uh fairly short paths um okay, well anyway um let's see, I'm looking at the comments here, okay, so the fact that classical mathematics are not constructive if we use intuitions like logic, I don't think we can establish a strict correspondence, so essentially we canis that physics is a representation of all possible mathematics and that the physics that we experience is simply a portion of this rulial space that we talk about that corresponds to the portion that we can easily understand with our senses, so to speak, and I think that is directly analogous, we can do the same thing with mathematics, there is a space of all possible mathematics and, you know, any given axiom system is just a cut through that space so that there is a mathematics of all possible mathematics, but We can already study this limit before reaching the limit of all possible mathematics, we can study the limit of a particular system of axioms in mathematics at its limit and, by the way, I do not think that the limits are going to be so different because all these systems of axioms are universal computation, among other things, um, uh, I don't think Bob is commenting, it's the mathematics that you know related to the universe that we live in I think the answer is um uh that the answer is if we allow hypercomputational mathematics , which basically means oh things like um well, it's like uh Turing logic based on ordinals, etc., if we allow for hypercomputational mathematics then we get something different that we can't concretely reach from our universe, but yeah, anyway, okay, I can see that people have a lot of questions and comments, but I need to go to a different meeting and I need to have some dinner, um, it's that time. of the day in my part of the world um and yeah, yeah, I would love to keep talking about this, this is me and you guys have um like I say, one of the assignments that I've been given is this

### metamathematics

thing and I hope someone sent me a link to a version where we can download the structured form and then maybe we can grind it up and see what the math looks like, just like good old Euclid.
Alright. Better go, but um uh thank you to our local people and um thank you to the people on the live stream um and um uh God, I wish I was waiting to get to the end of my metamath study, but I'm realizing that it's a big and deep topic and um uh, I'm a little afraid, I mean, we have these applications of the physics

#### project

to distributed computing to import mathematics, potentially to biology and potentially even more strangely to economics, and I'm understanding I have a kind of afraid that each of these will turn into a giant monster, um, uh, but I find this really interesting, so, um, that's good, okay, thank you very much guys, and you see.