Science Research Session: Hyporuliad

Hi everyone. Welcome to something we haven't done for a while, which is a science discussion with Jonathan Borod and myself and Nick Merzin and and so on. We thought this time we would talk about a subject that just came up recently that is kind of abstract, but I thought it'd be interesting. It's the hypo ruliad, so otherwise known as do we live inside a black hole? Or what would it be like to do so maybe. And so reminder to people, the ruliad is this thing which is the entangled limit of all possible computations, but computations. One of their features is that they can kind of run as long as they want. There are Turing machines that run arbitrary along and don't halt and so on. And the question is, well, what happens if we lived in a world where all computations are a finite length, where all computations eventually halt? What would that be like? And so I kind of view this as being a little bit like the Flatland analog of three dimensions. Normally we live in three dimensions, but imagine what it would be like if you were a two dimensional critter. And there's a book called Flatland that mostly seems to be about the social order of polygons, but that's a different thing. So meanwhile, Jonathan has a hypothesis that he's floated, which is the kind of the rule relativity hypothesis that basically says whether you're in the ruliad, the hyperrulliad, where one can do, where one has oracles for computations as well as ordinary computations, or whether one's in the hyporuliad where all computations terminate, that somehow it will look the same to some type of observer. So Jonathan, do you want to start off commenting, attacking, disagreeing with, agreeing with, or whatever the things I was just saying?

No, I think that's broadly correct. I mean, it's okay, I guess I would phrase it, which is quite similar to how you formulated it as, you know, if we assume that we exist in a whatever, that our universe is some Turing computable fragment of reality, and we look at some hyper, you know, we look at some hyper computational process, would what we see be fundamentally different to what a, you know, a primitive recursive, an observer living in a primitive recursive fragment of reality would see, looking at our physical universe, you know, and if there is a difference, what would be the essential difference between what that observer would perceive, but between what those two observers would perceive? And I guess so the thing you're characterizing as the kind of rule of relativity hypothesis would be the hypothesis that actually what they would see is the same. Okay.

But the important thing is we have a tractable example here because in the world of a hypo observer, a primitive recursive observer, that is something we can actually, we with our own computational hands can actually get our hands on, so to speak speak. The hyper case, we don't think we can explicitly get our hands on. So I think the thing we want to do is to look at what is the answer to that question. What does the ruliad look like to a hypo observer, so to speak, and what is the world of the pure hypo world? What is it like? So I guess we can imagine. Let's imagine that we have. Okay, let's do a little bit of a review of what happens in the hypo world. My assumption is that every causal graph terminates. Do you think that's correct?

Yeah. Yes, I think so.

But now let's just play that out for a moment more because in metamathematics we are thinking of the hypo rulial case as being decidable theories only. But yet there's a bit more to decidable theories, I think, than the termination of causal graphs. Is that true or not? So I mean, let's say you have Boolean algebra, which is a decidable theory. What does that mean? It means that there is a guaranteed finite computation that determines rather boringly truth or falsity for a Boolean expression. But can we think about that in terms of the same kind of dynamics?

What I think you're hitting on there is this fundamental distinction in term rewriting systems between normalization between normalizing systems and strongly normalizing systems, Boolean algebra, decidable theory, notably. That certainly does not mean that if you build a multi way system just applying Boolean algebraic transformations on some initial expression, that every multi way path terminates in finite time, that every multi way path normalizes. All it means is that there exists a normalizing path, not that every path normalizes.

Maybe we should see this explicitly. Should we look at this?

Yep. Do you want me to run something?

If you can drive.

Yeah, yeah, sure. Hang on. Sorry, let me change my setup here. I've got a bunch of random notebooks open. But.

By the way, I just want to comment for folks on the live stream. If people, if we talk about things where nobody knows what the heck we're talking about, you're welcome to ask and we'll try and explain the.

Oh, whatever. Okay. I don't think I have it as a Documentation example. That's what I was checking, but. Oh well. One second. Okay, can you see a Untitled 6? All right, cool. Right. So. Do we want to use your axioms or do we want to use standard axioms?

I don't know. I mean, it might be faster with standard axioms.

I think standard axioms are more. It's certainly more understandable with standard axioms.

Okay, let's do it.

So if we do that and. Okay, so. All right. This is a slightly painful process. I suspect there's. Nick might know better ways of doing this than that, but I'm an old fashioned kind of person.

Nick, do you have a. Do you want to comment on this? Do you have a.

No, I don't think.

Go ahead.

We had a bunch of functions, but they didn't end up in some resource function repository. But for the mathematics piece, we had a bunch of functions that turned axioms into rules and all of that, but we can't really make it simple.

Okay, Jonathan, this is going to be very painful. What are you doing here? You're not going to do all of them, are you?

No, I'll do some fragments. What should we do? Maybe we just copy paste from the mathematics book. We had exactly such examples there. Okay, that's what I was checking, but.

Okay, Nick, if you can just put something in.

You want me to check? I'm sure I produced this as a documentation example of a multi way operator system. Oh yeah, yeah, I did, I did. Okay, I've got it, I've got it.

Okay, go ahead. Let me stop sharing here. Okay, go ahead.

Right, There we go. Okay, so if I run this, that should produce some kind of states graph. And so this is just the axioms of Boolean algebra being applied to an initial expression and xy. So it's certainly not the case that every path terminate. You know, there will exist non terminating paths in this multi way system. The point is that there's a difference between saying there exists a terminating path and saying that every path terminates. And decidability is simply saying there exists a terminating path.

Well, a terminating path that does what in this case?

Well, that relates to expressions. These are all equivalent.

It would be nice if we had something where we could see that it goes to true, for example. So I mean, in this particular case it won't. And if we say and xx, it'll rather trivially go to true because it'll evaluate.

Right, but okay, I could inactivate these operators, I guess, but I should just.

Transform them all to lowercase or something.

Sure, let's do there and goes to and or goes to or not goes to not. And then let's do. Let's just check that that works.

If I do this.

Yep. Okay. And then if I say end of.

One comma one that's rather trivial because you have that as a rule. Why don't you do of X comma X which doesn't obviously, which I don't think there's a rule for there.

And of X comma X equals X.

Well, how about equals Or. Is that right? Yes.

Yeah, that's correct. We may not be able to see the true.

Well, how. Okay, so let's figure out how we see the true. How is true going to be represented in this thing?

It'll just show up as true. I mean, hang on. We can take this thing and I can just get the all states list, I guess. Any of these true.

Well, just, just ask it. Just ask it. Say position of true in this thing. I'm suspicious. They'll all be equal signs. I don't think any of them. I think they're all equalities. Oh gosh, you have your crazy strings thing. We are getting rid of that crazy strings thing. This is a slow progress towards multi evaluate which won't have any crazy strings.

Well, the crazy strings are simply a byproduct of the fact that we never made a general state rendering system.

Oh, I know, I know, I know, I know, I know. There's no, there's no. Just say flatten that and then do it.

Yeah, yeah, sorry. But it, it, for some reason it wasn't finding the closed bracket. But if I do. But in particular, if I take this and I say.

Oops, That's not going to be there.

Yeah, it's not there.

Okay, so let's understand what happened. Is that because we didn't go far enough in the proof or what?

Yeah, probably. I suspect this takes more than two steps.

Well, can we just. Well, let's just do. For. Let's do exactly the same thing for three steps and just don't print out the output.

Yeah, yeah. If I do that, let's see what happens. We are facing combinatorial explosion. But anyway, look, so the, the underlying point is that. Yeah, decidability is not the same as the causal graph being finite. And so the point is that saying that the thing is like a primitive recursive function is strictly. That's a stricter condition than being analogous to a decidable theory.

I agree. Because let's just review that. I mean.

Yeah, sorry.

The primitive recursive function is like we have a recursion and the recursion always, the N in F of N is always going down towards zero and we have our initial conditions at zero, right? And then so for example, I was just playing around with nestedly recursive functions that, you know, you could have a case where the function could go up instead. In other words, the n value could go up like it does in induction, for example, rather than going down, at which point you're no longer in the primitive recursive world and you need mu operators and things to know what's going on. So that was. Go ahead.

So the important point is that with a primitive recursive function, it doesn't matter what the evaluation order is, it doesn't matter which path in the multi way system you take, you're eventually going to hit a normal form. And so for instance, if you look at terminating fragments of combinatory logic, or you look at terminating fragments of lambda calculus, those are all strongly normalizing, right? Because of the Church Rosser theorem, if there exists a path that terminates, then all paths will eventually terminate, but with a primitive recursive. And so that's analogous to a primitive recursive function. This ran and didn't find it with merely a decidable theory that's not guaranteed to be true.

Okay, so let's be concrete. In the case of combinators, for example, which are actually probably. Which are not a decidable theory.

No, but if you look at only the terminating fragments of a combinatory logic, then you get strong normalization. All I'm trying to say is this basic fact that confluence. So confluence plus normalization implies strong normalization.

Yes. Well, you're saying. Wait a minute, let me check that I understand that claim, because in combinators, for example, it's perfectly possible to have some branches terminate and others not.

Right?

And why is that not? So that's saying some are normalizing. I mean, and let me, I mean to pull up an example. I can just show an example here. Let's see. Okay, so here, where is it? Okay, so this is a multi way system for a combinator and there it terminates happily, right? But now it's pretty common to have. So these are ones that all just terminate, but then somewhere here there are ones. Oh, come on, where is this? Here we go. Maybe I think this is a case where it terminates where many of the paths terminate but some do not. Right? Which is. Let me try and find a more explicit example here. I think I had some nice examples where for some evaluation orders it Terminates, and for some it does not. Probably there are examples here of that. Yeah, I mean, It must have reached its fixed point. But I don't know whether this is. Is this contradicting anything you're saying? I don't think it does. I mean, I think we're just saying that there exists. I mean, this is just a concrete example, which. I thought I had a better example than this where. Oh, I see. I see. Every red path here isn't. Oh, actually, wait a minute. Wait a minute, wait a minute. I was confused here. Every path that has a red dot is an unterminated path. And so this dot here is a nice normalizing terminating thing. And the Church Rosser property, I believe, says if it's going to terminate, there has to be a unique termination point. If it doesn't terminate, all bets are off. Okay, so can you restate your thing about the relationship between confluence, which is the Church Rosser property, isn't it?

Yeah. Okay, so the important. Okay, the point with strong normalization or the relationship between strong normalization and confluence is that you have to distinguish between two things, right? There's local confluence, which is. So, okay, the basic point is when you talk about confluence, you can either say I have two paths and they diverge after one step, and any such branch pair eventually merges. But I can also take what in general is a stronger condition, that I can have two branches and they diverge for an arbitrary amount of time, and then they eventually reconverge.

Okay? And I assume that the converge after an arbitrary time is the confluence property.

Right? Right. And so this is the distinction between confluence or local confluence. And the point is that when you have strong normalization, then you have. There's this result which is in mathematical logical, Newman's lemma that says that confluence and local confluence are the same in that case.

So strong normalization means everything always terminates. Is that right?

Yeah, exactly. So normalization means that there exists a normal form, there exists some rewriting path where you get an expression that can't be rewritten any further. Strong normalization says that all rewriting paths lead to a unique normal form.

Okay, so your claim is if all paths lead to a unique normal form, then local confluence is true, which means that every branch pair must resolve after one step. Is that correct? Is that obvious, If all paths are going to converge? Oh, yes, it's sort of obvious that you have to have some way to converge after one step if all paths will always converge. Well, I mean, let's look at an example. Let's look at an example. Here's an example. No, what the heck's going on here? Well, here, let's look at this example. This is an example where all paths converge eventually, right? So now the claim would be something that frankly doesn't appear to be true, which is that every path, every branch pair. Well, it doesn't seem to.

Oh.

No, it doesn't seem to be true. I mean, look at that node there.

That node, sorry, localhopolis doesn't say that they have to converge after. Says that if they diverge after one step, they have to converge.

Well, okay, but I mean like the 15 here, right? Let's zoom in a bit. Or we can actually recompute this if we want to. Oh, for goodness sake. The 15 here. Okay, so 15 diverges to 21 and another 15. Oh no, I'm sorry, the 15 is just the number of elements. It doesn't tell you what's there. Each one of these nodes is unique, so it diverges there and then it converges after one step. We're all happy. It's kind of a lattice like thing. Now let's look at this one here. Which diverges after one step, but it takes two steps to converge again. Right, so is that consistent with what you're saying or not?

Yeah, it's irrelevant to what I'm saying.

It's irrelevant.

Yeah, As I say, the divergence, the convergence time is not relevant to Newman's lemma or to local confluence. It's the divergence time is what's relevant.

I'm sorry, could you explain that again? Because I didn't understand it.

You have an expression, it can be rewritten in two different ways. So you can enforce the condition that, okay, you have an expression A and then you say I've got arbitrary rewriting paths that lead from A to B and A to C, and therefore B and C can be resolved. They can be resolved at some point in the future to some common expression D. That's regular confluence. Local confluence is the weaker condition that if I rewrite A to B and A to C in one step, then B and C can be unified at some common expression D. In an arbitrary rewriting system. Those are not the same property. Local confluence is a weaker condition in a strongly normalizing system. They are the same. They are logically equivalent.

I haven't quite got this, but maybe I don't need to. But let's continue with what you were saying. So you're saying the conclusion was there is the Perniv case where everything always terminates after a bounded number of steps. And then there is the decidable case where there exists a way to get to a terminating state, but there might be a way you might get lost, basically, right? What does it mean for physics that you can get lost, so to speak. In other words, it means you can.

Escape to future null infinity would be the analog.

Okay, well let's talk about that in the context of a black hole. So in the context of a black.

Hole, that means the order on the exterior of an event horizon.

What's that?

It means the on the exterior of an event horizon. The way you define an event horizon is by saying you can't escape the future null unflincing.

Okay. So the claim would be that.

The.

Failure to be normalized is the failure to be crushed at the singularity inside the black hole. And that if you choose the path that doesn't normalize, that doesn't terminate. I see. Then that's the analog of being.

Okay, so the point is, okay, if we assume a, if you'll pardon the expression, if we assume a strong form of the weak cosmic censorship conjecture, namely that there's a one to one correspondence between singularities and event horizons, right? That you don't get naked singularities, but you also don't get just like naked event horizons, then any path that you have through spacetime has one of two, one of two futures to it. Either it terminates at a singularity or it can escape to future null infinity. And that's the analog of saying that in a multi way causal graph, either I terminate at a normal form or I run forever.

Okay, all right, so now what does it mean if we're living so in the hypo ruleiad, what is our definition of that? Is our definition primitive recursion or is our definition decidability? Because you're saying so. I mean the issue is. Okay, hold on. Because the ability to escape to future null infinity, it feels like the thing that makes that not a complicated undecidable theory is some kind of simple inertial motion that is not too much can be happening. In other words, if what happened is you could hide in that infinite future, you could hide an arbitrary computation, which is not what should be happening in a decidable theory. You shouldn't be able to hide an arbitrary computation. It's as if you either go to the terminating state or you go to a trivial loop of some kind that doesn't terminate.

Hmm.

Because otherwise I don't understand what you know by the time you have a non terminating computation, right? A non terminating term Rewriting. Why can't you just do any computation there?

Because that's not the way round the implication goes, right? If you can do arbitrary computation, you'll get non terminating rewriting sequences. But the fact that you get non terminating rewriting sequences does not mean you can do arbitrary computation. Computation. And so the point is that theories that are decidable but not primitive, recursive are in that intermediate space where you have non terminating. You can have infinite paths, but you're not doing arbitrary computation.

Okay, well, so a good example of this concretely will be the S combinator case where we know where it's been my sort of claim that there is the possibility of arbitrary computation purely with the S combinator, but it could be the case that the S combinator is a boring infinite system.

Well, it's interesting, right? Because in a sense, okay, if I can characterize it this way, a lot of your intuition seems to be that if something can do, if you get non terminating rewriting paths, then it's a kind of computational equivalence story, right? PCE would suggest if you get non terminating rewriting paths, then there's probably arbitrary computation going on there. The fact that we have decidable theories that are not primitive, recursive in that sense, that are not strongly normalizing, tells us that can't be true always. But in a sense where I guess what you're probing with things like the S combinator prize is just the empirical question of how common are those intermediate cases, how common are those cases where you get non strongly normalizing, but non turing complete.

Okay, so let's be even more extreme about it. Let's say that, you know, we have a tale of non termination, okay? Tail T T of non termination, right? Where, where, you know, where basically the thing is, in some trivial loop, that's one way it would not terminate. And, but then my question is, can we. If we can number all the ways it cannot terminate, then we don't have universal computation. So, for example, something we're just thinking about in a different connection context is if you can put ordinal numbers in correspondence with. Well, maybe this is true, maybe it isn't. If you can put ordinal numbers in correspondence with all the possible non terminating behaviors, then the claim would be. It would be non terminating but not universal. Maybe. What do you think about that claim? So let me give an example that.

Sounds plausible, but I need to know what. Yeah, well, let's think about it.

I mean, let's think about. The question is, and this is relevant to the hyperruliad because this is life in the hyperruliad. If it is the case, the definition of the hyperruliad is decidability, but not necessarily finite termination. I mean, my initial assumption. Go ahead.

Maybe it's worth us taking just a moment to be a little bit pedantic about what we mean by termination. Right? Because I know that you're not a fan of halt states in Turing machines, but in a sense halt states, or accept states, whatever, were invented to resolve exactly this problem. Because you've made the claim, which I of course agree with, that in a sense, in the kind of bare metal multi way system there is only branching. Merging is something that we have to somehow impose after the fact. But the point is, the notion of termination requires a notion of merging. It requires us to be able to say we did nothing to this state and it stayed the same.

Yes, it's merging, that's a time merging, right? In many cases branch hill merging, but in the case of termination, it's merging in time and saying nothing happened.

Well, so in the multi way system function we make this distinction between states graphs and evolution graphs. States graphs merge across branches and time, whereas the evolution graphs merge only across branches, but on the same time step. So in a states graph, so in an evolution graph there is no termination really. Whereas in a states graph you have termination because the graph just, you end up with vertices that don't have an out degree, don't have an out component. And so I think it's just worth being a little bit careful in stating that in exactly the same way as notions like confluence or merging or causal invariance only make sense if you impose an equivalence function, the same is actually true of termination. And in a sense, although yeah, you have.

Okay, so what you're basically saying is to talk about termination as termination you need time equivalence. But really what termination is is things become boring. It's just the same, the same, the same all over again. It's just exceptional.

But that's the same with branch equivalence, right? That's not a different problem. I would say when we say that these two branches are equivalent, what we're really saying is they're only different in a boring sense, right? They're only different in the sense that for instance, the vertices in the hypergraph are labeled differently or something. Yes, and it's the same with termination. You're saying that these two states that are time like separated are only different in a boring sense.

I agree, I agree. And that same will be true with a time like singularity. Right. Where you have, where space has been crushed, what you're saying is in the case where you can apply equivalences, space has been crushed. In the case where you don't apply equivalences, you've got. Everything is fine, except that there's no difference between every point in space.

Right, Right.

So, so in other words, what you're saying is that, okay, so really the definition of the, the hypo, the hypoexistence, you know, entity, it's not a question of the fact that they. That time ends, it's really a question of the fact that nothing different ever happens in time or nothing different ever happens in space.

Yes. And so the reason I'm being a little bit pedantic about that is because I think that therein lies a, a component of what I think makes the rule, your relativity claim plausible. Because the observer in the hyporuliad, so to speak, has not just their state evolution function be, say, primitive recursive, but also their state equivalence function be primitive recursive. So we, from the God's eye, from the Turing's eye view, looking down at the hyper ruleiad, we can see, oh, those two states are the same. The observer in the hyper ruliad can't necessarily make that determination.

I understand.

So they can see. They might see a path that appears to go on forever and be doing what to them them look like interesting stuff. And it's only to us that we can look down and say, actually no, it's not interesting, it's boring. Because this, that and the other.

That's an interesting point because probably it's the case that to prove equivalence between two atoms of space, for example, requires the hyper ruliad. In other words, even in the standard Turing case, to prove equivalence. I think you can't prove equivalence from within the height, within the rule of itself.

Well, okay, I. This, this. Now I have to confess, I. I never know exactly what, what it is you mean when you talk about equivalence between atoms of space.

Okay, fair enough. I mean to say. Okay, so your point is equivalence is in the eye of the beholder. The question is only, is it consistent to say that these two things are equivalent?

Right. So. So I think. Okay, so one important point that I think again is a little bit subtle is that the evolution. Okay, we typically assume that the evolution function and the equivalence function have kind of the same level of computational sophistication, but there's really no reason to make that hypothesis. Right. That you could imagine that the state evolution function is either obeying a Completely different model of computation to the state equivalence function. Or you know, they're functions with different kind of bounds on their time complexity or something. And you know, in a sense how. Okay, sorry, this is a separate discussion.

Actually, you're making a good point there because the fact is part of the claim about many of the things we're doing is the observer is computationally bounded relative to the actual underlying physical processes. So there is a different. So the equivalence function is a different level of computational sophistication to the evolution function.

Right, right. And so, okay, there's a paper I wrote like last year that kind of looked into this in a little bit more detail. But so there are definitely cases where you can fix the complexity of the evolution function and vary the complexity of the equivalence function and vice versa. It is not yet clear whether they are completely independent. And the question of whether or not they are completely independent turns out to be an alternative way of formulating the P NP conjecture. That in a sense, what P versus NP is really asking, if you think about it in these terms, is if you fix a complexity of your evolution function, how much, if at all, does it determine the complexity of the valid complexity or the complexities of the valid equivalence functions and vice versa?

Okay, let me pull that apart for a second. So I mean, the basic issue you're saying, well, okay, walk through what NP means. In the case of equivalence functions. You're saying that NP comp. You know, in NP you've branched out into many different possible non deterministic paths. But at some point you have to find out what the answer is. You have to determine did you get an answer? And that requires essentially something like an equivalence function. Some kind of observer action.

Yeah. So the point is that choosing a branch in a multi way system is a computational equivalent process to defining a function that equivalences all branches to that branch. You can recast it in those terms. And so really the thing that puts the N in NP is really the definition of an equivalence function on an arbitrary multi way computation.

Let me think about that for a second. How does that relate to the quantum computation case? So that's defining the complexity of the observer in quantum computation too.

Right? Right. So, so the, the point is that in quantum computation you have this slight muddle because typically when people talk about complexity classes in quantum computation, they are considering only the multi way behavior.

I know, which is just wrong. But I've known That's wrong for 40 years. It's wrong.

And thinking about it in those terms helps clarify that point. In a Sense, the conjecture would be that we've talked about many times that any speed up you get in the branching is canceled, likely by slowdown that you get in the merging.

Okay, but how does that relate to the P versus NP case? Because the P versus NP case, you're saying the merging in the case of np, the question is, is there a polynomial time merging function for np? Is that right? Is that the P versus NP question?

Yeah, and so. So in a sense. So P equaling NP would be the most extreme case of quantum computation not working in. In that way of thinking about it.

Right. Why is that? But P, why was P equal to NP would, would suggest that, that, that NP re choosing the output in NP would then be np.

Right, Right. Which in particular would mean that there's no benefit. You know that.

Oh yeah, there's no benefit.

But there's unconditionally no benefit in branching. Right.

But you would be able to reconnect the branches in polynomial time, which you might think was good. The only problem is, well, in the end the branching wouldn't do you any good. But let's come back. Okay, you're saying, let's come back to the hyper ruliad question because you've got. Okay, first point is you're reformulating things by saying that in the sort of ultimate God's eye view, the hyper, hyper, hyper God's eye view, everything just branches. That anything that equivalences has to be in, that's in the eye of an observer at some level of the rulial hierarchy. And you're saying that the hypo ruliad are observers who can successfully. Okay, well what is the characterization?

Sorry again. I think it's worth making the following pedantic point, although I'm not as confident about this, but I'm pretty sure this is the case. I think the branch like and timelike versions of merging work in opposite directions with regards to complexity of observers. So what I mean by that is if you consider the limit of an observer becoming arbitrarily computationally sophisticated, then you know which is your hyper hyper hyper ruliad view as you were describing earlier, then you wouldn't need to equivalence, you wouldn't have branchial equivalences. You could distinguish everything.

Right.

But equally, as your level of computational sophistication increased, you could begin to see patterns in computational threads that exist in the multi way system that you wouldn't have been able to see if you were more computationally limited. So the amount of branchial equivalencing that you need to do goes down, but the amount of kind of temporal equivalencing that you can do goes up.

Okay, when you say see a branch, you're talking about sort of the breaking of computational irreducibility rather than the breaking of multi computational irreducibility.

Yes.

Right. So you're saying, Let me try to understand that. You're saying as your computational sophistication increases, you can jump ahead in time.

Right.

But why, why can't you also merge branches?

You, you can, you just don't need to, so to speak.

Because.

So, okay, the, the question is what? All right, let's, let's distinguish between equivalences that you are kind of voluntarily imposing. I'm somehow I'm reminded of you, of the, what was it? The autonomous events that you. That occurred in those, that came up in those, in that distributed consensus thing.

Right.

But equivalences that you are voluntarily imposing versus equivalences that are imposed for you by virtue of your own limitations, so to speak. Okay, so the point is that the, that the. Traditionally when we've thought about these things, we tended to think about branchial equivalencing as being an equivalence that's imposed on us by the fact that we're computationally bounded, whereas the termination notion is subtly different. Termination is something that is voluntarily kind of imposed by us where we say, oh, I can see ahead that this path is going to be uninteresting and whatever. I don't need to evaluate this.

Yes, well, wait a minute. So why is the branch hill equivalence? The branch hill equivalence. The claim is we simply can't tell the difference, therefore we have to equivalence the thing. So that would be the case for spatial branching even in space. In the space like case. Because we're extended, so to speak, in space, we can't distinguish nearby events even in the space like case.

Right. And there probably is a version of that. I mean, sorry, not probably, there is a version of that for time as well. Right. That you could imagine seeing a system and it's like we're so big and coarse graining so much that to us the system looks like it's not evolving, it's reached a fixed point. But actually there's a bunch of stuff going on that we don't have. Right.

Our brains are really slow. Takes milliseconds for us to form the next thought, so to speak.

Right. You can perfectly well have a situation where you think that a multi way system has terminated, you've reached some normal form. But actually it's just because you can't distinguish between the evolution steps that are going on.

Let's be concrete about that. The claim would be, let's imagine that for example, every tenth step it returns to the same. I don't think it can do that. It can't return to the Same state every 10th step and still make it out.

Not if the state is the entire.

So then the question is, what would it mean? Let's say we had. We're coarse graining in time. What does it mean, coarse graining in space? It's, you know, the entropy of things in space. We still have many possible states consistent with some fixed coarse graining in space. Is that different in time?

No, I don't think fundamentally so. I mean there's an interesting question there about like it's sort of similar somehow in style to things like Poincare recurrence. Right. There's a question of.

If you impose.

A non trivial equivalence function, can you have a system that returns arbitrarily close to its initial state and still manages to diverge and not.

Right. So back in cellular automata I had this notion of spatial entropy and temporal entropy. So spatial entropy is the number of possible configurations horizontally, temporal entropy is vertically. And there are a bunch of inequalities, but they're both, they're both valid things to look at. And so I would imagine that suggests.

Yeah, someone should derive new inequalities that also account for the branchial case.

Yes, that's a good point. I mean. Well, let's just think about those inequalities are pretty trivial. They just involve an apple and exponent. Because they're basically saying if you have a certain number of configurations in the spatial, you know, this temporal sequence can have an effect on this spatial sequence. Based on just how far you can get from that sequence of temporal values. So I would claim it's going to be the same thing, except that there might be a different layer upon an exponent in the branchial direction and the spatial direction. Why is it, is it, is it not the same same argument?

It might well be the.

Let's see. Okay, but, but so now let's come back to the, the sort of life in the hypo ruliad. So what you're, what you're claiming is if you are an observer, we're saying you're an observer incapable of, of determining certain kinds of equivalences. I see. So for you life's still interesting because you can't see that in fact it's just going xxxxx all the way down.

Right. In other words. Yeah, there's some regular pattern or something in the evolution that you're observing. But knowledge of that pattern involves evaluating functions that are not accessible to you, be they're only accessible in some higher level of the hierarchy.

Okay, so let's look at that in a. Okay, let's ask the question how that relates to proof theory and how that relates to kind of. Because I think there must be an analogous thing that is that there are things that look interesting to you if you can only do piano arithmetic, but as soon as you can do set theory, it's like this is boring.

Right? Right. I mean, okay, so like, like a slightly trivial version of that might be. So if you imagine that you're in a finitely axiomatizable fragment of arithmetic, and so in particular you don't have an induction axiom, then induction. Then if someone comes along, and if someone, you know, if someone comes along who knows how to do induction, then they can produce an infinite set of what look to you like non trivial interesting theorems. And it's only if you know the induction axiom that you realize they're not.

Right that you knit them all together and realize they're all the same thing.

Right, exactly. But to an observer in the finitely axiomatizable fragment, they all look like distinct interesting theorems because the statement that they are all consequences of the same result is not accessible to you.

Right. Okay, so let's take the example I'm still looping on the looping question of these tails, where the thing doesn't terminate, but it doesn't terminate, where the way in which it doesn't terminate, just like in the trivial case we've talked about. One way in which you don't terminate is it just goes xxx all the way down. Right. But there are more non trivial non terminations that we see in Turing machines. They can loop, they can go X, they can have all kinds of behavior that loops, but not in such a trivial way. So I have a question which is, can one classify the looping behaviors and how does one relate the looping behaviors to, for example, PR arithmetic? So this would be a claim. The claim would be that the looping behaviors, certain looping behaviors, are classifiable by ordinal numbers, say ordinal numbers below ε 0. Let's say so. For example, let's imagine you have the loop. I was thinking about this in terms of evaluation sequences. The loop X equals x plus one. So the claim would be that that loop is omega. Do you believe that claim?

That sounds right.

And Then that the loop X equals list xx is 2 to the omega.

Right?

So the question would be, you know, what kinds of looping behavior? I mean, is there a way of seeing that you can, you can, you know, you can kind of, you can get your arms around certain kind of looping behavior with. Well, again, this is not okay. This is okay. One level is what level of looping behavior can you decideably get your arms around? What level of looping behavior can you get your arms around with? Piano arithmetic. What level of looping behavior can you get your arms around with? Arbitrary Turing computation.

Mh. So, right.

Decidable. What level of looping behavior can you decidably? Let's think. Yeah, the case of so, so if you've got just the xxx looping behavior, what does it mean? To get your arms around? Means that. What does it mean? It means you have an equivalence function that correctly responds that it is equivalent.

Right? So, okay, let me start by making a slightly philosophical point, which is that, so this is, I would argue this is precisely this question is what an awful lot of debate around in the kind of modern foundations of mathematics is really all about. These ideas, like, you know, the, the central idea of category theory that you shouldn't talk about equality, you should only talk about natural ISO, you know, you shouldn't talk about objects being equal, you should talk about them being equal up to natural or equivalent up to natural isomorphism or the ideas in hamatomy type theory around, you know, you shouldn't talk about equality, you should talk about identity types or something. All of these discussions are really different ways of graph. The foundations of mathematics are found to grapple with exactly this question of, like, how do you classify different kinds of equivalences that you can place on expressions? So in the construct, in our case, We ultimately fall back to same Q, right? We assume sameq as our kind of ground truth base. And all of the other equivalence functions that we build on top of it that we develop are really built on top of sameq. Because in a sense, when we do hypergraph isomorphism checking, we reduce the things to our canonical form and then we apply sameq on. And really this idea that in category theory that you should, for instance, that you shouldn't talk about equality or identity of objects. One way that you can recast that idea is the question, well, what if you couldn't do sameq? What if sameq wasn't there? Then you have nothing to fall back on. And so you have to use somehow a, a less rigid notion of equivalence.

Well, the Equivalent notion of equivalence you probably use is a genealogical notion of equivalence. You say if things came from the same thing, then they're the same thing.

Which is the notion of a critical pair lemma. Right? So when you define a critical pair lemma precisely what you're saying is these two expressions came from a common expression and therefore by transitivity of equality, they are the same. And so that's another way of defining equivalence.

But isn't that also in homotopic type theory, isn't that this sort of equality is the same as equivalence type, you know, univalence axiom type thing?

Right, right, exactly. And again, the kind of multi way of recasting that would be to say that the notion of equivalence in time, that is where you're connecting states by evolution edges, and the notion of equivalence in branchial space, that is where you're connecting states by branchial edges. There's no a priori reason to believe that these two notions of equivalence are the same. Univalence is the statement that they are in some sense. And so things like the notion of critical polymers, the idea that if I rewrite A to B, that's a proof that A equals B, but also if I rewrite A to B and A to C, then that's a proof that B equals C and that those two are the same notion of equality, that is a non trivial statement that necessitates the notion of equality induced by temporal evolution and the notion of equality induced by state equivalence to be the same notion of equality. And that statement is really the univalence axiom.

And how does that relate to the intuition about P versus np?

Well, the point would be that in general you don't have to make that assumption. Right? You could imagine. So if you're particularly if your state evolution function and your state equivalence function, as I mentioned before, come from different, if they're at different levels in the arithmetical hierarchy, for instance, they use different models of computation, then they're definitely not the same notion of equality that they're inducing. You can have pairs of states that are branchly equivalent, but not kind of temporarily equivalent, because your equivalence function can witness more equivalences than your evolution function can witness, for instance. And so the recasting I made of P versus NP is really asking, you know, how different can they be? And you know, yeah, P not equaling NP is just not equaling NP is just the kind of, in a sense is the weakest case of those two notions of equivalence being different. I'm not explaining that very well, but I Can.

Right. Well, I mean what you're saying is that branching is that somehow. The question is whether you get the same, you know, whether going a certain distance in time with as much branching as can happen in that time is the same as just going that distance in time without worrying about the branching. That's basically the P, a single fiber for P versus the whole, you know, whatever you call it, the whole kind of. I don't know what it is. I'm thinking of the whole paintbrush or something in the case of. Okay, well let's see. I'm supposed to go off and do a different live stream here, but this is. We're having fun. So, So I, I want to come back again to this, you know, the experience. Your claim is basically that we have the experience of being able. We, although we don't know it, we're doing certain equivalences and that in a world. So for example, we could be in the hyper ruliad capable only of doing incredibly weak equivalences. We could be in the hyper ruliad and not even be able to do the XXX equivalence. What would the world look like to us then?

Wait, wait, hang on. Sorry.

You're saying in the hypo ruliad.

Yeah, yeah, we.

I see, I see. The basic point is if we are, if we're embedded in the hypo ruliad, then we don't get to do the XXX equivalence potent, you know, potentially.

Yeah, or maybe, you know, or some, some, maybe not. Maybe we could do one that was that trivial, but we couldn't do one of these more non trivial, you know, Turing machine normalization.

I want to come back again to this question about ordinal numbers and so on, whether that's a useful dividing line. What are the dividing lines basically? Is there primitive recursion? Is there ordinals? Is there pearl arithmetic? What are the dividing lines for? The capability, the characterization, which would be a little bit like the computational complexity characterization for the capabilities of the observer. That is, you know, in, for example, I don't know, in thinking about statistical mechanics, the question of whether the observer can only do polynomially bounded computations is a reasonable distinction. But the question of whether we're asking the observer in the hyperrulead, the basic defining claim is the observer can only do decidable computations.

Right.

Actually the observer, even in our universe, the observer can only do decidable. Go ahead.

But the hyper. There isn't just one hypo rulead. Right. Like there's. So the point is that, okay, even so there's one version where you just have primitive recursive functions, but you could also imagine one where you have primitive recursive functions where the recursion is bounded by some number. And in that case it's like, okay, if you can only recurse 10 levels deep and your computation takes you on a loop that's 11 steps long, then now, okay, you have no way of determining the original loop.

Right. So your claim is that just as there's an arithmetic hierarchy above Turing computation, so there is a. I don't know how meaningful. I don't know how robust the hierarchy is below.

Right. But I mean, even with the arithmetical hierarchy, it's still not known because the arithmetical hierarchy is incredibly coarse and you can have intermediate Turing degrees and so on, and there's still a bunch that's unknown about exactly how robust those intermediate levels are. And yeah, I think the same is true for the kind of sub arithmetical hierarchy as well.

Well, but the claim is there are robust levels in the arithmetic hierarchy. So the question is, are there robust levels in the sub arithmetic hierarchy?

I would suspect that primitive recursion is one.

Yeah.

But, yeah, I mean, this. So I don't know. All of this is kind of. It's always bothered me that, you know, the fact that Turing computation is treated so is given this very special status in the hierarchy. Right. That it's the kind of unique minimal element where I don't. I think the. The reason why I like the. I find the idea of. Of like rulial relativity appealing is precisely because it gets rid of the specialness of Turing computations, like Turing computations.

Right. It's kind of goes all the way down. We don't want anything special about anything.

Right.

The only thing that's special is that we are who we are and not. Not who we are, so to speak. In other words, that in the whole ruliad idea, you know, that becomes the only specialness is our feature, you know, because the same kind of specialness as the fact that we're at this place in physical space rather than another. And presumably that we're in this place in branchial space rather than another. Right, But I mean, so in other words, the only thing that's special is the particularness of our history, which leads us to be where we are rather than somewhere else, so to speak. But I mean, and your claim is, See, it might be the case that there's simply a reformulation as soon as you say the observer has this level of computational sophistication at this level of the arithmetic hierarchy, it could very well be that the whole mechanics of what you can conclude is exactly the same, that is that, that, that which is essentially your statement of rule relativity, that, that with respect to that the robustness of what you can say. So that will be kind of like a statement like, oh, I don't know, there are plenty of these where, where it doesn't matter. What is another example? Something where you can be at a different level like, like in computational complexity theory. Are there statements that you can make at any level in the hierarchy of computational complexity that are true at that level, that are true statements that you can make relative to that level and around that level that will be true for any level?

I mean, there are definitely. So in the polynomial hierarchy, which is the computational complexity theoretical analog, I know there are certainly sort of recurrence relations and things that like somewhat non trivial recurrence relations that apply, that allow you to relate like you know, sigma IP to some product of delta I plus two thing, you know, for arbitrary I or whatever. I can't remember off my help locally.

In the polynomial hierarchy that there are local statements that can be made near you in the polynomial hierarchy that are kind of translation independent about where you are in the polynomial hierarchy.

Right, exactly, exactly.

Yeah. Well, that seems like probably what will happen with the rule of hierarchy as well.

Yeah. And okay, so to be a little bit cryptic, I mean, I know I made a similar comment at the summer school. So I think the very fact that we have a notion of geometry, that we have a notion of continuum geometry, is itself evidence of that fact. Because so if you imagine an observer anywhere in this hierarchy and they look up and they look at all the stuff that's above them, what do they see? Well, they can't determine in detail what's actually going on. At best they can make some coarse grained determination and say this has this bulk feature or whatever. And in the case of computational models of physics, those bulk features are essentially geometrical features. And so my suspicion is that again to be a little bit cryptic about it, that wherever you are in the hierarchy, everything above you kind of looks like geometry, everything above you looks like space and everything below you sort of looks like algebra. It looks kind of discrete because you have the level of computational sophistication to actually tease apart what's going on.

Okay, so your basic claim is that which you can't understand, you consider to be the continuum, so to speak.

Right.

So in other words, that which you can't understand, you necessarily kind of Conflate things to the point where you get the continuum, whereas that which is below you in this hierarchy, you can tease apart the elements and see that there's lots of cogs going on there.

Exactly. If there was hyper computation going on in any of the. Whatever it is, hundred or so orders of magnitude between the elementary lens scale and things that you see, how would you know? How would you be able to distinguish that hyper computation from just the background noise of space?

Right, but wait a minute. So you're saying. Yes, I mean, so you're saying anything that is more computationally sophisticated than you are, if you're looking at it without that computational sophistication. Yeah, I mean, another way to say it is it will look random, and the randomness will lead to an ergotic type hypothesis, which will lead to the continuum.

It's really just extending. It's just extending the bounds of PCE in one direction, if you see what I mean.

Right.

You're saying. So rather than just saying anything that's of equivalent sophistication to you will look complex, which is of course true. It's also saying anything that is also greater complexity than you will look indistinguishable from something that is of equivalent complexity to you.

Well, wait a minute. Will look indistinguishable, will look random to you. And that randomness, you will be able to say about that because you can't say anything about it. The thing you will be able to say about it is that it looks like geometry.

Right. But it looks random. But then, so would something that had the same level of computational sophistication as you. The point is that if you have. I guess. Okay, so, you know, you have an undecidable statement in some axiomatic theory, and you have an undecidable statement in the stronger axiomatic theory. In some stronger axiomatic theory. Well, they're both still undecidable. They both look, you know, they're both equally intractable to you, and so to you as an observer. I would hypothesize they will both look random even though one of them is strictly more sophisticated.

Okay, but so do you get geometry at your level, or do you only get geometry when you're looking above your level? I would get geometry at your level.

Yeah. Yeah, exactly. I mean, that would be the PCE claim, which I think is nice. Yeah.

And it's also the claim of thermodynamics. I mean, it's basically the claim that as soon as your. Right. Equivalent. So, okay, so Your claim is all that we would see if we look above us is we would see, all we'd be able to distinguish is geometry. We wouldn't be able to distinguish any of the machinery because by definition we can't distinguish the machinery because we're not sophisticated enough to do that. And the what it will look like to us, it will look random, it will look geometrical, it will look continuous.

Maybe another way to say it would be so, you know, again, PCE just says, like anything that's of equivalent sophistication to you, you can't reduce. But I guess the rule of relativity view would be to say, well, and anything that was more sophisticated than you, that you couldn't reduce anyway, so it would look like something of equivalent sophistication to you. And so you can just. So from the vantage point of view as an observer just caring about things that you can observationally access, you might as well truncate the rule, your hierarchy at that point, because anything above you is not going to make any difference.

Right? So this is an argument that I actually made in NKs about real number computation, which is if you can't prepare the initial state or recognize the output as a perfect real number, what's the point of talking about the perfect real number? And the question then is, could the machinery inside do computations that you can distinguish at the two ends, so to speak. And the claim, which is not an obvious claim, and I'm not sure, you know, I don't think I could prove that claim is that machinery in the middle. It isn't the case that the machine. Okay, so the assertion would be with that machinery in the middle, you can't solve the halting problem. There's no real number machinery in the middle where you can prepare the initial state. You can say, here it is, it's an integer specified Turing machine. Now the analog computer inside can solve the halting problem and give you a discrete answer. So the claim is that that's not possible. Can you prove that?

Well, okay, I think this is. Let me just try and make sure that I have correctly abstracted the idea. So would a valid way of formulating that be to say, okay, so suppose we've got a hyper computational universe, but we have the restriction that we can only build computation. We can only build like computational experiments, right? Experiments are all Turing computable. So and this is really a question about what experiments you can and can't build. But so, you know, in that case, could you imagine building a computable experiment that was able to somehow mine the non computability of the outside world as an oracle to produce non computable observables. Right.

It's a reservoir computing for hyper computation.

Right, Right. Well, I mean you would need a formal definition, you would need some formal model for experiments which I think we don't yet have.

Well, but okay, but we can now play that same story again against the hyporuliad because while we can't verify things for the hyperruuliad, we're kind of out of luck in verifying it. So it seems. But for the hypo ruliad, we can just run a computer experiment on a Turing like computer and see what happens. So the analogous statement would be given a. I mean, I agree that we don't have enough of observer theory to be able to really formulate this well, but given a poor equivalence function in the hypo ruliad, can we ask, can we mine, you know, can we mine the ruliad?

Yeah, My intuition is the same as yours. I think so for the following reason that again, if you look at the PCE case, it seems unlike. So if you have the experiment mining a computational process that is the same sophistication as the observer, then the observer sees essentially random bits or can just see what appear to them to be random bits. For the claim that you could do the reservoir computing thing, you would need to have something that was somehow stronger than randomness. Right. The random bits they would see from the Oracle would somehow need to be more random than the random bits they see from just an ordinary Turing computable thing, which I don't know, I don't.

That's Greg Chaitan's claim that this is the long running discussion that we've had. Is the universe like PI or like omega? Because the randomness that you would get, algorithmic randomness there is a notion of bits that are more random than random. It's bits that are algorithmically random rather than PI. Like random. Sure. And.

Right. But the thing, you know, the reason why it's non trivial, I would argue, is because, you know what's coming out of this general way of thinking about the world is that there's two notions of algorithmic information and therefore algorithmic randomness that you have to care about. There's the algorithmic complexity of the system and there's the algorithmic complexity of the observer. And those two trade off in a fairly non trivial way. And so my hypothesis would be, and I think I can even probably formulate a formal version of at least Part of this is that there's, you know, you can cast the universe in a way that makes it like PI, but the observer becomes like Omega. And you can cast the universe in a way that the universe is like Omega, but the observer becomes like PI.

You don't think that's our actual universe? I mean, probably in our actual universe, it's both the observer and the universe are like PI.

Probably.

I mean, that will be the mystical claim of Godel, Penrose, whoever, that we are like Omega, but the universe is like PI.

Right? Well, yeah. Okay. Actually, and let me correct what I just said. Yeah. I think what is more likely is, actually. Yeah, sorry, is that you could make the universe trivial by making the observer like. You could make the universe like one if you made.

I agree, yes. Yes. If the universe, if the observer is at a higher level in the arithmetic hierarchy, the universe will look trivial to them.

Right? Yes.

Just as the ruliad level, the hyporuliad, like the XXXXX behavior looks trivial to us.

Right? Yeah. Okay. So to correct my earlier statement, so I guess the claim would be more like 1 and omega are like the two extremes and PI is this intermediate case where you can balance and have both be like PI, or you can unbalanced and have one be like one and one be like Omega or something.

Yes. Yeah. And that's the way in the case of Omega, in the case of everything is like PI, you've got observers and the universe sort of at the same level. In the case of Omega versus one, you've packaged up everything that could happen in the universe into Omega and there's nothing more to say about what happens in the universe.

Right, right.

It's all, it's all over, so to speak. I think I'm supposed to go to my other live stream, but maybe we made, we made some progress here. This is good. This is. We should continue and.

Sure.

Yeah. No good. I mean, is there anything else we might reignite on a different. On a different thing here? But is there anything obvious? I mean, we've covered. I think we've gotten some distance in understanding this kind of rulial relativity, thinking through the rule of relativity thing. I think this idea of sort of. Well, okay, to ask one last question, which isn't such a simple question. This idea of inevitable geometry and the idea of looking up in the computational hierarchy, so to speak, to see inevitable geometry, how does that relate to Grothendique's inevitable geometry ideas?

Right. I mean, so it's a good question. So my view would be that, okay, so we have a way of formulating the, you know, the kind of infinity groupoids thing that appears in the context of Croton Duke's hypothesis in terms of multi way systems. Right? So if you, if you start from a non terminating, non confluent multi way system and then you add progressively more completion rules that let you bind paths together, that's equivalent to adding higher homotopies in some, in some initial, to adding progressively higher morphisms than the initial homotopy category. And if so, if you keep, that process won't terminate. And so if you keep doing it, eventually you build up to some infinity proof point. And so then. Yeah, so Grundy's hypothesis would be the claim that thing has inevitable geometry and when you take global sections through it, those global sections inherit geometry from that infinity groupoid. I guess the rule of relativity viewpoint would be to say, well, okay, yes, but that infinity category is just quotes, just the ruliad. There's a whole hierarchy of these things starting from multi way systems that embody different models of computation. And the point is that all of them will look geometrical from that starting point, so to speak. In other words, there's a finer grained notion than one gets in standard category theory when one starts to consider computability classes and complexity classes of the underlying morphisms and equivalences that allow you to kind of formulate a whole infinite class of Gross and Dieck like hypotheses. And so I guess the point is that the Grossen Diecks hypothesis thing is then relativized to a particular model of computation. It's like this version of the ruleier, this version of the infinity group point looks geometrical if you are an observer obeying this particular computational model. But if you had a stronger computational model, you could look at that thing and it would not look geometrical. You could actually tease apart its algebraic structure.

It's in some way. Okay, so one way to say that would be maybe the ordinary growth and Deek hypothesis with growth and Deek's infinity groupoid works for a category theoretic like observer. What is a category theoretic like observer? That's presumably an observer who has the belief that aggregating morphisms is free.

Yeah, I mean, and again, this relates to some stuff. I've written that, and we discussed this a few years ago now about how category theory kind of implicitly assumes computational reducibility. Right.

I know this is one of my favorite Jonathan things that you go and start a center for applied category theory and the next week you write a paper about why category theory is limited anyway.

Right, Right. But fortunately you can extend it. And so there's this generalized notion of category theory where the morphisms have complexity of information or have information about their computational complexity, and there's an algebra for how those complexities comp. And then as long as those complexities compose sort of, If they compose purely additively, then that's kind of like maximal computational irreducibility. And anytime they compose subadditively, you have some degree of computational reducibility in that corresponding computation. And there's an analogous thing you can do when you tensor different computations together. If you have a monoidal category, you can ask how additive are the. How additive or sub additive are the complexities under the tensor product? That gives you multi computational complexity. And then. Yeah, exactly. The thing we've been discussing about the trade off between the sophistication of the evolution function versus the sophistication of the equivalence function can be recast as an algebraic question about how do these two composition operations, how do they interact? What are their coherence conditions? And so a classical category theoretic observer is something that is much more like a primitive recursive function. It's one that has no computational irreducibility that we can look at and we can just immediately kind of decode because we kind of have to be able to in order to be able to do morphism composition. And so Grothen Dieck's hypothesis, in that sense, yeah, you could say it's the hypothesis for the case of a category theory like observer. And that's really more like the hypo ruliad case. I would argue this more fine grained kind of computation informed version of category theory allows you to also consider versions of that hypothesis that have categories that are, you know, Turing computable or whatever.

But so. Right. So the point being that this statement about what is more computationally, what is as computationally sophisticated as you, or more computationally sophisticated to you, looks inevitably geometric. Would be a simple case of that would be the growth and deep case. A more sophisticated case of that will be the Turing case, so to speak, the Turing computation ruled type case. And so I mean this notion of inevitable geometry, it is a nice, it's nice. I mean that, that's, that's very, you know, the concept of the presence of inevitable randomness implies the presence of inevitable geometry is a, I mean, that's pretty, I don't really know what, you know, the connection between randomness and inevitable geometry. I suppose you could take that as being some kind of ergolic hypothesis type statement about the fact that as soon as it is random, you're kind of filling the space and then you've got a space that you're filling and then you've got geometry, so to speak. Something like that.

Right, exactly. And I mean, I know we've discussed this in other words forms, but I think it's a nice new thing that the physics project gives us that goes beyond the NKS view. This idea that in the NKS view you'd say, okay, laws of nature correspond to computationally reducible pockets of some computation. In my opinion, the slightly non trivial fact that seems to be true about physics is that the laws of physics that we know are exactly the opposite of that. They correspond to statements that you can make as a consequence of irreducibility, that general relativity and in a different way, but related way, quantum mechanics are both instances where you have maximal kind of computational irreducibility, but by virtue of that irreducibility, that allows you to make certain coarse grained macroscopic statements.

Well, in effect, geometric statements.

Right, exactly.

Which is. But I mean, so what it sort of is, is the trade off when you talk about geometry. It's the trade off of you can get very, very complicated in the integers, but that's like being in the reels, so to speak, right? That's like saying, you know, okay, you've got all this fancy stuff you're building in the integers, but now we can just put a big blanket over the whole thing and that's the reals, right? Is that the way to. But I mean that, that's sort of interesting because it says, What does that say about things like the continuum hypothesis and the question of whether there are, I mean, that claim that you have inevitable geometry which is associated with the reals, what does that say about the. The lack of an intermediate level?

Yeah, that's interesting. All right, so I mean, when one thinks about the classical constructions of the reals, right? Things like Cauchy sequences or Dedekind cuts or continued fractions or something, one way you can think about them is in exactly the terms you said, right? They're just ways of kind of saying we take the integers or by extension the rationals and we package them up in some way and we define some non trivial operation on some non trivial computational operation on them, like the ability to add Cauchy sequences together or something. And then you can do operations just at the level of those packagings of Integers and rationals. But what you're doing is really isomorphic in some deep sense to real number, the arithmetic to continuum arithmetic. And so then I guess the analog would be. So now suppose that we were in the hyper ruleiad where real numbers were to us, like what integers are real numbers were to us, what integers are to us now, right. For instance, there's a bunch of things that are special about the integers and the naturals, like the fact that they are the basis of how we do recursion and so on. So you can imagine if you were a hyper rule mathematician, you build mathematics in a way that treats the reals on the same level. And then to you, you could imagine doing a similar thing. You could imagine packaging up real numbers together and defining complicated operations on them. And those are then isomorphic to doing arithmetic in whatever 2 to the R in some 2 to the RF null model. And so really I guess the continuum hypothesis is about, I guess the way you could recast it is that it's about what mathematics is possible at different level. What kinds of mathematics is possible at different levels of this rule, your hierarchy. Right. And you know, in other words, does there exist an intermediate mathematics that you can do that involves packaging up, you know, real numbers together in some, in some non trivial way that gives you a structure that's not isomorphic to the, you know, to a, to, to a, to, to the RF null model?

Right. Well, I mean, I suppose one thing is, you know, we've got all of these rules that have, you know, we've got multi way systems and all this kind of thing. Multi way systems are very integer like, like what on earth is the real number? I thought about this for a while actually and didn't really come up with anything very good. What is the real number analog of a multi way system? Because a multi way system crucially has this idea of branching. And you could imagine instead of branching you have a fan of stuff that is parameterized by real number, but that doesn't quite get you what you need because then you no longer have kind of the duality between steps of, you know, that branching step. I mean it feels more like a PDE because instead of, it's like some kind of generalized construct, maybe like a path integral actually that where at every step you have a continuum number of possibilities, at every epsilon away step you have a continuum set of possible branches.

Yeah, exactly. So if you imagine a hyperbolic system of.