Hello. Welcome to the Active Inference Lab. This is Mathstream 1.1, our first Mathstream, and we're really lucky today to be able to have some time with Shauna Dobson. And we're going to be talking about emergent time and chromatic types. And hear a lot more on this topic and also check in with everyone's questions that are in the live chat or in the comments later. But first, we'll just quickly introduce ourselves and then hear a presentation and then have a discussion. So again, I'm Daniel and I'm here with my co-organizer, Sarah. So maybe Sarah, say hi and then pass it off. Hey, I'm Sarah, calling in from Walk Down Forever Berlin, Germany. And I'm looking forward to talking to Shauna Dobson. Hey, Sarah, thank you. Thank you for a nice intro. Hello, everyone. My name is Shauna. And I'll be a graduate student in the fall, but I've studied a whole bunch of stuff in the middle. I'm kind of a, I guess, a polymath that got a bunch of degrees at once and did a master's. And I was hired right away as a full-time lecturer kind of everywhere. And found a couple topics that I'm pretty interested in. So let's see if this can be exciting for someone. Shauna Dobson. Okay. Daniel, can you see this? Okay. Yep. All right. Am I ready to go? Okay. Good. Great. All right. So I want to talk about emergent time and chromatic types. And it, you know, maybe, I mean, it's up to y'all if you want to ask me questions as I go or maybe hold them because I'm trying to constolate what's going on here. You're going to see there's some heavy topics involved, but I tried to make them as beautiful and palatable as I can. So let's just get right to it. What are these and what am I talking about? So I have a few questions. Kind of driving this independent of the mathematics. I'm trying to make this not super math focused. I was trying to make this kind of more applicable to everyone because the math that I do is super niche. And I was trying to not do that and be like non-inclusive. So the big questions, the philosophy questions that are driving me is what is time? Is there anything that isn't time? Time is always modeled as either just like a Geiger counter or it's this sort of accepted background phenomena. And I don't really agree with that. I feel like we haven't focused too much on it. People think like, where is it? You know, I asked like, sometimes I give talks and my first question is like, where is yesterday?  But it's like phenomenologically really interesting. Where is it? And, you know, will it always be here? And I remember I was talking to Rafael Buseau and he's got this great paper about like, you know, I mean, is it, will it always be here? And maybe it's not. Maybe it sort of just maybe, you know, it came at some point as well. Hence, I think it's sort of emergent. So and I like, was it always here? And here is probably not the appropriate word. You're going to see that I love language. So again, I'm super polymath and a lot of my math papers are super math philosophy. And so I think there's a lot of problematic terms that we need to actually go into. And so I actually say, where's non Archimedean time? You know, so non Archimedean, the, you know, to be Archimedean is to believe in infinitesimals that you can always have a number bigger or lower. And in quantum mechanics, you really have a cutoff, right? So if quantum mechanics is modeling, you know, cutoffs as, you know, spatial cutoffs as non Archimedean, why is time not also non Archimedean? So I'm very interested in that. What would that actually look like? I'm going to go heavy into that. So, you know, Sarah, I had the Ilias, that's Derrida and the French continental philosophy saying that there's always a there is and in mathematics, we always go, there exists, you know, there exists an X such that whatever. And so that's very powerful, the existential quantifier, right, but in qualifier, but I'm, you know, what are the conditions for existence? This is something Chris Fields and I are working on like you, what are those, not just like metabolic conditions, I mean, all of this, you know, so I think the big bang is a real problem.  But I'm really interested in like, what are the conditions to even have existence, right? So what would the conditions be to have a discretization of time? What does that even mean? So what is entropy in a temporal discretization? So we have this like very vague notion of entropy. And I agree with like max tag market, there is no objective notion of entropy.  So there seems to be a calibration there, you know. So my whole point is to categorify the stop mechanism, just like Chris talked about in his last talk, you know, biologically speaking, like when to stop, like, you know, cellularly dividing the stop mechanism on the bio level is very, very, very intricate, in my opinion. And so I'm wondering like, how would you categorify that? I mean, like using category theory. So I'm my whole vibe here is to try to actually upgrade the mathematics that's used in modeling some of these things. Again, my head is pretty abstract. So I see things abstractly, but I think it's a gift because it's allowing me to see connections in a nonlinear way. So my brain is definitely nonlinear. And I try to use that in linear phenomena. So I say, you know, how can we say, you know, there exists a time asymmetry, but it uses this like phrase, well, when there is no objective notion of entropy, I think it's really, really tricky. There's also no proven structural causality. All right. And Sarah, you had this, you know, the light cone, it's a little vague, right? There's some problems with GR. There's some problems with the formulation of light cone and things like that. And light cones wouldn't withstand quantum mechanics and stuff like this.  And, you know, the world was made five minutes ago. And there's really no logical impossibility in saying that. So I have not seen a structural causality. And so again, Chris Fields and I agree with this. And that's why we're writing this cool stuff. Number eight, why do we actually experience memories? This is actually like why? A lot of this is very, again, taken for granted. And so I'm really interested in like why that is.  And that's not what I'm talking about, right? I'm actually talking about how why do you experience them? Somebody like me, like a total pre-cog, like I remember everything. And sometimes I can get, you know, haunted by thoughts, like memories that just come in. And I know there's a lot of people like that.  Or maybe they're really good memories. But the experience of that is very strange. Nobody knows what a thought is, right? And I remember, Sarah, I think you asked in Chris's talk, you know, what is a thought? And I haven't heard anything about what those are. People say, oh, quantum fluctuations and things like this. Yeah, but there's something else going on because, you know, for instance, like the speech that I just said, I read number eight, what, like two minutes ago. And so the speech is gone, right? It dies, but something remains. And so what is that? It seems like there's a hologramic copy of everything that you say that's actually stored in the brain. And you're doing some kind of recall with that. But I don't understand storage if everything is discreet. And so that's why this is very complex and very beautiful. So I say, what are they? Like, what is the fundamental unit of thought?  And I'm not really sure what the bio people are going to say. What is it? You know, what is not a thought? You know, so for me, you're going to see that I'm going to invoke this profinite condition. A profinite set is a compact, totally disconnected space. It's it's like a canter set. So I think that what's happening is time is like a canter set. And somehow you are recalling over a canter set. What is a thought that no idea, right? No idea how we're even speaking to each other. So I think we actually need to go over what that is. These are just the questions having begun the talk, right? So questions too, right? What is causality? Like, what is it? Again, if I don't believe in structure, you know, oh, A came before B. That's very reference frame dependent and reference frames are flimsy, right? And so I say, well, causality per him. And then I say, well, what is a measurement? That's very tricky, right? Of course, the quantum problem and just measurement in general. Yeah. And I say, well, what is a reference frame? That's even worse. You know, according to me, my world according to me and like, who is me? And so that's going to be tricky. So Chris and I, you know, both agree, like, what is this personal identity over time? And you've got big branches of philosophy, like pedurantism and endurantism trying to just trying to figure out what that is, right? Who is this, right? Is is me today the same as me yesterday? Well, clearly not because the cells are doing a whole bunch of things, right? We don't have any of the same cells that we used to have.  But, you know, you have a boundary. And so what's happened? So I, you know, I don't understand what over time means. I don't know how anyone can be over anything. So then, you know, Sarah asked me, well, isn't, you know, shouldn't you talk about space time? And I'm like, that's just a manifold. It's like one of the most basic structures in mathematics. And we have more sophisticated mathematics. So I don't really vibe with the manifold thought. I think you can really, really sparkle and dazzle that up to see what's actually happening. I don't know if you actually have to unify them like that. You know, you could just be spatializing time instead of leaving time as it is. So then I'm like, okay, if you want to just use like manifold and differential geometry, you could also go number theory. So like, what would a number theoretic time look like? And what would like a number theoretic space look like? So you don't have to just like, my whole point is that physics, like the framing of physics is based on the mathematics at the current time. So a lot of quantum mechanics is like linear algebra and operator theory, but you could also up that and make it like algebraic number theory. And then you'd get new physics. So with me, if you give new math, you get new physics.  That sounds a lot cooler to me than the, you know, the, the framing of the black hole, the old model has a lot of problems. Can you all still like see me okay? Yep. Okay, cool. All right. So let's go through this outline. And again, I'll just keep the talk at like an hour and then all right. So here we go. So those are my like big questions kind of funneling everything. So I'll talk about discrete time. I'm going to really firmly say that there is no synchronous reference that everything that's happening now already happened. You do have like a lot of time in the brain to process. So there is nobody's experiencing anything as it is and things like that. I'm also going to say that there really is no self reflexivity because I don't know if that there's a self. And so Chris and I are trying to actually formulate all this together, right? So I really don't think there is a predurantism or an endurantism with those two philosophies that are trying to say, you know, that you exist as a four dimensional being, and you kind of exist through time. And I think there's some problems with that. So problems within predurantism or endurantism is like, Oh, I get cut and I have a reaction later. Those those pieces of you were supposed to be independent, but they're not we're in touch with our pain. But they would say that it's like not so it's a little tricky. I think there is no personal identity over time. I have no idea how this is happening. There's like a reset happening every every every whatever you may not want to say plank time. But if there's not personal identity over time, there's definitely, you know, so I think the memory whatever it is has a singular support. The support is the the the the, you know, places where a function is not zero. So if you only have singular support, then that means you're not zero on a singular set. So it's actually again, I'm trying to upgrade the mathematics used in all of this to fractals and cantorsets things like that. I'll introduce chromatic types. And kind of types is going to be this new spin that I'm working on. That's sort of paralleling what's called a homotopy type. A new holographic principle based on these diamonds that I really like and emergent time. Okay, so for the why how and to what extent that I was asked to kind of fill out and I said why because this day there does not exist a coherent formalism of emergent time nor a discretization of time. Imagine there is no continuum, right? That's a huge, it's a huge statement. What exists actually between the discretization? What is emergence? How could we again our complexity class complexity class in the sense of Turing degrees? That's what I'm referring to. So not equivalence classes, Sarah, but like the Turing degree equivalent, that you still have this like there is and that's the physics of time equivalent of a meaningful president, but it's the absent of all kazoo history and kazoo history is like a sophistry. We have a we have lots of physics sophistry and mathematical sophistry and I'm like can we actually try to figure out how we could actually have existence with this discretization of time independent of any kind of conniving, you know. So how well we're going to, you know, hopefully have a wondrous introduction discretization mathematically all that is commensurate with it. Okay, so I'm into a lot of mathematics where you can construct spaces based on algebraic spaces. So I'm going to try to go really lightly through this because this is pretty heavy, but so you can quotient out a scheme which is like a higher dimensional notion of an algebraic variety by a pro etel equivalence relation. Pro etel is a certain topology and equivalence relation is going to be a gluing. So you get this diamond, which is a sheaf on a category and a category called perfectoid spaces. So this diamond has these quote impurities, which are these geometric points I refer to. So this is a notion from Peter Schultz and I really like the diamond idea. And so what happens that you have this geometric point, which is a morphism of schemes. And it's almost like, so it's sort of like this that there are mathematical impurities in this object. And so when you have a normal diamond, you have, you know, different colors on it that are reflections of the impurities, but you don't ever actually see the impurity itself, you just see the reflections of the impurities. And so you're never actually getting the thing, you're actually only getting shifted versions of a reflection. That's what I think is going on here, right? So you have your, you know, your interface, and no one's ever actually experiencing the thing, just a profinitely many copies. So you get, you can pull this morphism of a scheme back, which is a topological move. And you can get this, what's called a profinitely many copies of spaC. This is the attic spectrum of an algebraically closed curve. So you can make these products spaC cross S underline for S underline being a profinite set, which I described earlier. So you can get these awesome products. And this is this is the new like model that I used to write this cool paper with my awesome colleague, Robert Prettner, about a new model of consciousness where it's like, okay, this is what's going on. You know, you have profinitely many copies of this of perception. It's, you're not really perceiving. So but that's pretty deep. That's the hardest, some of the hardest math on the planet. It's like looking at sheaves on a category. So discretized time, right? So something I came up with just a bit ago is like there, you know, there really is no, you can't really say A equals B, whatever you want to say that. Really, what you're saying is A equals B up to some condition X, where X is what is called a homotopy type. So type theory is like pretty rampant in like lots of like awesome models. And so a homotopy type is going to be like it's an equivalence. It's an equivalence based on homotopies, which are deformations between two spaces. So you can say A is equivalent to B of criterion X is considered. So you have to be really careful with the statement equals. Yeah. And so really, we really only have intersubjective agreement on certain metrics. So I really think there is no equality for anything. You just have a bunch of agents agreeing on these metrics, right? The brilliant Don Hoffman has a bunch of work on that too. And I was working in his research group for a bit. So, you know, I think that mentation, these biological processes, I just give a talk on like mentation and a certain homotopy type that it actually works in discrete time. So how are you actually going to do that? How can you actually model memory and biological functions over a discretized time that looks like a bunch of dots? So that looks like memory of a singular support. And you're going to have to really figure out how you could get words and things like that. So what I'm going to do is like introduce this chromatic types. So again, there's no personal identity over time. I think you only have chromatic towers of localization. I'll go over that. But in your singular support, you're doing what's called a localization. Localization is a way of adding inverses. You know, for instance, the integers, right? If you want to add like inverses to the integers, you need to get something like the rational numbers. So to add inverses is a way of making a space more full, I'll say. And then also, you can take categories, which have morphisms, and you can make the morphisms isomorphism, which means they're invertible. So localization is pretty powerful. So again, there really is no personal identity. You know, there's only personal identity up to time. I also think, I don't know, time may be just the up to condition for you to exist. You are in essence localizing at each point of the singular supports. And I think I wrote, yeah, wrote a little paper about this that should be out in the Journal of Cognitive Science. So hopefully it's there. So what is this? So homotopy type is a topological space regarded up to weak homotopy equivalents. Okay, so it's a topological space regarded up to, you know, how you know, how you can actually equate various objects in the space. And what's called an infinity group void, you know, you can get an infinity group void where all the morphisms are actually so all the maps are actually invertible. So we call a weak homotopy equivalents a map between topological spaces, which induces isomorphisms on all the homotopy groups, the homotopy groups are encoding the homotopy behavior. And so, Sarah, that's in that book that I recommended to you.  But a group void, you have symmetries acting between more than one object, as you can see in this, like this visual. So I've got A, B, C, D, these are multiple objects, and you have maps going everywhere. So a group void is a category where all the morphisms are isomorphisms. So if I have an infinity group void, it means I have an infinite morphisms. So infinite isomorphisms. So the big idea for the chromatic type is to actually put a chromatic tower over a temporal logic, we're all familiar with temporal logic. And you're going to see what I mean by this tower. So I think you can actually create chromatic types instead of a homotopy type. So we're going to be interested in creating these chromatic spaces that have their own notion of localization. And then you can get a new idea of time. So localizing a geometric homotopy, so again, I told you localization is a way of adding inverses. And category theory, that means you want to make your maps isomorphisms. So we say, given a category C and a class W of morphisms, we seek the localization C W inverse. You can get a reflective localization. What you do is you localize, which means add inverses, you complete the space, that's a Cauchy space, like if you want to get sequences to converge. And then you take the inverse limit, which is gluing. You're going to see I do a lot of gluing. So for a category, the category is a collection of objects and morphisms. And you have an associativity condition and an identity arrow. So homotopies, again, are maps, ways to get objects to be equivalents. And there's all kinds of ways that you can do that. So I'm interested in something called topological localization. So this is another type of localization. So we say a topological localization is a left-exact localization. And again, I'm going to go very gentle with some of these definitions. Left-exact localization of an infinity one category. So that's a category of infinite morphisms. In the sense of passing to what's called a reflective sub-infinity one category, add a collection of morphisms that are monomorphisms. So with the details are, of course, they're important. But the whole point is that you have a reflective sub-category. What's that? Reflective sub-category. It's a full sub-category such that the objects D and the morphisms have reflections in the category. So that's very cool. So the objects and the morphisms have reflections. So every object in D looks at its own reflection by a certain morphism, D to TD. And the reflection of an object C and C is actually equipped with an isomorphism. So that's great. The inclusion creates all limits and it has all co-limits which D admits. So to say that you have limits and co-limits means that your category is sort of very, very nicely well behaved. So this is very neat that in the reflective sub-category, it seems very apropos to pair that with the diamond work that I'm trying to do because the diamond is dealing with profilingly made copies of these reflections. And the reflective sub-category contains reflections of the actual object. So it's a very, very special sub-category. So again, the whole idea is to get, okay, so maybe at every moment you are actually topologically localizing.  There's a lot of maxims that say you see what you are, absolutely. So I'm using all of that beautiful mathematics to actually justify a statement that would only, without the mathematics, a statement would just be a sentence and I can actually try to justify it structurally. So I think you're actually localizing time as these kinds of equivalences. So there really is no well-defined notion of metric. You really just have, you know, why are we working in topological spaces? Because the metrics are very subjective. And so you work topologically where you don't really have a notion of metric, but you have a closeness condition. So everyone works topologically, and I'm trying to categorify the notion of time and space means to throw it into category theory. Let's talk about its morphisms and how you localize it and how you invert the morphisms. So you need to ask, huh, you know, so topological spaces, you have a topology on the space. So here, right with like the diamond, I put it, you know, like, you know, Peter puts a sheaf on a, you know, on a category, like, okay, so what are open sets in a category? That's actually not trivial, right? Like, so, you know, I say the construction of the topological localization seems appropriately meta in its construction of a reflection of what is already profinitely many copies. I say the word V stack, it's a higher notion of a diamond. And I'll go into that a little bit later. So the second meta question is, how would you actually put a topology like on an infinity one category? That's hard, right? You have to actually figure out what the open sense are in a category, right? And so a basic point set topology and like anything in Hatcher is going to go over, you know, topological spaces and topologies, but to actually get one on a category, that's not so easy. So there is something called an infinity one site on an infinity one category. It's the data encoding and an infinity one category of infinity one sheaves, the sheaves seed pre sheaves inside a certain category of pre sheaves. So these are just like I said in Chris's talk, these are like ways of monitoring kind of local data around a space. Sheaves are attached to certain spaces. And so we call a topology on a category of site. So what's this chromatic tower? Well, the chromatic tower looks like this. So it's actually, there's a whole bunch of work on chromatic homotopy theory, just plug it in. But what is that, right? It's a way of identifying spaces using these chromatic towers. So given this LN, this is a boost field localization of what's called a Marava K theory. So these K theories are ways of looking at isomorphism classes of vector bundles. You have a wedge sum there. And so this localization gluing, right localization is adding inverses. So given that we call the homotopy limit over the chromatic tower. So there's a tower of these things called spectra, which are objects that behave like co homology theories and co homology theories are exploring holes in spaces, and just kind of data nice invariance about space because you want to classify these spaces. And so you have these towers based on the levels of spectra. So this is, this is what I'm, when I say the P localization of a spectrum that's, it's pretty detailed, but we're saying like it's localized with respect to a certain P. And so without without like super going into the details of that, you have towers of equivalences. And so based on your layer, based on where you are on the chromatic tower is what you're going is how you're going to be navigating. So that's memory space to me recall memory space. It's all the same navigating your Turing degree is, is in a sense a chromatic tower like this. So to create a chromatic type, I'm going to make equivalences based on the layers of this tower. So for instance, you know, if you're on the top layer making an equivalence between the top layer, and you know, so replacing home, never replacing but upgrading the homotopy type with chromatic type. Okay. So we say for memories and computability. I said, again, at that mentation talk that I just gave not too long ago at the biological conference. What else besides computing might be involved in biological mentality, right? I'm just going a little phenomenological so we can pull this together. But you know, it's a it's a homotopy type that could be a chromatic type. It proceeds by weak equivalences. It far exceeds there's a lot of like, expandographs and graph theory that model it. I just think it's insufficient. And I think that you can get into these towers and get a little more of what's going on. So I also say memory proceeds like an almost apology. That's a very strict word from faultings, what it means to be almost. But we can say that memory has a singular support, like I told you, like a canter set. And it's what's called a descent condition, in my opinion, which is a gluing, right? So memory is recall and it's putting everything together so that you can actually localize at this specific instant, whatever that is. So it behaves more like a descent condition. So again, a weak equivalence is a class of morphisms in a specific category. Computability, right?  There's a huge MSRI awesome MSRI conference or program during 2020 about decidability, definability and computability. So computability, right, is the ability to solve a problem in an effective manner. You've got all these calculuses, Turing computable, weaker, automata, stronger, hyper computation. I say, you know, I think it's the actual ability to construct descent conditions. So I think we need to take these very strict terms about how we navigate and categorify them. So you see me kind of adding, like, well, why don't we upgrade that, upgrade the mathematics use there, you know, define abilities about structuring knowledge and concept formation, decidability, say, binary decision problem is decidable, if there is an effective method for deriving the correct answer, you know, and this is very, this is very kind of clearly not, you know, this is not what I was giving you in the chromatic tower. I think there's more ways of getting around and saying things are actually effective if you use the sort of weak equivalence kind. So memory. So what else besides information algorithms are involved in biological mentality? Well, a lot. You've got geometric logic, girdle and completeness, singular support, right. So I call this, in my paper with Robert, we come up with a nested hierarchy of awarenesses and I'll go over that in about one second. So I call it a one, one awareness is capacity to construct morphisms between other one awarenesses. So we're all relating consciousness, whatever it is, is geometrical and relational, I don't think it's a computational. So you can get this categorical, I'm trying to figure out a way, oh, you know, we know the notion of turing degrees, let's get a categorical notion of that. So, you know, I say recall, your ability to recall is what's called a localization sequence. This is a coal fiber sequence, where you have these maps and you have a condition between functors and what pops out is a fiber. And so you given, so this is what I think is happening when you actually recall, you look for a certain state, out pops a memory, right. So there's a bunch of category theory going on. So what are the fundamental units of life? That's very tricky, right. There's a lot of, I think there's a story about like a beautiful monk that went up to a hill to just sort of self desiccate and everyone thought he passed away, but maybe he had one heartbeat a year. So my question is, like, what is that? What are the fundamental units of life? Could it be that someone slips under the radar of what we would call metabolic life, but there's another set of brain waves that are maybe mere symmetries on the other side. So like you see, I have this chromatic tower with everything on the right side, I want to build a, what's on the left side. So all of this, I have chromatic towers with right sides. Yeah. So maybe this is the condition of life, but what if you reflected on the other side? What does that even mean? So again, I'm, you know, playing around, trying to figure out levels of consciousness. So for instance, Clive wearing the famous case, awesome pianist, right, super brilliant pianist, got an, I think an HPV and got an H, like, got a retrograde amnesia. And so he's got a seven second memory and he only has motor memory and no personal memory. Right.  So it's very interesting with someone like that, you have to ask the conditions of what are Clive wearings fundamental units of life. And I literally think what you could, you can more easily categorize what's going on rather than just saying retrograde amnesia, maybe some layers in the tower, layers in the chromatic tower have been gone. And that's what's happening. So if layers in the chromatic tower cut, then you actually have, you know, so just like a geometric way of interpreting all of this. Okay, so let's get to discrete time, right. So discrete time, why do we model time as continuous? Like, why? Again, I think it's an overlook. And I just think we haven't focused on it. So why do we model time as a counter or as the background for some kind of background dependent model, right? What I have pictured here is what's called a spectral sequence. And so if time looks like these series of dots inside the dot is encoded a whole bunch of information about a space. So you could model time is like a modular space. That's what I said when Chris gave his talk, right? So that points, they're not just points, right, they actually encode other information. But you can see that there are gaps between the points. So if you model time as discontinuous or, you know, or discretized, what's actually in between the gaps? So then we say, okay, well, if quantum mechanics model space is a non-archimedean field, why is time also not not Archimedean? Why not? So I opened with that question, what would it mean to phenomenologically have a discrete time? What would exist between like T1 and T2? That's really rich. What's actually there? Maybe Susskin would say that's the pure emergent, right? He's got the two black holes entangled. He thinks emergent comes from the entanglement of black holes. So everyone's looking at that. And then you get paternalism and temporal parts.  So I, you know, I asked, why do we experience memories? Well, is time entangled with memory? Where does entanglement end? That's my question. How do memories actually exist in discrete time? That's tricky. That's tricky, right? So again, you have someone like me who thinks there is no personal identity over time, but yet you have a memory, right? Somehow. So anyway, so how would you actually define memories in end time? I don't think there's just one timeline here. So I put together this theory where I think you have the levels of time you have is depending on the levels of category that you are. And so I'm going to go over that. So the levels of time that you are able to navigate depends on your complexity class and the level of space that you can actually navigate. So how would we experience memories in an end time? Imagine you had in levels of time in a tower. So with Robert's in our Floralismanus paper and also the Perfecto Diamond paper, right? We say, you know, awareness is a, it's, I, we can do it representationally. Say it's a representation of a numinon, but we also want to throw consciousness into levels and layers. So right here, you have a basic notion of a one category where I have objects and I have morphisms between objects. That's it. So one category is contains objects and one morphism. So just a quick category theory, like category theory is beautiful and I cannot do it justice in two seconds, but I'm giving just a little recap of what we need. In the middle here, you see a two category. Two category contains objects, one morphisms and what we call two morphisms. So two morphisms are morphisms between the morphisms. You see, I have a map here from F1 to G2. So this fee and you see I have a gamma here from F2 to G1. So that's, this is getting at what I think, right? So basic, basic complexity class can only, oh, hi, Daniel. Hi, Sarah. We have an interaction. That's it, right? But super precogs or anyone with total recall like me, I can put a map between this interaction and this interaction. This is what I'm talking about. A three category. Three category looks like this. You have objects. You have morphisms between the objects. You have morphisms between the one morphisms and you have a three morphism, which is a morphism between the two morphisms. So you can see this F2 going this way, right? So this is what I'm saying, right? Depending on your level of category, your level of awareness, if you are a three awareness creature, you're able to navigate this way.  If you're a one awareness creature, you live in one time. And again, I just, I just went on to this because people like me are either haunted by, you know, sad memories or blessed by amazing memories and time for me did not seem to go linearly. It goes like this. So we have this first subjective experience. We have this nested categories of representation. You've got a one awareness. You've got a two awareness. So that means a category of categories of representation, the three awareness, a category of category of categories of representation in awareness is our top categories of categories of categories. So you can see that what, what, what does this sort of correspond to is like, well, me giving you this awesome presentation, being welcome in your environment and us also, you know, being at the Cambridge apostles and having like tea, you know, so not just being able to imagine something like that, but being able to partition your brain and split your consciousness and to actually be able to do that. So I'm all, I'm really excited about splitting consciousness and trying to physically be in multiple places because I don't think, I don't think there's only one timeline here. But to split consciousness, you'd have to be able to do this in the brain, you know, which is cool. So we come up with, so along with end time. And so again, you can see that with the mathematics, you're able to do a lot of stuff here. So with end time, I came up with this, well, you know what, I think, according to end time, you'd have a profiling language like what is that? Well, if you, you know, any dairy doggies or anything like in his of grammatology, it's going to show you that the, the gloss semantics or the gloss seems, you know, that your fundamental units of language, if you open them up, there's not a lot there. But actually the process of reading is a trace. It's a wild phenomenon because you're pulling together so many pieces. So we actually, I came up with this awesome language scheme that was like, well, you know, in the one space, we just have your normal I, we, you know, me, my, mine, but in two, you'd have like a one I, a two eyes. This is if you're in two awareness, right? If you're in the two category, then you would have actually like a one eye, a two eye, a one me, a two me. If you're in a three category, you'd have a one eye, a two eye, a three eye, a one me, a two me. So instead of just she, her, there'd be like three, she, three, her. And I think that's the level of intricacy in the new language. So getting to some of the physics stuff and going into the emergent time. So I came up with this idea for a perfectoid dictionary, again, perfectoid spaces or Peter Schultz's brilliant work. And I just think it's amazing. And so I do think quantum physics needs to be a little revolutionized by replacing Hilbert space with a perfectoid space replacing state vectors, the geometric points. And I just gave this talk at UCLA's IPAM entropy conference not too long ago. And I think it was pretty well received. Replace the tensor product with the diamond product, place non locality with the finite and many copies of spa C, replace superposition with these pro etel sheaves and pro finite sets. The collapse of the wave function was such so tricky. Replace it with what's called a tilting operation, which takes us from characteristic zero to characteristic P. You can get these also perfectoid modular curves. Replace, upgrade the holographic principle with the six functor formalism with those, which is a growth and deconstruction. And I'm about to show you that you can replace quantum topology, upgrade it with this etel co homology, replace operator out again, it's great, but you can just upgrade stuff. So I shouldn't say replace but upgrade. Let's let's let's let's replace operator algebra with a non noetherian version and unitarity with a certain descent data. So you have this. This is Peter's six functor formalism where in a six functor formalism, it is a way of making a theory, I would say really it's a it's a notion of replacing Poincaré normal Poincaré duality with Abelian sheaf co homology and it's something called Verdeer duality. So there's dualities going on here, which is why I thought it'd be great to pair it with the holographic principle because the holographic principle is really fortified by Juan Maldecina's ADS CFT proposed duality between you know, like the gauge gravity duality. So you have this FM off K theory thing that I'm working on of a diamond and you have these the six operations. And so with these six operations, these are just image functors between sheaves. And you get this notion, this is what I'm thinking of. So in quantum gravity, right, you have the holographic principle and it's still very kind of loosely defined. And it says it's given an only a negative condition, like a defined by what it's not. And so I think the holographic principle instead of having ADS, you know, anti to sitter space in the middle and the conformal field theory in the bulk that you could actually have the six functor formalism in the middle, and the profoundly many copies like the diamonds looking like the conformal field theory. This is what I'm working on right now. So emergent time. So these V stacks, these are higher dimensional diamonds. And so there's certainly holographic in a sense that they encode already profiling many copies of data that's already multiple on two fronts. So a V stack is at two sheaves. So this is like, you know, the sheaves that hold you encode local data. So this now it's, it's, it's one category up. It takes values in categories rather than sets. So, you know, you're going to have to shift to V stack if you want to do this profiling language. So here's a conjecture with Chris and I, you know, we're trying to generalize the distinction between separable and entangled. This is not easy. So once again, recall that the geometric points in the diamond amorphous is a schemes, there's something called a bear wise silicon and infomorphism. And this is an adjoint pair. And so, okay, so you have an adjoint pair, there's a sort of like inverses, like inverse functions, but they're functors. And so we extend this idea to the two adjoint pairs. This is f upper star, or f lower star for the derived category, and then f lower shriek and f upper shriek shriek is a kind of slang for the punctuation. So those are referring to once again, the Peters operations. So that's the grand correspondence, right, is going to be this, this derived category, which consists precisely of the atel sheaves of these certain modules for a small diamond. Okay, so let's get to emergent time. So how what am I, I'm going to use all this, how do I actually pull this together? So if you consider the simplest case, we say an event, what is an event? It's a, I'm going to call it a topological localization of any particular reference frame. Right, event is normally like, I don't know, simultaneity or says something like that, try not to use the word simultaneous. We'll say a topological localization of somebody's reference frame. Let's consider that to be a point in the diamond topological space. On that point is a certain site, I told you right this how you're going to get a topology.  So this is from more Peters work on condensed mathematics. So global time emerges as a set of continuous maps from all pro finite sets s to t. So again, you have these chopped up notions of time, and you can get global time to emerge as the set of continuous maps from again, the, all the pro finite canter sets back to t, back to the topological space. So global time can be considered as a sheaf of sets on this pro etel site, and that's called a condensed set based on his work with the flossing. So emergent time results in passing to the larger category of she's to consider a condensed version of, this is an F theory construction. I don't want to go over too much, it's a little deep. So a second conjectural way to get an emergent time, you can actually use diamonds and take equivalence relations of those to actually get the V stacks. So again, when I showed you the definition of diamond, you took equivalence relations of, you took pro etel equivalence relations. Well, let's upgrade, and you can take actually equivalence relations of diamonds to get modular spaces of V stacks. That's pretty big. So there's new work being done about how do you actually, you know, how about a new algebra? This is Peter and Dustin's new work. How do you do algebra when rings and modules carry a topology? I think this stuff is great. And this is where I'm at. It's like, hey, we need to like rev up mathematics, seem to upgrade some physics stuff. And I think we can all learn from it. So if you can turn functional analysis into a branch of commutative algebra, that's very nice. Build a version of algebraic geometry allows rings of convergent power series as basic building blocks. Because my question is, how do you actually do analysis in non-archimedian geometry, right? That's actually kind of tricky and exciting for a mathematician, right? So you want to put some kind of completeness condition on modules, you know, which are like, they mimic vector spaces, but the scalars are not fields, they can come from the rings, the coefficients of the rings. So in such a way that the free objects behave like some kind of power series instead. So you can construct algebraic geometry, you can start with an abelian category of abelian groups. You have a tensor product, which is just a way of organizing. And then you can consider rings in this sort of category. You can associate spec r with the same spa r that I showed you in the diamond. And basics and open subsets can correspond to localization. So look how cool this is. You're using localizations to define what an open subset is. You can glue these spec rs together along these open subsets. You can form these schemes. And accordingly, the category of r modules glues to form the category of quasi-coherent sheaves. So again, this is all Peter's construction. I think it's great. How do you construct analytic geometry? We'll do the same thing. But topological abelian groups don't form an abelian category. I'll just go over this. You can use the his notion of condensed mathematics to make it make things a little better. You can replace topological groups with these condensed sets. So here's the whole point. So this definition, right, the proatell site of a point is the category of profinite sets. So these are canter sets-ish. With finite jointly-surjective families, a map is covered. So we say a condensed set is a sheaf of sets on the proatell site. So again, we're working profinite and proatell because of the diamond condition. So similarly, a condensed group ring, whatever on the proatell site is called that. You can put a category together, condensed, or condensed objects. So this means that this whole thing is a functor. So you can get a functor from certain categories. That's fine. I don't want to go into this too, too much. So given a condensed set, you can refer to T star as its underlining set. So there's a canonical example that illustrates how you would actually change from a topological space. So let T be a topological space. To T, there's associated a condensed set, T bar. And that's defined by sending any profinite set to the set of continuous maps from S to T. This satisfies both properties. It means that for any map for which the composite is continuous, it's continuous. So T underline is a condensed this profinite set. Yeah. So why, like, why are we using all this profinite condition? Well, because once again, it's in the language. I don't want us to get lost in the detail. So it's in the language, right? It's in the awareness, the level of awareness that you are, right? Dictates your whole complexity class. So if, and again, if these spaces are profinite spaces, right, then you need to make sure that what do we mean when we say localize? We need to make sure we know what the open subsets are. And that's why I was using like the pro hotel site. That's just one idea. There's lots of ideas how you could model this. But the main takeaway that I want to kind of leave everybody with is like, okay, so what are memories? Like, there is no synchronous reference. How are we actually getting through time? Yeah, if every point is discretized, you must be getting through time through your localization. But I'm trying to try to upgrade the consciousness level and go, well, you know, I think you can actually upgrade the level of awareness if you can start thinking in terms of morphisms and morphisms between morphisms. So you get the emergent time phenomena by using the pro hotel site. And then you get the diamond holographic principle by using the six operations and the profinitly many copies like that. So once again, you get the chromatic types from looking at these towers of localization, and you can put this over temporal logic to maybe to maybe, you know, figure out what's going on with my wearing. Once again, like, where is the meaning in that? Yeah. Okay, so I put up some of my references and I think I will just stop there. Awesome. Thank you so much. Maybe you can unshare and then we'll be really happy to ask you some questions. Sure. Learn more. All right, Sarah, maybe take it away. Yeah, so so that was a great brain scramble. Like, I guess I'm wondering actually this really comes directly from philosophy before I went on this live stream, I was looking at just where topology fits into philosophy by way of the standard insight or Stanford encyclopedia. And one thing that stood out to me was was modal logic. And I found myself asking, you know, with all of these new topologies or set theory or however you want to do it, like, we always kind of run into this tension between these really complicated, homeomorphic, whatever kind of morphic, you know, our ability to conceptualize these morphisms and our ability to like make use of them. Like, or, you know, and we make these really complicated math constructs and then but at some point we have to communicate with each other. And so I'm kind of wondering with any math construct and like, maybe even specifically yours, like, how do you see the construct converging, like looping back, you know, conceptually such that it doesn't just run away from, you know, from us in our ability to understand. Yeah, so well, that's why I said so for me, thinking like those constructs are not difficult, I would say the communicating part is the difficult part, you know, like, I mean, but you should watch children, they're all over the place, right? I think, I think you are here to imagine and that you actually I'm like a super sinister. So for somebody like me, right, I just rattled that off. And you saw that that was probably the easiest thing for me. But asking me to maybe replace a cartridge in an ink, like a printer, that might be really difficult for me. So it's interesting that when you say, how do we not lose it? So I think and that's why I was saying I think it depends on where your brain thrives, like somebody like me, I love the thought of like, Oh, yeah, I'm definitely a free awareness. I want to think about morphisms between morphisms and morphisms. And so I actually don't think they're difficult. I think you you know, it's something like it's just a it's just a math is just another language, you know, and if you're a person that like likes learning new languages, I think there is no difficulty around it. And so I don't think I for me, I don't think there's anything hindering it. Because I think what's worse is to leave it in a word statement. If you didn't have that math justifying it, you would just have a sentence saying, what, like the world is unknowable, or something like that, right?  Like, okay, but let's say, for example, because in modal logic, they were just talking about knowing versus assuming versus, you know, so they had these kind of conjunctive sections in a sentence.  And so actually having math such, you know, that works with all of these subtleties is like, I don't know. Anyway, that's that's where I'm coming from is like, how do you get a math that doesn't that doesn't get like punked by language? That's funny. Yeah, but isn't it exciting? Because you didn't know much about topology till I said it, right? But you're on an exploration, and you're learning more. So I don't think it's actually detracting at all, right? Like, no, I'm not not saying detracting. I'm saying facilitate help, like, compressing, compressing language, you know, compressing concepts. Right. And I think on my end, I'm not, I think that's might be deleterious. I think it might be open them, open all the concepts. What's, you know, I don't think time exists anyway. So it's like, what's the rush, right? And one note on that is on the linear strings, spoken or podcast or live stream or text, right, they're often ill specified, but also it enforces a so called temporal structure of local syntactic in a semantic structure. And whether one chooses to see these these associations as foundational, like the roots that this is growing out of, like the previous literature threads, is it running off away into the dark forest of the literature? Or is it actually anchoring in vast bodies of knowledge? So really interesting this discussion. Oh, yeah, see, that's a dance. So how cool that what's natural for me is like math and literary, right? So you've got some heavy philosophers that were onto some great things. And I'm like, Hey, why don't we just fortify what they're saying with some diagrams? And you see, it's easy if I say, Oh, maybe Clive wearing chromatic powers cut that in a certain way, like the math part there is actually, while there is a steep learning curve, you're able to actually get inside of Clive wearing in a way that you're not in a way that just like Daniel saying, you know, retrograde amnesia. Okay, great. That's a linear word, you need a sentence, you actually need to go into that. What is that? And so I actually think language is deceiving, if it's actually too compressed, and if it's actually not transparent, there's a beautiful transparency in mathematics, it's all there. You know, and again, math is not perfect either, right? There's, what is that point?  And that's why it's like, Oh, we should just have all the languages, you can see, I talk about everything at once. And I'm like, that that's my synesthesia, right? I think there's actually the beautiful thing about the synesthesia is that maybe you can see like an aerial view of things. And I think for me, I can do a local global just because of the way my brain works, you know, but then I was, you know, I was trying because I love everybody's like differences, and I don't want anybody to be like anybody else. And I was trying to say like, Hmm, it seems that, you know, maybe sometimes we're one awareness is and sometimes we're 12 awareness is, but I really, I'm really interested in why, like, I don't know how you two are, but if like, if you have total recall, like, or memories, like, have you ever had memories just come in or future memories or deja vu, I'm really interested in stuff like that. Yeah, I just wanted to clarify the compression thing, though, I want to defend compression for a minute. Normally, I would agree, you know, like the connotation with that, I'm like, no, that's a throw away a lot of nice subtleties. But I've definitely felt in my limited experience with philosophy, I've definitely bumped up against some spots where there have been words that have, you know, somebody, once somebody hits it, what somebody puts a noun to a concept to a word, it's like this fantastic compression algorithm where it's like, Oh, that's a concept somebody had, and they put it into a word. And now I can pack it up and like make some more space in my brain for other stuff. And I think that is like that practice, whether it can be in math or or in words, like in whatever means you do, it's just, it's the best. So I'm always now I'm finding myself like looking for that, for that ratchet, you know, then you would love, well, then you would love math, but it's like then because it's all symbols, right? But I'm just saying, yeah, I don't see it compressing, it doesn't, at least not from my brain, like I can't, I can't like it doesn't package something up so I can make more space for other things. But it does because it's like, Oh, there's a tower. So you can just keep the tower in your brain and not the 800 pages that got there, just like when somebody puts a noun on a concept like anti to sitter space. I think, yeah, I mean, for me, it would be I would be deceiving myself to think I'm just storing the word, you're not you're storing all the physics and everything. And what do you what do you think about that? Right? You know, this nesting is a topic that comes up a lot. So we talk about nested Markov decision processes, how within the active inference model, just only looking at the mathematics within this one little clade of math, you can nest those models. But then there's also nesting that's based upon category theory, not my area, but it seems like there's very nuanced ways of doing nesting and relationships. And so what I heard with this discussion around like a sort of a whole total system shift upgrade with a dictionary, it's like synergetics and Bucky Fuller in that if you understand the interfaces interfaces between what these things are currently doing in the system, you can take them out and like hot swap them. It's like, okay, it's a graphics card, we can make it a different graphics card, or we could have the same interface, and then improve what's on the other side of the interface. And then it turns out to be really meta because we're working on active inference, which is in a way, the theory about how the interfaces work. And definitely in the active inference lab, we're taking a different approach, but it's like Mount Everest because we're meeting at different base camps together to talk with people who have different perspectives. Yeah, yeah, yeah, which is, but that's where everything thrives is in the intersections, right? So that's, yeah, it's like, look how much stronger math is when you hit it with philosophy also vice versa, or any of this stuff, or any of the bio, right, upgrade Richard grade amnesia to a permanent tower. And it's like, whoa, you know, so I think, I think nothing has lost in the intersection. And what did you say earlier about what Bucky called emergence? You said something cool. Emergence through emergency. Yeah. And as emergency like the, every time I like survival or like what to actually connect, that's what you said about your string of natural language being effortless. In active inference, we think about how the action is following in some sense, actually on a geodesic, a path of least resistance, like the chemical reaction, even if it releases a lot of heat and light in the context that it's in, it actually is the most probable thing or the most energy releasing thing, the free energy is negative. And so it dissipates and understanding how that dissipation can be brought into the informational regime with semantic flow and like a delta G for semantic information, rather than just syntax or even chemistry. That's about this mapping of mappings with free energy as a nomadic concept in a way or a nexus that we connect to free energy and the chemical we develop certain experiments, maybe unique predictions. And then by building the math that's general enough to interface nicely, we actually we nest the concepts so that we can deal with the nested systems that are more syntactic at the lower level going down into the quantum and then more semantic at the higher level into the teleological. Good, yeah. And then the semantic at the higher level, right, then you can invoke a homotopy type, chromatic type, and then you can actually, you know, all the types are nomadic as well. So I like that, right? Nothing is until it actually like spreads. So I also think, you know, it's a false conception of movement.  Hey, Shayna, can we, I wonder if we can anchor some some some exploration around one of your questions that's on the list. Yeah. So like, but I don't remember what they are. And it might be too much of a technical hassle to post them. So if you could like pick one of the questions that you're bringing up. Yeah, let me just share again. And then you tell me which one you like. Okay. Okay. Yeah. This will be like the who wants to be a millionaire of. Yeah, right. This is cool. Yeah. Who wants to be the millionaire? Should we do a quick random number generator? No, pick one. Yeah, just look at a few and then let's anchor them and then go wild, whatever one. Okay. So so actually now this helps trip my memory. So and then I had these two. So pick which one maybe I can like make it. Can you see that? Yeah, I can. I can. Do do. Oh, oh, I want to do I want to do I want to do what is measurement. I this is something I've been tripping out on a bit. So and see so measurement to two. Yeah, no, I know where it is. I thought I might pick a second one, but that's definitely should I go with that and then you bring it or do you want to go? Let me on show. I can see your lovely faces. We're saying what is the what is measurement right? Yeah, go for it. Yeah. Well, I tend to take a lot of these. I'm finding that I have a attraction to taking a lot of these questions and putting them in some embodied form that seems to be I don't know how I'm going to do that in a thesis in such a way that I actually graduate. But this is it's really interesting to me. I, you know, we have affordances, we have tools and all these other things, these kind of prosthetics that allow us to make sense of the world that I really just think of them as prosthetics. And in a sense, we're, we're making sense of the world by feeling and and I mean feeling in the one sense of like feeling, you know, whether or not that actually relates to emotions. That's another really interesting question. But to be conservative, you know, I'm just talking about feeling like our tactile sense. And then we use all these affordances to augment our tactile sense. But so it's really interesting to think about, like, all this objective measurement in science and whatnot, and how that's actually related to our embodied, like, it won't be true to us at some level, unless we can make sense out of it at an embodied level. So that's, there was another kind of hook I had in philosophy, but I can't really remember what it is right now. But that's something that's alive for me. So I don't know if you have any thoughts about it. Yeah, well, I'm like super empath to write. So to, you know, I, I'm going to dissociate the emotional from the rational, like, I don't believe in that. And they both are equally there, right? The rationality is like the emotional guidance system. So yeah, when I say, like, when I say what is a measurement, I'm just saying there's a lot that goes into that that says, you know, this is the fixed speed of light. And this is the this and so measurement seems like it only holds up to a certain condition, right? This is the this is the speed of light until I change your ambient space, right? If you lived on a Taurus, or if you lived on a Klein bottle, if you lived on something else, I promise you you'd have a different speed of light, because that's just the illumination problem in mathematics, given a strange space, right? And I put a point source in it. What if you trace out all the rays, right? Well, every point in the actual shape be hit, not really, it won't, if you have a very complicated space. So for, for me, how far, how far does the measurement hold and metrics, again, since I'm such a fan of like, topology and sheaves and things like this, the metric seems like it is not substantial, right? It is it is based on, like I said, intersubjective agreement among agents. So when you say, you know, like, I'm not sure if your question is like, is it objective or subjective? I think all measurements here are subjective, right? And so also, there's an actual problem in the measurement. If you're like strict Copenhagenist and stuff like that, right? So nothing should quote exist, right? If everything is in its free form of superposition, what does it actually mean to measure something without like, actually limiting the thing? And who's the measurement for? If you crush the superposition into like one state, I've measured it, right? I don't know what that means. And I also think that the minute all your observers go away, the thing that was measured goes back into its original state. So that's why I said, like, what is a measurement? And Chris and I are always talking about this, right? That assumes that you are somehow independent of the thing that you're measuring. And I don't know how to actually do that, right? I don't really know what a boundary is, because if you zoom into your arm, we could take like an electron transmission microscope, and I'd be able to zoom into your arm in like two seconds. So where the boundary go? You know, where is the boundary? And that I'm very perplexed because the boundary in mathematics is like some kind of asymptote, right? What is an asset? Like, you know, so are you dealing with sort of asymptotes? And that's that's what we're calling a measurement. You're like, a bus is this long, true, until I subject the bus to a lot of time, or I pour gasoline on the bus and burn it, you know what I mean? So how long does the measurement hold? You know? Yeah, I agree with that. Yeah, so I hadn't thought of it so much about context and feeling. I was definitely like, it's a fixed thing. So that's that's useful to think about. Not really like fixed according to whom fixed for that second, but like everything is in free fall. And like you see, I really think there is nothing periodic here. I think there is no repetition, right? We have to actually account for like, what leap year or something, right? Because the calendar is off. So I'm I'm really, I'm really impressed that in our, like in everybody's free fall, and you know, what the earth is going 1000 miles an hour east, right? And that everything is rotating, your own cells are popping in and out. Where is the measurement? I'm not sure nothing is fixed. So it's very silly to think that I have this measurement. It's like, if you only knew that you were actually working over discrete time, in that moment, the moment that you were measuring the bus is no longer there, right? So that's what I'm very, very interested in. I measured the bus five minutes ago, but is it the same bus? No. And I don't know what that means. So that's like, what are you actually measuring? We could measure profinitely many copies of the thing, but you need to say that the measurement is of the reflection of the thing. And that's, that's the one thing I do like about the temporal parts work is that the bus yesterday, is it really the bus today? I don't know. And I've always been, I've always been perplexed that, you know, you wake up to the same space, but do you really, I'm not really sure. And so that's, you know, boundaries, I'm really interested in boundaries and like how they actually, how they work, because I'm not really sure they do. I mean, clearly, like I can hit my hand on the, on the, on the glass, but that's just, that's at this scale, right? If I, so I'm just saying that a measurement, if you're going to say a measurement, then you need to be real precise about there's no time involved. There's no this involved. It's only at an instant. Boom. This measurement holds at an instant, because tomorrow, who knows. And what's up with the measuring stick? Like there's so many, like quantum mechanically, the meteor stick doesn't exist anyway. So, so I don't even know how to use all these frigging philosophical labels, but I want to ask if your ambition in, in trying to apply topology, a kind of a paradigm is to arrive at some form of realism, or are you interested in, I mean, I'm kind of wondering what your ground is in terms of like, yeah, what would be the, what's the end goal in terms of like, what would be the perfect math in your mind? Like, where would it take you? Right. Yeah. Well, I'm after those grand unified theories, right? So I'm after the, I'm after the unifying the grand unified theories of physics with the grand unified theories of math, because I actually think the math one encompasses the physics. That's why you see me going, Hey, why don't we look at like a number theoretic version of a black pole, something like that? Because I think that in the grand unified theories of math, I'm going to get everything because I'm a super tech mark like a level four. You're going to, you're going to get everything. You're going to get, you're going to get, what are you going to get? Oh yeah. Well, that's why I say like, well, are you calling reality like just this little glit? You got to remember that I believe in like, in consciousness, right? So my goal is to actually partition the brain, partition the consciousness so that you can actually experience everything at once. Again, I don't believe in linear time. I don't think it goes linearly. And I'm really baffled why the brain only does that. And I think it does not.  I'm like, where did you come from? You know, like the very act of birth is something interesting, because it seems like it's something that got through the interface. Right. Where did this little portal come from? And I'm actually, I'm not so interested in the mechanics of the interface. I'm interested in breaking through the interface, taking off the interface and trying another one. Because again, I think the 20 fundamental numbers and everything that you're subjected to here and the laws of physics or whatever, are just purely interface space, which I, which I think is a mathematics concept. It would be so rad if we were all able to take off the interface of this complexity of our current complexity, and quote, go somewhere else. I don't mean Milky Way. I don't mean that. I mean, leave this interface altogether of three time of three space and one time and go find a different modular space. Right. And that's what I'm trying to do with. I'm like, Oh, if you could actually partition consciousness, right? Because it seems like, I don't know if y'all have had like general anesthesias or like if you've ever fainted or anything. I have a like a few times just from like kind of intense things. But I remember something remained. It's like, you might have wiped out the consciousness by a general anesthesia, right? But you didn't take away the awareness. And that's what I was very curious about that something was still there. And so, you know, Don Hoffman and like me, it's like, I'm actually interested in what that space is, right, the kind of talky sensory thing, like I can see that as an apparatus. But I'm sort of interested in like the divinity, whatever the infinity, not no religious sense, but like, what actually transcends this, because I don't think anything here is like real at all. You know, so when people say, what's your baseline reality? Well, it's not this, I'm really always curious to get in the dream space. I'm really always curious to quote sleep, because when I'm asleep, who's asleep? You know, your body is not asleep, right?  So I'm really curious in the states that are actually, I guess I could say it like this, I'm trying to take subjectivity out of the interface and put it in a different setting. I'm not gonna have subjectivity limited by just what the body can do, because there's a lot of pain. A lot of pain keeps people in the body when I think you could actually be like somewhere else, you know, so if you could take subjectivity on its own exploration, which is what I think mathematics is really good about doing, like, if I can have, you know, people like, oh, let's think about simultaneously, like, what if, you know, what if we were actually here and also, you know, at a Cambridge Apostle meeting, that baffles people because, you know, you have this imagination, right?  You can go there just like that. What is that? Why can you do that? What is that imaginary space? So I actually think we're actually really gifted creatures and you have this beautiful tool called the imagination, which has not been fortified. So what I'm trying to do is actually put the imagination in its own little Hogwarts school, right? Like when you're young, you're able to do plays and you're doing all this thing, then you go up and it's like taxes and like that. And it's like, no, I actually think we're here to be creative. And you can see all the philosophers that I like are vitalists, right? And so I think being is creation. So you can see that I'm here to, I think we're here to create new theories and to break through the interface.  That's not what I'm saying, right? I'm saying it'd be cool to peek through it, just sort of like how a new baby arrives. I am perplexed at birth. Nobody can explain it. And you have no idea like how something broke through the interface. So where I'm after with that, you see I'm using very beautiful mathematics. I think life is very sacred and very beautiful, and it should be supported and explained by an equally sophisticated mathematical structure. So I don't really abide by this. The simplest version is the best. I don't know about that. I actually think that what is sophisticated philosophically should be equally matched in sophistication mathematically. And then you get a synergy and a harmony. And actually, then you can actually explain a grain unified theory, because I really think, you know, if everything is a diamond, which I really think it is, you know, there's a lot of, I think if this reality is like a modular space of diamonds, then I can just easily jump to a new modular space. It's not that difficult, you know. I don't know if that sounds like, you're like, who's this little person? Oh, yeah, I'm totally like an explorer. I'd be like, let's jump in the consciousness bus and go look at stuff. Because that's not just this. It's not just what is.  Yeah, because I don't know where one skill ends, and the brain seems to be, you know, a hologram itself. So when everything, when your central processor is a hologram, it makes me really curious about what this space is, you know.  But it's freaky ability to rewrite, right? It's like a super hamming codes, you've got like air correcting code in the brain, obviously, to keep the organism alive, you know.  So I'm just kind of trying to upgrade, you know, find what is the essence of things, and it's like, Oh, is it math? What is this, you know? Ah, the essence. I'll ask one question, which is how would you imagine that this insight or this set of upgrades potentially being taught to advanced or younger students could change research or practice? Because I noticed words like descent condition, definability, decidability, completeness, of course. So where do the sort of everyday usages of these terms come into play with how we can rethink them based upon a new level of mathematical rigor? Oh, that's such a great question. Yeah, I'm such an advocate for like new models of pedagogy, right? Because I think the way mathematics is taught now is horrible, and it makes, you know, everyone frightened. It's like you're taught calculus first before logic, before topology, before any of it, right? And so it seems like, you know, there are people are not using the apostle book, that should be the only book used for calculus. Yeah, so if you actually, if you taught category theory straight, you know, from the bottom up, imagine the beautiful connections that would be able to be made cross disciplinary, you can see that one of my main goals is to get people to talk with each other, get the mathematicians to talk with the physicists to talk with the philosophers. You know, I think through category theory, you would naturally embody this synesthete thinking, which is what I'm trying to be an advocate for. Because there's people like nonlinear thinkers like me that can be easily shunned and didn't say, well, she works at the intersection, she's not, you know, serious about the one topic. And I'm like, but the one topic is a reflection of the other topics when everything is a diamond. So for me, I see that if you were introducing these beautiful words like, oh, descent condition. And like, if you start from category theory, which is so minimally constructed and only and allows you to add what you need to build your own space, right? You are, you are able to build, I would say you're able to build in accordance with your own idea, versus taking another idea and shizzling in a way and adding things artificially. The nice thing about category theory is that these limits, everything is built into the actual theory itself. I'm not starting with something and adding and then adding and then taking away and being like, well, is it analysis? Or is it like this?  So I'm such an advocate for inclusivity and welcoming so many trains of thought. And if everyone was like, oh, hey, though that looks like just a morphism of this and this looks like a morphism of this, it would actually aid research because everybody would be working together.  And I'm like, you know, if you actually were using the morphisms or the chromatic tower idea, maybe we'd be able to understand what's actually going on in somebody, you know, and we'd be able to relate to them empathically, instead of just like this like clear cut medicinal thing, well, they'll never be good again, they lost their memory. Well, we can, you know, no, I don't think it's something that can be lost. Let's just try to reconstruct the morphisms, you know. So there's like, there's so many things that I'm after with this, one is unifying all these branches that are working against each other. It's like when the mathematicians and the physicists honestly work together, right, if they can and not a bunch of egos in the way, then the beauty that results, you get these beautiful grand unified theories because it's like, we're meant to be inclusive and everybody to help each other.  And like, I don't know why we test people to be better than everybody, we could actually be working on things like this. So I think it like, it would completely revolutionize pedagogy and it would offer a future model of like what a what a Hogwarts school could look like, because then you're actually like, okay, hey, we have the imagination as a gift. It is a gift that you have this. And so through what I'm trying, what I'm doing here, it's like a very creative take at learning mathematics and things like that. So then you've actually welcomed into mathematics those underrepresented who identify as creatives. You know, I also teach at this art school, and you know, I was like, well, I don't have the best math gifts. And I'm like, but look at your visual skills. And at a certain sense, they're almost the same. You know, so I think there's a lot of beauty, like if like diamonds and things like this were taught, you know, very early or even to like a lot of people, you know, if you could integrate this into the research, then you would, I just think there's beauty in everyone working together. I'm not a very competitive person, you know. So what is it reminded me of how could you imagine a way that that art could actually inform math rather than the reverse, like not not just art, but like improv improvisation, improvisational arts upon somebody's drawing just freeform. And if you could figure out a dynamic mapping of like, oh, you're actually talking about this, this kind of logic and you don't and you're drawing it, but you're not, you're not putting it to, you know, you don't even know that you mean that, but you mean that that is actually kind of fascinating. Look how beautiful it is. Yeah, I just got accepted to what you what you just said is, you know, I'm an advocate for pedagogy in the creative settings.  And I want you to talk about adinkras, nor adinkras or James Gates air correcting codes. These students, right, have minimal mathematics skills, but they are in their learning analysis, geometry, and topology to actually make a structure that can hold flying lessons. They're actually having to teach themselves a four dimensional coordinate grid. That's not Euclidean. This is not Euclidean at all. It's differential geometry, right. It's bumpy.   So what I do is I take the Harkness method and I speak it everywhere, right, I teach it at Cal State, I teach it here and all over the place. And that method of showing of giving a concept synergetically like that, and showing, you know, just giving minimal instruction, they're learning math themselves. And so it's like, you know, I so I'm interested in other people who want to experiment like that, and you don't learn the chapter one, this is a triangle, you know, nobody learns like that because the way that you actually learn is to integrate. So you know, wider students have so much problem, so many problems solving work problems, because they've been taught these subjects separately, these subjects are not separate at all. They should be taught integrated in a holistic manner. And you're totally right, Sarah, the way to get holistic manner, like easily digestible, isn't a creative way. I showed my students like a, this awesome computer programmer solving a 40 Rubik's cube. I said, recreate that. And they actually had, oh, the four dimension does this. Oh, it's a test. Oh, it has that many sides. So much easier than a very linear lesson.  And I really think it's shunned. So there's mathematicians like Ramanujan and stuff like that who is saying, I don't know where this idea came from, it just came from this. And, you know, that can be really shunted. And so, but I think a lot of this visionary stuff, you know, comes from like a nonspecific space. And oh my God, I will nurture people's creative space. Look how many ideas come out when you do some kind of like fun icebreaker like that, right?  So I had people change the space if you change it.  What would you do with these claws, you know? And so it's like, if you can kind of shake up, you know, the real and turn it into something kind of imaginary, I think the imagination is more real. So for somebody like me, I privy the imaginary space because like this pin, I think there was an idea of a pin before the pin. So the pin is okay in my opinion, but I still pick the imaginary space trumps everything, right? You had to have the idea of the car. I want to have the idea of the thing. So I put a lot of my efforts into nurturing like students imaginary space, you know? So oh my God, yeah, infusing like having, you can teach math in a creative way. And that's why I'm so excited that there are these people like me, you know, I write books and it's like, oh, it's like, I'm going to write, I'm going to write, you know, I'm going to write a girl and a diamond. I like my opening line and this is like my new book is like she slept in an hourglass. Why not? Right? So that's what you're doing anyway. And it's like, oh wait, through glass, what is that? So I can actually give you a shape. I can introduce the shape to you creatively. And then I can actually tell you theorems about the shape, but it didn't start with the theorems. Right? I started with this rough intuitive idea that actually had a texture to it. So people go, what does it like to sleep in an hourglass? God, does Dwight get eaten by the sand? Am I the sand? Does somebody turn me over? What happens when the time runs out? You know? And so through people's own introspection, you know, I get some really solid thinkers. And that's what I'm after. I'm not after people wrote memorizing and like wrote tests, tests, of course, are difficult, right? But it's how many people remember the material after it? You know what I mean? So memorizing through rote is not what I'm into. I'm into actually building the inner space, the inner space of independent learners who can think and who can cross reference. Because again, I will, I'll just always say that your strongest thinkers or thinkers who can think along all of those intersections, right? You know, if there's a problem like the rover crashes, yeah, you need your CS people, but you also need your physics people, you need your math people. And that's why, you know, normally those research groups have that. I don't know. And then you may have that too, right? In your labs, I bet you have people from all disciplinary is working on the inference stuff, right? Species of every type of insect. Well, our approach with Active Inference Lab is provide the affordance and the opportunity to co design. And wherever people are at in learning about active inference or curious about related topics or wanting to apply it in some domain, we just sort of meet in the middle and work to use Active Inference as a unifying topic and a bridge because it and that's really that's why you're here is because we actually got to intersect with Active Inference coming to me all these other ideas. And what you said about the book reminded me of Bucky Fuller's Tetra scroll, which is about the goalie locks and the three bears. But the four of them are a tetrahedron. And so it talks about how the tetrahedron is the minimal system. And there's all these tetrahedral jokes that are embedded. And so it's like learning by doing and through stories and through characters and by drama. So all these creative dimensions that allow us to actually be generative within so called alternate. But today's alternate fashion is tomorrow's, you know, out of date. And then retro it just it's historical. So how do we respect rigor, as well as this very radically inter subjective, like you're talking about with agents with different perspectives. But then also we need local logics that aren't just totally discordant, because then when we're talking about the systems that sustain us, we need those to also persist. So how do we how do we that was my question is how do we unify the rigor and the sort of almost transcendent or at least if not universal something like that with inter subjective definitions on the mortal plane. Right. And you can actually see right in my talk, I didn't give you some kind of loose map, very particular. Right. So I think I'm actually rather gifted in that regard. Probably why I got hired so quick with all this stuff is that I'm able to introduce an idea in a creative way. But I give you absolute rigor so that you know it through and through. Right. But I give you enough rigor to know that object. But I also endow you with the creativity to be able to intersect it somewhere else. Else. So we all say the mortal plane, right.  Right. So I'm like, yes, respect your life force. Yeah. But don't ever think that you're not infinite, because there's no proof that you aren't. Right. So like there's this structure, like there is no structural causality, you know, you go somewhere, but you will never be aware of it. And I always say, when people ask me that, how do we how do we take your your lovely, you know, your lovely, you know, kind of goodies from the higher planes and like, work with them here. And I'm like, well, the higher planes are here, you're in them. Right. It's not like it's outside of you, you know, like you're born from the very universe. It's all in you. And it's sort of like, I remember now was trying to say, like, would you actually be aware when you pass what you want? Right. And that's that's strange. That's very strange. So it seems to be this like impasse limiting thing, all the more to value your imaginary space. Like I know when I'm trying to go to sleep and I'm like, I'm like, I'm going to actually be aware when I actually go to sleep. You're not. Right. When you go to sleep, you just you wake up. Right. You're never actually like aware of the moment you go to sleep. Right. And I'm like, Oh, that's very strange. What is this awareness phenomenon? Right. So yeah, there is the I and so when people call it mortal plane, I guess I call it the interface. That's one possible interface. But again, babies can breach through it. And I also think there's other ways of getting other interfaces. Yeah. So how you balance the rigor is that, well, I'm giving you some of the top mathematics like on the planet. Yeah. So that rigor you have to be for your own integrity.  So you already, you have an integrity to do honor to your subject. So to do honor to your subject, you learn it perfectly. Yeah. But you can also know that there's stories around it. You learn the rigor, but you can also how you learn the rigor could be creatively. And it doesn't have to be through a very rote memorization, right, a category as an object with, if you learn it in a very beautiful way, then I'm also going to honor that. So that's what I mean by like a creative setting. I don't ever go, oh, yeah, here's this like diamond thing. And that's it. No, it's actually very specific. But it's according to who I'm talking to, if it's like super, you know, you know, mathematicians that are already this level, then I just start with that, right? If I'm introducing it to undergrads, right, I will start with an intuitive and intuitive, kind of, I don't know, derivative of the thing and then actually introduce them into that, because I want them to retain it. I don't want them to shut down. I don't want to invoke their cortisol system, right? So I'm really an advocate for like metacognitive techniques. So once I, I am getting a really good feel about how people learn. And so the minute you know how you think somebody like me is super abstract, throw me the intersection of something and I'll get it, right? The way that you balance rigor and creativity is to know who the actual learner is. Yeah. So who they are.  So the way that you're going to actually balance the rigor and the creativity is for your own self to be harp like so that I can tune to the actual student. Oh, they're more of a creative. I'll try it like this. Oh, they're more of an engineer. Well, I'll try an engineering. I'll try a system way. Right.  And I turned it into data structures. So if I can align it with something that they already know, for me, I think that learning is about having constellations in your head. How many constellations do you have? Can you actually link everything together? And so if I'm, if there's too much like, quote, rigor without, you know, without a creative setting, I think I have, I will fail the people that don't have the constellations yet. A bunch of experts, I could do that as well. But, you know, I also, nobody knows, nobody knows the consciousness realm. Nobody knows what this actually is. And at least I'm open to saying that every theorem I come up with them. It's just a theorem. Some people think they're absolutely right. And I really don't agree with that, like at all. We have rough ideas of what it means to do this and this. But that's why I'm asking, like, what actually are the fundamental units of life? You know, you can tell me when the interface breaks, but you can't tell me about awareness. And like, if I asked a neuroscientist, like, why can I not split the consciousness? Right?  And you just have like two, it's like, like you would see like a split screen, like those old movies, I'm not talking about that, right? I'm actually talking about double experience, double experience, triple experience, but not just sensory, you know. So I think, I think the way that you balance the rigor with the creativity is to also have integrity with yourself, integrity to the subject. Anyone who actually wants to learn it wants to learn the specificities, right? But what's what, if you don't have creativity, you'll never be able to go anywhere with the rigor I give you. If you're just rigorous, you're just going to be there. But if I can instill in you creativity, then you're actually going to be able to take the rigor and link it and intersect it. And again, if, so it's all about like, why you think, you know, your phenomenological like state, I think we're here to create new things. I think being is creation. So if I can give you rigor, but it can also hold your imaginary space, then you'll be able to take that idea like nobody else but me would take diamonds and apply in the consciousness, you know what I mean? Right? So we're getting close to the end of the hour, but maybe as a closing question, I'm kind of curious about what started you in math, like how old you were. Oh, totally young. Well, I was like, hmm, you know, I was like, what is this thing? You know what I mean? I would stare at mirrors and I'm like, that's really interesting. And then I was basically asking my parents, like, why am I not falling through the floor? You know, because I, I got pretty early on that all this stuff sort of looked like it was made out of the same thing. And I'm like, why are they like, why am I not falling through the floor? Like, what is holding all this together? It's also very, very curious as to why I woke up in the same thing. I was like, why am I not a lobster tomorrow? So there was some kind of stability here that I was very perplexed by. I'm also an avid reader.  And I was like, who's that? Who is that when people are talking to themselves? Who is that? You know, I was also highly aware that when you're sleeping, who was asleep? Nobody was asleep if I was up remembering dreams and things like this. So it was very vivid, you know, I would have lots of connections with like animals and, you know, not like body swap, but it was super, super empathic. And I would visit, you know, kind of divine places and be like, oh, this is exciting. And so what I found is that, you know, avid readers already had the literary space, but I was like, what are these? I always thought that there was something else. And so what I love about math, right, is that it was a it was this beautiful language and a gift that allowed me to actually think about higher worlds, you know, in a, you know, in a structured way. And so I was like, oh, wow, this is an actual language. And I thought that there was something universal about it, right, that there's not a lot of dialects for math, you know, and it made it really simple.  Versus like I was teaching myself Mandarin and mine, you know, I still have this California kind of accent thing, right, it didn't work so well. And so the dialects can actually hurt language, but math seemed to be this like universal way.  Again, math is not, it's like my version of math you can see is very philosophical, right. So it's like, also my favorite writer was like, you know, Alice in Wonderland. And I'm like, oh, through the looking glass, like that's cool. And at the end, he's doing all the syllogisms.  And I've just messed with causality, you know, paradox and things like that. So super avid reader getting my hands on big philosophy, very young and also staying open to like, to what is because I could tell that time wasn't linear. And I was like, I need a subject, but I can actually figure all this out, like time, it's not linear, like how can I do that? And then I, oh, topology and spectral spaces and stuff like, Oh, it looks like that. I would say it made my synesthesia like comfortable, you know what I mean? Yeah. Yeah. Cool. Well, I'm, I don't think I have anything else and we're getting curds in. So what do you think? I would be remiss not to ask how you thought active inference plays into this or what you thought about active inference. Yeah, with how the like, that's what you're saying, right? Have the interfaces actually intersect? Or just what what what comes to mind when you think about active inference and cybernetics and control theory and where some of this like agent in the niche goes from there's the analytical, there's the philosophical and then not getting into the theology of what's real, but just like, there's the agents, the model about how the agent is operating in the niche. So where does, what is active inference? Well, I guess from like, you know, the way that you've defined it. So again, I'm interested into, okay, so here's like, I guess my knowledge of it, right?  And so in my opinion, the way that things are interacting is through how many morphisms they actually generate. So it's actually categorical. So I don't do this sort of like analytical thing, like philosophy philosophical, because again, I am from this higher space. So if you just, if you look at active inference, maybe through category theory, now what's really happening, it's like, oh, oh, interesting, at what, at what instant does my interaction with this like opposite agent or whatever throw me into a reflective subcategory where I'm actually just interacting with myself. So I'm really interested in where the interface theory can actually meet topological localization, like I was talking about. So I think what, see, what I like about this is that it sounds like you're doing the same thing that I'm doing with which different language. So topological localization, right, I'm interested in how do you find the inverses and relate more and make my space stronger, you know, bypassing through these reflective subcategories. And so if it's like, oh, if the active inference is how these agents are swapping and actually interacting, because that's very curious. It's very curious how what I'm taking your information into like my data structure. And I don't know like what receiving looks like on that end. So I would just say that categorically, it sounds like we're doing the same thing. And then it gets pretty meta for me when I'm sort of like, I take your info, and then it becomes part of my system. And it's part of my own reflective subcategory. And I like, I don't know, like keep moving on or something. But you know, if it's about if it's about how interfaces work, I still don't know how you got the original interface. So I don't know if that means anything, you know, some of these ideas, we talked a little bit about in active inference 17, the dot zero dot one and dot two, because that was the paper that we focused on for those weeks by fields in Glazebrook. But I think what you have pointed to today really also gives us a lot of things to go over again and think about. So maybe we can stay in touch. And if you ever want to join for another conversation or participate in any other way, it'd be awesome because it's really thought provoking. And I hope our lab enjoys it. Yeah, so fun. Thank you so much. I had a great time. Thank you. Bye.