Science & Technology Q&A for Kids (and others) [Part 149]

Foreign. Welcome to another episode of Science and Technology. Q and A for kids and others. I gather we just added Facebook as a streaming platform. So hello to anybody watching this on Facebook. All right, let me see. Well, we have all kinds of questions that, that came in. Let's start off with one from Nuno here. Can machine learning find a pattern in all animal languages? It's an interesting question. There are a number of groups that are trying to do this and have been trying for a few years. I have thought about this for, oh, I don't know, 15, 20 years or so that, gosh, we should be in a position to finally figure out what are the animals talking about. I remember there was a company, oh, must be 20 years ago now, that was marketing kind of a dog interpretation app where your dog and it was segmented for different breeds of dog and, you know, your dog would go woof in some way and it would say, that means I'm hungry or something like this. There's that. There was another company that I have a bad feeling didn't make it. That was a tail wagging company which had got accelerometers of the kind that are used in fitness watches and so on, and had been set them up so you could put them on the tail of a dog. And it was supposed to interpret kind of like in sign language or something. What is that tail wagging? What does the tail wagging actually mean? So there have been a variety of initiatives like that. There's also now two different initiatives trying to deal with whales and whale songs. So first question is, What creatures make kind of sounds that might be interpreted as languages? Then we have to talk about what it means to have a language. Then we have to talk about what it means to interpret the language. And then we have to talk about how would you actually do this with machine learning? Well, so first of all, different kinds of, of species, whales seem to have particularly rich vocalizations. Probably helps that sound travels really well in the deep ocean. So whales on opposite sides of the Pacific with some time delay, can be communicating with each other. And they can have, in a season or something, they'll have these pretty complicated sounding whale songs that they communicate. And it seems like other whales respond and do things like this. You know, knowing what these mean and being able to correlate what is being said by the whales to what the whales are doing. I don't think that's been easy to do at this point. Well, you take other kinds of critters a little bit less sophisticated in terms of the vocalizations you know, cats and dogs are, you know, making noises, but they're not as diverse, I think, as whale noises. Then there are birds, songbirds and so on. I mean, there are parrots, obviously that can make human ish sounding noises. I mean, they don't make quite the same. If you looked at the kind of the trace of the sound from a parrot, it wouldn't look the same, even if it sounds like words that humans say, because we humans have a single set of vocal cords and so on. Birds have syrinxes, which are a more elaborate two stage kind of thing that generates sound, but it generates sounds in a different way than our vocal cords generate sound. So even though it might sound a bit like hello there or Pretty Polly or whatever the bird says, if you looked at it sort of in terms of the sound, it does have a somewhat different structure than the human analog of that. So there are those kinds of things and occasionally there have been people who have sort of the champion parrot that supposedly has lots of elaborate things to say. Well, then there are things like primates, apes, so on that are less good on the vocalization side, but seem to do pretty well on, you know, typing on a screen and so on. I know there are things for dogs where in principle you can have a dog kind of with its paw type things on lots of big buttons. I gather from people who I've encouraged to get these kinds of tools for their dogs that many dogs don't seem to manage it and don't seem terribly interested in it. But there are certainly videos of dogs that seem to be able to tap out elaborate messages. But you know, it's a complicated subject because one thing is you've got the ape, you've got the dog and you present it with some means of communication that is very human, as in it's like we've got human words, human terms, human goals and so on, which we're putting on the screen for the dog or monkey or whatever it is to press. And then it's not surprising that you can sort of decode the language because the language was already set up to be a human language. The more interesting question is when it's kind of the creatures in the wild talking to each other, what are they saying to each other? And that's a very challenging thing to figure out because the kinds of things whales might talk about, I don't think we have any idea, we don't know what's sort of in common in the thought, in the things that are being thought about between these different kinds of creatures. Now, one thing that is true, and at least among mammals, emotional states seem to be different. Mammals have them, and Charles Darwin was famous for pointing out that if you look at the kind of expressions on the faces of dogs and things, they're snarling, they're angry, the kind of. The lips curl in the same kinds of ways that they do when humans have the same types of emotions. So his point was that the kind of emotional states seem to be things which are shared between humans and at least mammals. And possibly the reason for that is that those states are associated with particular chemical neurotransmitters that are getting produced in the brain and that are leading to that overall state or are associated with that overall state, and that those things are kind of in common across all those different kinds of animals. So it's possible that that's a place where we can readily kind of have a common type of common thinking with various kinds of animals. But if you are having a discussion about philosophy or this discussion that we're having right now, I'm not sure whether this discussion would really work with a cat. Even if we could sort of decode cat language, it's not clear that the types of things that a cat would be thinking about are at all the types of things that we humans think about, in fact. So that's one of the challenges. Now, what you might think and what I've sort of imagined for a long time, is that if you correlate what the animal does with what the animal says, that somehow or also what the animal sees with what the animal says, then that should be a good way to be able to decode what the animal is saying when it's saying things. I mean, it's a little bit like the way we all acquire language as young children. We kind of see things and we hear language associated with the things we see, and sort of doing sort of the inverse of that. We should, one would think, be able to decode what kind of animal language is like. People have not done as much of that as I think one could do. And, you know, I think that the part of the challenge, again, is what does it decode into? Does it decode into something that is like human language and that we can understand, or does it decode into the dog describing the details of what it's smelling in a way that we humans just don't have? Because our sense of smell is not as good as a dog, and we don't tend to think about the world as much that way. We tend to think about it more visually and so on, that we'd end up with kinds of types of descriptions that we really don't have a good translation for. Now there are efforts where people have said we can take kind of the meanings of words in human languages and we can kind of make this space of meanings. Like there are words that are close in meaning and we put them close together. Words far away meaning, we put them far apart. That's often called an embedding space for a language, and that's in modern machine learning and with things like LLMs and ChatGPT and so on. That's a kind of common construct to make in Wolfram language that's called feature space plot, to create kind of this kind of distribution of features like words in some space. And people have made the claim that there's some sort of correlation between the feature space plot that you get from words in human languages and a feature space plot that you get from, for example, snippets of whalesong. I find these claims deeply. I'm very skeptical about these. I don't think that seeing that there's a correspondence between where different things are placed doesn't seem promising to me, because typically, that's much more about the algorithm that's used to figure out how the placement works than it is about the real content. By the way, the way the placement basically is determined is by saying, well, two words are near, close together if the context in which those words occur are similar. So, for example, cat and dog might be close together because people might say, I took my blank out for a walk, and that might be either cat or dog. And the fact that those words are sort of interchangeable in those similar sentences means that in this way of sort of assigning. Assigning things to feature space, that those words get assigned nearby because they typically occur in similar context. And you can certainly imagine doing the same type of thing for whale songs or something like this, but being able to sort of the fact that the overall shape is similar, that seems really tough to conclude because it's something where the shape tends to be much more associated with the method of extracting things from contexts and laying things out and so on than it does the real underlying kind of meaning. It's also worth saying that the seemingly unique aspect of human language is the way that we get to put words together to make sort of more complicated concepts and to make sentences and so on. And that just doesn't seem to be a thing, at least in the kind of proto decoding that we've been able to do of animal languages. It seems like those are much more just, you know, like one might talk to a dog, you know, sit, fetch. Those kinds of things. Just boom, one word. Boom, one word. Et cetera, et cetera, et cetera. Rather than the kind of thing that we humans are able to do, where we build up sort of more complicated concepts from. From these primitives of words and so on. Let's see, Looking at this, truth is commenting, the cat says meow. The translation, they say, is, what a glorious day to be alive. I think I will frolic in the fields and catch myself some mice for my tea. Then I'll sit down and ponder the nature of the universe. And that's all packaged in the 1:1 vocalization. Meow. You know, it reminds one of kind of. In human settings, there are cases where when everything has been primed correctly, just that one word communicates a whole bundle of meaning, but you have to prime it correctly. Let's see. Okay, we have a question here from double. The question is, what is time? Okay, I think I finally more or less understand the answer to this famous last words. The thing that I would say is time is associated with the progressive evolution of the universe. What does that mean? If we think about a computer program, a program has some state. The computer has some arrangement of bits in memory. And all this kind of thing, as the program runs, it is doing operations that change its state, that progressively change its state as the computation runs. The state of the computer has progressively changed. Doesn't have to be a computer. We can be applying any kind of rule. My favorite tend to be these rules that are applied to, let's say, arrays of black and white cells. And the rule just says, if the black and white cells have this configuration, change them like this. If they have that configuration, change them like that. And you get these kind of patterns of behavior where you start from, let's say, one black cell or something, and you get this whole elaborate pattern. This particularly happens in the cellular automata that I've studied for the last 40 something years. So this idea that you apply rules, you keep applying these rules, they might be the rules inside an actual practical computer. They might just be conceptual computational rules. You get applying these rules over and over again. You change the thing you have. Well, what I think is happening in our physical universe is it's doing the same exact thing that the state of our physical universe can be thought of. It has some particular structure. And time is the progressive rewriting of that state, the progressive changing of that state as a result of Computations being done on that state. So in our model of physics from our physics project in the last four and a half years or so, the story is that we can think of the state of the universe as being this giant hypergraph. Basically, there are just these atoms of space represent points, and these points are related to each other. And this builds up this giant kind of network. You can think of it as like a friend network of the atoms of space. And this giant sort of friend network, at every moment, every piece of that network that has some particular form is getting rewritten into a piece that has some other form. And this keeps on happening all over this network. But the important thing is that that progression is happening everywhere. That it has this form, it is rewritten to this other form that just keeps on happening. And so I think that the progress of time is really the progress of that computation. The progress of time is sort of the progressive application of rules, the progressive doing of a computation that is the state of the universe. Well, there are a number of things to say about this. One thing is, does sort of that progression of time, is it the case that something is sort of achieved by that progression of time? Or is it just. Well, the universe does all these things, but if we were looking outside the universe, we could just say, I know where it's going to end up. There's nothing really happening in that progression of time. It's just the universe is doing all these complicated things, but it's sort of, in a sense, it's wasting its time. Well, there's this idea of computational irreducibility, which I've been sort of talking about for the last 40 years or so. That is this phenomenon where even though you know the rules by which something operates, to know sort of what will happen as a result of applying those rules is something. It's irreducible. You have to actually follow those rules, apply them each step to find out what's going to come out. There's no way to kind of be smarter than the system and kind of jump ahead and see what the result is going to be. So one of the important features of a lot of the progression of time in the universe is, is that there's a lot of computational irreducibility in it. So in a sense, there's a lot of sort of inexorability of the progress of time it is progressing. And we don't get to jump ahead and sort of say what's going to happen before it actually happens. Well, there are different kind of versions of time, I think that the core idea of time is time is this progress of computation that represents the state of the universe. And that progress of computation affects us as it affects everything in the universe. And our kind of. We are perceiving the universe as progressing through time because we ourselves are also progressing through time in the same way. The insides of our brains and minds and so on are similarly updating. So we are sort of having different thoughts at different moments corresponding to our which we can think of as being perceptions of what's happening in the universe at different moments. But if there was nothing going on inside us, all kinds of stuff could be going on in the outside universe, but it wouldn't have any effect on us because there's nothing going on inside us. So there's this kind of psychological time where. Which is the progression of things happening in us. And that psychological time can run even when there's nothing happening in the outside universe. We could have nothing that we're sensing. We could be completely cut off from our senses. Yet we could have had a sense of the progression of time just as a result of the computational processes that are going on in our brains, Just that sequence of thoughts that are happening. And correspondingly, if our thoughts are stopped, you know, maybe we're asleep, maybe we're anesthetized. Who knows what? If our thoughts are stopped in that way, then our internal sense of time does not run. Even though time has been running for the outside universe, because there's still computation going on in the outside universe. There isn't computation going on in our minds. Oh, there are all sorts of things. I mean, kind of the thing that happens sort of if the organism dies, time effectively stops for it. The computations that were going on in its mind aren't going on anymore. So time has basically stopped. And the same thing, by the way, happens in center of. The most common kind of black hole has a spacetime singularity at which time stops. And time stops because the progression of computation that normally happens in the universe at that kind of space time singularity, it's kind of defined, at least in our models of physics, by the fact that there's just no more computation that can be done. The computation has ended, the computation has terminated. So the time has died. In effect, the universe has died there, and there's no more time to progress. And so that's a place where time will just stop because there's no more computation that can happen in the universe. Now you can ask things like, well, could time ever run backwards? Well, the thing is that in the Idea that time is this progression, progressive kind of running of. You know, there are rules for what the universe does, and those rules just keep on being applied. You might ask, by the way, about the universe, what makes those rules be applied? Is there some giant sort of motivator outside the universe that's kind of driving the universe, that's saying, okay, apply this rule, apply it again, apply it again, and so on. That's not how it works. How it works, I think, is something a little bit more philosophically and abstractly complicated. But this is the whole story of this ruliad object that came up with, oh, what is it? Two, three years ago. Now, let's see if I can explain something about that. The thing is that in the end, I'm describing a particular rule being applied to make the universe progress. But the ruliad is a thing where all conceivable computational rules are being applied. So it's something where each different possible rule is being applied, and you might end up with the same states from different rules and so on. It's a big entangled mess of everything that can possibly happen from any possible computational rule. Now, one thing that's interesting about the ruliad is it's unique. Once you have this idea that it's sort of talking about computation, then this is just saying, well, let's put an object together that represents all possible computations running. And so that sort of there's only one of it. So there's no discussion about, oh, maybe I'll have a different ruliad, though. That's not possible. There is one abstractly defined ruliad, and that's the only one there can be. Now, the question of how do we perceive time and other things in the ruliad? The thing is that we are sort of observers within this ruliad, and like everything else, we have to be part of the ruliad. So what ends up happening as we think about how do we perceive the universe? It's like, how does this piece of the ruliad that represents us, how is it affected by how does it perceive the rest of the ruliad? And the first thing to understand is that for observers like us, we only get to sense certain rather coarse aspects of the ruliad. There's a lot of the ruliad that we never get to see. Just like even in physical space, we're sitting here on this particular planet, in this particular place, there's a lot of physical space that we don't get to directly experience or see. So that's One thing in the ruliad and there are other aspects of, you know, we don't directly get to see the molecules bouncing around in the air. We are just seeing some aggregated version of that. And there's a bunch of different things that are the result of attributes of us as observers that determine sort of what in the ruliad we see. Well, the ruliad is this unique object that is sort of the entangled limit of all possible computations. And in a sense, the ruliad, the whole ruliad does not have a notion of time. The whole ruliad is just all of these computations. And the ruliad contains every step in every one of these computations. It contains the whole of all these computations. But the thing is that we are, in a sense, we are observers who have very limited pieces of the ruliad in our minds. Our minds are quite finite. The ruliad is very, very huge. Our minds are sampling just some tiny piece of it. And one feature of the way that our minds sample ruliad is it's inevitable that we sort of are moving through sampling different parts, different times, different pieces of the ruliad that we interpret as different times. In a sense, it's, we are progressing through these computations in ruliad, but we are doing certain computation as we do our perception, as we do our thinking. And we only get to sort of sample the ruliad at the rate that that computation happens. And so it is not the case. While the ruliad in a sense has the whole of all time encoded in it, we only get to sample it kind of one step at a time, so to speak. And so we are still perceiving progression through time. Now, one thing that happens in the ruliad and our physics project in general, and with this whole thing about all these computations being applied, is there isn't actually really just one thread of time. Time. One usually says, oh, it's a one dimensional thing, there's just a timeline, it's just a progression of this time to the next time to the next time and so on. But actually our physics project says that time is multithreaded, that there are actually many possible paths of time and they branch, so that when they branch, it means that we're in this state now, but there are two different states we could be in later on. And really both of those states are happening, then we can also have merges where two different states end up winding up evolving to the same state, so to speak. So we have this complicated Branching, merging, kind of set of threads of time going on. So now the question is, well, that's really complicated. How come we don't perceive that most of the time? Well, the answer is, I think that it is actually a very core aspect of the way that we are, that we believe we have a single thread of experience. We believe that we are persistent in time. We believe that our minds perhaps believe that the thing they're thinking at this next moment, it's the same mind doing the thinking. Even though the sort of atoms of space from which our brain is made will be different at every successive moment in time, it is still we have the perception that we have this consistent existence through time and that our thoughts now are directly connected to our thoughts in the past, that our even physical instantiation of our brain and so on is what it was in the past. Now, of course, we know perfectly well that, you know, there's sort of turnover of the actual material. The atoms are different. When we look down at the atoms of space, they're completely different at every moment in time. But yet we perceive this continuity in time. That's a very core aspect of, kind of the nature of consciousness for us is that we perceive this thread of time. But more importantly, we perceive a single thread of time we believe most of the time. And it sort of drives us crazy. If we start thinking we have multiple threads of existence going on in our minds, we really keep consistent. We have this one thread of existence, and that is just our way of perceiving the ruliad, our way of perceiving what's happening in some kind of reality, a sense. If we were sort of the infinitely powerful minds, we could potentially perceive all these different threads, branching and merging threads of time and so on. But for us, with our comparatively limited ability to perceive things, we perceive a single thread of time. And it's an interesting question whether the AIs we build have to work the same way they do it when they're communicating with humans. They, they are similarly, you know, the ChatGPT is having the one conversation, so to speak. It's not kind of having the multi threaded kind of possible conversation. You can easily make a large language model which is like picking at every step the next word to produce. It's very easy to give it a multi threaded conversation, so to speak, where it's picking many different possible words and following all those different paths. But that's not something that we humans find really understandable at all. It's not the way we think about things, the way that we Think is in this very sort of single thread of experience. Our languages that we're used to writing out that linear chain of text. Not this whole network of, oh, it's this word, and then that one could come next, and then that one, and so on. It's a core aspect of our kind of way of being. So that's how we kind of end up with a sort of single thread of time. It's of our own making. By the way, the fact that there are these multiple threads is what leads to quantum mechanics and the phenomena of quantum mechanics. They're kind of the little pieces of slight deviation from this idea that there's a single thread of time. And so we get phenomena like quantum superposition and so on. From sort of multiple threads of time and things like that. So you can ask questions like about the nature of time. There are different kinds of arrows of time that we perceive in the world. So one arrow of time is that we remember the past. We can only predict the future. We don't know the future most of the time. And that is a consequence of the fact that we're doing this irreducible computation. And what we have is we can have sort of stored the results of the previous computation. But to know what will happen, we have to do that computation. Now, in certain cases, that computation is simple enough. There are enough. They're kind of pockets of computational reducibility where it's simple enough to do the computation that we can readily predict what will happen. You know, we drop something, we can readily predict that thing is going to fall. And if we are bouncing something back and forth, we might be able to readily predict the next bounce is going to look like the previous bounce. And it's just going to go back and forth. That's a case where we perhaps know, in the case of dropping something, we actually may know more about the future than we do about the past. In that case, we don't know where it came from. We just see it falling through the air type thing. But in general, there's this sort of time where we are at this moment in time. There's this sort of irreducible computation to get to that moment. And that's kind of how it progresses. Now, there are other aspects of time. Like, for example, one type of sort of time that we perceive as thermodynamic time. What does that mean? If we have something where we take a thing and we smash the object. Or we take some water and we let it fall and it makes a big splash. But let's say we're smashing something. We are used to the idea that we can take a thing which is nice all together, and then we drop it on the floor and it smashes into lots of pieces. We don't tend to see it spontaneously reconstituting itself from all those little pieces smashing. Sort of go from all those pieces back to, you know, the light bulb before it broke type thing or something like that. The. So it's a. And why is that happening and how is that related to these other forms of time? Well, if you looked down at the absolutely microscopic level of particles and electrons and protons and things like this, if you looked at things at that microscopic level, any process that can go forwards in time can also go backwards. In other words, you can't really tell when these two particles interacted in this way. They could as well have collided. The opposite collision could have happened. There is microscopic reversibility in the universe. There's sort of footnote. There's a slight difference to do with this phenomenon called time reversal violation, where there is a slight difference in the detailed things that happen with particles going forwards in time and backwards in time. But there's always a unique way, given that you're down at the level of particles, to say, that's what happens going backwards. That's what happens going forwards. But the thing is that what we perceive is that there are lots of kinds of. Once the glass has shattered, so to speak, the. It doesn't unshatter, I think. Why is that? Well, what's happening is that this is, again, a phenomenon of computational irreducibility, that what's happening is the sort of detailed motions of all these particles are sort of. They correspond to a computation. And the thing is that when the thing has shattered, there is still all the information needed to make the original glass. It's all still there in all those little shattered pieces. It's just that as computational observers, we don't tend to recognize, oh, all those shattered pieces. Of course, they all go together to make this glass that we started from. From the point of view of observers like us, it's just they're a bunch of shattered pieces. And we say, oh, it looks kind of random, but we can't tell what's going on. It is because we are kind of limited as computational observers that we think that if we were more powerful as computational observers, we just immediately say, what do you mean? That the glass is all randomized? It's not. I can see that those pieces fit together in just exactly this way. So the thing that's happening is we are preparing a state. The glass, we put it together, and we're preparing a state that we can perceive with our sort of limited computational abilities. Has some regularity. And computational irreducibility is causing that initial regularity to be degraded into something which is sort of irreducibly difficult for us to invert that we end up saying, oh, it just looks random because our computational abilities can't decode it. And so we're kind of. That's the sense in which we are. We are kind of perceiving because we are starting with states. We are preparing things that are sort of computationally simple for us to describe. And the irreducible evolution of the universe Is causing them to degrade into these irreducible things. That's why we kind of perceive that arrow of time. We can tell that something is sort of irreducibly happening in that case. And there are other kinds of things we can talk about. We can talk about things like the expansion of the universe, which is going in a definite direction. And for example, another thing we can think about is when we do something like, oh, I don't know, we get some light. We have a flashlight. And the light from the flashlight is just going out into wherever it goes to. It dissipates. There is light coming from the flashlight, but there isn't light coming back from the universe. And sort of being. It's like when we radiate things from our flashlight or from our radio antenna or whatever else, it goes out to the universe. And it doesn't tend to be the case that that stuff is coming back. Now, of course, there is radio emission from the universe. Even from the beginning of the universe. There's kind of a radio emission from every side. But so it's sort of a complicated issue that things tend to radiate the Earth, you know, we radiate heat into space and so on. We radiate more heat out Than we get heat back from from what is there in the universe. And there's sort of a question of how in the universe that is expanding and how does that kind of arrow of time relate to these other arrows of time. That's sort of a slightly separate thing. Because the fact that the universe happens to be expanding at this point in its history Is not, I think, a the fact that the universe is kind of irreducibly doing things to progress to later states of the universe. That's kind of the progression of time. But to know that sort of the universe. If we were trying to tell Are we kind of perceiving the forward direction in time? Knowing whether the universe is expanding or contracting doesn't really tell you that, because the universe can expand and then contract. When the universe contracts, it is not as if it is running itself backwards. It is merely that it is doing something like what happens in the formation of a black hole, where things are just sort of crushed together. Rather than that sort of everything that happened that came from one thing is reversing time and going back the other way. That's a long answer that. Sorry. You know what, Let me see. There's a question here from Tree. If the rules repeat a state, does that mean time has gone backwards? It means time isn't really moving. It isn't really. In other words, the fact that we perceive a progression of time is a consequence of the fact that we are seeing kind of irreducible computational processes going on that we can't readily predict. We can see what happened before, but we can't say what's going to happen in the future that is sort of creating this progression in time. If what's happening is we just have a pendulum clock going back and forth, back and forth, that doesn't create the same kind of progression of time. That really is something where time is sort of looping because nothing's really happening. If that was the only thing in the universe is the thing going back and forth, back and forth, we wouldn't really think of there being a progression of time. Progression of time is associated with this irreducible computation going on. So let's see. Okay, Okay. A well named handle, the rulad is asking. So less computation allowed for the progression of time, Time slows down. Yeah. Okay, so let me mention a few things. So one of the phenomena of relativity theory is phenomenon of time dilation. That an object that is going faster seems to have time effectively going slower. So for example, if you're going in a spacecraft at 99.9% of the speed of light, your sort of the perceived passage of time for you is much slower than the perceived passage of time on Earth. So that means that when you, for example, if you've got the so called twin paradox is you've got two twins, two identical twins, one of them stays on Earth, one of them goes off in a starship, goes and accelerates to a large fraction of the speed of light, eventually comes back again. The twin that went on that starship trip might have only aged a few years, while the twin on Earth is a thousand years later. Why is there that asymmetry there's that asymmetry because the twin who went off in the spaceship went and accelerated to the speed of light and then decelerated to come back again. And that acceleration and deceleration sort of makes the things not really equivalent. But the point is that you can sort of figure out how long the. How much the passage of time has been for the twin who's going close to the speed of light. And time is going a lot more slowly for that twin than for the one who stays on Earth. Now, this is a. Why does that happen? Well, there's sort of a mathematical derivation from the structure of relativity theory, but in our model of physics, there's a very mechanical kind of derivation of this. You think of the progression of time as being this progressive computation going on. And that progressive computation is kind of measuring. You could think about that computation after you've done a whole bunch of computation. That's one tick of your clock, another whole bunch of computation, another tick of your clock, and so on. Well, now imagine that you're moving. Well, when you move, what that means is you have to recreate yourself at a different place in space. And that process of recreating yourself takes computational effort. So if you're going to spend computational effort to get to the next tick of your clock, you can't apply as much computational effort to that as if you were staying still. If you're staying still, all the computation you have is you can apply to. So figuring out the next state of the clock. If you're moving, some part of the computation that you have is applied to recreating yourself at a different place in space. And that means that the progression of time, the computation, you have to allocate to the progressive ticks of the clock. There's less computation to allocate. And so you go. So time effectively goes more slowly. Less happens because some of the happening is associated with motion. So that's kind of a mechanical derivation of the concept of time dilation. Now, there's another phenomenon which is in a strong gravitational field or in a gravitational field, the. Oh, gosh, let me get this the right way around. Yes, in a gravitational field. Boy, the fact that I have to think this through is a bit embarrassing, but I think in a gravitational field, time. Runs more. Which way around is this? Oh, my golly gosh. Yeah, I think time runs more quickly in a. In a high gravitational field. I think that's the right way around. Let me think about this. It means that, for example, let me think. In. Gosh, it takes me a Little while to think this through. And it has, again, a very mechanical explanation in our physics project. Because gravitational field. And gravitational field is associated with the presence of mass, energy, momentum, and so on. And the presence of energy momentum is associated with a larger number of. Of a larger amount of activity, a larger number of computations getting done. Yeah, so that means. That's right. So time runs more quickly in that case. Because the very structure of space time has more computations going on. And if you're kind of existing in that, you get to do more computations. So that means that, for example, if you compare, the gravitational field at the surface of the Earth is higher than the gravitational field at the center of the Earth. Because at the center of the Earth, kind of gravity is pulling in all directions. So there's effectively no perceived gravity there. And so what that means is that at the surface of the Earth, we have aged a bit more than we would have done at the center of the Earth. I think the effect over the lifetime of the Earth is like, oh, maybe a few seconds at most. It's a very small effect. But that means that we're just by virtue of being in this gravitational field, we are aging faster. And that's a whole phenomenon you can apply. It leads to a thing called gravitational redshift and a bunch of other different kinds of phenomena. So, okay, let's see. Lots of questions about very different kinds of things here. All right, let's do another one about time here from Young says communication speed between Mars and Earth is not instantaneous. So wouldn't both the center and the receiver never have real time data of what was happening on both planets? We on Earth may only assume that something terrible happened on Mars after it's too late. Is that right? Yes, that is right. And that's sort of an essential feature of the way that relativity, the sort of structure of relativity and the way we think space time works. So let's explain how that's set up. The fundamental point is no signal can travel faster than the speed of light. That's a thing believed to be true about the universe. And the speed of light is about one foot in a billionth of a second. So in one second, light has gone a billion feet. That's, my gosh, 186,000 miles. So it's gone a long way. So light gets to the Moon in about a second. Light gets to Mars. Depending on where Earth and Mars are in their relative orbits, it takes 20 minutes, 10 minutes, 20 minutes, whatever, to. To get to Mars from Earth. Why is the Speed of light. Why is there that limit? Well, in our models of physics, we actually can see why there is that limit. Essentially what's happening is there are these little. The space is represented by this big network of all these different points that represent sort of atoms of space. And to be able to send a signal from one place in this graph, one place in space, to another, there's little rewrites that are happening, but they're always local. They always affect just things that are connected to each other. And that means that to get to some other place, you have to have a rewrite here. And then for that rewrite to affect another rewrite, you have to have a rewrite that overlaps with that one. And it keeps going, it keeps going, it keeps going until eventually the that chain of rewrites goes from one place to another place. So in the way this is set up in traditional sort of 20th century physics, there's this notion of a light cone. So you're plotting kind of space in the horizontal axis and time going down, let's say. And then you have this event, this flash of light that happens at a particular place in space at a particular time. And, and the effect of that light it is, you know, that there's a shell of light effectively going out in space over time. But if you look at that plotted as a function of time, you'll see this cone that starts from this event, this flash here, and this cone in space time. This is the light cone of what can be affected by that event that just happened there. Okay, so what happens in Earth versus Mars? Okay, an event happens on Mars, it takes, let's say, 20 minutes, depending on exactly where Mars is in its orbit. It takes 20 minutes from the light from that event to reach Earth. Okay, we on earth respond, takes 20 minutes from the light from Earth, the radio signal, light signal, whatever from Earth to reach Mars. Now the question is, what does it mean to say that? If we say, well, did this happen at the same time on Earth and Mars? How do we set up our time? How are we determining time? When is noon on Mars versus noon on Earth? Because we'll never know noon on Mars. It'll take 20 minutes from anything that happened at noon on Mars to reach Earth. It'll take 20 minutes from anything to happen at noon on Earth to reach Mars. So we end up with something where to say, we can set it up so that because we know we've exchanged lots of signals, we know how to line up noon on Earth with noon on Mars, but we'll Never know that even though we can say this is what we count as simultaneity between Earth and Mars, we'll never be able to. The fact that things are happening at the same time, we'll never know that it'll always be, well, exchanging signals. And it will be 20 minutes delayed. In what happens now. The notion that we set up a time coordinate system, so to speak, so that noon on Earth and noon on Mars are lined up that way. So it's kind of a 20 minutes one way, 20 minutes the other way, and so on. But we've kind of lined it up in some sense. That's a particular choice of reference frame, as it's called in relativity that. A particular choice of what one calls a simultaneity surface. So imagine this light cone. You've got this cone in space time. You could say, well, what counts? As we try and label what counts as a constant time across space, all we know is that things happen at this moment and then they'll happen at a subsequent moment and so on. We know the sequence in time of events happening, but we don't know the relative times at different places in space. So we can end up with this whole set of layers of what are called space, like hypersurfaces that are the kind of the simultaneity surfaces, the things which we say, we're going to treat them as being the same time in different places in space. But it's a kind of arbitrary choice. Now, in actuality, that really bites one. I mean, when you try to. If you're interested in interplanetary spacecraft, for example, it bites one. So, for example, in Wolfram language, we have all sorts of elaborate astronomical computation capabilities. And it matters how the time coordinate system is set up. So if we say, where's the Moon at this time? Well, it depends on whether you mean that time, the time when we would know on Earth from a light signal for the Moon that the Moon was in that place. Or do you mean something where we've kind of back calculated where the Moon would have been so that at that subsequent moment, a second later when the light has reached the Earth, we find out where the Moon is. And that leads to these different coordinate systems for describing time in the solar system. Somewhat complicated. And they're always dealing with the fact that we can set up these simultaneity surfaces by just arbitrarily saying noon on Mars, noon on Earth. It's going to line up in this way. But in a sense, we'll never have sort of a correlation between what happens in those different places until light has had time to get from one to the other. It's a bit of a complicated story. And people say, well, gosh, a supernova might have happened in our galaxy, A supernova might have happened within 20 light years of us, but we don't know it yet. Now it's even not clear what it means to say that, because it depends on the time coordinate system, the simultaneity surface you're using to say it's already happened, but we didn't know it. That assumes a particular way of assigning these simultaneity surfaces, a particular assumption about how we define time, which is not. It's not the only one. And there are other ones where we wouldn't really say it had happened until the light from it reaches us, and so on. It's a complicated subject. Let's see. Well, let's see. DES is asking, is there an objective time in our physics project? No, there's no more an objective time. You still have to define. You have these different rewrites happening at the smallest scale. But, gosh, it's a complicated subject. There is a fundamental rate of the progression of time in the universe. There is a fundamental processing speed of the universe that it is, in a sense, a fundamental constant. It is a similar fundamental constant to the speed of light. The speed of light tells you, for a given time that's passed, what is the extent of space that can have been covered? There is a similar constant in our models. The two others, actually, one is the maximum entanglement speed, but the other is this maximum rate of processing in the universe, which determines kind of the underlying rate at which these computational operations happen. But when it comes to the progression of time as perceived by an observer, that has kind of the same set of stories as in traditional relativity theory. All right, I'm going to turn to a quite different question here from hh. How do I prove that matter is made of atoms? Interesting question. Let's see what we can do. Well, back in antiquity, philosophers were arguing about this forever. You know, is matter discrete? Is the world made of lots of identical, potentially atoms all put together in different ways, or is it a continuous thing? It took an unbelievably long time for people to figure it out. The first kind of really, I would say serious mechanical views of atom, like things had to do with thinking about gases. What is air? What is going on in a gas? A chap called Robert Boyle in the 1600s was talking about the spring of the air, by which he meant if you put air in a box and you try and squash it. What is going on? Why is there something that is. And people began to think what might be going on is that the air is made of some kind of little molecules or something, some kind of little discrete things that keep on bouncing around, hitting the walls of the box, hitting each other and so on. And that's what leads to the spring of the air, the kind of the pressure of air and so on, but nobody knew. And then by the 1800s, there started to be theories about. Actually about thermodynamics, about heat. People didn't know what heat was either. People assumed that heat was a fluid that just sort of seeps into things and that's why they're hot. But then there started to be the idea that maybe heat was. If you believed in molecules, if you believed that matter was discrete, that heat was a feature of those individual molecules running around in different ways. So if you were clever enough, in a sense, you would have known there were molecules by the phenomenon of heat. And some people, particularly Ludwig BOLTZMANN in the mid-1800s, kind of really pushed that idea, but couldn't really prove it. You could work out the mathematics of what that would mean about the behavior of heat and temperature and so on, but there were other ways to get those results. And it wasn't clear that that was a reason to have molecules. The other thing that happened was that people started realizing that if you did chemical reactions, you would get sulfuric acid plus whatever you would end up getting. The multiples of amounts of stuff that you got tended to be simple numbers, like 2 or 1 or 3 or something like this. And that kind of gave one the sense that there was something discrete going on, that there was something that was kind of not. There was something where you were just counting things. It wasn't like that these reactions were like exchanging, oh, we've got two hydrogen atoms or something like this. That was a sort of circumstantial evidence for the existence of atoms. The thing that really, in the end, was the thing that convinced people about the existence of atoms and molecules was a phenomenon called Brownian motion. Brownian motion had actually been observed in the 1830s and by a chap called Robert, I think, Brown. And what he saw was he saw these little pollen grains, and he saw these little tiny particles associated with pollen grains and was in water. He was looking under a microscope and he saw that these little tiny particles, they just kept on getting kicked, kicked around. Why was that happening? Why were there discrete kicks to those Particles and the thing that. And so nobody knew. And so for 70 years that was kind of a mystery. What is Brownian motion? And then around 1900, chap called Smolakowski and then Einstein kind of got into the act and said Brownian motion is the effect of molecules. What's happening when those things are kicked is that an individual water molecule hits that thing and made it bounce in the other direction. And that was the first really clear evidence for the existence of discrete molecules. There were other pieces of evidence that came in later that were a bit circumstantial. I mean there are things about the so called specific heat. This relates to the whole heat temperature business for materials. That was another piece of evidence. Now a question you can ask is in modern times, how do you know that there are discrete atoms? That's a good question. I don't think it's super easy. I think that there's endless kind of circumstantial evidence. But with our own eyes, how do we see that they're atoms? Good question. For example, for photons, particles of light, maybe there's a little bit of a way of seeing that there are light amplification systems, photo multiplier tubes and so on, where eventually in a dark enough room, you will see kind of those things will see individual flashes. Actually, if we were king crabs living deep in the ocean and so on, our eyes, which are slightly better architected than mammalian eyes, they just have things better arranged. And there's a different separate way in which eyes evolved. But it is said that those creatures can observe a single photon falling on their retina. For us, the photoreceptors on our retina do sense single photons, but we don't recognize that. We only sort of collect together the effects of many, many photons. So if we were king crabs, we might have a better idea that yes, at least light is made of discrete things. Now in terms of, let's see, I mean, you know, by the time you're doing fancy physics experiments, it's not hard to know that there are discrete particles. But in terms of everyday experience, it's a good question because most of our everyday experience just doesn't get to a level of sensitivity where, where we can be, be aware of that. Let me think the other, Let me see. I have to say I, I saw Brownian motion at least once in my life. But that might be like was when I was a kid. It's been a really long time since I saw Brownian motion with my own eyes. What are some other things that we see in the modern world that, well, I mean, again, it's very circumstantial. I mean, the, the fact that, yeah, I mean, when we have, There are discrete particles from cosmic rays that we can sense like the. But again, it's very circumstantial. It's not really a direct. This is, you know, matter made of molecules. Now there are certainly fancy physics experiments which can readily see things like that. There are, for example, atomic force microscopes where you can sort of sense individual atoms and the placement of individual atoms and so on. But in terms of a sort of everyday experience thing, even with all the modern technology we have, I'm hard pressed to come up with a, a good sort of in your face piece of evidence for the existence of atoms. And, and by the way, the big, the big thing there is in our models of, of physics now we believe space is discrete as well. Something was believed 100 years ago, but people didn't really manage to make that work until our current physics project. And the question is, how can we tell that space is discrete? And so that's a big thing that I'm thinking about is to sort of go back and think about what phenomenon. It's going to be a fancy physics phenomenon. It's going to be something with telescopes and astrophysics or particle accelerators or maybe some careful quantum computing experiment, those kinds of things that is going to reveal the discreteness of space. But it's sort of the same problem as the existence of molecules. And I consider that people in the beginning of the 20th century were lucky that molecules are big enough that the techniques like Brownian motion that they had to perceive their existence actually worked. The atoms of space may be too small for the things that we can do in this century to be able to detect them. Hopefully not. But one has to sort of think cleverly about what's the kind of, what's that weird effect like Brownian motion that can perceive the presence of galaxy discreteness in space? I have a slight guess that dark matter may actually be that phenomenon. And dark matter kind of says, oh, that must be matter. But actually there's no evidence that it is matter. It might be a feature of the structure of space. Dark matter. Was that the phenomenon of rotation of galaxies, which is sort of the big clue about dark matter was discovered a hundred years ago and it's been still, it's mysterious what it is, but. And I'm sort of somewhat guessing that it has to do with the structure of space and that it eventually will be a sign of the discreteness of space, we'll see just as heat. We could have if we'd known better. 150 years ago, we could have known that the existence of heat is a sign of the discreteness of matter. But nobody put those. Well, people did start putting those things together in the mid-1800s, but nobody really believed that. Let's see. Question. Oh, my gosh. There's a question here from Kathy. How would you solve a hundred slash four open paren two plus three? You mean how would I compute the answer? Well, it's a good question because. Write it out. 104 open paren. What does that mean? You type it into orphan language. I can tell you the answer will be 104, which I know is 20 times 5, which is 2 plus 3. And so I know the answer is 100, but I'm assuming there that that 104 open paren 2 plus 3 is 100 divided by 4. Work that out, then multiply that result by 2 plus 3, and that's the way you figure it out. But that makes some assumptions. I mean, the way that's written out, you could mean 100 divided by 4 times 2 plus 3. In other words, work out 4 times 5 20, and then divide 100 by that and get the answer 4. Get the answer 5. Sorry. That's why I make big systems for doing math by computer, is because they're my tool for not doing that in my own brain, in any case. But. So the question is, why does 104 open paren two plus three mean to most of us 100 over four times, you know, times two plus three? Well, that has to do with the assumed precedence of these operators. What does that mean? Well, if I say something like two plus three times four, there is the convention that you do the three times four. You work that out first and then you do the two plus. And that's really just a convention. It doesn't have to be that way. There's nothing intrinsic about the operations of multiplication and addition that say that's the way you should do it. It is merely a convention of the way that we write things out that we choose to do things that way. Now, in. It's like in a human language, when we write it, when we say, oh, gosh, what would we say? We'd say something like, Well, we're thinking about, let's see, the cat slowly ran. Yeah, the cat slowly ran across the room. The fact that that's not a great sentence, but the fact that we associate slowly ran that we. That that's a thing rather than cat. Slowly, that. That would be a thing. That's an example of kind of the binding of words. When we interpret that sentence, we interpret it with the word slowly bound to the word ran rather than being associated with the word cat. That's a convention of the way that English is set up. And if we diagram the sentence, we'll make this sort of tree that shows that we have this verb phrase, slowly ran, that's a thing. And then we could say the black cat, that will be a noun phrase. Slowly ran, that will be a verb phrase. And those things are sort of lower down on the tree. And then those things together make up the piece of the sentence. Okay, so when we get to do math, we're sort of imitating those kinds of things, and we have these conventions about what gets worked out when. Now, sometimes those conventions really break down. Like If I say 2, 4, 8, what does that mean? Does that mean 2 divided by 4, that whole result divided by 8? Or does that mean 2 divided by 4 divided by 8? The answer will be different. Now compare that with 2 plus 3 plus 4. Well, you can put. You can do 2 plus 3, and then you can do plus 4, or you can do 2 plus 3 plus 4. It doesn't matter. Plus is what we call a modern language flat. What's called in math, usually associative. That means for plus, you can do the addition in any order you want. For division, that's not true, but division, it matters whether you say two divided by three all divided by four, or whether you say two divided by three divided by four, you get a different answer. You can also say, well, plus, you can say we can do plus of any number of things. Division intrinsically is just pairs of things. But you can think of plus as just being pairs of things as well. It's just that plus is flat in the sense that any kind of nesting of pluses will always be. It doesn't matter. You can flatten the whole tree out. It doesn't matter in which order you do the pluses. So these different operations in math have these certain conventions about the order in which they're done. Now, if I write out two divided by three divided by four, and I want to make sure that everybody knows which order I want that done in. I can specify that order by putting in parentheses. That's the role of parentheses is to kind of mark off what things should be done. First, you do the parenthesized thing. It's a way of defining how this tree should be constructed. That Says what gets done first. The parentheses kind of define the structure of the tree. And by the way, when you think about parentheses kind of nested within each other, you've got open paren, open paren, open paren, close, paren, close paren, open close, et cetera, et cetera, et cetera. That sequence of open closes is effectively defining, if you write out a sequence of matched open close parentheses, that that is defining kind of how you go on a tree. So every new open parenthesis is a new tree element that will turn into these two branches that correspond to those two parentheses and so on. But parentheses are the way of specifying, you know, this is the default precedence. You could decide, as we do in multim language, that 2 over 3 over 4 means 2 divided by 3 divided by 4. So that a type of operator, the slash operator there is what's called left associative. It associates to the left, does the left handed stuff before doing the right handed stuff. Now you might say, well, is that, Is left associativity what you always want? Well, no, actually. So for example, let's say you're saying you're writing X equals y equals 2. Okay? So you've got x equals y equals 2. So the question is, which of those equal signs do you deal with first? Which of those assignments do you do first? You better not do the left hand one, because that would say X equals Y as a thing. And then that's being equal to two, which is kind of, well, somewhat meaningless. What you actually want to do is you say X Equals y equals 2 is you break off the y equals 2. You do that, the result from that is 2, and then that result you feed back into the X equals and then that becomes x equals 2. And then you set that x equals y equals 2. We'll set both x and y to 2. So that's an example of an operator. The equals operator is a right associative operator. It's convenient to make that right associative. Then there are ones where nobody really kind of knows which order they should be done in. Like power. Well, if you write 2 to the, well, let's say right, you say 2 to the 3 to the 4, you write it 2, 3, 4. Which do you do first? Do you do 2 to the 3 all to the power 4? Or do you do 2 to the power 3 to the 4? The answers are different. So I made the choice when we built wolfram language back 38 years ago. Now I made the choice to make power right associative. And I think that choice actually has flowed into a lot of other places. And the reason I did that is because 2 to the 3, all to the 4 is the same, just by standard math, as 2 to the 3 times 4. So why would you write it as 2 to the 3 to the 4? But if it's right, associative, 2 to the 3 to the 4 is something distinct that you can write out. And so it seemed more sensible to have that thing be such that you're explicitly writing it out that way. Now, when it comes to precedence of operators, that is, which binds tighter? Do you do the times first or the plus first or the divide first or whatever? What you find when you build a computer language, you have to define all those precedences. And I did that for Wolfram language a long time ago. And what was interesting about it was you might have thought you've got 50 precedences to assign of all these different kinds of operators and all these different things that can happen. You might have thought that it will be very difficult. There would be very. You just wouldn't know. But it turns out that people have a very strong intuitive sense of what binds tighter. Actually, there's a hint in math books, because usually things that bind looser have just a little bit of extra space around them when they're typed out. And so just by the conventions of the systems that type out math, and we certainly use that convention too, and that gives kind of an extra clue that those things bind less tightly because they're actually further away from the next thing. So that's kind of a. But the thing that's sort of surprising is that across like 50 different kinds of things, there's a definite order of this binds tighter than that binds tighter than that. So, for example, let's say plus versus equals or something. Or if I say x equals 2, 3, I definitely want the plus to bind tighter because I want to work out the 23 before I say the X equals part of it. A lot of things. So generally multiplication binds tighter than addition. And you see that same phenomenon in and. And. Or so, for example, and it's kind of like multiplication. If you've got a true and a false, it's like a one. True is like one, false is like zero. That's the way Boolean algebra works. And so and is, you know, P and Q is true only if both P and Q are true. So P the and is like one and zero true, and false is false. It's like one times zero and one times one is True and so on. Or is kind of like addition. It's like one plus zero is still one. Okay, there's a tricky thing. If you have true plus true, true or true, that just gives true, not twice true, but roughly. Times is like sorry, and is like times, and or is like plus. And when you think about writing out in English, for example, if you say, You know, the cat and the dog or the elephant did this, I admit it's a bit subtle, but there is a tendency to say the cat and the dog or the elephant and the rhinoceros did this, and to group things that way. And if I say the cat or the dog and the elephant, to tend to group the dog and the elephant together. And when you do this in computer languages, uniformly the and binds tighter than the or. And that's direct analogy to the times and the plus. In ordinary arithmetic, there are other ones like that, union and intersection. So we have two collections of things, A, B, C, and C, D, E. For example, the union of those sets is all the elements that occur in either of those sets. So A, B, C, D, E. The intersection of those sets is the elements that occur in both of those sets. So it turns out that intersection is the analog of multiplication, and union is the analog of addition. Once again, the precedence of the union, intersection operations which work the same way as multiplication or addition. So there's some sort of intuitive sense that people have of how this should work, and that's what's captured. So in the traditional kind of way, one writes out math, one has kind of a small slice of that tradition of how one writes things out. Now, there are some very tricky cases. The division case, nobody usually talks about that. Is it left associative? People don't talk about it. They just don't write that out very often. And if they do write it out, they put parentheses in. Another one that's tricky is subtraction. So if I say 2 minus 3 minus 4, what is that equal to? Well, it's not totally obvious. It probably means 2, subtract 3, get minus 1 minus 4. Subtract, subtract 4, get minus 5. Okay, but it could mean 2 minus 3 minus 4, which would instead be 2 minus minus 1, I.e. 2 plus 1 or 3. So subtraction like that is again, usually treated as left associative. You're just keeping on subtracting things, not doing it from the right, doing it that way. So I'm trying to think there are other good examples of right associativity. I mean, there are lots of Wolfram language where there's right associativity, left associativity. Now, occasionally painful things happen. Occasionally there are things where by the time you've got 50 different things you've got to assign precedences to, occasionally it's like you just can't do it so that you don't have to use parentheses. You would like to have it be the case that most of the time the thing you write out just does the right thing without parentheses, that you don't have to put parentheses in to say what you mean. There are a few cases where there's just no way to win, where the ordering of things is forced by various different things and you just can't get the ordering to be such that it does the right thing locally, but it also does the right thing with respect to other operators. That happens in Wolfram language for the rule operation xaroy and the function application thing, ampersand coming at the end. And that forces. In certain cases, you want the ampersand to bind less tightly so that you can have a bigger thing as sort of included in your function, yet you want the arrow to sometimes bind less tightly than the ampersand. And so we've actually done, even quite recently, we've got more and more tricks about the way that sort of parentheses flash in as you're typing things and so on to give you a clue as you type about what's going on in how things group and so on. And to know that, oh, it grouped the way you didn't expect it to group and such like. But that's some. That's that story. And I think when you, you know, if you put in parentheses, you can always say what you mean. It's just that then you put in a bunch of parentheses and things look ugly. And you want to set things up so that most of the time the thing you want to mean is the thing that gets meant even when you don't have parentheses. So that's, that's that story. I think people become. It becomes kind of one of the. One of the. Kind of, Sort of one of the mystical features of mathematics. This kind of, you know, I don't even know the acronyms, but I mean, it's like parentheses followed by multiplication. I forget the. I mean, I know the ordering from the precedence order, but. But I don't remember the mnemonics. Oh, somebody is saying PEMDAs, PMDAs, which I assume stands for parenthesis, multiplication, division, addition, subtraction. Really? That's pretty weird. That will get you into trouble. Yeah, that'll get you into trouble with some of these things about left and right associativity I think doesn't really give you quite enough information. But in any case, the notion that you can write out these things and it's kind of a mystical language which you have to kind of know these rules to be able to decode is sort of an optimization. Just like in English we could have markers of how which word goes with which other word, but we don't. It's a convention that, you know, the adverbs slice off and go with the verbs and the adjectives go with this and the nouns and so on. It's just part of what we learn when we learn a language. But it's part of kind of what has become the tradition of mathematics. All right, I should probably wrap up for now and lots of interesting questions unanswered here, but I look forward to going on another time. Thanks very much and bye for now.