Stephen Wolfram: Can AI Solve Science?

Okay. Hello everyone. So last week I posted a piece called Can AI Solve Science? And so today what I wanted to do is go through that piece, read through it, and then open things up for Q and A about that topic. So this is something I've been wondering about for a while. Kind of AI is this big success doing all kinds of things. How far is it going to get? How much is it going to let us solve science? So that's my topic, can AI solve Science? So, first section here was, won't AI eventually be able to do everything? Particularly given its recent surprise successes, there's a somewhat widespread belief that eventually AI will be able to do everything, or at least everything we currently do. So what about science? Over the centuries, we humans have made incremental progress, gradually building up what's now essentially the single largest intellectual edifice of our civilization. But despite all our efforts, there are still all sorts of scientific questions that remain. So can AI now come in and just solve all of them? To this ultimate question, we're going to see that the answer is inevitably and firmly no. But that certainly doesn't mean AI can't be important, can't importantly help the progress of science at a very practical level. For example, LLMs provide a new kind of linguistic interface to the computational capabilities that we've spent so long building in Wolfram language. And through their knowledge of kind of conventional scientific wisdom, LLMs can often provide what amounts to very high level autocomplete fulfilling in conventional answers or conventional next steps in scientific work. But what I want to do here is to discuss what amounts to deeper questions about AI and science. Three centuries ago, science was transformed by the idea of representing the world using mathematics. And in our times, we're in the middle of a major transformation to a fundamentally computational representation of the world. And yes, that's what our Wolfram language, computational language, is all about. So how does AI stack up? Should we think of it as essentially as a practical tool for accessing existing methods, or does it provide something fundamentally new for science? My goal here is to explore and assess what AI can and can't be expected to do in science. So I'm going to consider a number of specific examples, simplified to bring out the essence of what is or isn't going on. I'm going to talk about intuition and expectations based on what we've seen so far. And I'm going to discuss some of the theoretical and in some ways philosophical underpinnings of what's possible and what's not. So what do I actually mean by AI here. In the past, anything seriously computational was often considered AI. In which case, for example, what we've done for so long with our Wolfram language, computational language would qual as with all of my rule, logical study of simple programs in the computational universe. But here, for the most part, I'm going to adopt a narrower definition and say that AI is something based on machine learning and usually implemented with neural networks that's been incrementally trained from examples it's been given often. I'll add another piece as well that those examples include either a large corpus of human generated scientific text and so on, or a corpus of actual experience about things that happen in the world. Or in other words, that in addition to being a raw learning machine, the AI is something that's already learned from lots of human aligned knowledge. Okay, so we've said what we mean by AI. So now what do we mean by science? And by doing science? Ultimately, it's all about taking things that are out there in the world, and usually the natural world, and having ways to connect or translate them to things we can think or reason about. There are several rather different common workflows for actually doing science. Some center around prediction given observed behavior, predict what will happen, find a model that we can explicitly state that says how a system will behave given an existing theory, determine its specific implications. Other workflows are more about explanation. Given a behavior, produce a human understandable narrative for it, find analogies between different systems or models. And still other workflows are more about creating things. Discover something that has particular properties, discover something interesting. So in what follows, we'll explore these workflows in more detail, seeing how they can or cannot be transformed or informed by AI. But before we get into this, we need to discuss something that looms over any attempt to solve science. The phenomenon of computational irreducibility. The hard limit of computational irreducibility. Often, in doing science, there's a big challenge in finding the underlying rules by which some system operates. But let's say we found those rules, we've got some formal way to represent them, say as a program, then there's still a question of what those rules imply for the actual behavior of the system. Yes, we can explicitly apply the rules step by step and trace what happens. But can we, in one fell swoop, just solve everything and know how the system will behave? To do that, we have, in a sense, to be infinitely smarter than the system. The system has to go through all those steps. But somehow we can just jump ahead and immediately figure out the outcome. A key idea, ultimately supported at a foundational level by our physics project, is that we can think of everything that happens as a computational process. The system is doing a computation to determine its behavior. We humans, or for that matter, any AIs we create, also have to do computations to try to predict or solve that behavior. But the principle of computational equivalence says that these computations are all at most, equivalent in their sophistication. And this means we can't expect to systematically jump ahead and predict or solve the system. It inevitably takes a certain irreducible amount of computational work to figure out what what exactly the system will do. And so, try as we might with AI or otherwise, we'll ultimately be limited in our scientific power by the computational irreducibility of the system. But given computational irreducibility, why is science possible at all? The key fact is that whenever there's overall computational irreducibility, there are also an infinite number of pockets of computational reducibility. In other words, there are always certain aspects of a system about which things can be said using limited computational effort, and these are what we typically concentrate on in doing science. But there are inevitably limits to this and issues that run into computational irreducibility. Sometimes these manifest as questions we just can't answer. Sometimes there's surprises we couldn't see coming. But the point is that if we want to solve everything, we'll inevitably be confronted with computational irreducibility, and there won't be any way, with AI or otherwise, to shortcut just simulating the system step by step. There is, however, a subtlety here. What if all we ever want to know about are things that align with computational reducibility? A lot of science and technology has been constructed specifically around computationally reducible phenomena. And that's, for example, why things like mathematical formulas have been able to be as successful in science as they have. But we certainly know we haven't solved everything we want in science. And in many cases, it seems like we don't really have a choice about what we need to study. Nature, for example, forces it upon us, and the result is that we inevitably end up face to face with computational irreducibility. As we'll discuss, AI has the potential to give us streamlined ways to find certain kinds of pockets of computational reusability. But there'll always be computational irreducibility around, leading to unexpected surprises and things we just can't quickly or narratively get to. Will this ever end? No, there'll always be more to discover. Things that need More computation to reach pockets of computational reducibility that we didn't know were there and ultimately AI or not. Computational irreducibility is what will prevent us from ever being able to completely solve science. There's a curious resonance historical to all of this. Back at the beginning of the 20th century, there was a big question of whether all of mathematics could be mechanically solved. The arrival of Godel's theorem, however, seemed to establish that it could not be. And now that we know that science is also ultimately a computational structure, the phenomenon of computational irreducibility, which is in effect a sharpening of Godel's theorem, shows that it too cannot be mechanically solved. We can still ask her whether the mathematics or science that humans choose to study might manage to live solely in pockets of computational reducibility. But in a sense, the ultimate reason that math is hard is that we're constantly seeing evidence of computational irreducibility. We can't get around actually having to compute things, which is, for example, not what methods like neural net AI, at least not without the help of tools like Wolfram language are good at. Things that have worked in the past. Before getting into the details of what modern machine learning based AI might be able to do in solving science, it seems worthwhile to recall some of what's worked in the past, at least as a kind of baseline for what modern AI might now be able to add. I myself have been using computers and computation to discover things in science for more than four decades now. My first big success came in 1981 when I decided to try enumerating all possible rules of a certain kind, elementary cellular automata, and then run them on a computer and see what they did. So this is what I saw when I did that. Throw this up here. I'd assumed that with simple underlying rules, the final behavior would be correspondingly simple. But in a sense, the computer didn't assume that. It just enumerated rules and computed results. And so, even though I never imagined it would be there, I was able to discover something like Rule 30 that's highlighted here. Over and over again, I've had similar experiences. I can't see how some system can manage to do anything interesting. But when I systematically enumerate possibilities. Oops, did I forget to share this? I forgot to share that. There we go. Okay, these are the rules that I looked at back in 1981. And that one there is Rule 30, the exciting and unexpected one. So over and over again, I've had similar experiences. I can't see how some system could manage to do anything interesting. But when I systematically enumerate possibilities, there it is, something unexpected, interesting and clever, effectively discovered by computer. In the early 1990s, I wondered what the simplest possible universal Turing machine might be. I would never have been able to figure it out myself. The machine that had held the record since the early 1960s had seven states and four colors. But the computer let me discover, just by systematic enumeration, this. Let's see if I could this two state three color machine that in 2007 was proved universal. And yes, it's the simplest possible universal turing machine. In 2000, I was interested in what the simplest possible axiom system for logic ruling algebra might be. The simplest known up to that time involved nine binary NAND operations. But by systematically enumerating possibilities, I ended up finding a simple six operation axiom, which I proved using automated theorem proving was correct. So once again, I had no idea that this was out there, and certainly I would never have been able to construct it myself. But just by systematic enumeration, the computer was able to find what seemed to me like a very creative result. In 2019, I was doing another systematic enumeration now of possible hypergraph rewriting systems that might correspond to the lowest level structure of our physical universe. And when I looked at the geometries that were generated, I felt like as a human I could roughly classify what I saw. But were there outliers? I turned to something close to modern AI to do the science. Making a kind of feature space plot of visual images and seeing what kinds of patterns there were in needed me as a human to interpret it. But yes, there were outliers that had effectively been automatically discovered by the neural net that was making the feature space plot. I'll give one more example of a rather different kind from my personal experience back in 1987, as part of building version one of what's now Wolfram language, we were trying to develop algorithms to compute hundreds of mathematical special functions over very broad ranges of arguments. In the past, people had painstakingly computed series approximations for specific cases. But our approach was to use what amounts to machine learning, burning months of computer time fitting parameters in rational approximations. Nowadays we might do something similar with neural nets rather than rational approximations. But in both cases, the concept is to find a general model of the world that one's dealing with here, values of special functions, and try to learn the parameters in the model from actual data. It's not exactly solving science, and it wouldn't even allow one to discover the unexpected. But it's a place where kind of AI, like knowledge of general expectations about smoothness or simplicity, lets one construct the analog of a scientific model. Can AI predict what will happen? It's not the only role of science. And in the section what follows, we'll explore other ones. But historically, what's often been viewed as a defining feature of successful science is can it predict what will happen? So now we can ask, does AI give us a dramatically better way to do this? In the simplest case, we basically want to use AI to do inductive inference. We feed in the results of a bunch of measurements. Then we ask the AI to predict the results of measurements we haven't yet done. At this level, we're treating the AI as a black box. It doesn't matter what's happening inside. All we care about is whether the AI gives us the right answer. We might think that somehow we can set up the AI so that it isn't making any assumptions and it's just following the data. But it's inevitable that there'll be some underlying structure in the AI that makes it ultimately assume some kind of model for the data. Yes, there can be a lot of flexibility in the model, but one can't have a truly modelless model. Perhaps the AI is based on a huge neural net with billions of numerical parameters that can get tweaked. Perhaps even the architecture of the network can change. But the whole neural net setup inevitably defines an ultimate underlying model. So let's look at a very simple case. Let's imagine our data is the curve. Let me put this up here. Let's imagine that the data is the blue curve here, perhaps representing the motion of a weight suspended on a spring, and that the physics tells us it continues according to the red curve. Now, let's take a simple neural net. Here it is. And let's train it using the blue curve data above to get a network with a certain collection of weights. Here it is. And now let's apply this trained network to reproduce our original data, what was originally the blue curve, and extend it. There's the result we get. What we see is that the network does a decent job of reproducing the data it was trained on. But when it comes to predicting the future, it basically fails. So what's going on here? Do we just not train long enough? Well, here's what happens. With progressively more rounds of training, it's cut off. I have no idea. But anyway, doesn't seem like it helps very much to just do more training. So maybe the problem is that Our network is too small. So here's what happens with networks having a series of sizes. And yes, it is true that larger sizes help. They let us kind of track the trained part of the curve better. But maybe I should describe for people who might be listening to this and not being able to see pictures. What we've got here are a series of pictures kind of showing a wiggly sine curve and showing the prediction based on a bunch of networks, neural nets of progressively larger size, predicting the last part of those wiggles. And what typically happens is when you get to the part that the network hasn't seen yet, the curve just diverges and starts going in more or less a straight line in some direction. But so what we see is that things track better in the part where we trained. They don't have any kind of little perturbations from that, but they don't really help solve the problem of making the predictions successful. So what else can we do? Well, one feature of the network is its activation function, how we determine the output at each node from the weighted sum of its inputs. So we can look at results from various popular activation functions. Here it is. And so we've got relu and gilu and swish and soft plus and sigmoids and even sine and cosine. And there's something notable here which highlights the idea that there are no modelless models. Different activation functions lead to different predictions. And the form of the predictions seems to be a direct reflection of the form of the activation functions. So things like relu, that's a rectified linear thing, that the prediction is just a straight line, things like sine, the prediction is a wiggly thing like sine. And so there's sort of no magic here. It's just that the neural net corresponds to a function whose core elements are activation functions. So if we look just a very small network here, this corresponds to an actual mathematical function that we can write out. And if phi is the activation function, we can just write it out as a series of nested phi functions and so on. And so this is kind of what we're doing here. So the idea of approximating one function by some combination of standard functions is extremely old. I mean, think epicycles. And before neural, lets allow us one to use more complicated and hierarchical combinations of more complicated and nonlinear functions and provide a more streamlined way of fitting all the parameters that are involved. But at a fundamental level, it's the same idea. So as an example, here's some approximations to our sine wave Data constructed in terms of more straightforward mathematical functions. So we've got some curves here and we're fitting them to ratios of polynomials. And the advantage here is that it's pretty easy to state what each model is. We can just write it down as this formula as a ratio of polynomials. But just as with our neural nets, there's a problem in making predictions. These curves kind of track more or less the sine wave at the beginning and then diverge off when we get to the region that where we actually have to make a prediction, so to speak. Not quite sure why these gray lines show. Oh, I see, show. They just show the region over which we are successfully able to fit the function. So there are actually a whole bunch of methods for things like time series prediction involving ideas like fitting to recurrence relations in modern times using transformer neural nets. While some of these methods happen to be able to capture a periodic signal like a sine wave, well, one doesn't expect them to be broadly successful in accurately predicting functions. Okay, so I might say perhaps we're trying to use and train our neural nets in too narrow a way. After all, it seems like it was critical to the success of ChatGPT to have a large amount of training data about all kinds of things, not just some narrow specific area. Presumably, though, what broad training did was to let chatgpt kind of learn the general patterns of language and common sense, which it just couldn't, wouldn't be able to pick up from narrower training data. So what's the analog for us here? It might be that we want our neural nets to have a general idea of how functions work, for example, to know things like continuity of functions, or for that matter, periodicity or symmetry. So, yes, we can go ahead and train not just on a specific window of data like we did above, but on whole families of functions, say collections of trigonometric functions, or perhaps all the built in mathematical functions in the Wolfram language. Needless to say, if we do this, we'll surely be able to successfully predict our sine curve above, just as we would if we were just using traditional Fourier analysis with sine curves as our basis. But is this doing science? In essence, it's saying, I've seen something like this before. So I figure this is what's going to happen now, and there's no question that can be useful. Indeed, it's an automated version of a typical thing that a human experienced in some particular area will be able to do. We'll come back to this later, but for now, the main Point is that at least when it comes to things like predicting functions, it doesn't seem as if neural nets and today's AIs can in any obvious way see further than what goes into their construction and training. There's no emergent science, it's just fairly direct pattern matching. Predicting computational processes. Predicting a function is a particularly austere task, and one might imagine that real processes, for example, in nature would have more ambient structure, which an AI could use to get a foothold for prediction. As an example of what we might think of as an artificial nature, we can consider computational systems like cellular automata. So is an example of what a particular cellular automaton rule does with particular initial conditions. There's a mixture here of simplicity and complexity. And as humans, we can readily predict what's going to happen in the simple parts, but basically can't say much about the other parts. So how would an AI do? Clearly, if our AI can just run the cellular automaton rule, it will be able to predict everything though with great computational effort. But the real question is whether an AI can shortcut things to make successful predictions without doing all that computational work. Or put another way, whether the AI can successfully find and exploit pockets of computational reducibility. So, as a specific experiment, let's set up a neural net to try to efficiently predict the behavior of our cellular automaton. Our network is basically a straightforward, though modern convolutional autoencoder. This one with 59 layers and about 800,000 parameters. Let me show you a picture of it. There's a gap in the middle there, so there's a lot more units than I'm showing here explicitly. So this is trained very much like an LLM. We get lots of examples of the evolution of our cellular automaton. Then we showed the network the top half of each one and tried to get it to successfully continue this to predict the bottom half. In the specific experiment we did, we gave 32 million examples of 64 cell wide cellular automaton evolution. And yes, this number of examples is tiny compared to the 2 to the 64 or 10 to the 19 possible initial configurations. Then we tried feeding in chunks of cellular automaton evolution, 64 cells wide and 64 steps long, and looked to see what the probabilities the network assigned to different were to look to see what the probabilities were that the network assigned to different possible continuations. So this is what happens with the sequence of different initial conditions. And basically what we see is what we might expect when the Behavior is simple enough. The network basically gets it right even when it's periodic and keeps going here and so on. But when the behavior is more complicated, the network usually doesn't do so well with it. It still often gets it vaguely right, but the details aren't there. And so we can see here, when we have a cellular automaton evolution that has just a piece that becomes periodic, the network correctly continues it in a periodic way. But when the behavior is more complicated and there are lots of little, kind of little things that look like little plants and so on growing around, the network kind of gets the rough big pieces, but doesn't get the details right. So perhaps one might think the network just wasn't trained long enough or with enough examples to get some sense of the effect of more training. Here's how the predicted probabilities evolve with a successive quarter million rounds of training. So each different stage here is another quarter million rounds of training. And what we see is that it goes from things being very fuzzy at the beginning and not even the periodic behavior being correctly reproduced, to something where the periodic behavior gets nailed. But there's still fuzziness in these more complicated areas. So we can compare this to the exact result. If you kind of remember what this looks like, you can see the exact result here. I wish that thing that keeps on showing up on the, on the slides wasn't showing up well. Okay, so with more training, definitely gets better. But in the end, it seems like it probably won't get much better. Though if you look at the loss curve of how accurately was the prediction happening, it does show some sudden downward jumps during the course of training, presumably as sort of discoveries get made by the neural net. But essentially pockets of computational reducibility get discovered, and we can't be sure there won't be more of these. It's extremely typical of machine learning that it manages to do a good job of getting things roughly right. But nailing the details is not what machine learning tends to be good at. So what one's trying to do, when what one's trying to do depends on that, machine learning will be limited. And in the prediction task we're considering here, the issue is that once things go even slightly off track, everything basically gets worse from here on out. Identifying computational reducibility so computational reducibility is at the center of what we normally think of as doing science, because it's not only responsible for letting us make predictions, it's also what lets us identify regularities, make models, and compressed summaries of what we see. And develop understanding that we can capture in our minds. But how can we find computational reducibility? Sometimes it's very obvious, like when we make a visualization of some behavior, like the cellular automaton evolution we're just talking about, and immediately recognize simple features in it. But in practice, computational reusability may not be so obvious, and we may have to dig through lots of details to find it. And this is a place where AI can potentially help a lot. At some level, we can think of it as a story of finding the right parameterization or the right coordinate system. So as a very straightforward example, let's consider a quite random. Let me just bring up here. A quite random cloud of points. So if we just turn this particular cloud of points to an appropriate angle, here's what we'll see. Obvious regularities. But is there a general way to pick out regularities if they're there? So we, we had a bunch of points that just sort of seem to be randomly distributed in a box. Then we rotate the bo and suddenly we see that these points actually lie on a bunch of planes. And we can see this as a series of stripes. So how are we going to. Is there a general way to pick out these kinds of regularities? Well, there's traditional statistics. Ask questions like is there a correlation between A and B? There's model fitting. Is this a sum of Gaussians? There's traditional data compression. Is it shorter after run length encoding? But all of these pick out only rather specific kinds of regularities. So can AI do more? Can it perhaps somehow provide a general way to find regularities? To say one's found a regularity in something is basically equivalent to saying one doesn't need to specify all the details of the thing. That there's a reduced representation from which one can reconstruct it. So, for example, given the points lie on planes, regularity in the picture we're looking at right now, one doesn't need to separately specify the positions of all the points. One just needs to know that they form stripes with a certain separation. Okay, so let's imagine we have an image with a certain number of pixels. We can ask whether there's a reduced representation that involves less data from which the image can effectively be reconstructed. And with the neural net, with neural nets, there's what one might think of as a trick for finding such a reduced representation. The basic idea is to set up the neural net as an autoencoder that takes inputs and reproduces them as outputs. One might think this will be a trivial task but it's not because the data from the input has to flow through the innards of the neural net, effectively being ground up at the beginning and reconstituted at the end. But the point is that with enough examples of possible inputs, it's potentially possible to train the neural net to successfully reproduce inputs and and operate as an autoencoder. But now the idea is to look inside the autoencoder and to pull out a reduced representation that it's come up with. As the data flows from layer to layer in the neural nets, it's always trying to preserve the information it needs to reproduce the original input. And if a layer has fewer elements, what's present at that layer must correspond to some reduced representation of the original input. So let's start with a standard modern image autoencoder that's been trained on a few billion images typical of what's on the Web. Feed it a picture of a cat, and it'll successfully reproduce something that looks like the original picture cat to cat, but in the middle, there'll be a reduced representation with many fewer pixels we see here that somehow still captures what's needed of the cat. So we can separate out the four color channels that happen to be used in that intermediate representation. And this data represents enough to reproduce the whole cat, so to speak. We can think of this as kind of a black box model for the cat image. We don't know what the elements, the features in the model mean, but somehow it's successful in capturing the essence of the picture. So what happens if we apply this to sort of our artificial natural processes like solar automata? So here's an example where we get a successful compression. The image on the left looks like the image on the right. Here's a case where it's not quite so successful. The middle sort of compressed representation. We're going from a kind of nested image on the left here to an image on the right that's kind of a slightly damaged nested image. In some cases, like if we look at a kind of a Rule 30 image, for example, or another image that has sort of a lot of apparent randomness in it when there's computational irreducibility, this, the autoencoder has trouble in reconstructing what the image was. But there's a bit more to the story. You see, the autoencoder we're using was trained on everyday images, not these kinds of scientific images. So in effect, it's trying to model our scientific images in terms of constructs like the eyes and Ears that are common in pictures of things like cats. So what happens if, like in the case of cellular automaton prediction above, we train an autoencoder more specifically on the kinds of images we want. So here are two very specific the very simple neural nets that we can use as an encoder and a decoder to make an autoencoder. So let's take the sort of standard MNIST training set and then try to use these to train the autoencoder. So each of these images has 28 by 28 pixels. But in the middle of the autoencoder we have a layer with just two elements. So this means that whatever we ask it to encode must be reduced to just two numbers. So that's what happens in the case of these digits. What we see here is that, at least for images that look more or less like the ones that the autoencoder was trained on, the autoencoder manages to reconstruct something that looks at least roughly right even from this incredibly radical compression of turning everything about the image into just two numbers. So if you give other kinds of images, however, it won't be as successful. So here I started by trying to get it to reproduce handwritten ones and twos and threes and did a fairly good job here. Now I'm giving it just a, B and C, and I'm asking it to reconstruct those. And it doesn't do such a good job. It reconstructs the A as a bit of something like a 3, and the B also something like a 3, and the C is something like a 0. It insists on reconstructing them as being things that look like the data that came from its training set. Okay, so now we can try and do the same thing with cellular automaton images. Let's take 10 million images generated with a particular cellular automaton rule. It has fairly elaborate behavior here. Now we train our autoencoder on these images. Then we try feeding it similar images. And what we'll see is it's trying to reduce every one of these images to just two numbers. And the results are, at best, approximate. We've got a whole elaborate pattern that's going into the autoencoder. It's turning it into two numbers, and then on the right, just kind of this blurred gray and white big blobs type output. So the results are, at best, very approximate. The small neural net didn't manage to learn the detailed ways of this particular cellular automaton. If it had been successful at Characterizing all the apparent complexity of the cellular automaton evolution with just two numbers, then we would have considered this an impressive piece of science. But unsurprisingly, the neural net was effectively blocked by computational irreducibility. But even though it can't seriously crack computational irreducibility, the neural net can still make useful discoveries, in effect by finding little pieces of computational reducibility and little regularities. For example, if we take images of noisy letters and we use a neural net to reduce them to pairs of numbers and use these numbers to place the images, we get a dimension reduced feature space plot that separates the, the images of different letters. So we've got A's and B's and C's on noisy backgrounds. And we can see that the neural net successfully figured out what's the A type object from the B type object, and so on. But if we consider, for example, cellular automata with different rules, here's a bunch of pictures of, of, of cellular automata, Some of the very simple behavior, some with complicated behavior. Here's what happens if we ask a neural net to arrange these in kind of a feature space. And what we see is that the complicated ones are off in one corner. The ones that are. I have to get it to stop doing that. Sorry. The thing that I'm using to put slides up keeps on spraying garbage into the slide area. But in any case, this is kind of arranging the neural net which is trained on cellular automaton images as an auto encoder, is arranging these images. And yes, it sort of makes the complicated ones go in one corner. The ones that are mostly white mostly go in one another corner. But then it's very keen on the fact that other ones are, are mainly black. And it's not noticing though those are to me, as a human, those mainly black ones are kind of uninteresting and like the mainly white ones. But it chose to make a distinction between the mainly black and the mainly white. And so what we can see in this picture is that in a sense the, the autoencoder managed to automatically discover, almost managed to automatically discover the four classes of behavior that I identified in cellular Automata in early 1983. But it's not quite there, though. In a sense, this is a difficult case, very much face to face with computational irreducibility. And there are plenty of cases, think arranging the periodic table based on element properties or similarity of fluid flows based on Reynolds number and so on. But one can expect a neural net to key into pockets of computational reusability and at least successfully recapitulate existing scientific discoveries. AI and the non human World in its original concept, AI was about developing artificial analogs of human intelligence. And indeed, the great recent successes of AI, say, in visual object recognition or language generation, are all about having artificial systems that reproduce the essence of what humans do. It's not that there's a precise theoretical definition of what makes an image of a cat be of a cat versus of a dog. What matters is that we have a neural net that will come to the same conclusions as humans do. So why does this work? Probably it's because neural nets capture the architectural essence of actual brains. Of course, the details of artificial neural nets aren't the same as biological brains. But in a sense, the big surprise of modern AI is that seems to be enough universality to make artificial neural nets behave in ways that are functionally similar to human brains, at least when it comes to things like physiologic recognition or language generation. But what about questions in science? At one level, we can ask whether neural nets can emulate what human scientists do. But there's another level. Is it possible that neural nets can just directly work out how systems, say, in nature, behave? Imagine we're studying some simple, some physical process. Human scientists might find some human level description of the system, say, in terms of mathematical equations. But the system itself is just directly doing what it does. And the question is whether that's something a neural net can capture. And if neural nets work on human like tasks only because they're architecturally similar to brains, there's no immediate reason to think that they should be able to capture raw, natural processes that aren't anything to do with brains. So what's going on when AI does something like predicting protein folding? One part of the story, I suspect, is that even though the physical process of protein folding has nothing to do with humans, and the question of what aspects of it we consider significant does. We don't expect the neural net will predict the exact positions of every atom. And in natural environments, the atoms in a protein don't even have precisely defined positions. Instead, we want to know things like whether the protein has the right general shape with the right identifiable features, say, like alpha helices, or the right functional properties. And these are now more human questions, more in the eye of the beholder, and more like a question such as whether we humans judge an image to be of a cat versus a dog. So if we conclude that a neural net solves the scientific problem of how a protein folds, it might, at least in part be because the criteria of success that our brains subjectively apply is something that a neural net with its brain, like architecture, happens to be able to deliver. It's a bit like producing an image with a generative AI at the level of basic human visual perception. It may look like something we recognize, but if we scrutinize it, we can see that it's not objectively what we think. Here are some examples of that. There's a picture that looks like a knight in armor on a horse. There's another picture that looks like kind of a tower on the side of a lake. But if we start looking in more detail, we'll discover that the horse is made up of something that looks like a piece of a feather and a bone, and this and the knight is really just a bunch of lines, and the background is just sort of a general dappling pattern. And when we look over at the lake, it's really just a bunch of lines, and the tower is really a bunch of rectangles and so on. And what looked like a forest is just a bunch of wavy lines and so on. So although we had the human impression, you know, a knight on a horse, a tower by a lake, what's really there is something rather different from that. Back to protein folding, it wasn't ever really practical with first principles physics to figure out how proteins fold. So the fact that neural nets can get even roughly correct answers is impressive. So how do they do it? A significant part of it is surely effectively just matching chunks of protein to what's in the training set and then finding plausible ways to stitch these chunks together. But there's probably something else, too. One's familiar with certain pieces of regularity in proteins, things like alpha helices, beta sheets. But it seems likely that neural nets are effectively plugging into other kinds of regularity. They've somehow found pockets of reducibility that we didn't know were there. And particularly, if just a few pockets of reducibility show up over and over again, they'll effectively represent new general results in science, say, some new kind of commonly occurring metamotif in protein structure. But while it's fundamentally inevitable that there must be an infinite number of pockets of computational reducibility, in the end, it's not clear at the outset either how significant these might be in things we care about, or how successful neural net methods might be in finding them, we might imagine that insofar as neural nets mirror the essential operation of our brains, they'd only be able to find pockets of reducibility in Cases where we humans could also readily discover them, say, by looking at some visualization or another. But an important point is that our brains are normally trained only on data that we readily experience with our senses. We've effectively seen the equivalent of billions of images, and we've heard zillions of sounds. But we don't have direct experience of the microscopic motions of molecules or of a multitude of kinds of data that scientific observations and measuring devices can deliver. A neural net, however, can grow up with very different sensory experiences, say, directly experiencing chemical space, or, for that matter, metamathematical space, or the space of financial transactions or interactions between biological organisms or whatever. But what kinds of pockets of reducibility exist in such cases? Mostly, we don't know. We know that the ones that correspond to. We know the ones that correspond to known science. But even though we can expect others must exist, we normally don't know what they are. Will they be accessible to neural nets? Again, we don't know. Quite likely. If they are accessible, then there'll be some representation or, say, visualization in which the reducibility will be obvious to us. But there are plenty of ways this could fail. For example, the reducibility could be visually obvious, but only, say, in 3D volumes, where, for example, it's hard even to distinguish different structures of fluffy clouds. Or perhaps the reducibility could be revealed only through some computation that's not readily handled by a neural net. Inevitably, there are many systems that show computational irreducibility, and which, at least in their full form, must be inaccessible to any shortcut method based on neural nets or otherwise. But what we're asking is whether, when there is a pocket of computational reducibility, it can be captured by a neural net. But once again, we're confronted with the fact that there are no modelless models. Some particular kind of neural net will readily be able to capture some particular kinds of computational reducibility. Another will readily be able to capture others. And yes, you can always construct a neural net that will approximate any given specific function. But in capturing some general kind of computational reducibility, we're asking for much more. And what we can get will inevitably depend on the underlying structure of the neural net. But let's say we've got a neural net to successfully key into computational reducibility in a particular system. Does that mean it can predict everything? Typically, no, because almost always the computational reducibility is just a pocket, and there's plenty of computational irreducibility and surprises outside. And indeed, this seems to happen even in the case of something like protein folding. So here are some examples of proteins that we perceive as fairly simple structures. They're kind of things that look like a coiled old fashioned telephone cable or something. The. Not sure anybody even sees those things anymore. The kind of helical coil that used to go from a telephone receiver that you would hold to your ear to the, the telephone that's sitting on your, on your desk. But proteins often form those kind of alpha helices that look like that. And the neural net prediction, we can see here the sort of comparison between the yellow neural net prediction and the actual physical experiments, which are kind of the gray tubes. And we can see that in the case of where it's just a few little alpha helices, it does a reasonably good job. But for proteins with a more complicated structure, the agreement is often not nearly as good. Here are proteins that are more sort of globular and just sort of random looking. And you can see that the prediction from the neural net is kind of vaguely in the right direction, but certainly not in detail of the same shape. But these are all proteins, even the ones we're looking at here, at least similar to ones that were used to train the neural nets. How about very different proteins, say ones with a random sequence of amino acids? So here are some examples of those predicted as predicted by a neural net. It's hard to know how well the neural net does here. It seems likely that particularly if there are surprises, it won't successfully capture them. Of course, it could be that all reasonable proteins that normally appear in biology have certain features. And it could be unfair to apply the neural nets to unbiological random ones, though, for example, in the adaptive immune system, biology does effectively generate at least short random proteins. Solving equations with AI in traditional mathematical science, the typical setup is here are some equations from a system, solve them to find out how the system behaves. And before computers, that usually meant one had to find some closed form formula for the solution. But with computers then it's an alternative approach. Make a discrete numerical approximation and somehow incrementally solve the equations. To get accurate results, though, may require many steps and lots of computational effort. So the question is, can AI speed this up? And in particular, can AI, for example, go directly from initial conditions for an equation to a whole solution? So let's consider as an example a classic piece of classical, classic piece of mathematical physics, the three body problem. So given initial conditions and velocities of three point masses interacting via inverse square law, gravity, what trajectories will the masses follow? There's a lot of diversity. I'm showing pictures here. Some of them are pretty regular, many of them look like scribbles. The and, and the fact that these look so complicated is why the three body problem has been such a challenge over the years. But what if we train a neural net on lots of sample solutions to the three body problem? Can it then figure out the solution in any particular case? We can use a rather straightforward kind of multi level perceptron network to try and study this and we feed it initial conditions and the problem is can it then generate a solution? So here are some examples of what it does and where we're indicating the correct solutions with a sort of, with the background, background sort of lighter paths. And what you see is when the behavior is fairly simple and things are sort of going in simple kind of repetitive ways, the neural net does a pretty good job. When the behavior looks more like a scribble, the neural net really doesn't do a very good job at all. So it seems like the neural net has sort of successfully memorized the simple cases, but doesn't know what to do in more complicated cases. And in the end this is very similar to what we saw in, in the examples before, like predicting cellular automaton evolution and presumably also protein folding. And yes, once again, this is a story of computational irreducibility. To ask to just get the solution in one go is effectively to ask for complete computational reducibility. And insofar as one might imagine that if one only knew how to do it, one could always in principle always get a closed form formula for the solution, one's implicitly assuming computational reducibility. But for many decades I've at least thought that something like the three body problem is actually quite full of computational irreducibility. Of course, had a neural net been able to crack the problem and immediately generate solutions that would effectively have demonstrated computational reducibility. But as it is, the apparent failure of neural nets to provide provides another piece of evidence for computational irreducibility in the three body problem. It's worth mentioning, by the way, that while the three body problem does show sensitive dependence on initial conditions, that's not the primary issue here. Rather it's the actual intrinsic complexity of the trajectories. We already know that discrete computational systems like cellular automata are rife with computational irreducibility. And we might have imagined that continuous systems described for example by differential equations would have more structure that would somehow make avoid computational irreducibility. And indeed, insofar as neural nets, in their usual formulation, involve continuous numbers, we might have thought that they would in some way be able to key into the structure of continuous systems to be able to predict them. But somehow it seems as if the force of computational irreducibility is too strong and will ultimately be beyond the power of neural nets. Having said that, though, there can still be a lot of practical value to neural nets in doing things like solving equations. Traditional numerical approximation methods tend to work locally and incrementally, if often adaptively. But neural nets can more readily handle much larger windows, in a sense knowing longer runs of behavior and being able to jump ahead across them. In addition, when one's dealing with very large numbers of equations, say in robotics or systems engineering, neural nets can typically just take in all the equations and do something reasonable. And whereas traditional methods effectively have to work with the equations one by one, the three body problem involves ordinary differential equations. But many practical problems are instead based on partial differential equations, PDEs in which not just individual coordinates, but whole functions F of x and so on evolve with time. And yes, one can use neural nets here as well, often to significant practical advantage. But what about computational irreducibility? Many of the equations and situations most studied in practice, say for engineering purposes, tend to avoid it. But certainly in general, it's there, notably in phenomena like fluid turbulence. And when there's computational irreducibility, one can't ultimately expect neural nets to do well. But when it comes to satisfying our human purposes, as in other examples we've discussed, things may look better. As an example, consider predicting the weather. In the end, this is all about PDEs for fluid dynamics. And yes, there are also other effects to do with clouds and so on. And as one approach, one can imagine directly and computationally solving these PDEs. But another approach will be to have a neural net just learn typical patterns of weather, as old time meteorologists had to, and then to have the neural net have the network, a bit like protein for protein folding, try to patch together these patterns to fit whatever situation arises. How successful will this be? It'll probably depend on what we're looking at. It could be that some particular aspect of the weather shows considerable computational reducibility and is quite predictable, say by neural nets. And if this is the aspect of the weather that we care about, we might conclude that the neural net is doing well. But if something that we care about, will it rain tomorrow, let's say, doesn't tap into a pocket of computational reducibility, then neural nets typically won't be successful in predicting it, and instead there'll be no choice but to do explicit simulate computation and perhaps impractically much of it. AI for multi computation and what we've discussed so far, we've mostly been concerned with seeing whether AI can help us jump ahead and shortcut some computational process or another. But there are also lots of situations where what's of interest is instead to shortcut what one can call a multi computational process in which there are many possible outcomes at each step and the goal is, for example, to find a path to some final outcome. As an example of a multi computational system, let's consider a multi way system operating on strings, where at each step we apply a particular set of rules. A goes to bbb, BB goes to A, let's say, but we apply these in all possible ways and we get this whole network starts from the string A go to bbb, then it branches to AB and ba, then it merges again, branches, et cetera, et cetera, et cetera. It's a complicated network. Given the setup, we can ask a question like what's the shortest path from A to baba? So in the case shown here, it's it's fairly easy to compute the answer, say by explicitly running a pathfinding algorithm on the graph. And we get this particular result, A BBB ab, B, B, B, B, B, abb, aa, et cetera, et cetera, et cetera. And it goes on eventually gets to baba. There are many kinds of problems that follow the same general pattern. Finding a winning sequence of plays in a game graph, finding the solution to a puzzle as a sequence of moves through a graph of possibilities, finding a proof of a theorem given certain axioms, finding a chemical synthesis pathway given certain basic reactions, and in general, solving a multitude of NP problems in which many non deterministic paths of computation are possible. In the very simple case that we been looking at here, this one, we're readily able to explicitly generate a whole multi way graph, but in most practical examples, the graph will be astronomically too large. So the challenge is typically to suss out what moves to make without tracing the whole graph of possibilities. One common approach is to try to find a way to assign a score to different possible states or outcomes, and to pursue only paths with, say, the highest scores. In automated theorem proving, it's also common to work downward from the initial propositions and upward from the final theorems, trying to see where the paths meet in the middle. There's also another important idea. If one has established the lemma that there's a Path from X to Y. One can add X goes to Y as a new rule in the collection of rules. So how might AI help? As a first approach, we could consider taking something like our string multi way system and training what amounts to a language model AI to generate sequences of tokens that represent pods or what in a mathematical setting will be proofs. The idea is to feed the AI a collection of valid sequences and then to present it with the beginning and end of a new sequence and ask it to fill in the middle. So we'll use a fairly basic transformer network here. Then we try to train it by giving it lots of sequences of tokens corresponding to valid paths. We might have E being the end token, and here we'll just have a sequence where we give the initial and final strings A, B, B, A, B, A and then a colon and then bbb ab, B, B, B, B. That's the path ending with an E. We also might give it negative examples of paths that don't exist. For example, BABA doesn't ever go to A. We'll just say that. Now what we can do is after we've trained the network, we can prompt it with a prefix of the kind that appeared in the training data and then iteratively run LLM style effectively at zero temperature, ie choosing the most probable next token. And here's what we get if we do that and we get A, we're saying go from A to B, A B, A and it starts off bbb ab. It manages to get that right. It's filling in one character at a time. After a while it starts going wrong. We see that in red at the bottom here. After a bunch of characters it put down, it puts down aab, but that should have been an AB etc, et cetera, et cetera. So as I say, eventually it starts making errors and there's different performance with different destinations. Here's some examples where sometimes it just goes off track right at the beginning, sometimes it manages to figure it out all the way to the end. So how can we do better? One possibility is at each step to keep not just the token that's considered most probable, but a stack of tokens, therefore in effect generating a multi way system that the LLM controller could potentially navigate. One can this think of this sort of whimsically as a kind of quantum LLM that's always exploring multiple paths of history. By the way, we can also imagine training with many different rules, then doing what amounts to zero shot learning and giving a pre prompt that specifies what rule we want to use in any particular case. One of the issues with this LLM approach is that the sequences it generates are often even locally wrong. The next element just can't follow from the one before according to the rules that are given. But this suggests another approach one can take. Instead of having the AI try to immediately fill in the whole sequence, get it instead to just pick where to go next, always following one of the specified rules. Then, a simple goal for training is in effect to get the AI to learn the distance function for the graph, or in other words, to be able to estimate how long the shortest path is if it exists from one node to any other. So given such a function, a typical strategy is to follow what amounts to a path of steepest descent at each step, picking the move that the AI estimates will do best in reducing the distance to the destination. How can this actually be implemented with neural nets? One approach is to use here two encoders, say, constructed out of transformers that in effect generate two embeddings, one for source nod, one for destination nodes. Then the network combines these embeddings and learns a metric that characterizes the distance between the nodes. So training a network on the multi way system we've been discussing, by giving it a few million examples of source destination distances, plus an indicator of whether this distance is infinite, we can use the network to predict a piece of the distance matrix for the multi way system. And so what we find here, this is a picture of the predicted well. On the left is the actual distance matrix, on the right is the predicted matrix. And what we see is there are a bunch of places where indicated with red boxes here where it definitively didn't get the right answer. But we can imagine trying to build a path where at each step we compute the estimated distances to destination predicted by the neural net for each possible destination, then pick out the one that goes furthest. And here's an example of how we do that. And this is showing a path. And at each step there's kind of a stack, just like in an LLM, which is saying the next word might be this or that or the other. It's saying the next token might be a BBB or B A B B, and they each have a different probability, a different weight. So each individual move that we get by doing this is guaranteed to be valid. And we do indeed eventually reach our destination Baba, although in slightly more steps than the true shortest path. But even though we don't quite find the optimal path, the neural net has managed to allow us to at least somewhat prune our search space. So here we can see what it managed to do. In the original graph of all possibilities, only the red edges are the ones that the neural net told us to bother looking down. Down. So, by the way, a technical point is that the particular neural net we've used here has the property that all paths between any two pair of nodes, pairs of node given, any pair of nodes have the same length. So if a path is found, it can always be considered the shortest. You can look at other rules like A goes to aab, BBA goes to B that don't have this property, and a neural trained for a rule like that can end up finding paths that reach the correct destination but aren't as short as they could be be. There's something about that that doesn't make sense to me because we just had an. Yeah, okay, yeah. Still, as is typical with neural nets, we can't be sure how well this whole process will work. The neural net may make us go arbitrarily far off track, and it might even lead us to a node where we have no path to our destination. So if we want to make progress, we'll have to resort to something like traditional algorithmic backtracking. But at least in simple cases, the approach potentially performs well and the AI can successfully find a path that wins the game, proves the theorem, and so on. But we can't expect it to always work. And the reason is that what's is that it's going to run into multi computational irreducibility. Just as in a single thread of computation, computational irreducibility can mean there's no shortcut to just going through the steps of the computation. So in a multi way system, multi computational irreducibility can mean that there's no shortcut to just following all the threads of computation, then seeing, for example, which end up merging with which. But so even though this could happen in principle, doesn't in fact happen in practice, in cases of interest to us humans in something like games or puzzles, we tend to want it to be hard, but not too hard to win. When it comes to mathematics and proving theorems, cases that we use for exercises or competitions, we similarly want to be hard, but not too hard. But when it comes to mathematical research and the frontiers of mathematics, one doesn't immediately expect any such constraint. And the result is then that one can expect to be face to face with multi computational irreducibility, making it hard for AI to help too much. There is, however, one footnote to this story, and it has to do with how we choose new directions in mathematics. We can think of a metamathematical space formed by building up theorems from other theorems in all possible ways in a giant multi way graph. But as we'll talk about in a moment, most of the details of this are far from what human mathematicians would think of as doing mathematics. Instead, mathematicians implicitly seem to do mathematics at a higher level in which they they've coarse grained this microscopic metamathematics, much as we might study a physical fluid in terms of comparatively simple to describe continuous dynamics, even though underneath there are lots of complicated molecular motions. So can AI help with mathematics at this fluid dynamics style level? Potentially so, but mainly in what amounts to providing code assistance. We have something we want to express, say more from language, but we need help. Help LLM style. In going from our informal conception to explicit computational language, and insofar as what we're doing follows the structural patterns of what we've done before, we can expect something like an LLM to help. But insofar as what we're expressing is truly new, and in as much as our computational language doesn't involve much boilerplate, it's hard to imagine that an AI trained on what's been done before will help much. Instead, what we in effect have to do is some multi computationally irreducible computation that allows us to explore some fresh part of the computational universe and the ruliad. Exploring spaces of Systems can one find a system that does X? Say, a Turing machine that runs for a very long time before halting? Or a cellular automaton that grows, but only very slowly? Or for that matter, a chemical with some particular property? This is a somewhat different type of question than the ones we've been discussing so far. It's not about taking a particular rule and seeing what its consequences are. It's about identifying what rule might exist that has certain consequences. And given some space of possible rules, one approach is exhaustive search. In a sense, this is is ultimately the only truly unbiased approach that will discover what's out there to discover, even when one doesn't expect it. Of course, even with exhaustive search, one still needs a way to determine whether a particular candidate system meets whatever criterion one set up. But now this is the problem of predicting a computation where the things we said above or before apply. Okay, but can we do better than exhaustive search? Can we, for example, find a way to figure out what rules to explore without having to look at every rule? One approach is to do something like what happens in biological evolution by natural selection, start, say, from a particular rule and then incrementally change it, perhaps at random at every step, keeping the rule or rules that do best and discarding the others. This isn't AI as we've operationally defined it here. It's more like a genetic algorithm. Them though it's a bit like the inner training loop of a neural net. Will it work? Well, that depends on the structure of the rule space. And as one sees in machine learning, it tends to work better in higher dimensional rule spaces than lower dimensional ones, because with the more dimensions, there's less chance that one will get stuck in a local minimum, unable to find one's way out to a better rule. And in general, if the rule space is like a complicated fractal mountainscape, it's reasonable to expect that one can make make progress incrementally. And perhaps AI methods like reinforcement learning can help one refine what incremental steps to take. But if instead it's quite flat, with, say, just one hole somewhere, sort of golf course style, one can't expect to find the hole incrementally. So what is the typical structure? Rule spaces? There's certainly plenty of cases where the rule space is altogether quite large, but the number of dimensions is only modest. And in such cases, an example being finding small Turing machines with long halting times, there often seem to be isolated solutions that can be reached incrementally, and they can't be reached incrementally. And when there are more dimensions, it seems likely that what amounts to computational irreducibility will more or less guarantee that there'll be a random enough landscape that incremental methods will be able to do well, much as we've seen in machine learning in recent years. And I realize that what I wrote here is there's some words that make it slightly muddled. So what about AI? Might there be a way for AI to learn how to pick winners in rule space without any kind of incremental process? Might we perhaps be able to find some embedding space in which the rules we want are laid out in a nice simple way and thus effectively sort of pre identified for us? Ultimately, it depends on what the rule space is like and whether the process of exploring it is necessarily multi computationally irreducible, or whether at least the aspects of it that we care about can be explored by computationally reducible process. By the way, trying to use AI to directly find systems of particular properties is a bit like trying to use AI to directly generate neural nets from Data without incremental training. Well, let's look at a specific example based on cellular automata. Say we want to find a, Let's see. Say we want to find a cellular automaton rule that when evolved from a single cell, initial condition. Grows for a while but then dies out after a particular exact number of steps. We can transolve this with a very minimal AI like evolutionary approach. Start from a random rule, then at each generation produce a certain number of offspring rules, each with one element randomly changed. Then keep whichever is the best of these rules. So if we want to find a rule that lives for exactly 50 steps, we define best to be the one that minimizes a loss function equal to the distance from 50 of the number of steps a rule actually lives. So let's take a look here. Let's see, to arrange myself here. Okay, so let's say we start from this randomly chosen 3 color cellular automaton rule. So our evolutionary sequence of rules showing, showing here only the kind of the three cubed sort of outcome values. Let me show that this is the kind of evolutionary sequence of rules showing how the rule changes from one generation to the next. So now let's look at the behavior of these rules. So we produce will kind of live for exactly 50 steps. And we can see it starts off rather inauspiciously just running far beyond the 50 steps. Then it gets, then it dies out very quickly. But progressively we manage to to tune the rule. We managed to evolve the rule to the point where eventually it exactly nails, produces a pattern, lives for exactly 50 steps and then dies out. So with this particular kind of path of evolution, that's what we get. We could sort of try and summarize what happens with other paths so we can plot the loss, the distance from living exactly 50 steps that we get with different paths of evolution over the course of, let's say 100 generations for a collection of paths. And we can see that much of the time it doesn't make it after 100 generations, it's still up here with a loss of 40 or 50. It's far away, but there's at least one case in which. Yep, it got it. That's the case. We saw it nails it. It managed to actually find a rule that lives for exactly 50 steps. There's only one winner here that achieves zero loss. All the other parts of evolution get stuck. However, if we have more dimensions, one's less likely to get stuck. So for example, if we look at a four color cellular automaton, there are now 64 other than 27 possible elements or effectively dimensions that one can change. And in this case many more evolution, many more paths of evolution managed to get much further. And there are many, in a sense, more winners. So here are some examples of winners which all make it in the end to living for exactly 50 steps. So how could something like neural nets help us here? Insofar as we can use them to predict teleautomatin evolution, they might give us a way to speed up what amounts to the computation of the loss for each candidate rule. Though from what we saw earlier, computational irreducibility is likely to limit this. Another possibility is that much as we talked about before, we could try to use neural nets to guide us in which random changes to make at each generation. But while computational irreducibility probably helps us in making things effectively random enough that we don't get stuck, it makes it difficult to have something like a neural net successfully tell us which way to go. Science as Narrative in many ways, one can view the essence of science, at least as it's traditionally been practiced, as being about taking what's out there in the world and somehow casting it in a form we humans can think about. In effect, we want science to provide a human accessible narrative for what happens, say, in the natural world. The phenomenon of computational irreducibility shows us that this will ultimately often not be possible. But whenever there's a pocket of computational reducibility, it means that there's some kind of reduced description of at least some part of what's going on. But is that reduced description something that a human could reasonably be expected to understand? Can it, for example, be stated succinctly in words, formulas, or computational language? If it can, then we can think of it as representing a successful kind of human level scientific explanation. So can AI help us automatically create such explanations? To do so, it must, in a sense, have a model for what we humans understand and how we express this understanding in words and so on. It doesn't do much good to say here are 100 computational steps that produce this result. To get a human level explanation, we need to break this down into pieces that humans can assimilate. So, for instance, this is an example of a mathematical proof generated by automated theorem proving. Got lots of different steps, 100 steps here. And a computer can readily check that this is correct in that each step follows from what comes before. But what we have here is a very non human thing about which there's no realistic human narrative. What would it take to make such a narrative? Essentially, we'd need waypoints that are somehow familiar, perhaps famous theorems that we readily recognize. Of course, there may be no such things, because what we may have here is a proof that goes through uncharted metamathematical territory. So, AI assisted or not, human mathematics as it exists today may just not have the raw material to let us create kind of a human level narrative. In practice, when there's a fairly short metamathematical distance between steps in a proof, it's realistic to think that a human level explanation can be given. What's needed is very much like what WolframAlpha does when it produces step by step explanations of its answers. Can AI help with this? Potentially using methods like the ones we talked about a little while ago in AI assisted multicomputation above, particularly the method where we're saying which move do we make, which move does the AI make next? Rather than which character should be produced next. By the way, our efforts with Wolfram language help too, because our whole idea of our computational language is to kind of capture common lumps of computational work as built in constructs. In a sense, the process of designing the language is precisely about identifying human assimilable waypoints for computations. Computational irreducibility tells us that we'll never be able to find such waypoints for all computations. But our goal is to find waypoints that capture current paradigms and current practice, as well as define directions and frameworks for extending these. Though ultimately what we humans know about is something that's determined by the state of human knowledge as it's historically evolved. So proofs and computational programs, language programs are two examples of structured scientific narratives. A potentially simpler example, aligned with a mathematical tradition for science is just a pure formula, like saying it's a power law, it's a sum of exponentials, and so on. One Can AI help with this? Well, there's a function like find formula language that's already using machine learning inspired techniques to take data and try to produce kind of a reasonable formula for it. Like let's say we feed it the first hundred primes. That's the result that we get some formula involving n to the 1.2, we go to 10,000 primes, it'll produce a more complicated and bizarre formula. Or let's say we ask what's the relationship between GDP and population for countries, we'll get some weird formula about population to the power 0.033 and population growth percentage per year, et cetera, et cetera, et cetera. What, if anything, do formulas like this mean? Well, it's a bit like with proof steps and so on. Unless we can connect what's in the formula with things we know about, whether in number theory or economics, it'll usually be difficult to conclude much from them them except perhaps in some rare cases where one could say yes, that's a new useful law. Like for example, we can just go ahead and use fine formula to take the data on. Planets that we have in our knowledge base and derive Kepler's third law, where we just work out this formula and we get something which says the period in days to PowerPoint 7. Actually the true result is 2/3 not too far away. So there's an even more minimal example of this kind of thing in recognizing numbers. So if you just type a number into orthmalpha, it'll try and give you what possible closed forms for the number might be. So we typed in 12.1234 and said that's about 7 times square root of 3 or it's about 7 PI minus 31 over PI, or it's about Champanone's constant plus 12 or its various other kinds of things. There are all sorts of trade offs here, some very AI informed. What's the relative importance of getting more digits right compared to having a simpler formula? What about having simple numbers in the formula compared to some obscure mathematical constants like PI versus Champanone's number? Well, we set the system up wolf math of 15 years ago. We use the negative log frequency of constants that as they appear in the how often they appear in the mathematical literature as a proxy for their kind of information content. With modern LLM techniques, it may be possible to do a more holistic job of finding what amounts to a good scientific narrative for a number. But let's return to things like predicting the outcomes of processes like cellular automaton evolution. Before we talked about getting neural nets to do this prediction, we viewed this essentially as a black box approach. We wanted to see if we could get a neural net to successfully make predictions. But we weren't asking to get a human level understanding of those predictions. It's a ubiquitous story in machine learning. One trains a neural net to successfully predict, classify, whatever, but if one looks inside, it's very hard to tell what's going on. So here's an example of the the final result of applying an image identification network to a picture of a cat. It says, yes, it's a domestic cat, but. But here are the sort of the intermediate thoughts generated after going through about half the layers in the network. And it's a bunch of kind of sort of cat cut out type pictures of pieces of ears and, and, and things that highlight the eyes and so on. Maybe something here is a definitive signature of catness, but it's not part of our current scientific lexicon. So we can't usefully use it to develop a scientific narrative that explains how the image should be interpreted. But what if we could reduce our images to just a few parameters, say using the autoencoder we talked about before? Conceivably, we should set things up so that we'd end up with interpretable parameters, or in other words, parameters where we can give a narrative explanation of what they mean mean. So for example, we could imagine using something like an LLM to pick parameters that somehow align with words or phrases, pointiness, fractal dimension, and so on, that appear in explanatory text from around the web. And yes, these words or phrases could be based on analogies. Cactus shape, cirrus, cloud like. And something like an LLM could creatively come up with these names. But in the end, there's nothing to say that a pocket of reducible computational reducibility picked out by a certain autoencoder will have any weight to be aligned with concepts, scientific or otherwise, that we humans have yet explored or so far given words to. Indeed, in the ruliad, this kind of entangled limit of all possible computational processes that we've talked about a lot in connection with fundamental physics, it's overwhelmingly likely that we'll find ourselves in inter concept space, unable to create what we would consider a useful scientific narrative. I had a thing I wrote a few months ago, sort of showing interconcept space of sort of images that start with a cat and a party hat and look at all the kinds of images that serve around that image in concept space. And what we discover is that the vast majority of images are not described, but have no sort of human narrative description. They are an interconcept space. Maybe all but one part in 10 to the 600 of the space of images that I explored, there has a human concept associated with it. The rest is unexplored interconcept space. But so as I say, when we're trying to figure out sort of is there a scientific narrative for things we'll often find from a neural net that we are in the. The internal thoughts of the neural net are sort of operating in interconcept space? Well, this whole story depends a bit on just how we constrain what we're looking at. We might implicitly define science to be the study of phenomena for which we have at some time successfully developed a scientific narrative. And in this case it's of course inevitable that such a narrative will exist. But even given a fixed method of observational measurement, it's basically inevitable that as we explore computational irreducibility will lead to surprises that break out of whatever scientific narrative we were using. Or in other words, if we're really going to discover new science, then AI or not, we can't expect to have a scientific narrative based on pre existing concepts. And perhaps the best we can hope for is that we'll be able to find pockets of reducibility and that AI will understand enough about us and our intellectual history that it'll be able to suggest a manageable path of new concepts that we should learn to develop a successful scientific narrative for what we discover. In fact, in our recent efforts, which are, it's a very hard problem of building a kind of AI tutor to teach people things, something that we're sort of somewhat uniquely able to do because of the whole technology stack we have, we get to kind of live this question very much in the trenches, so to speak. Finding what's interesting. So a central part of doing open ended science is figuring out what's interesting. Let's say one just enumerates a collection of cellular automata, as I seem to do so often in my life. They're ones that just die out, make uniform patterns. They don't seem very interesting the first time one sees a nested pattern generated by solar automaton. It might seem interesting, as it did to me in 1981 1, but pretty soon it comes to seem quite routine. Like the array of pictures we have here has a whole bunch of nested examples, as well as ones that look more complicated than that. So at least as a matter of sort of basic ruliology, what one ends up looking for is a surprise, qualitatively new behavior one hasn't seen before. If one's concerned with specific applications, say to modeling particular systems in the world, then one might instead want to look at rules with certain structures, structure whether or not their behavior abstractly seems interesting. The fact that one can expect surprises and indeed be able to do useful, truly open ended science at all is a consequence of computational irreducibility. And whenever there's a lack of surprise, it's basically a sign of computational reusability. And this makes it plausible that AI and neural nets could learn to identify at least certain kinds of anomalies or surprises. And Thereby discover some version of what's interesting. Usually the basic idea is to have a neural net learn the typical distribution of data and then to identify outliers relative to this. So, for example, we might look at a large number of cellular automaton patterns to learn their typical distribution. Then plot a projection of this onto a 2D feature space indicating so of where certain specific patterns lie. But the patterns that lie kind of at the show up in parts of the distribution where their probabilities are high. Others show up in where their probabilities are low. The cases where the probabilities are low, they're the outliers. Here are a few examples. Are these outliers interesting? Well, depends on your definition of interesting. And in the end, that's in the eye of the beholder here. The beholder is a neural net. And yes, these particular patterns wouldn't be what I would have picked. But relative to the typical patterns, they do seem at least somewhat different. And presumably it's basically a story like the one with neural nets that distinguish pictures of cats and dogs. Neural nets make at least somewhat similar judgments to the ones we do, perhaps because our brains are structurally like neural nets. Okay, but what does a neural net intrinsically find interesting? If the neural net is trained, then it'll be very much influenced by what we can think of as the cultural background it gets from this training. But what if we just set up neural nets with a given architecture and picks their weights completely at random? Let's say they're neural nets that compute functions that say f of x. Then you can ask, well, what functions do they compute? And here are examples. These are neural nets with completely random weights and activation functions that are like ramp, amp or tant or sine. And what we see is the whole, a whole kind of mess of curves here. And, but, but roughly these curves reflect the underlying activation functions like nets that randomly chosen where with randomly chosen weights that have ramp activation functions have kind of straight lines in the function, because ramp has straight lines. Tanch has kind of these sort of of progression from one level to another kinds of things. Sine has wiggliness, like a sine. So the functions that come out very much reflect the underlying activation functions that appear at the nodes of the neural nets. But what we see, a bit like a random walk process, more extreme functions are less likely to be produced with neural nets with random weights. So can be thought of as sort of intrinsically more surprising for neural nets. But okay, surprise is only one potential criterion for interestingness. There are others. And to Get a sense of this. We can look at various kinds of constructs that can be enumerated and where we can ask which possible ones we consider interesting enough that, for example, we've studied them, given them specific names, recorded them in registries, and so on. So, as a first example, let's consider a family of hydrocarbon molecules, alkanes. Any such molecule can be represented by a tree graph with nodes corresponding to carbon atoms and having valence at most 4. There are a total of 75 alkanes with 10 or fewer carbons, and all of them typically appear in standard lists of chemicals and in our Wolfram knowledge base. But with 10 carbons only, actually, it should be 11 carbons only. Some alkanes are interesting enough that they're listed. For example, in our knowledge base. Aggregating different registers, one finds more alkanes limited listed, but I guess this should be 12 carbons out of at least 42 out of 159 always seem to be missing. So here's an example. The ones that are not highlighted here are the alkanes that you don't find, for example, in our knowledge base. They were not interesting enough to make it into our knowledge base, even though in principle, those are molecules one can construct sucked. If you just sort of start enumerating molecules and say, which is going to be a great drug for curing this disease, that one would be considered interesting. But here, when we look at ones that have already been considered interesting, we've only considered some of these molecules to be interesting. So what makes some of these alkanes be considered more interesting in this sense than others? Operationally, it's a question of whether they've been studied, say, in the academic literature. But what determines this? Partly, it's a matter of whether they occur in nature, sometimes in petroleum or coal. Alkanes form through what amount to random reactions where unbranched molecules tend to be favored. But alkanes can also be produced in biological systems through careful orchestration, say by enzymes. But wherever they come from, it's as if the alkanes that are more familiar are the ones that seem more interesting. Interesting. So what about surprise? Whether a surprise alkane, say, made by explicit synthesis in a lab, is considered interesting probably depends first and foremost on whether it's identified to have interesting properties. And that in turn, tends to be a question of how its properties fit into the whole web of human knowledge and technology. So can AI help in determining which alkanes we're likely to consider interest? Traditional computational chemistry perhaps sped up By AI can potentially determine the rates at which different Alkanes are randomly produced in a quite different direction. Analyzing the academic literature, say, with an LLM, can potentially predict how much a certain Alkane can be expected to be studied or talked about. And this is particularly relevant for drug candidates, whether there are existing hints of if only we could find a molecule that does blank think that one could pick up from things like the academic literature. So, another example, let's consider mathematical theorems. Much like with chemicals, one can in principle enumerate possible mathematical theorems by starting from axioms and then seeing what theorems can progressively be derived from them. So if one starts from the axioms of logic and goes just a couple of steps, this is what happens in deriving all possible theorems. There's a big mess of points here. There are vast number of uninteresting and often seemingly very pedantic theorems here. But among all of these, there are two that are interesting enough that they've typically been given names. In this particular case, they're the idempotence laws in textbooks of logic. So is there any way to determine whether a theorem will be given a name? One might have thought that would be a purely historical question. But at least in the case of logic, there seems to be a systematic pattern. Let's say one enumerates theorems of logic, starting with the simplest and going on in lexicographic order. Most theorems in the list will be derivable from earlier ones, but a few will not, and these turn out to be basically exactly the ones that are typically given names. This is diagram that kind of shows that. So, in other words, at least in the rather constrained case of basic logic, the theorems considered interesting enough to be given names are the ones that surprise us with new information. If we look more generally in metamathematical space, we can kind of get some empirical idea of where theorems that have been considered interesting lie. We're seeing all kinds of Ramsey's theorem and the sum of a geometric series and so on, all sort of laid out in mathematical space. Could an AI predict this? We could certainly create a neural net trained from existing literature of mathematics and its few million stated theorems. And we could then start feeding this neural net theorem is found by systematic enumeration and asking it to determine how plausible they are as things that might appear in mathematical literature. And in our systematic enumeration, we could even ask the neural nets to determine what directions are likely to be interesting. Like when we were talking about finding paths for theorem Proving in traversing multi way systems before. But when it comes to finding genuinely new science or math, there's a problem with this because a neural net trained from existing literature is basically going to be looking for more of the same. It's sort of much like in the typical operation of peer review. What it'll accept, that is what's mainstream and not too surprising. So what about the surprises that computational irreducibility inevitably implies will be there? By definition, they won't be easily reducible to what's been seen before. Yes, they can provide new facts and they may even have important applications, but there often won't be, at least at first, a human accessible narrative that reaches them and what it'll take to create that for us humans to internalize. To internalize some new concept that eventually becomes familiar earlier. And yes, as we discussed above, if some particular new concept or say new theorem seems to be a nexus for reaching things, that becomes a target for a concept that's worth adding but in Oops wanted to do. But in the end there's kind of a certain arbitrariness in which new facts or new directions we want to internalize. Yes, if we go in a particular direction, it may lead us to certain ideas or technology or activities, but we abstractly we don't know what direction we might go in. That is right, at least in the first instance. It seems like a quintessential matter of human choice. There's a potential wrinkle though. What if our AIs know enough about human psychology and society that they can predict what we'd like? Well, at first it might seem like they could then successfully pick directions, but once again, computational irreducibility blocks us because ultimately we can't know what we'll like until we get there. We can relate all this to generative AI, for example, for images or text. I mean, at the outset we might imagine enumerating images that consist of arbitrary rays of pixels, but an absolutely overwhelming fraction of those won't be interesting at all to us. They'll just look like random noise. By training a neural net on billions of human selected images, we can get it to produce images that are somehow generally like what we find interesting. Sometimes the images that are produced will be recognizable to the point where we'll be able to give kind of a narrative explanation of what they look like. But very often we'll find ourselves out and interconcept space with images like these. It's hard for me to describe these verbally because they're an interconcept space. And we don't have kind of human words to say what these images are like. It's kind of stripes and colors and blotches and, I don't know, hard to say. Are these interesting? Hard to say. Scanning the brain of a person, looking at them, we might notice some particular signal, and perhaps an AI could learn to predict that. But inevitably, that signal would change if some type of interconcept image became popular and started, say, to be recognized as some kind of art that people are familiar with. And in the end, we're back to the same point. Things are ultimately interesting if our choices as a civilization make them. So there's no abstract notion of interestingness that an AI or anything can go out and discover ahead of our choices. And so it is with science. There's no abstract way to know what's interesting out of all the possibilities in the ruliad, that's ultimately determined by the choices we make in colonizing the ruliad. But what if, instead of going out into the wilds of the ruliad, we stay close to what's already been done in science and what's already been deemed interesting? Can AI help us extend what's there as a practical matter, matter at least, when supplemented with our computational language as a tool? The answer is at some level, surely yes. And for example, LLMs should be able to produce things that follow the pattern of academic papers, with dashes of originality coming from whatever randomness is used in the LLM. But how far can such an approach get? The existing academic literature is certainly full of holes. Phenomenon A was investigated in system X and B and Y, but not vice versa. And we can expect that the AIs and LLMs in particular, can be useful in identifying these holes and in effect, planning what science is, by this criterion, interesting to do. And beyond this, we can expect that things like LLMs will be helpful in mapping out sort of usual and customary paths by which science should be done. You know, when you're analyzing data like this, one typically quotes such and such a metric. When you're doing an experiment like this, you typically prepare a sample like this. When it comes to actually doing the science, though, our computational language tools, together with things like computationally controlled experimental equipment, will presumably be what's usually more central. But let's say we've defined some major objective for science. You know, figure out how to reverse aging, or a bit more modestly, you know, solve bionics. In giving such an objective, we're specifying something we consider interesting, and then the problem of getting to that objective is, at least conceptually like finding a proof of a theorem or a synthesis pathway for a chemical. There are certain moves we can make, and we need to find out how to string these together to get the objective we want. Inevitably, though, there's an issue with multicomputational irreducibility. There may be an irreducible number of steps we need to take to get to the result. And even though we may consider the final objective interesting, there's no guarantee that we'll find the intermediate steps even slightly interesting. Indeed, in many proofs, as well as in many engineering systems, one may need to build an immense number of excruciating details built on an immense number of excruciating details to get the final interesting result. But let's talk more about the question of what to study, or in effect, what's interesting to study. Study Normal science tends to be concerned with making incremental progress, remaining within existing paradigms, but gradually filling in and extending what's there. Usually the most fertile areas are on the interfaces between existing, well developed ones. At the outset, it's not at all obvious that different areas of science should ultimately fit together at all. But given the concept of the ruliad as the ultimate underlying structure, this begins to seem less surprising still. To actually see how different areas of science can be knitted together, one will often have to identify, perhaps initially quite surprising analogies between very different descriptive frameworks. A decidable theory in metamathematics is like a black hole in physics. Concepts in language are like particles in rulial space, and so on. Well, this is an area where we can expect LLMs to be helpful. Having seen the linguistic pattern of one area, one can expect them to be able to see its correspondence to another area, potentially with important consequences. But what about fresh new directions in science? Historically, these have often been the results of applying some new practical methodology, say, for doing a new kind of experimental measurement that happens to open up some new place to look where people have never looked before. But usually one of the big challenges is to recognize that something one sees is actually interesting. And to do this often in effect, involves the creation of some new conceptual framework, framework, paradigm. So can AI as we're talking about it here, be expected to do this? It doesn't seem likely. AI is typically something trained on existing human material, intended to extrapolate directly from that. It's not something built to go out into the wilds of Riad, far from anything already connected to humans. But in a sense, that's the domain of arbitrary computation and of things like the simple programs we might enumerate or pick at random in ruliology. And yes, by going out into the wilds of the ruliad, it's easy enough to find fresh new things not currently assimilated into science. The challenge, though, is to connect them to anything we humans currently understand or find interesting. And that, as I've said before, is something that quintessentially involves human choice and the foibles of human history. They're an infinite collection of paths that could be taken. And indeed, in a society of AIs, there could be AIs that pursue a certain collection of them. But in the end, what matters to us humans and the enterprise we normally call science is our internal experience. And that's something we ultimately have to form for ourselves. Beyond the exact sciences. In areas like the physical sciences, we're used to the idea of being able to develop broad theories that can do things like make quantitative predictions. But there are many areas, for example, in the biological, human, and social sciences, that have tended to operate in much less formal ways, and where things like long chains of successful theoretical inferences are largely unheard of. So might AI change that? There seem to be some interesting possibilities, particularly around the new kinds of measurements that AI enables. How similar are those artworks? How close are the morphologies of those organisms? How different are those myths? These are questions that in the past, one mostly had to address by writing an essay. But now AI potentially gives us a path to make such things more definite and in a sense, more quantitative. In some sense quantitative. Typically, the key idea is to figure out how to take unstructured raw data and extract meaningful features from it that can be handled in formal, structured ways. And the main thing that makes this possible is that we have AIs that have been trained on large corpora that in effect represent what's typical in our world, and which have, in effect, formed definite internal representations of the world, in terms of which things can, for example, be described, say, by lists of numbers, as we talked about before in connection with autoencoders and so on. What do those numbers mean? Mean? At the outset, we typically have no idea. They're just the output of some neural net encoder. But what's important is that they're definite and repeatable. Given the same input data, one will always get the same numbers. And what's more, it's typical that when data seems similar to us, it'll tend to be assigned nearby numbers. In an area like physical Science, we expect to build specific measuring devices that measure quantities we know how to interpret. But AI is much more of a black box. Something is being measured, but at least at the outset, we don't necessarily have any interpretation of it. Something we'll be able to do. Sometimes we'll be able to do training that associates some description we know, so that we'll at least get a rough interpretation, like in a case like sentiment analysis. But often we won't. You know, it has to be said that something similar can happen even in physical science. Let's say we want to test whether one material scratches the surface of another another. Presumably we can interpret that as some kind of hardness of the material, but really it's just a measurement that becomes significant if one can successfully associate things with the results of that measurement. One thing that's particularly notable about AI measurements is how they can potentially pick out small signals from large volumes of unstructured data. We're used to having methods like statistics to do similar things on structured numerical data data. But it's a different story to ask from billions of web pages whether, say, kids who like science typically prefer cats or dogs. But given an AI measurement, what can we expect to do with it? None of this is very clear yet, but it seems at least possible that we could start to find formal relationships. Perhaps it will be a quantitative relationship involving numbers. Perhaps it will better be represented by a program that describes a computational process by which one measurement leads to other others. It's been common for some time in areas like quantitative finance to find relationships between what amount to simple forms of AI measurements and be concerned mainly with whether they work rather than why they work or how one might narratively describe them. That's the story of many hedge funds is operating with those kinds of measurements. In a sense, it seems rather unsatisfactory to try and build science on black box AI measurements that one can't interpret separate. But at some level, this is just an accelerated version of what we often do. Say with everyday language, we're exposed to some new observational measurement and eventually we invent words to describe it. You know, it looks like a fractal, for example, and then we can start reasoning in terms of it. But AI measurements are potentially a much richer source of formalizable material. But how should we do that? Formalization? Computational language seems to be the key. And indeed we already have examples in Wolfram language where functions like image identifier, text cases, or for that matter, LLM function can effectively make AI measurements. But then we can take their results and work symbolically with them. In physical science, we often imagine that we're working only with objective measurements, though I recently worked a lot on what I call observer theory. That implies that actually even our nature as observers is crucial. Even that can. But AI measurements seem to have a certain immediate subjectivity, and indeed their details associated with the particulars of a neural net encoder will be different every. For every different AI we use. But what's important is that if the AI is trained on very large volumes of amounts of human experience, there'll be a certain robustness to it. In a sense, we can view many AI measurements as being like the output of a societal observer that uses something like the whole mass of human experience, and in doing so gains a certain centrality and inert. What kind of science can we expect to build on the basis of what a societal observer measures? For the most part, we don't yet know. There's some reason to think that, as in the case of physics and metamathematics, such measurements might tap into pockets of computational reducibility. And if that's the case, we can expect that we'll be able to start doing things like making predictions, albeit perhaps only for the results of AI measurements, which we'll find hard to interpret separate. But by connecting such AI measurements to computational language, there seems to be the potential to start constructing sort of formalized science in places where it's never been possible before, and in doing so, to extend the domain of what we might call the exact sciences. By the way, another promising application of modern AI is in setting up sort of repeatable Personas, entities that effectively behave like humans with certain characteristics, but on which large scale repeatable experiments of the kind typical in physical science can be done. Done. Okay, we get to the last section of what I wrote, which is called so can AI solve science? @ the outset, one might be surprised that science is even possible. Why is it that there is regularity that we can identify in the world that allows us to form scientific narratives? Indeed, we now know from things like the concept of the ruliad that computational irreducibility is inevitably ubiquitous. Ubiquitous, and with it, fundamental irregularity and unpredictability. But it turns out that the very presence of computational irreducibility necessarily implies that there must be pockets of computational reducibility where at least certain things are regular and predictable. And it is within these pockets of reducibility that science fundamentally lives, and indeed that we try to operate and engage with the world. How does this relate to AI? Well, the whole story of things like trained neural nets that we've discussed is a story of leveraging computational reducibility, and in particular, computational reducibility that somehow aligned with what human minds also use. In the past, the main way to capture and capitalize on computational reducibility was to develop formal ways to describe things, typically using mathematics and mathematical formulas formulas. AI, in effect, provides a new way to make use of computational reducibility. Normally, there's no human level narrative to how it works. It's just something that within a trained neural net that we manage to capture certain regularities that allow us, for example, to make certain predictions. In a sense, the predictions tend to be very human style, often looking roughly right to us, even though the level of precise formal detail, they're not quite right, right. And fundamentally, they rely on computational reducibility. And when computational irreducibility is present, they more or less inevitably fail. In a sense, the AI is doing shallow computation, but when there's computational irreducibility, one needs irreducible deep computation to work out what will happen. And there are plenty of places, even working with traditional mathematical structures, where what AI does won't be sufficient for what we expect to get out of science. But there are also places where AI style science can make progress even when traditional methods cannot. If one's doing something like solving a single differential equation, say, and precisely, AI probably won't be the best tool. But if one's got a big collection of equations, say for something like robotics, AI may successfully be able to give a useful rough estimate of what will happen, even when traditional methods will get utterly bogged down in detail details. It's a general feature of machine learning and AI techniques that they can be very useful if an approximate sort of 80% answer is good enough, but they tend to fail. One needs something more perfect and precise, and there are quite a few workflows in science and probably more that can be identified where this is exactly what one needs. The kind of 80% level. Pick out candidates for something, identify a feature that might be important, might be important, suggest a possible question to explore. Flaw. There are clear limitations, though, particularly whenever there's computational irreducibility. In a sense, the typical AI approach to science doesn't involve explicitly formalizing things, but in many areas of science, formalization is precisely what's been most valuable and what's allowed towers of results to be obtained. And in recent times, we have the kind of powerful idea of formalizing things computationally and in particular in using computational language to do this. What I've seen spent many decades working on. But given such a computational formalization, we're able to start doing irreducible computations that let us reach discoveries we have no way to anticipate. We can, for example, enumerate possible computational systems or processes and see fundamental surprises. In typical AI, there's randomness that gives us a certain degree of originality in our exploration, but it's of a more fundamental, it's of a fundamentally lower level than we can reach with actual irreducible computation. So what should we expect for AI and science going forward? We've got, in a sense, a new and rather human like way of leveraging computational reusability. It's a new tool for doing science destined to have many practical uses in terms of fundamental potential for discovery, though it pales in comparison to what we can build from the computational paradigm and from irreducible computations that we do. But probably what will give us the greatest opportunity to move science forward is to combine the strengths of AI and have the formal computational paradigm, which, yes, is part of what we've been sort of rigorously pursuing in recent years with Wolfram language and its connections to machine learning and now LLMs. And that's all I wrote last week. And so now I'd be happy to turn this over to some Q and A here about, about this topic. Gregory comments. AI is a great tool, but can't replace the critical thinking involved in science, mathematics or computer science. Just my opinion says Gregory here. Yeah, I think that's, that's true. It's, it's a tool. And, and the thing about tools is what are you going to do with them? And this is where kind of the, the what problem are you trying to solve? What question are you trying to answer? That is a deeply human thing because there's an infinity of possibilities and it's up to us to decide we want to go this way rather than that way. And that remains something that is, is, needs a human to be involved with. I think also my feeling is the, the biggest accelerator to science right now is the kind of computational paradigm. I like to think our computational language is an important part of that. As a. Having defined a sort of computational way to describe the world, the thing that is critically important there is to sort of do the human computational thinking, to try and take the things you're vaguely thinking about and formulate them in a kind of precise computational way so that you can get kind of the superpower of computational language to kind of let you figure out what's what. What let the Computer help you figure out the consequences of what you've come up with. And that's, that's this sort of process of, of formalizing the world in computational language and with computational thinking. I think that's sort of the biggest accelerator and the kind of linguistic interface that we have with LLMs can certainly help provide an on ramp to that. But in the end it's for us to decide what direction to go. What inventions do we want to make? What things about science do we care about exploring what, what kinds of patterns of pixels do we think of as interesting art and so on. Atori asks, is tool calling enough for LLMs to do deep computation? That's an interesting question. I mean we've been doing a lot and other people have been working on this also in terms of knitting together kind of our computational language with the kind of innards of a neural net AI. AI. We don't really know how to do that in a fantastic way yet. So we're getting better and better technology for having the AI say, and here's where I need to use a tool and taking the results from that tool and ingesting them back again. But you know, I think one of the things to realize about LLMs, for example, is that the, you know, LLMs are trained on all of this human text and so on, and that text contains lots of things like quantitative facts. You know, what is the density of tungsten or something it really isn't necessary to encode somewhere in some collection of neural net weights the density of tungsten. That's something you'd be much better off asking Wolfram Alpha, going to our knowledge base and just get, basically getting the computational knowledge of that density of tungsten or how the, or the, or the density of water at a certain pressure and so on that we can compute with some algorithm. Algorithm. You'd be much better off going to our sort of computational system to get that answer that the particular one with pressure of water. Density of water you just won't be able to get by just having read it in some piece of text. But the question is, how do you untangle the parts that the LLM uniquely does the kind of linguistic interface it provides to kind of flowing human language, how do you disentangle that from the raw facts? How do you make it so that you have a thin linguistic, linguistic interface and then interfacing to kind of the, the use of the tool to actually get computations done and, and get computational knowledge? That's a big challenge. When that's successfully done, we will have very small, thin LLMs that provide thin linguistic interface. I don't think language is that complicated. In fact, I think LLMs kind of show us the extent to which it's not that complicated and it'll be that thin interface that then goes down to kind of computational knowledge where you really get to compute things and build kind of big towers of results on that. That's, that's what I think the future is, but I think it's difficult to get to that point. Actually. I just had some ideas recently that might, might help us get there, but, but they're very embryonic so far. So. Let's see. Hashir asks, couldn't we use AI to mine patterns and common paths across diverse disciplines? This sort of thing is not possible for humans now because each domain has become highly specialized. Yes, I, I mentioned that a little bit. I think one of the, the wows that we will increasingly see with, with LLM is this, you know, there's a pattern of how things work in field A. It can be mapped into field B. I have to say, some part of my life has consisted in knowing enough about different fields to be able to sort of import the ideas of fundamental physics, let's say, into thinking about distributed computation or import the ideas of metamathematics into physics, things like this. In a sense, one of the issues with science as it exists inside of the academic world right now is that, yes, it has become hyper specialized and people know about only their small corner. And much of the most fertile things to figure out come by, for example, making analogies between different fields. So you have to know a bunch to be able to make those analogies. You have to know about a lot of different fields. The LLMs know about it in the sense that LLMs know about things. I will say that I think, think for humans, the value of kind of knowing broadly about things has gone way up in modern times as we have more ability to sort of access knowledge computationally to compute things, to have LLMs and so on. The value of just the human knowing about a bunch of different areas and being able to define what's interesting, pick up these analogies is, is, is something that I think is, is really valuable and it's sort of a. There's been this trend towards specialization in education. I think that trend at least should reverse with sort of the rise of computation and AI and so on. And I think, But I think that getting the hint from, from LLMs and AI and so on, of you should try looking at this analogy between this and this that's something that I can imagine having happen, and I, I haven't, it hasn't yet happened to me. I've tried a few times, times looking for those kinds of things. I, I haven't. Maybe I haven't known where to look. Maybe the, the challenge is even. What should you ask the LLM? Is there an analogy between this and that? Because you just ask the LLM find all analogies in science. I don't think you'll get a very good result. Maybe one day you'll get a better result. Maybe one day you'll be able to do essentially a systematic enumeration. You know, given all these fields of science, systematically enumerate what, what the analogies that the LLM might find between these different areas are. Maybe something really interesting. That's a, that's a good experiment to try, actually. I should try that. Let's see. Okay. HP asks, do you think AI researchers can make a smarter AI if they stop trying to mimic human thought with AIs? No, I don't think so. Because I think the whole story of what makes us think AIs are smart is that they're doing things like what we do. It's very easy to go out into the computational universe, find a program, run it. It does incredibly complicated things. But we say, that's not smart, that's just doing complicated things. That's just doing a computation. We don't care about what that's doing. For it to be smart, in a sense that we humans care about, it needs to be something that's aligned with the way we humans think about things. In the science that I've done over the course of many years, things like the principle of computational equivalence of have shown up. And that kind of tells us that if we look at the kind of processes that go on in systems, in nature, we compare them with the processes that go on in our brains. They're really all of equivalent computational sophistication. We look at all those electrical neuron firings in our brains. We look at the sort of fluid motion in the atmosphere, the weather. We can think of having a mind of its own that's sort of computationally just as sophisticated as human minds. Thought are driven by kind of the electrical impulses in our brains. And so it's the sort of the idea of having something computationally sophisticated that's easy to get. The question is whether that computationally sophisticated stuff is aligned with the things that from our history, our civilization, even our biology are things that humans have decided we care about and that's the thing that really is, is that that's sort of how we go, go beyond that. There's this sort of big universe of possible computation. Then there's the part we care about it. In a sense, what we've been trying to do in our computational language is to kind of capture those parts of, of kind of the computational universe that we can, that we humans have chosen to colonize, have chosen to, to think about, give names to, and so on. So I think that it is very easy to make a very computationally smart quotes AI, but it will be one that we don't care about. The challenge is this alignment with things that we humans care about. Okay, Educon asks, what about the neural nets perception of time? Do you feel it could be leveraged for new discoveries ways? How do humans perceive time? How does time work in the universe? I think we kind of know from the stuff we've done in fundamental physics in just the last few years what time fundamentally is. Time is kind of the progression of computation. So the universe starts from some state, it computes the next state, it computes the next state. That progression of computations that represents the passage of time in the universe. And it's the same thing in our brains. The successive states of the neurons in our brains as those successive states are computed one from the the previous. That is our perception of the passage of time. So we can think of an AI as having the same kind of perception of the passage of time from its sort of internal computational processes that are going on now. You know, will we be able to run, run the things that. Will we be able to run things like our human thinking, but just a lot faster? The answer is yes. You know, a typical brain sort of cycle time is maybe milliseconds. The typical cycle time of modern computers is nanoseconds. In some sense, we might be able to make things go a million times faster. What will it be like light, when there are kind of AIs that do even human like things a million times faster than we do, for example, their perception of the world will be different. The speed of light will be a lot more important to an AI thinking a million times faster than we do than it is to a human, It'll be the. It's a good question. You know, as we sort of have bigger brains and so on, as we have the ability to access tools that give us sort of the equivalence of bigger brains, what different things will happen happen? You know, for example, we humans, typical languages might have, I don't know, 40 50,000 words that we use for kind of the concepts that we typically think about. As we got, as we get a bigger brain, maybe we can have a million words. Maybe that's a thing that lets us think in a somewhat different way. We don't know. You know, the big invention of our species was language. This idea that we can take kind of the things in the world and, and present them using this kind of compositional modular of structure that is language. And, you know, we don't know whether something qualitatively different might happen when we get to something which is, is kind of the, the bigger brain, so to speak, where there are more concepts that we can kind of hold in our mind. My guess is it won't be that exciting. But there's computational irreducibility in the way and I don't know what to expect. And there will inevitably be surprising is which I can't predict. Let's see. Wyatt is asking, I wonder if AI has tried to define a perfect number base for math. I Wonder if base 10 is holding us back. You know, there is a sense in which that kind of thing happened in, in the course of mathematics. People would, they kind of had the idea of variables like X, but it was difficult to disentangle the X from Roman numerals. An X in roman numerals means 10. It's, you know, and, and, you know, V means 5 and so on. People for a long time wrote numbers with letters and that made it hard to have to break off the idea of an algebraic variable that was a. It required a notational innovation of having Hindu Arabic numerals separate from sort of the things you, the letters you might use to represent variables, to discover variables. So, you know, it is very likely that there are things about the way that we kind of represent knowledge and so on that hold us back. Whether the AIs can give us kind of different representations which we understand, which we'll understand is not really cool. Clear. And I think in, you know, if you. If. But in terms of the specifics of, of numbers, a few things to understand there. I mean, computers are fundamentally based on binary. You know, is there. Are there a bunch of electrons here or not 1 or 0? And neural nets actually are based on numbers with many digits. Like the weights in a neural net are, you know, 2.758246. And so. And as you change even the high order digits there, eventually you change that six to a seven in the last digit of that weight. Eventually that will cause the neural net to make, to say, red rather than blue. At some point these small changes will eventually become, will eventually affect the output of the neural net. And so it's an interesting question whether you need all of those precise numbers in a neural net. One of the funny things about neural nets is, is training them is hard. Well, a couple of things to say neural nets, you have a certain architecture of neural net. You might say let's be very careful and let's build this in a very special way. But it turns out if you just bash the neural net hard enough and you give it enough quintillion of training examples and so on, it'll kind of learn almost anything. One exception to that is that the, the overall architecture might have to represent sort of the geometry of the original data. If you're dealing with images, it's good to have sort of two dimensional arrays of neur. If you're dealing with language, it's good to have this kind of one dimensional sequence which has a lot of potential, look back in the sequence and so on. But beyond that, the meta learning from machine learning in the last decade or two has been just bash it harder and harder and it'll learn almost anything. But once it's learned, it's really hard to get it to learn. It kind of has to find these little paths, find these pathways to a place where it has learned something. But then once it's learned, you can often throw away a lot of the neural net. You can prune down the neural net, kind of take these numbers that are very precisely specified and say, ah, throw out most of that. I'll just keep a few bits of that number. And that's, that's been a thing that's been, been found. There's an interesting question whether you can just go intrinsically from something that starts with just a few bits of information rather than having these, these continuous numbers at every, in every weight of the neural net. It's been sort of a tradition of neural nets different from the tradition of digital computing. That one sort of starts with something which intrinsically involves real numbers that can have arbitrary numbers of digits. So there's sort of some interesting things there about the extent to which brains are continuous and neural nets are continuous or not. My guess is that, that, that, that really all that's there, that's important is something discrete, that the continuous thing is just there because it makes some math easier. And it's been the path that allows, allows us to see how to make this stuff work, but it's not necessary to actually making it work. Let's see, the 1 and 0 asks, would the question of what's interesting be relevant even when computational resources are virtually endless? If the AI can reliably create valid theories, then maybe it can be just sent out randomly to do science about everything. And then a meta AI can be used to find the interesting ones. Well, the, the, the problem is that what's interesting is defined by what we find interesting. And it's up to us. We'd be the ones who have to go through that infinite collection of things that the AI dug up and say, oh, that one we think is interesting. I have to say, in my own life I've been involved a lot in doing basic science. I now call it rulology, studying simple computational rules and seeing what they do do. And I kind of feel like, you know, I, I do a lot of different things and in the end a lot of them boil down to I just make a whole bunch of little pictures on the screen and I look and I see which ones seem interesting. It's kind of. And I know that if I've got, I don't know, 10,000 objects, I can do okay at figuring out which ones are interesting. If I have 10 million objects, well, maybe I'll get some AI assistance and maybe I'll be able to let the AI do some of 10. Take my version of what I have so far found interesting and automate my version. Capture my version of what I've so far found interesting. So at least the things I've so far found interesting I'll be able to identify. Or more to the point, the things I've so far found boring. It'll be able to throw out and say, you don't need to look at those, those can't be interesting. But interestingness in the end is for us in the eye of the human behavior beholder, that what is abstractly interesting. There is nothing that you can say is abstractly interesting. As the AIs go off and kind of build their civilization, maybe there'll be all kinds of AI art that exists that the AIs will find wonderful and interesting. We look at it and it's like, this does not relate to anything we know about. It's just out in the computational universe, just sort of hanging out there, not connected to anything of human interest. Interest. Prime asks about quantum computers. Let me not dive into that here. I will, I will only say that that sort of the big idea of quantum computers is this ability to sort of run many threads in parallel. But there's sort of a big problem which is that in the end humans want to get the answer, they have to knit those threads together. And I've done a whole bunch of work on, on trying to understand that trade off and I, I think it's more complicated than one might have imagine. Let's see. P asks, may there be a way to, to improve storing information in neural net? In which ones? Ones I don't quite understand the scripting things to store data as a library in AI cluster memory. So I think this point I was making earlier that, that I think the future is to sort of disentangle what is really about sort of really AI suitable really neural net suitable. The sort of layer of a linguistic interface from the things which are much more. Just store it in a database, just store it as part of a computational knowledge base. Just do the computation using kind of precise things. It's kind of like the trade off between human thinking and, and the formalizations that we've had in mathematics and science. I mean there, there was a lot of human thinking that got blocked that we managed to figure out how to solve problems using mathematics and science starting about 300 years ago. But then there are things that yes, we can figure it out with human thinking. Is it a picture of a cat or a dog or something? And we haven't in the past been able to do that in a, in a kind of automated way. Now we can do those things in an automated way, but there's kind of a, there's this kind of what the greatest power comes from combining kind of the. Oh yes, humans could do that quickly and easily with the. Yes, that involves the deep computation because we have the means now to do deep computations. And combining that with the now more automated human like thinking is a very powerful thing. Let's see, 1 and 0 asks, is it a good idea to try to get rid of the human bias and training data for LLMs? Would an attempt actually be problematic for its internal consistency or maybe even create new biases? So here's the problem. Let's say you generate, you just generate lots of data and you train the neural net on that generated data. That's what I was trying to do in the things I showed predicting cellular automata, things like this. First point is how well does the neural net learn from arbitrary, not human like stuff? It's not obvious how well it learns because a neural net has an architecture that is similar to human brains and so it can learn things that are kind of like what human brains learn. A lot of what happens in the natural world seems to be of a different kind or the mathematical world seems to be a different kind. That isn't like what human brains, what intrinsically happens isn't like what human brains are learn. Now the slices of that that we humans have chosen to study, those are affected by how our human brains are set up. And those have more of a potential to be accessible to things like neural nets. But if you just say let's go train the neural net on all possible cellular autometer, all possible computational processes, I don't think that will be successful. I think the neural net will essentially it won't. In order to operate, it needs to find pieces of computational reducibility. If we give it all of that irreducible stuff, it will sort of just get confused. I think more to the point in, in the end what we humans are interested in is things that we find interesting and going out into the sort of computational universe and training and all these different things, things. The, the AI is not going to give us kind of human, it's not going to be able to tell us human type things from that. It's just going to say this is, it's, it's at best going to sort of be able to do its exploration of the computational universe without knowing what it should bother to report to us without knowing what, what sort of creatures it should bring back to us and what creatures it just is like, well that's just a thing that happens in the computational universe. Universe. Now there's, it's. I think, I think there's a problem. What I see is in very practical terms is it's sort of a different problem which is that if you generate training data from AIs then there's a certain amount of practical benefit in that. Just because this question of sort of shaking out the right weights in the AI may be something that's easier to do. You know, if you, if you train the self driving car are from a bunch of sort of generated generative AI video game like scenes of cars driving. That may be useful for the practicality of managing to find the right weights in the neural net, but it's not going to allow you to discover new things. That's just help in getting the neural net trained. And I think in the end it's kind of a, you're sort of reverting to the mean of everything by saying, saying let's just generate data from the AI on which the AI will be trained that it will then use to generate more data. That that's not a thing which, which in and of itself is Going to allow you to kind of discover new things. It's. This is a little difficult to untangle, and I was not doing quite so well there, sort of untangling this in real time. All right, let's see. There's a question here from a character set that I can't decode. If AI starts creating, testing and automating new axiom systems to solve mathematical problems, will human scientific research still hold meaning? Fundamental answer is it will still hold meaning to us humans. There's lots that goes on in nature, nature that is very, you know, very rich. But we humans just look at it and we say, we don't know what to make of this. So similarly, in kind of the metamathematical space of possible axiom systems and so on, there's lots that goes on. And it's just a question of, of, you know, we, we don't. We look at it. I, I've done this. A whole bunch of enumerating possible axiom systems and saying, well, that's an axiom system that one could look at at. I don't know why we care. And the, the thing that is kind of you should think about mathematics as more like art. It's more, what do we choose to do? Where do we choose to build things? What is our aesthetic in building things rather than a, a machine, an industrial machine that's grinding out mathematical theorems because, because all those theorems that it grinds out, most of them will be ones that we humans just don't care about. It is really a matter of sort of. It's an artistic question, what, you know, what do we build in mathematics that we find interesting, so to speak? Let's see, should wrap up very soon here. But. Let's see. D asks the classical binary computing systems limit our observational modalities. And is a quantum biological interface essential for enhanced scientific efficiency without the constraints of digitization? Well, gosh, you know, one of the things about us humans is we think definite things happen. We have a thread of experience. We think that we live a definite single thread of experience. The universe, sort of viewed from the outside, doesn't work that way. The universe has many threads of experience, many parts of history. That's kind of the story of how quantum mechanics works. But we humans somehow conflate all these together and say there was a definite threat of history that happened. And that's something I've been working on quite a bit in the last few years of understanding quantum mechanics as kind of the interface between sort of human brains that have many threads of history, going in them and the universe with many threads of history. But human minds that assume that we can conflate everything into a single thread of history, that's. And so sort of what you could imagine. You could imagine a quantum brain that is. That actually is operating at the level of multiple threads of historical history. It wouldn't be like our human brains. I wouldn't be able to be sitting here talking to you and. And going kind of yak, yak, yak and saying a single sequence of words. We'd be dealing with kind of this multiple parallel sequences of words which we humans don't consider to be our form of experience. It seems to be fundamental to what we think of as human consciousness that we have this single thread of experience. That we don't have this kind of incoherent collection of different threads of experience. That we manage to conflate everything down to a single thread of experience. Now, needless to say, the AIs, as they're set, don't necessarily have to do that. That is a feature of the humanization of computation that we're doing that. And so I think that it's an interesting thing that, for example, in an LLM, an LLM in the current way they're architected, it takes the text so far, the prompt or whatever, and it's trying to add one token at a time. One, one. One word or whatever at a time. And it's sequentially doing that. And at every step it's. It's figuring out what are the probabilities for what the next word might be. And it's picking one of those probabilities and just saying, okay, that's what the words next word's going to be. Let me go on and do that again and again and again and build up my essay or whatever. But in fact, it has many possibilities. But it's only. It's only picking one. You can kind of imagine a quantum LLM in which it is actually picking all those possibilities. It's making many paths of. Of. Of, of. It's making an essay of many paths. Humans looking at that, we're not going to understand it. It's not the way our brains work. It's not the way our minds work. It might be the way that. That actually the individual neurons in our brains could operate. But somehow we manage to get that single thread of experience out of the operation of all of those underlying neurons. It's kind of like when we look at, you know, an ant colony or something and it decides a definite thing to do, or even human society and sort of deciding a definite thing to do. But that's a feature that seems to exist in us humans and sort of the essence of our consciousness, this going down to kind of a single thread of history. And that isn't a feature of the universe, nor does it have to be a feature of the computational systems we build. But it's an interesting question what it will be like to sort of experience the sort of the, the AI, the quantum AI that isn't, isn't, doesn't really have to use quantum mechanics. It's just using parallel computation. And it has many threads of experience at the same time. And it's like it's doing what it's doing and we're looking at it just like we look at things in nature. And we might say this is completely incomprehensible to us. And we humans don't, you know, don't connect with that. All right, we should wrap up there. So thank you all for, for joining me. And as I say, the. This was based on a piece that I wrote last week, which you can find on my website. You can find it at writings.stevenworthon.com and it's, it's a thing, the experiments that I did there, we're all done with WOLM language. And in fact, every picture that you'll see in the thing that I wrote, you can click on, you can get the WOL language code, you can run it on your own computer. I will say that you can if you want to train the neural nets that we trained for this piece that's a little bit more computer intensive. Some of them took a while to train on fancy GPUs and so on. But all the code is there. Just click, you get it it. And you should be able to explore these things for yourself. And there's a lot of exciting things to do in this, in this direction. I think sometimes that there's been a certain amount that people have sort of said, oh, AIs can do everything. They'll just solve science. I wouldn't stop learning about science just yet. The, the humans, it's. We humans still get to define what it is that we want science to be doing. And, but nevertheless, the AIs can help us. I like to think that kind of computational language and the whole computational paradigm can help perhaps even more than AIs. And we're just in the, the early decades of the real development of sort of computational X for all X and thinking about things computationally. But that's kind of the effort that we've been doing with, with Wolfram language and so on all these years is to provide the notation and the framework framework for thinking about things computationally in a precise way. And by the way, that framework now has both, well, millions of human users and millions of AI users at this point, as well as the AI start to use our systems as, as tools for what they do. So thanks very much and bye for now.