From here on, we’ll be livestreaming what we’re doing—sharing whatever we discover in real time with the world. The particular way I’d set up my rules seemed a little too inflexible, too contrived. The rule tells us what they all are—and we can represent them (as we discussed above) as a multiway system—here illustrated using the simpler case of strings rather than hypergraphs: Each node in this graph now represents a complete state of our system (a hypergraph in our actual models). But it’s more than that. Or could it be that this is a kind of question that’s just outside the realm of science? But pretty soon I got swept up in building Wolfram|Alpha, and the Wolfram Language and everything around it. So what should this language be like? But later we’ll see that it’s also the direct analog of something completely different: the process of measurement in quantum mechanics. A book, A Project to Find the Fundamental Theory of Physics, was published about the project in June 20… So today marks the debut of the Wolfram Physics Project. But in a curved space, you won’t: And essentially what’s happening in the uncertainty principle is that you’re doing exactly this, but in branchial space, rather than physical space. But now we need to finish the job. OK, but now let’s come back to causal relationships. And our foliations that represent motion are the standard inertial reference frames of special relativity. As I mentioned earlier, there’s potentially a big problem here with computational irreducibility. More details can be found in this paper (see page 49) by Jonathan Gorard and in this Wolfram Physics Project bulletin by Stephen Wolfram. Official website of Stephen Wolfram: Founder & CEO of Wolfram Research; creator of Mathematica, Wolfram|Alpha & the Wolfram Language; author of A New Kind of Science Wolfram Physics … Wow. To a language designer like me, this is something interesting in its own right, with its own scientific and technological spinoffs. I was worried this was going to be one of those “you’ve got to throw out the old” advances in science. OK, but we also know something else about what is supposed to be inside our light cones: not only are there “background connections” that maintain the structure of space, there are also “additional” causal edges that are associated with energy, momentum and mass. We have such a simple rule. Thank you for sharing your incredible journey and making this accessible to a humble secondary school science teacher. (To clarify: these are not the strings of string theory—although in a bizarre twist of “pun-becomes-science” I suspect that the continuum limit of the operations I discuss on character strings is actually related to string theory in the modern physics sense.). Here are some of the things they do: Somehow this looks very zoological (and, yes, these models are definitely relevant for things other than fundamental physics—though probably particularly molecular-scale construction). And it looks like these causal edges have an important interpretation: they are associated with mass (or, more specifically, rest mass). Here’s roughly how this works. To an observer far from the black hole, it’ll seem to take an infinite time for anything to fall into the black hole. But it’s still, in my view, quite spectacular: from the basic structure of our very simple models, we’re able to derive a fundamental result in physics: the equation that for more than a hundred years has passed every test in describing the operation of gravity. And it has to be a language that can actually represent the underlying structure of physics. And what this means is that in the rule-space multiway graph, we can expect to make different foliations, but have them all give consistent results. We’ve had mathematical idealizations and abstractions of it for two thousand years. In other words, from the property of causal invariance, we’re able to derive relativity. We reproduced, more elegantly, what I had done in the 1990s. Instead it’s π, where a is the radius of the sphere. spacelike) to “vertical” (i.e timelike) distances on the causal graph. And for this to be the case we actually have to freeze time for that qubit. It’s computation, but it’s computation sampled in a different way than we’ve been used to doing it. But as entities embedded in the universe, we’re picking a particular foliation (or sequence of reference frames) to make sense of what’s happening. And here there’s an interesting possibility that’s relevant for understanding cosmology. And for example it tells us that all those causal graphs we get by taking different branchtime slices are actually the same when we project them into spacetime—and this is what leads to relativity. That’ll correspond to trying to describe the universe using some computationally reducible model—and over time it’ll get more and more difficult to maintain this as emulation cones effectively deliver more and more computational irreducibility. And the answer is that there is. But this usn’t true with subtraction, for example. The picture above makes it plausible that we’ve got something where things can go in, but if they do, they always get stuck. In the abstract, it’s a familiar idea that given any particular description language, we can always explicitly program any universal computer to translate it to another description language. But let’s look more closely at our light cones. And whatever we consider to be “energy” corresponds to the fluctuations of that flux around its background value. And—much like in the spatial hypergraph case—an excess of these causal edges will have the effect of producing what amounts to curvature in branchial space (or, more strictly, in branchtime—the analog of spacetime). And see if we can finally deliver the answer to how our universe fundamentally works. This is again somewhat complicated. (Note that that’s very small compared to the Planck length ~10–35 meters that arises essentially from dimensional analysis.) By the early 1990s I had a definite idea about how the rules might work, and by the end of the 1990s I had figured out quite a bit about their implications for space, time, gravity and other things in physics—and, basically as an example of what one might be able to do with science based on studying the computational universe, I devoted nearly 100 pages to this in my book A New Kind of Science. Will we be able to bring together physics, computation and human understanding to deliver what we can reasonably consider to be a final, fundamental theory of physics? And then there’ll be the physics experiments. But even though the structure is well represented in the Wolfram Language, the “use case” of “running the universe” is different from what the Wolfram Language is normally set up to do. But then there are both spacelike and branchlike relationships, where the event affects elements that are either “spatially” separated in the hypergraph, or “branchially” separated in the multiway system. But I wouldn’t say that anymore. In Wolfram models, Hawking radiation occurs as a consequence of branch pairs in the multiway graph that are unable to converge due to the presence of a multiway disconnection, where the latter corresponds to an entanglement horizon, from where this radiation is emitted. But for now let’s just recall that particles (like electrons) in our models basically correspond to locally stable structures in the hypergraph. In the effort to serve what people normally want, the Wolfram Language is primarily about taking input, evaluating it by doing computation, and then generating output. I expected that we’d start exploring simple rules and gradually, if we were lucky, we’d get hints here or there about connections to physics. Most of the example models come from a rewriting system on ordered graphs; some concepts are illustrated by examples pertaining to string rewriting systems. (And, yes, this part of what we’re doing is basically following what Einstein did when he originally proposed special relativity.). In fact, in what perhaps can be viewed as some sort of endorsement of the structure of the Wolfram Language, the models are in a sense just a quintessential example of transformation rules for symbolic expressions, which is exactly what the Wolfram Language is based on. Holy matrix! [1][2] The project builds on Wolfram's previous research into computational systems, as explored in his book, A New Kind of Science. But what about causal edges that are “more vertical”? Can’t wait. I was personally struggling with “rules” as the fundamental way the universe works. What if all conceivable rules could be used? through timelike hypersurfaces). But note that this picture shows the whole multiway system—with all possible paths of history—as well as the whole network of causal relationships within and between these paths. It will take some more discussion to explain how this all works. One of the great achievements of the mathematical sciences, starting about three centuries ago, has been delivering equations and formulas that basically tell you how a system will behave without you having to trace each step in what the system does. Stephen Wolfram is a cult figure in programming and mathematics. (There could still be other universes that do various levels of hypercomputation.). How We Got Here: The Backstory of the Wolfram Physics Project April 14, 2020. In designing a computational language what one is really trying to do is to create a bridge between two domains: the abstract world of what is possible to do computationally, and the “mental” world of what people understand and are interested in doing. [12] Stephen Wolfram and Jonathan Gorard have posted preprints on the topic to the arXiv; Gorard's was submitted to the journal Complex Systems, which was founded by Wolfram in 1987. And to say that this can only happen at a finite speed is to say that there’s computational irreducibility: that one rule cannot emulate another infinitely fast. But then there’ll be lots of causal edges associated with the particle, defining its particular energy and momentum. And that suggests the bizarre possibility that—just maybe—something like the angular structure of the cosmic microwave background or the very large-scale distribution of galaxies might reflect the discrete structure of the very early universe. We want to make it as easy for people to get involved as possible, whether directly in our centralized effort, or in separate efforts of their own. Like Wolfram told us in the NKS, it is truly a matter of reading the graphs correctly. It didn’t help that there was something that bothered me about my ideas. And there is difficult work to do on both sides. The rule defines how to take two connections in the hypergraph (which in this case is actually just a graph) and transform them into four new connections, creating a new element in the process. If true, and there is a path to solving this problem, however we try to tackle it, we’ll get there eventually! And we can translate this into saying that we imagine a series of “moments” in time, where things happen “simultaneously” across the universe—at least with some convention for defining what we mean by simultaneously. (It might approximate a projective Hilbert space.) And when we draw the graph, all that matters is what’s connected to what; the actual layout on the page is just a choice made for visual presentation. And a crucial idea in our model is in a sense just to do all of them. What kinds of primitives should it contain? So this gives us a way to measure the effective dimension of our hypergraphs. But if you are for example trying to construct a quantum computer, it’s not just a question of having a qubit be perceived as being maintained in a particular state; it actually has to be maintained in that state. Then follow r hyperedges in all possible ways. Doing new measurements is equivalent to getting entangled with new quantum states—or to moving in branchial space. The answer in our setup is basically no. OK, so let’s talk about setting up a model for the universe. So how does an observer deal with that? Amazing stuff. This is a big, complicated object. This is so flawless, beautiful and ‘intuitively trustworthy’, I feel like scientists will learn how to use that theory not just to explain, but to find new discoveries, in my lifetime. Where we go from there..! But for now, we can just say there are cones in our causal graph, and in effect the angle of these cones represents the maximum rate of information propagation in the system, which we can identify with the physical speed of light. One of the big predictions of general relativity is the existence of black holes. Thanks for putting this together. So you have a follower in me, jajaja. The speed of light c is a fundamental physical constant that relates distance in physical space to time. A very fundamental fact about space as we experience it is that it is three-dimensional. As time progresses we are in effect seeing the results of more and more steps in a computation. But at least in the way I have done it, the essence of language design is to try to find the purest primitives that can be expressed this way. What an incredible moment to witness! The model doesn’t tell us. That something similar is the multiway causal graph: a graph that represents causal relationships between all events that can happen anywhere in a multiway system. And so in a sense the multiway causal graph is where relativity and quantum mechanics come together. We were doing zillions of computer experiments, building intuition. And it’s not easy to bridge that gap. With our estimate for the elementary length, this quantum of mass would be small, perhaps 10–30, or 1036 times smaller than the mass of the electron. So when something is deflected going around the Sun, that happens because space around the Sun is curved, so the geodesic the object follows is also curved. Yes, a lot of what has to be done requires top-of-the-line physics and math knowledge. What does turning correspond to? So what this means is that, for example, a particle like an electron or a photon must correspond to some local feature of the hypergraph, a bit like in this toy example: To give a sense of scale, though, I have an estimate that says that 10200 times more “activity” in the hypergraph that represents our universe is going into “maintaining the structure of space” than is going into maintaining all the matter we know exists in the universe. Thank you for sharing. When the causal graph gets complicated, the whole setup with light cones gets complicated, as we’ll discuss for example in connection with black holes later. Actually, it’s rather straightforward. It could be that essentially everything we can see just expands too—so in effect the granularity of space is just getting finer and finer. It’s just like what happens in general relativity. And I have to say that our recent success in getting conclusions just from the general structure of our models makes me much more optimistic about this possibility. It helped that—after a lifetime of developing them—we now had great computational tools. But a possible (though potentially unreliable) estimate might be that the “elementary length” is around 10–93 meters. A geodesic is the shortest distance between two points. The form of the whole multiway system is completely determined by the rules. And let’s try to make this the time in human history when we finally figure out how this universe of ours works! But if the paths go into different disconnected pieces of the causal graph, that can’t ever happen. These concepts have multiple values. Too much has worked. What needs to be true for us to have d-dimensional space, as opposed to something much wilder? But every new mathematical model pose new issues associated with itself, e.g. But the point is that—even though when looked at from “outside” the paths are different—causal invariance implies that the network of relationships between causal events (which is all that’s relevant when one’s inside the system) will always be exactly the same. So, OK, what might we see in the universe today that would reflect what happened extremely early in its history? And we can once again imagine identifying the flux of causal edges—now not through spacelike hypersurfaces, but through branchlike ones—as corresponding to energy. But here’s the thing: this derivation isn’t specific to the toy model; it applies to any rule that has causal invariance. The formalism of the model is that of a discrete spacetime in which space is represented by a hypergraph obeying the rules of an abstract rewriting system. But the surprising thing is that there’s a remarkable depth of richness before one hits irreducibility. But what then is time? Think about expanding out an algebraic expression, like (x + (1 + x)2)(x + 2)2. Assuming we’ve drawn our causal network so that events are somehow laid out in space across the page, then the light cone will show how information (as transmitted by light) can spread in space with time. And indeed the phenomenon of computational irreducibility implies that there is something definite and irreducible “achieved” by this process. And this is particularly common with the very structureless models we’re using here. Here I’ve used numbers, but all that matters is that the elements are distinct. The speed of light c in our toy system is defined by the maximum rate at which information can propagate, which is determined by the rule, and in the case of this rule is one character per step. In other words, to successfully make a qubit, you effectively have to isolate it in quantum space like things get isolated in physical space by the presence of the event horizon of a black hole. That this can work ultimately depends on the fact that sequences of our rules can support universal computation (which the Principle of Computational Equivalence implies they ubiquitously will)—which is in effect why it only takes “choosing a different reference frame in rule space” to “run a different program” and get a different description of the observed behavior of the universe. These causal edges are associated with events that in a sense reuse elements in the hypergraph, without involving new ones. (Somehow it’d be like the ultimate calculus epsilon-delta proof: you challenge the universe with an epsilon, and before you can get the result, the universe has made a smaller delta.). But in our models it just comes directly from the analogy between branchial and physical space. 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