The Universe's Rule Might Fit on a Postcard

Lex Fridman opens his second conversation with Stephen Wolfram by asking what breakthrough moments in physics actually felt like. Wolfram points first to the 1920s, when quantum mechanics was assembled in five or ten years by Schrödinger, Heisenberg, Planck, and, a little later, Dirac. The episode he lived through was the 1970s, when quantum field theory became usable for quarks and gluons and a crowd rushed in. He was a teenager in that circle, among people such as David Gross, Frank Wilczek, David Politzer, Richard Feynman, Murray Gell-Mann, and Steven Weinberg, who in his memory were everyday colleagues rather than statues.

From the outside, he says, history always looks as if everything happened instantly. From the inside there is usually a long buildup, then a method starts working and five or ten years of low-hanging fruit follow. He maps the same pattern onto deep learning: neural nets existed from the 1940s, but eyes lit up around 2011 and 2012, with AlexNet as a plausible spark. His own spark was more staged. In the early 1980s he began using programs rather than equations as models; around 1984 he saw that very simple programs could generate great complexity; later systematic experiments convinced him the phenomenon was general and led to the principle of computational equivalence.

That line of work became the 1,200-page *A New Kind of Science* in 2002, and Fridman asks why the book did not produce the revolution one might have expected. Wolfram says the shift already happened, minus the hero narrative. For three hundred years, mathematical equations were the respectable way to model the world; in the last fifteen years, new models have more often been programs. Whether that is a consequence of his book is hard to prove in citation chains, though he keeps hearing from people who started whole research programs after reading it. What still disappoints him is that the abstract study of the computational universe—something like mathematics that is not mathematics—never became a large field. Rule 30, a tiny cellular automaton that looks random, is for him a core question of science, and almost nobody sits down to study it systematically.

Fridman asks whether clean prediction may simply be impossible in real systems, or whether humans are not yet smart enough. Wolfram says the question is not open: even if you know the rules, you cannot in general be smarter than the system and jump ahead. That is computational irreducibility, a cousin of Gödel's theorem and Turing's halting problem. The principle of computational equivalence, which he extracted from experiments in the early 1990s, says that once a rule-following process is not doing something obviously simple, it has reached the same computational sophistication as everything else, including our brains. Weather, markets, and a tiny program are not lower grades of computation.

Science can still be done, he argues, because irreducibility comes with scattered pockets of reducibility, and those pockets are where we live. For three hundred years science has concentrated on the places its methods can predict: planetary motion is largely cracked; who wins a battle is not; weather is about halfway there. Ancient Babylonians tried to forecast all three and had no idea which would be easier. When we engineer systems we try to stay in the reducible zone; when the natural world throws something at us, we have no such guarantee. Rule 30 itself may contain pockets, but their size is an empirical question, and a randomly chosen question is most likely irreducible.

The COVID-19 pandemic, then at its height, is his working example. He tried to be useful with data and models and concluded it was genuinely hard: we lacked the human contact network, the virology, and the immunology. Big narratives—T-cell immunity, a sea of asymptomatic infection—are signs that a reducible pocket has been found. From first principles you are stuck simulating each step. One of his children studied spread on different graphs and found few robust conclusions; a small number of large gatherings being worse than many small ones was one of them, while finer questions just depended on details. What frustrated Wolfram was the public expectation that science must have a definite answer and that we must be able to know it. After playing with little programs, he says, you lose that intuition: "the computational animals are always smarter than you are."

He is still an optimist about civilization. A kick to a system, unless it kills it, often leaves it better, and he suspects people are noticing what is worth caring about. Computational irreducibility, on this telling, is not only a limit. If science could always announce the result of a life in advance, living it would be pointless. Irreducibility is what gives a life meaning, because something is actually being computed as it unfolds. That is also why heat later in the conversation will not look like a void: the parts of the world we cannot summarize are still doing work.

One long-term fate of the universe is heat death: everything reaches equilibrium and becomes a uniform gas. People call that a terrible ending. Wolfram disagrees. Every molecule would still be bouncing in very complicated ways, doing very elaborate computation. It looks boring only because present mathematics has no interesting story to tell about that computation. The principle of computational equivalence is both humbling and hopeful: human intelligence is not a higher grade of computation than a gas or the weather. What is special is that we understand our own computations because they attach to our purposes. That specialness is human-centric, and he thinks it is real.

What, then, would a theory of everything even mean? For Wolfram it is crisp. You write down a rule; you run it enough times; you reproduce everything in the universe, down to the fact that two people are having this conversation. Enough times is perhaps 10^500 steps, so you cannot query the rule about whether lions will eat tigers. The analogy is the digits of pi: there is a definite procedure, and you still have to generate them. Physics would have been reduced to mathematics, and the mathematics may be irreducible. He had not expected, in his lifetime, to match the theory to present-day physics. He thought it would be a success merely to say what happened in the first 10^-100 seconds.

The surprise, which he calls embarrassing because it should have been obvious, is that large classes of these unstructured rules share the same pockets of reducibility, and those pockets are the pillars of twentieth-century physics. General relativity and quantum mechanics, he says, are not exotic add-ons. They are what you can still say when most of the underlying computation cannot be shortcut. Even more surprising to him: the two theories, long treated as incompatible, turn out in this framework to be the same theory seen two ways. The rest of the conversation is an attempt to show how that could be.

The Wolfram Physics Project asks for a simple computational rule that describes a whole universe. If everything must fit in that tiny rule, almost nothing familiar will be obvious in it: not three dimensions, not a parameter list, not tomorrow's lunch. Wolfram thinks a central assumption of the last few thousand years is wrong. Euclid's first definition said a point is that which has no part, and physics has treated space as a continuous theater you can label with precise coordinates. Water looks continuous too, until you reach molecules. Einstein, Wolfram notes, already suspected space would turn out discrete and said the mathematics of his time could not explore it.

In the model, space is a giant collection of atoms of space. There might be 10^100 of them. They have no coordinates, only connections—like a friend network with no street addresses. On a large enough scale those connections can look like continuous space of three dimensions, or 2.6, or something else; three dimensions are not fundamental. The elementary length might be around 10^-100 meters, against 10^-15 for a proton. Continuity from discrete pieces is, he says, no longer shocking: a computer screen is pixels, and from a normal distance it is a picture. The practical representation is a hypergraph, in which a relation can bind any number of nodes rather than only a pair. That move, which he made less than two years before this recording, is what finally made the model feel non-arbitrary.

The second key claim is more radical: space is all there is. There is no extra matter and no extra field living in a background. An electron would be a twisted or knotted feature of the same hypergraph; energy, mass, and momentum are other features of it. A rough estimate then lands with a thud. Everything we notice—particles, atoms, electrons—is only about one part in 10^120 of what the hypergraph is doing. The vast majority of the activity is just maintaining the structure of space. We live, in his phrase, as a little froth on top of that infrastructure.

If space is a hypergraph, time is the repeated application of a rewrite rule: whenever a small pattern of relations appears, replace it with another small pattern. Each replacement is an event. Events can fire anywhere, asynchronously, with only a local constraint: the inputs must already exist, so one event can be a prerequisite of another. That partial order is a causal graph. Special relativity, Wolfram claims, is a consequence of the fact that the order of updates does not matter so long as those causal relations are respected.

He calls the relevant property causal invariance. Multiply a string of polynomials in any order and the product is the same; sort a string of A's and B's by flipping BA to AB anywhere and you always finish with the A's first. The network of updating events is unique even if the timetable is not. Twentieth-century physics mixed space and time into spacetime; in this approach they come from different places—extension of the hypergraph versus progress of the computation—and still become Lorentz-invariant in the large. The analogy is fluid mechanics: molecules know nothing of the Navier-Stokes equations, yet those equations describe the continuum.

An observer has to slice the causal graph into reference frames, deciding what counts as simultaneous, just as Einstein did for moving clocks. Causal invariance says you always see the same physical processes. Wolfram offers a thought experiment: suppose the universe had only a single update head, like a Turing machine. That sounds absurd, because your interlocutor seems to be updating while you are. You cannot know they have been updated until you yourself are updated, so the causal graph you inhabit still looks ordinary. At human scales the speed of light is so large, and brains so slow, that the world feels like a simultaneous tick everywhere. Two descendants of apes, he and Fridman joke, are hypergraphs exchanging photons and experiencing that tick as a conversation.

Wolfram had expected the project to be a hunt through rule space, trying candidates until echoes of known physics appeared. The better news is that general statements are available. Given causal invariance, computational irreducibility, and a large-scale limit that is finite-dimensional, he says you can derive special relativity, general relativity, and quantum mechanics. Dimension is measured like this: from a node, count how many nodes you reach in r steps. In a two-dimensional grid that count grows like r squared; in d dimensions, like r to the d. Curvature is the next correction, the same way a circle drawn on the Earth has area a little less than πr².

The derivation does not depend on the specific rule, only on those general conditions. An infinite-dimensional hypergraph would look more like a tree, with exponential growth. He would like to know how common that is, and he suspects our universe may have started infinite-dimensional at the Big Bang and cooled toward three dimensions, perhaps with fluctuations to 3.01 or 2.99 that experiments could hunt. Energy, in a late-2019 realization, is the flux of causal edges through spacelike hypersurfaces—intuitively, how much activity stays in one place. That reading, he says, gives E = mc² and then the full Einstein equations with matter, not only the vacuum case.

A pure mathematician will not find a fully rigorous continuum limit; even the derivation of fluid equations from molecular dynamics has never been made rigorous. Wolfram's substitute is "proof by compilation": instead of discretizing Einstein's equations for numerical relativity, run a large discrete hypergraph and see whether black-hole mergers agree. The same minimal model, he adds, is a picture of parallel computation. Eventual consistency in industrial databases is essentially causal invariance. In the 1980s he tried to design parallel languages with graph rewriting and could not see how to make them useful; physics is now showing the way. Future programmers, he guesses, will announce a reference frame the way physicists do, then change frames and watch the same program look different.

Asked for the specs of the computer running the universe, Wolfram gives rickety but concrete estimates. Elementary time is one computational step; the ratio of elementary distance to that time is the speed of light. Because a rule involves only finitely many elements, other speed limits follow: a maximum entanglement speed in the space of quantum states, possibly measurable in black-hole systems, and a maximum translation speed between description languages in rule space. In Wolfram Language operations per second he puts the computation around 10^500; the number of atoms of space might be 10^400. Computational irreducibility means you cannot compress cosmic history into a short string. He keeps being surprised, though, at how deep the reducible shortcuts go: you do not need to know what an electron is in order to talk about quantum interference.

Classical physics says a definite thing happens. Quantum physics says many paths might, and we observe probabilities. In the hypergraph the sequence of updates is not unique, so the set of possible sequences is another graph: a multiway graph. Tic-tac-toe is the toy picture. Different move orders branch, then merge when they reach the same board. Causal invariance says the number of mergers equals the number of branches, which is his rough account of why we still believe definite things happen. Slice the multiway graph at a time and the leaves are possible quantum states. Distance between leaves—how far back their common ancestor sits—is entanglement distance. That geometry is branchial space.

Measurement, on this view, is the choice of a quantum observation frame: a way of slicing branchial space analogous to choosing simultaneity in spacetime. The observer is inside the system and computationally bounded, so only some frames are available. Wolfram says that is the same limitation that makes entropy increase. If we could track every molecule we could run the movie backward and would not see disorder growing; because we cannot, we talk in thermodynamic blobs. Because we cannot track every quantum branch, we conclude that definite outcomes happen. He used to agree with Feynman that "nobody understands quantum mechanics." He now thinks he is getting close to a middle-school explanation, with the caveat that the universe was not built for our understanding.

Einstein's equations say energy deflects geodesics in physical space. Wolfram says the same thing happens in branchial space: Lagrangian density deflects paths through the space of quantum states. The Feynman path integral writes an amplitude as a phase e^{iS/ℏ} built from the action, which is essentially energy; that phase is the deflection angle of a geodesic. Magnitude and phase, he adds, come from different places in the construction. Motion in physical space is general relativity; motion in branchial space is quantum mechanics; the underlying equation is the same, presented differently because one is asking different questions. For someone who spent part of a career inside twentieth-century theory, he finds that identity amazing.

The double-slit experiment becomes a story about the interface of the two spaces. A photon has two paths; in the model those paths end up at opposite ends of branchial space, so they cannot merge into something measurable in physical space. That, in work from only the previous few weeks, is the picture of destructive interference. Branchial space is probably exponential-dimensional and may limit to a Hilbert space, and the universe is likely expanding there as well as here, which would mean the number of quantum states grows with time. The mathematics that would make this fully rigorous—calculus in fractional and changing dimension—does not yet exist. Computer experiments show the shape of it.

Between 1981 and 1983 Wolfram worked with Feynman on a chip that would generate randomness at high speed from quantum mechanics. They found it was not really possible, with measurement as the damper, and never properly published the conclusion. The modern promise of quantum computing is intrinsic parallelism: Shor's algorithm, in the standard formalism, factors integers faster than a classical machine. The footnote is that you still have to read a classical answer. Standard quantum theory describes the branching and not the corralling. Their model does describe the pulling-together, and the present indication is that the parallel advantage is lost in that step, a phenomenon not unrelated to decoherence. They have only reached 2×3 in factorization, about as far as physical quantum computers. People should not stop the research, he says; using physics to compute in new ways is valid. He doubts the strong claim that exponential problems become polynomial.

One bizarre remaining hope is to piggyback on the expansion of branchial space. Energy is conserved in high-school physics and not simply conserved in cosmology; a spring between two receding galactic black holes would gain potential energy from expansion. That perpetual-motion cartoon is absurd in physical space. A laboratory analogue that mines branchial expansion for "quantum computing for free" might not be. They do not know. Asked how a beginner should enter the project, he points to a visual summary, to his long April 2020 announcement post—which readers grokked more deeply than he expected—and to papers by Jonathan Gorard, a mathematically strong collaborator. The code is on the Wolfram Physics Project site. High-school students can run experiments on day one; the scarce skill is taste in which experiment to try. Connecting the model to the Einstein tensor still takes something like graduate physics, but discrete pictures remove a lot of hand-waving about continuum limits.

In harvesting mode they are ticking through known effects. The quantization of spin—always integer or half-integer, a 1920s fact never really explained—looks within reach. The distinction between bosons, which pile into one state and make lasers, and fermions, whose exclusion principle keeps matter from collapsing, seemed to come together in the previous few days, subject to revision next week. The guess is that bosons correspond to merging in the multiway graph and fermions to branching, so that fermions occupy extent in branchial space and are forced apart. Spinors, which pick up a minus sign after 360 degrees and return only after 720, may be as simple as directed versus undirected hypergraphs, the way a Möbius strip has two twists. If that holds, there are only two cases because there are only two kinds of graph. He learned physics several academic generations after the founders, when the foundations felt inaccessible; the shock is that they may not be.

Suppose they find the rule and it is simple. Why that one, universe 317, rather than an arbitrarily complicated alternative? That is the Copernican embarrassment again. Wolfram's more bizarre move is to let every possible rule apply in every possible way, building a rulial multiway graph. It still has causal invariance, so slices—reference frames—remain equivalent after translation. An observer embedded in that object infers a rule from the frame they pick. All possible rules are in a sense giving the same answer; what you see can still be a completely different description language. The apparent simplicity of our laws is then a fact about us, not a prize the cosmos handed to Earth.

The immovable fact, he says, is that the universe is computational at the level of an ordinary Turing machine: not a hypercomputer that knows infinite-time answers immediately, and not less. Friendly aliens need not share our physics. He used to assume they would at least live in the same laws; now he thinks they could have a description utterly incoherent with ours, because they occupy a different frame in rule space. Intelligence, on the principle of computational equivalence, is not a rare peak. The weather has a mind of its own, in the sense that it is computationally equivalent to a brain; we simply lack a language for what it is "trying" to do. Superintelligence would not be a higher grade of computation. It would be another structure in the same multiway object, at some translation distance. He is not sure how to measure that distance yet, but he expects a maximum translation speed in rulial space, analogous to the speed of light and to maximum entanglement speed, and a physical version of algorithmic information.

The same picture, Wolfram argues, is a picture of mathematics. The standard axioms fit on about a page and a half in *A New Kind of Science*. Applying an axiom rewrites an expression into another; the possible rewrites form a multiway graph in metamathematical space. A proof is a path from one expression to another; a theorem is the fact that the endpoints are equivalent. Gödel incompleteness is the statement that some of those paths have no upper bound on length. Euclid started from ten axioms and derived 465 theorems. Mapping the dependencies, Wolfram finds the hardest is that there are five Platonic solids, a 33-step path and Euclid's last proposition.

Why is mathematics hard, and why is it doable? It is hard because of irreducibility: even arithmetic, unlike truth-table logic, does not compress. It is doable because mathematicians do not parachute into random equations. They extend an existing proof trajectory, the way Lewis and Clark followed a path, and they steer around patches of undecidability. Pick a random Diophantine equation and you fall into the ocean; Fermat's Last Theorem happened to have a long proof; the twin-prime conjecture and the Riemann hypothesis might be undecidable from the current axioms. Automated theorem provers search the same multiway graph; their critical pairs are the branch pairs of the physics models, and the proof graphs Wolfram Language already draws look uncannily like physical multiway graphs, even though the two visualizations were built years apart.

He suspects the univalence axiom of homotopy type theory—the idea that equivalent things may be treated as identical—is causal invariance in mathematical clothing. Paths between proofs are higher-order proofs, related to causal graphs; the infinite-order limit is the rulial multiway system, which he identifies with Grothendieck's infinity-groupoid. Nonconstructive proofs look a bit like event horizons. Because the model is so minimal, it also shows up in distributed ledgers: a global blockchain is a bad deal if country A's transactions need not reconcile with country B's for a while, and that reconciliation is the same story as light cones. The real power, he insists, is not to throw out existing physics but to reuse a century of physical intuition on parallel computation, databases, and proof.

Fridman raises the worry, associated with Sabine Hossenfelder's book, that physicists chase beauty instead of predictions. Wolfram's reply is that we do not intrinsically know whether Occam's razor works for the universe. Order itself is the fact that makes science possible; early theologians called that the role of God, and a formal theory is the modern stand-in. If the rule is simple, almost everything we see has to pack into it, which will look beautiful to some aesthetics, especially the surprising connectivity he likes. Fundamental physics as a profession, he says, mostly gave up. String theory no longer looks poised to deliver a final theory; quantum information made computational ideas about physics more respectable; and when he published a hundred pages on this in 2002, the harshest fork came from friends in fundamental physics. He even paused the project because it felt like building a product for a market that did not want it.

Why should anyone care? His analogy is Copernicus. The Ptolemaic system already computed the planets; the heliocentric formulation let mathematics overrule common sense. Once the world is known to be computational, computational irreducibility is no longer an optional metaphor. He has homework for a NASA workshop on faster-than-light travel and has not solved it. Imagining black holes in the backyard, he talks about "space tunnels," higher-dimensional regions that would look like a black hole at one end and a white hole at the other, and a "vacuum cleaner" that would strip out some of the activity that maintains space, producing negative energy density and repulsive gravity. His instinct is that this is extremely hard. Then he remembers people who said a copper wire could never carry data beyond a modest baud rate, until finer understanding raised the limit by orders of magnitude. Future centuries, he suspects, will find our shrug that heat is just random molecular motion as naive as geocentrism.

The last physics question is why the universe exists at all, even if the model is right. Wolfram's guess is Gödel-shaped: from inside an axiomatic theory you cannot prove the theory's own consistency, and for any entity inside the universe the statement that the universe exists is undecidable. Fridman asks whether that is peace or frustration. Wolfram says it is acceptance, like wanting a hypercomputer and being unable to build one here. "This is all we have." He still finds it strange that this definite physical body should be a reference frame of an infinite abstract object. The consolation is the same as the principle of computational equivalence. There will not be a more impressive superintelligence that shows we do not count. All such minds are equivalent as computation. What makes us special is the details of us: we are here, and we understand what we are doing. Two finite descendants of apes then have to stop, because darkness has come and the mosquitoes are biting, in the middle of a conversation about the fabric of reality.