Stephen Wolfram: Complexity and the Fabric of Reality | Lex Fridman Podcast #234

Rule space is a third fundamental space (beyond physical space and branchial space) where each point represents a possible rule or law of physics; observers have positions in rule space, and the Ruliad represents all points in rule space, with our perceived laws of physics corresponding to our particular position in rule space.

“in Ral space we could live in many different places in Ral space but we happen to live here and what does it mean to live here it means we have certain sensory input we have certain ways to parse the universe those are our interpretation of the universe what would it mean to travel in Ral space what it basically means is that we are successively interpreting the universe in different ways”

The Ruliad is the necessarily existent object that emerges from running all possible computational rules together in an entangled way, producing a definite structure that is not a mere chaotic collection but rather contains all of reality including our universe as one particular reference frame within it.

“the thing that had confused me for a long time was let's say we get the rule for the universe we hold it in our hand we say this is our universe then the immediate question is well why isn't it another one... the resolution of that is the realization that there that the universe is running all possible rules rules... the thing that is sort of the result of running all possible rules in all possible ways and you might say if you're running all possible rules why can't everything possible happen well the answer is because when you there's sort of this entanglement that occurs”

Simple deterministic rules without random input can generate outputs that appear random, and one cannot determine whether randomness-like output came from filtering random input through a structure or from intrinsic generation by simple rules, but if output is reproducible across repeated runs then it must be intrinsically generated rather than from random external filtering

“there are all these different kinds of dynamical systems theory sort of big era in mathematics that developed from the early 1900s...if you start off from an arbitrary random number which is quotes randomly chosen so all its digits are random then when you run that sort of Chaos Theory shift map all that you get out is just whatever you put in”

The universe at its most fundamental level is a hypergraph—a collection of atoms of space with only identities and relationships to each other (like a 'friend network')—not a continuous geometric space, and all features of reality including particles and forces are tangles or patterns in this hypergraph structure.

“space is made of kind of atoms of space and the only thing we can say about these atoms of space is they have some identity... and then all we know about these atoms of space is how they relate to each other... we say these three atoms of space are associated with each other in some relation... the sort of starting point of our physics project is that's what our universe is it's a giant friend network of the atoms of space”

Chemistry can be understood as a multiway computation system where molecules and their reactions form a dynamic network graph, and the different possible observers in this system (such as molecular-level observers sensitive to momentum and orientation, or human-level observers focused on concentrations) would perceive different effective laws.

“chemical reactions are like you've got this particular chemical it's represented as some graph of you know these uh these are this configuration of molecules with these bonds and so on and a chemical reaction happens... so that's kind of the the the abstract view of what's happening in chemistry... we've got this big network of all these molecules having all these reactions and so on and this is some whole multicomp computational story”

The immune system may be better understood through a dynamic network theory where immune memory and function arise from complex interactions between T-cells, B-cells, and antibodies (anti-antibodies, etc.) rather than from the traditional clonal selection model of individual cells, and this network structure may store information in ways not captured by concentration measurements.

“Neils yo who was the guy who invented monoclonal antibodies various other things um kind of had this network theory of the immune system where it would be like well we produce antibodies but then we produce antibodies to the antibodies anti- antibodies and we produce anti- anti-antibodies and we get this whole dynamic network of interactions... the question is what you know for example immune memory where is the where does the immune memory reside is it actually some cell sitting in our bone marrow that is you know living for the whole of Our Lives that's going to spring into action as soon as we're showing the same antigen or is it something different from that is it something more Dynamic is it something more like some network of interactions”

Multiway computation is a new computational paradigm where instead of a single deterministic sequence of states (as in classical computation) or a single probabilistic thread (as in traditional quantum mechanics), there are multiple asynchronous, parallel threads of execution happening simultaneously with no inherent ordering—and observers embedded in such a system can only detect causal relationships between events, not a global ordering.

“in this multic computation setup that it's this idea of these multiple threads of time and models that are based on that... there are all these different possible paths a nondeterministic touring machine can follow... but in the multic computational Paradigm after a thousand steps not even clear what a thousand steps means because you got all these different threads of time but there is no State there's all these different possible you know there's all these different paths”

Branchial space is the space of possible quantum histories and quantum branches, where motion in branchial space corresponds to changes in quantum state; the analog of the speed of light in branchial space is the maximum entanglement speed, which determines how quickly quantum correlations can spread.

“the space of quantum branches so in this in this thing we call a multi-way grow off of all of the sort of branching histories there's this idea of a kind of space where instead of moving around in physical space you're moving from history to history so to speak from one possible history to another possible history and that's kind of a different kind of space that is the space in which quantum mechanics plays out... the speed of light what's the analog of the speed of light in branchial space it's the maximum speed of quantum entanglement”

Time dilation in relativity can be understood as a consequence of the fact that the universe is computing, and objects moving through space must use some of their computational budget to maintain their identity at different positions, leaving less computational resource to update their internal clocks, thus causing time to run slower for moving objects.

“when you move from somewhere to somewhere it's you're having to sort of recreate yourself at a different place in space m when you exist at a particular place and you just evolve with time you're again you're you're updating yourself... the question is when you have a certain amount of computation in you so to speak when there's a certain amount you know you're Computing the universe is Computing at a certain rate you can either use that computation to work out sitting still where you are what's going to happen successively in time or you can use that computation to recreate yourself as you move around the universe and so time dilation ends up being... the time dilation is a story of the fact that as you kind of recreating yourself as you move you are using up some of your computation and so you don't have as much computation left over to actually work out what happens progressively with time”

Quantum mechanics is not an add-on feature to the universe but an inevitable consequence of multiway computation—since there are multiple possible update sequences of the hypergraph (multiple possible histories), quantum branching is fundamental; what we experience as definite outcomes results from our branching brains selectively conflating different branches of the multiway graph into consistent narratives.

“quantum mechanics is not some kind of plug-in add-on type thing you absolutely cannot get away from quantum mechanics because as you think about updating this hypergraph there isn't just one sequence of things one definite sequence of things that can happen there are all these different possible update sequences that can occur... these different Quantum histories and one of the things that's kind of surprising about it is they they Branch you know there can be a certain state of the universe and it could do this or it could do that but they can also merge... how does a branching brain perceive a branching universe and the key thing is as soon as you say I think definite things happen in the universe that means you are essentially conflating lots of different parts of History”

We perceive the universe to be 3-dimensional because at human scales the volume of a ball grows as r^3 (cubic growth), but the universe is not inherently 3-dimensional; in the early universe it may have been higher-dimensional, and as the universe evolved it became lower-dimensional, with dimension being an emergent property of the scale at which you're observing.

“in our model the universe is not fixed dimensional I mean we think we live in three-dimensional space but this hypergraph doesn't have any particular Dimension it can emerge as something which on in an approximation it's as if you know you say what's the volume of a sphere in the hypergraph where a sphere is defined as how many nodes do you get to when you go a distance R away from a given point... the very early Universe was essentially infinite dimensional and that as the universe expanded it became lower dimensional”

Decidable mathematical theories (like Boolean algebra where all questions can be answered in finite steps) are analogous to black hole event horizons where time stops—past a certain computational threshold, one can no longer ask new questions. Undecidable theories (like Peano arithmetic) are like open universes where computation can continue indefinitely with questions of arbitrary complexity

“in mathematics between decidable decidable theories and undecidable theories that's a that's an example and I think we're sort of the the attempt to understand so so another question is kind of what is the what is the general ativity of uh of metam mathematics”

Dimension fluctuations in the early universe (the universe having slightly non-integer effective dimension at the cosmic microwave background era) might be detectable through unusual photon propagation patterns predicted by Huygens' principle, providing a potential test of the Wolfram Physics model.

“is dimension fluctuation that is is there leftover Dimension fluctuation of at the time of the cosmic microwave background 100,000 years or something after the beginning of the universe is it still the case that there are there were pieces of the universe that didn't have Dimension three that had Dimension 3.01 or something and can we tell that is that possible to observe uh the fluctuations in Dimensions... when you try and do Optics you know a common principle in Optics is hen's principle which basically says that every piece of a wavefront of a of a of a light is a source of new spherical waves and those spherical waves if they're different dimensional spherical waves will have other characteristics and so there will be bizarre Optical phenomena”

The challenge with AI safety and formal verification is inherent to computation itself: as soon as you want computation to be genuinely useful and Turing complete, you must accept Gödel's incompleteness and computational irreducibility, which means you cannot box in what the system will do in advance—there is a fundamental tradeoff between computational power and provable safety

“girdles theorem tries to say you know piano arithmetic the axim of arithmetic can you box in the integers and say these axim give just the integers and nothing but the integers goodles theem showed that wasn't the case”

Relativity emerges necessarily in the Wolfram Physics model because observers embedded in the system can only know causal relationships between events, not absolute orderings of events, and causal invariance (the property that the causal graph is the same regardless of the order in which updates are applied) ensures that different reference frames will agree on these causal structures.

“the the fundamental point is if you are an observer embedded in the system that are part of this whole story of things getting updated in this way and that there are there's sort of a limit to what you can tell about what's going on and really in the end the only thing you can tell is what are the causal relationships between events... it it turns out that well there's this property of causal invariance... makes it be the case that it doesn't matter kind of if if you are uh sort of saying well I've got this hypergraph and I can rewrite this piece here and this piece here and I do them all in different orders when you construct the causal graph for each of those orders that you choose to do things in you'll end up with the same causal graph and so that's essentially why uh well that's in the end why relativity works”

Consciousness is fundamentally characterized by two constraints: computational boundedness (observers can only process limited information about the universe) and a single thread of time (sequential experience), which together distinguish consciousness from mere computational sophistication or intelligence.

“I think that Consciousness has two limitations I think one of them is computational boundedness that is that we're only perceiving a sort of computationally bounded view of the universe and the other is this idea of a single thread of time that is that we and in fact we know neurophysiologically our brains go to some trouble to give us this one thread of attention so to speak”

Mathematics can be understood as having two levels: an underlying 'molecular dynamics' level (axiomatic formalization) where Gödel's incompleteness applies and there are proofs of arbitrary length, and a higher 'hydrodynamic' level where mathematicians operate and experience mathematics as comprehensible and unified.

“what I'm increasingly uh coming to realize is that's similar to saying let's take a gas and break it down into molecules there's gas laws that are the large scale structure and so on that we human hum are familiar with and then there's the underlying molecular Dynamics... the axiomatic level of mathematics which we can access with automated theorem proving and proof assistance and these kinds of things that's the molecular dynamics of mathematics and occasionally we see through to that molecular Dynamics we see undecidability”

Rulology is the systematic study of simple abstract systems (cellular automata, substitution systems, register machines, combinators) and their behavior, distinct from mathematics, computer science, and physics, and represents a pure science worthy of institutional support similar to what has been given to complexity science.

“the foundations of complexity... are really two ideas two conceptual ideas... one is what I call Meta modeling the other is rology... rology is kind of the the okay you've got these simple rules you've got cellular autometer you've got turning machines you've got substitution systems... what do they actually do in the wild... it just hasn't had a home... it's a surviving field so to speak... it's something where you know one of the things I I find sort of inspiring about mathematics... this idea of studying simple rules and what they do it's a Timeless activity”

Time is not a coordinate that can be arbitrarily moved forward or backward (as in mathematical equations) but rather an irreducible computational process—a sequence of rewrites of the hypergraph—that necessarily advances in one direction, making time fundamentally different from space.

“time is this kind of this rewriting of the hypergraph and one of the things that's important about that time is this sort of computationally irreducible process there something you know time is not something where in kind of the mathematical view of of time tends to be time is just a CO ordinate we can you know slide a slider turn a knob and we'll change the the time that we've got in this equation but in this picture of time that's not how it works at all time is this inexorable irreducible kind of set of computations”

Four epochs can be identified in the history of scientific modeling: (1) ancient structural models (water, atoms, crystal spheres); (2) mathematical models with equations (17th century onward); (3) computational models where time is discrete and deterministic but sequential (early 1980s onward); (4) multiway computational models with asynchronous, branching time.

“I I've kind of concluded that I'm in the business of making kind of artifacts from the future which means you know... four epochs in the history of making models of things and um um and this multicomp computation thing is is the fourth is a new Epoch what are the first three the first one is is back in Antiquity ancient Greek times people were like what's the universe made of... the the sort of revolution of mathematics being introduced into physics... the computational idea that I kind of uh started really pushing in the in the 19 early 1980s... the multic computational Paradigm the kind of idea is instead of there being the single thread of time there are these kind of distributed asynchronous threads of time”

The history of philosophy contains insights (such as Leibniz's monadology) that align with modern computational theories but were expressed in period-appropriate language ('souls' for autonomous processes, 'consciousness' for spontaneous activity) that made them incomprehensible to later scientists.

“Leibniz had this idea that what exists in the universe is this big collection of monads and that they that the only thing that one knows about the monads is sort of how they relate to each other which sounds awfully like hypergraphs... but Leibniz had really lost me at the following thing he said each of these monads has a soul and each of them has a Consciousness and it's like okay I'm out of here I don't understand this at all... but I realized recently that in his day the concept that a thing could do something could spontaneously do something that was his only way of describing that”

Economics can be understood as a multicomputational system where elementary events are transactions between agents, economic space is knitted together by the consistency requirements of these transactions (similar to how physical space is knitted together), and the apparent simplicity of economic values emerges from observers sampling this complex transaction network through the lens of numerical currency (a numéraire)

“sort of the perhaps I don't even know if this is right yet there's sort of events in economics are transactions there are states of agents that are kind of the atoms of economics and then transactions are kind of Agents transact in some transact in some way and that's an event”

Physics-inspired concepts (event horizons, relativity, causal invariance, maximum entanglement speed) have analogs in other domains like economics, immunology, and mathematics, and can guide research by suggesting which features of a domain might be experimentally or observationally important.

“one reason that's extremely powerful is because physics has been very successful so we know a lot based on what we figured out in physics and if we know that the same model governs physics and governs I don't know economics Linguistics Immunology whatever we know that the same kind of model governs those things we can start using things that we've successfully discovered in physics and applying those intuitions in all these other areas and that's that's pretty exciting”

Wolfram's intellectual trajectory has involved repeated cycles between basic science and technology (~5 times), and each time he believed there would be no path back—yet surprising opportunities for application keep emerging, suggesting deep connections between pure theory and practice.

“the strange thing in my life is I've sort of alternated between doing basic science and doing technology about five times in my life so far and the thing that's just crazy about it is you know every time I do one of these alternations I think there's not going to be a way back to the other thing and like I thought for this physics project I thought you know with doing fundamental Theory of physics maybe it'll have an application in 200 years um but now I've realized um actually this multicomp computation idea is is applicable here and now”

Wolfram does not know whether elements of Leibniz's monadology (monads as atoms of existence, each with its own consciousness) refer to the same thing as the atoms of space in his physics model, but Leibniz may have been describing discrete computational entities that must spontaneously do something, which is precisely what the rules of the hypergraph do

“libbets had really lost me at the following thing he said each of these monads has a soul and each of them has a Consciousness and it's like okay I'm out of here I didn't understand this at all”

The shift from mathematical equations as the primary modeling tool to computational programs is an irreversible paradigm shift in science; however, this shift has been slower to adopt in physics than in other fields due to the success and entrenchment of mathematical physics.

“it is really interesting that just 20 years a span of 20 years it's gone from you know pitchforks and horror to yeah we get it... in physics people were like we've got our physics models we're very happy with them yeah in physics there's more resistance because of the attachment and the power of the equations”

Computational contracts and symbolic discourse language could enable legal and regulatory rules to be written in computational terms, allowing automated analysis and execution of contracts and laws, similar to Leibniz's historical vision of computational law but now technologically feasible.

“if you want to run the world you need you know with with with contracts and laws and rules and so on there are rules at a human level... back in gotfried liet back in you know 1680 or whatever was like um I'm going to you know figure out how to use logic to decide legal cases and so on... what I'm calling a symbolic discourse language it is just finishing the job of being able to represent everything like the conversation we're having in computational terms and one of the use cases for that is computational contracts another use case is something like the the Constitution that says what the AI what we want the AI to do”

Rotating black holes near critical rotation rates may show observable signatures of the discrete structure of spacetime because at those extremes the continuous approximation breaks down and the underlying atoms of space become relevant, potentially visible in gravitational wave signatures.

“a rapidly rotating black hole right at the sort of critical rotation rate um is it's it looks as if that's a case where essentially the the structure of SpaceTime is just about to fall apart and you may be able to kind of see the evidence of sort of discrete uh elements... there may be some effect in for example gravitational waves produced by rapidly rotating black hole that in which one could actually see some phenomenon where one can say yes those don't come out the way one would expect based on having a continuous structure of SpaceTime”

Molecular biology likely does molecular computing not through building nanotechnological constructors (original nanotechnology promise) but through dynamic network processes encoding information in reaction dynamics rather than molecular products.

“I don't think I I'm now increasingly concluding that's not the big point the big point is something more Dynamic that will be an interesting end point for any of these things but that's perhaps not the thing you know because the one example we have molecular Computing that's really working is US biological organisms and you know maybe the thing that's important there is not uh this you know what chemicals do you make so to speak but more this kind of dynamic process”

Metamodeling and rulology represent two foundational activities for understanding complexity: metamodeling asks 'what underlying Primitives operate beneath a model?' while rulology catalogs 'what do simple rules actually do?'—these are the bedrock beneath applied complexity research.

“the um about um uh kind of you know the foundations of complexity what really are they I think they're really two ideas two conceptual ideas that I hadn't really enunciated I think before one is what I call Meta modeling the other is rology so what is metamodeling So Meta modeling is you've got this complicated model and it's a model of you know hedgehogs interacting with this interacting with that and the question is what's really underneath that what is it you know is it a touring machine is it a cellular automaton you know what is is the underlying stuff underneath that model”

Natural systems (weather, biological networks, immune systems) may contain embedded intelligences with alien perception frames—'alternative parsings of the universe'—that are completely imperceptible to us because we do not share their computational constraints, yet they might be comprehending or computing something of significance.

“for all we know right here in this room you know in the in the details of the motion of these gas molecules there could be an amazing intelligence that we were like but we have no way of we're not parsing the universe in the same way if only we could parse the universe in the right way you know immediately this amazing thing that's going on and this you know huge culture that's developed and all that kind of thing would be obvious to us but it's not because we have our particular way of processing the universe”

Computational language (like Wolfram Language) is fundamentally about building a bridge between what humans can conceptually understand and the vast space of computational possibilities, similar to how physics models create a bridge between what we humans can think about and what the universe actually does.

“when I built wol from language and our whole sort of computational language story it's all about how do you take sort of raw computation in this ocean of computational possibility and how do we sort of represent pieces of it in a way that we humans can understand and that map on to things that we care about doing... when you add physics you're adding this other piece where we can you know mediated by computer can we get physics to the point where we humans can understand something about what's happening in it”

History matters fundamentally in physics (events' causal histories determine their effects), and Wolfram has spent 40 years recording his own life extensively in part because he has a personal proclivity to record information, but this turned out to mirror a deep scientific principle rather than being coincidental

“I like recording stuff you know I one of the things that's come out of of kind of my science I suppose is this this history matters type type story”