Wolfram Summer School 2022: Physics and Metamath Opening Keynote with Stephen Wolfram

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uh so what i wanted to do today here was just give some kind of general i would say theming and uh uh kind of um thematic comments about the physics project and the mathematics project and how i see them evolving right now and what i see as being sort of the key uh directions that can be taken at this at this point in these projects so i mean this is kind of a super exciting time you know i think as somebody who's watched the history of lots of kinds of science over the course of uh well now about five decades um it's it's interesting to see there's a certain rhythm that things seem to follow which is that some new idea methodology gets invented and suddenly lots of stuff opens up and there's this wonderful period of 5 10 maybe 20 years where there's just a lot of low-hanging fruit to be picked and then things get a bit harder and it maybe takes a century or so before you get to the next moment where there's lots of low-hanging fruit to be picked um we are at one of those moments where there is low-hanging fruit to be picked there's a bunch of new ideas new methods and uh it's kind of exciting i really didn't expect this to happen um i didn't think you know it was not obvious it was of this century kind of activity but uh we've been really lucky that uh this is this we seem to have kind of um hit something that is just extremely fertile and extremely fertile both for physics and now we realize from our mathematics and the same kinds of methodologies we think are going to apply to lots of other fields and we've started exploring that uh pretty much under the banner of multi-computation but um so you know it's it's always interesting to see i i have to say as sort of a student of the history of science and the sociology of science um there's this wonderful thing that happens when sort of new fields open up which is both there's a there's a set of people who entrepreneurially choose to be involved with those new fields and those people tend to be very productive but there's also the field itself allows for high productivity and the combination of those two things makes for just a really terrific time in these particular stages in in the opening of fields so another thing i say would say has happened with the physics project is i had been thinking about things that were sort of the precursors of this project from the early 1990s onward i had kind of stopped for quite a while after the nks book came out and so on um and the thing that is kind of like well i think it's going to work this way and i think it's all going to fit together like this and the thing that i think has been just amazing is to see that yes it really all these pieces really do seem to fit together it's not like you know when people say is this is this theory right um at this point it's beginning to feel like look there's no choice this has to be the way it works there's a huge amount of work to do to go from sort of the general idea of how it works to the specifics of what physics experiments do i do what specific things can i conclude about the general ideas of mathematics and so on but the you know is one on the right track it's like saying is calculus on the right track it's kind of there is a thing and it's it's something that we can kind of see has has sort of deep resonance with the way that things work and now it's a question of really working out uh how do things actually fit together so i think that some uh to just give you a little bit of maybe my personal sort of history with respect to this because it also i always find that um kind of a a tip that i've noticed about trying to understand ideas is that um it's uh understanding the history of those ideas often helps in understanding the content of the ideas as well so let me let me just offer a few comments about that um in my from my own kind of life and times uh back in the in the late 1970s mid to late 1970s i was involved in kind of traditional physics it was a time that was extremely an extremely fertile time for physics i mean i would say that in terms of fundamental physics we can see there has been various sort of epochs in its development i mean one epoch was the kind of 1915 to 1920s type period when general relativity emerged and then quantum mechanics emerged and then there was sort of a you know even by the end of the 1920s one had the outline of quantum field theory one didn't know the details of how to actually do quantum electrodynamics how to actually do other kinds of quantum field theories but the outline of how quantum field theories should work already existed at that time well then it wasn't clear you know it looked kind of bad you know maybe field theory wasn't really the right thing maybe there was some other kind of approach to strongly interacting particles that was going to be relevant that was sort of the story the 1950s early 1960s um and then by the 1970s with asymptotic freedom and qcd having been discovered in 1973 and so on and uh deep in elastic scattering but bunch of other things it really was the the sort of the the moment when it was realized yes field theory is actually going to work and so there was a whole bunch of low-hanging fruit to be picked it was also at a time when uh i was quite involved in this when uh cosmology and general artillery had kind of emerged as uh as sort of real things which you could do computations with and that was something that happened in the 60s but by the 70s that was by the late 1970s that was kind of converging with some of the things that were being thought about in particle physics and this kind of idea that you know there was there was uh there was commonality in what had to be done between cosmology and particle physics that emerged but so the 1970s were this kind of uh another period of kind of rapid growth for um for fundamental physics and and i happened to be involved in that and uh it was um uh i was kind of not uh you know i was uh just a a person operating in that zone and not the initiator of those things um and uh but it was an exciting time and then uh after that i well i had been using computers to uh uh this kind of a a personal tale that that you know i started using computers when i was kind of ancient by today's standards like 12 13 years old um because that was 1972 1973 when computers were very antiquated compared to what they are today um but uh i had this kind of idea you could do things like mathematical computation by computer you didn't have to do it by hand as most people who did physics did and the shocking thing was that yes i got to the point where i was using existing systems about my own system for doing sort of mathematics by computer the shocking thing was that other people doing theoretical physics weren't doing that and i just i you know at the time i just couldn't understand it it's like why wouldn't you use this tool you know there's a way to do this and um uh you know yes it took understanding a certain amount about how computers worked and and those kinds of things and you know that didn't uh maybe if you were just trained as a physicist or something you wouldn't know those things but it wasn't terribly hard um and uh but it turned out that gave one sort of a superpower that you know i was able to do all these calculations of people like how do you do these calculations you must be a brilliant uh you know calculator no i'm not i just use a computer um so i think the lesson from that is if there are tools that allow you to have a superpower learn them and you know i've now spent the last 40 years built trying to build our sort of super power tower of war from language and so on and the more fluently you know that the more you will be able to just take ideas that you have and kind of turn them into uh you know really work them out and that was the thing i started being able to do back in the 1970s um with computers for doing at first fairly traditional physics but then having built my first uh language called smp which was a kind of a forerunner of mathematical modern waltham language um that was in 1979 to 1981 or so um i kind of was interested in this question of okay so let's look at the natural world well there are things that we find by sort of drilling down and finding elementary particles and so on there are things that are more general questions about the natural world about uh for example the uh you know how does how do things get to be complicated in nature and so i got sort of interested in that and and um started exploring started okay so this was sort of the next step was so what kind of model would you use to study that so my first assumption was i know all this fancy mathematical physics let's just use fancy mathematical physics to study this didn't work don't you know make a model of a snowflake growing didn't really work had some pdes couldn't solve them didn't really couldn't really tell what was going on i said what's the essential feature of what's going on what is the the underlying kind of uh um in a sense meta model that one needs for what's happening in these kinds of systems and i thought well you know i've just built this this computer language and in this computer language i have certain primitives and what i've discovered is those primitives i can build up from these kind of somewhat arbitrarily chosen primitives i can build up this whole world of everything you can do in this language well maybe i can do that for nature as well let me just see you know let me just pick some primitives and see what happens it's kind of applying the same meta idea that i had from this um uh from developing this this uh computer language to what was and then natural science basically and again kind of i suppose a lesson from that is that uh it's kind of like you've got this this area you know you think you're doing physics but actually you have ideas that come from a quite different area and a way of thinking that comes from a quite different area in that case kind of the computational way of thinking now an important piece of that story was that in building smp i had tried to figure out what is the how should one think about computation what is the the right underlying stuff to think about in terms of computation and what i had come up with was this whole transformation rules for symbolic expressions idea which kind of harkens back to the early days of mathematical logic um back in the in the early part of the 20th century um and uh to the 1930s and 1940s and so on but it was kind of a practicalization of those ideas as applied to actually doing computation okay so so that led me to kind of start thinking about okay so if you just think about you know these primitives for the natural world what are the simple primitives okay let's look at simple programs let's look at cellular automata then because i could use computers i could immediately do experiments on cellular automata and my first uh the first thing was gosh these experiments don't work the way i thought they would work now another kind of cautionary tale i suppose is that like rule 30 my kind of all-time favorite cellular automaton i first generated that in 1981 i didn't really pay attention to it until 1984 because basically i didn't really have the intellectual framework to think about what was going on in rule 30 it's just like i sort of assumed that oh you know i could study what happened with random initial conditions i knew dynamical systems theory and a bunch of other kinds of approaches and i i could kind of engage with that but this thing about simple initial condition simple rule very complicated behavior was just sort of outside of my lexicon of what what i thought was possible and to happen so then what i think the um uh the the thing was uh uh kind of the okay so so another thing was going on at the same time was cellular automata they did different things well i guess by the time i saw really understood rule 30 i was realizing gosh it was the case that with simple rules simple initial conditions you could generate complicated behavior but i wasn't sure it was really complicated i thought oh i can use the renalization group i can use number theory i can use some other technique and somehow this little system that i've got will be able to be kind of crushed down to something simpler so i tried to do that and actually sort of a number of very good mathematicians who i knew i was at the institute for advanced study in princeton at the time um and was the sort of hotbed of top mathematicians got interested in the same kind of thing and they were like yeah we can we can solve this you know we know all these fancy mathematical methods we're going to solve it okay so we'll try doing this and nobody really got anywhere and so the thing that i only realized very recently is what was significant about that moment which was the the my mathematician friends just said look our methods don't work we give up we're done what what i realized was and again it took me an extra few decades to actually realize kind of the historical arc of what was going on there what i realized was the very fact that one had got stuck was itself incredibly interesting and in fact that was probably more interesting than any of the details of what one might have been able to pick away at that phenomenon that you know in this computational universe of possible programs you don't you can't crush the whole thing that was a very important thing and and that was what led to this idea of computational irreducibility which i kind of understood by around 1984 or so and um that led to uh that kind of led to the the realization that sort of computational ideas were critically important in in thinking about natural science well the uh uh then you know then i started building mathematica in 1986 and i kind of got diverted from basic science and then came back to it around 1991 and at that time i was like yes there are simple programs that can do complicated things i know one example it's cellular automata i've developed some general idea about how computation can be relevant and and things like computational irreducibility and so on can be more generally relevant and i certainly imagined that computational irreducibility was more generally applicable to you know who knows what the three-body problem all kinds of things in in in in physics and in in um in natural science in general i mean the the idea i'm sure you'll know this but but uh you know the sort of key point of computational irreducibility is once you have the you might think once you have the rules by which a system operates then you're done you can predict everything about what the system will do and that's kind of the the sentiment that you get from sort of traditional mathematical science but in fact computational irreducibility says that isn't the case that in general you have to basically follow each step you can't make those kind of shortcuts there's no way that you as an observer of the system can be computationally more sophisticated than the system itself you can't jump ahead and say and so the answer is 42 or whatever you just have to follow every step just like the system does so that's kind of the concept of computational irritability and again i sort of seen these pretty good examples you know the proof of that and that's a that's a complicated story of to what extent you can prove that phenomenon and how you kind of nail that down and so on um and that's a that's a long-running story but um uh the you know sort of seen how that worked in um uh in cellular automata didn't know how it worked elsewhere so 1991 i i started working on on the uh you know the nks book the the thing that eventually turned into this big book which is now 20 years old um just had its 20th anniversary and in fact i i'm now very aware of its history because i just for its 20th anniversary i really went back and looked in my archives and tried to remember how did that how did the book really evolve and and one of the main things i found was that that again a sort of story of pivoting a bit the book originally was about how does complexity happen in nature and by the by a little ways in i kind of thought i knew the answer to that and then i realized actually the much bigger picture is this new kind of science that you build from thinking about everything computationally from studying simple programs in the computational universe and so on but then the big result was that it you know i knew about cellular automata then i started looking at all kinds of other programs from from touring machines to tiling systems to even partial differential equations and the the first big discovery was this same phenomenon of simple rules complicated behavior happens everywhere and from that i was led to this principle of computational equivalence this idea of this sort of fundamental computational equivalence of all these different kinds of systems and the principle of computational equivalence and computational irreducibility these are two kind of core ideas of sort of the computational paradigm for science and i think that that's sort of the analog in our times of what might have been i don't know the idea of force um momentum and energy or something in in previous times these are kind of guiding principle ideas that uh uh are sort of core things that that i think one has to get sort of very conceptually familiar with so that one can reason in terms of them now you know i will say for those of you who are kind of mathematically oriented and so on that if you say okay prove this or that property of these things it's actually very hard and it's i don't feel quite as bad about the fact that we haven't been able to do it because all kinds of people have tried to do it for 150 years because this is basically directly related to the to the second law of thermodynamics and to something that for example david hilbert spent i don't know i i only learned this recently that he'd spent like a decade trying to understand how do you go from atomic type things to the continuum how do you go from molecular dynamics to fluid mechanics um the the the full mathematical sort of nailing down of that has not been done the big thing that i think is is now pretty clear is that the fundamental why does the second law of thermodynamics work which has always been a muddle um it is because of computational irreducibility and i'll talk about that a little bit later it's all because of the interplay between us as observers with our bounded computational abilities and the underlying systems with their intrinsic computational irreducibility that is the origin of the second law of thermodynamics but nailing down all those details requires that you be able to characterize things like what are we like as observers and you can that is not it's not obvious what we're like as observers that's a kind of thing that science hasn't really had to address and we can somewhat axiomatize we can start to think about accidentizing what are we like as observers that leads to the very modern stuff that we're working on about observer theory maybe i'll talk about that in a few minutes but any case back to the sort of uh historical narrative so to speak so sort of the the big point of um uh the the nks book um ended up being sort of okay there's this way of thinking about models and science which is computationally thinking about simple uh computational rules and what their consequences are and a kind of a different way of thinking about how you model things and i suppose the big point there was uh what i came to realize is that there have been a few epochs in sort of the history of of making models of things one could say that the very first epoch was kind of in antiquity where people were saying what is the world made of is it made of atoms is it made of this is it made of that it's kind of a structural period where where the big story is just sort of what is something made of so to speak then that was the dominant form of science for you know a couple of thousand years basically and in many fields of science that's still where we're at it's still you know a lot of areas of biology and things like that it's what is stuff made of and and that's a it's a perfectly productive kind of science i mean you know it's both what is it made of or make a flow chart for it these are the same kinds of things it's kind of the static picture of how are things constructed so then the big sort of surprise of the late 1600 mid to late 1600s was this kind of the mathematical you know as as isaac newton put it you know the title of his book was mathematical principles of natural philosophy the idea that you could wheel mathematics into natural philosophy or physics um and and and have something useful come out the idea that you could basically just have a formula an equation that describes how things in the world work you could have you know the universal law of gravity or whatever else it's an equation that describes how things in the world work it doesn't tell you structurally how the thing is made it's just an equation that tells you this is how the world works and at first people were very resistant to that idea but um in a little while it became clear it was a very useful idea you could compute all kinds of things whether it's orbits of comets or all sorts of other stuff and so over the next over the 300 years after that that became kind of the dominant form of science that it was kind of use mathematics or at least of exact science use kind of mathematics to essentially write down a formula write down an equation for how things will work and it works very well in lots of areas now it uh uh in a sense that but it gives you certain expectations for example one expectation it gives you is once you have a formula for how something works maybe time appears in that formula but you just fill it in as a parameter and you can work out what will the system do at some time in the future there's no notion of of actually having to grind through every moment of time it's rather just there's a formula just go fill in the value of the time parameter and you'll know what the answer is and that gives you the idea that exact science is about making these sort of jump ahead predictions or it's kind of effortless for for the science to make this kind of jump ahead prediction but so when you know with my sort of nks efforts and so on the big point was that okay there's a different paradigm for thinking about science it's a paradigm in which instead of saying we have the answer we can just work out this formula for the answer we say no we have the rules by which we can generate an answer by which we can generate a simulation of the behavior but we're not we're not immediately able to sort of solve it and say this is the answer we're not able to give some formula for the answer and computational irreducibility is the big sort of thing that stands in the way between uh of doing that and kind of the way i've seen it in in recent times it's kind of an evolution in the way one thinks about time in the structural paradigm for science there isn't really any time it's just what stuff made of in the mathematical paradigm it's like there's time but it's just a parameter and it can it can be filled in with any value in the computational paradigm time is something that has sort of a meaningful character it is the inexorable progress of computation that is sort of the measure of time so to speak and the progress of time is measured in in the progress of a computation and and the the existence of computational irreducibility is the thing that makes sort of time meaningful that allows that to be something that is a con that is sort of achieved by the by the passage of time okay so in any case that was sort of this this this effort to make a new paradigm for thinking about science and you can say well how did that go and the answer is although i think almost nobody noticed it was just amazingly successful because in the 20 years or so you know in the last 20 years-ish that there's been this transition from when new models are made of things before that they were made in terms of equations nowadays most of the time they're made in terms of programs and that's that's a kind of somewhat silent but very dramatic transition that's happened um in in science and so on over that period of time and i don't think people are fully internalized yet the significance of that transition in terms of what kinds of phenomena are important to think about you know computational irreducibility tells you that certain kind of this that this test of predictability of yes we can jump ahead we can work out what's going to happen in this or that system that's not necessarily what you're going to see um in this in this kind of paradigm and you know you don't have the full predictions just when you have the underlying rules you can you can run the simulation explicitly but you don't get to just sort of say and the answer is x um as as as uh as immediately as you do in kind of a mathematical paradigm so in any case that that um uh so that's sort of the the big picture is you've got this new raw material for making models of things it comes from computation and uh you also have a new raw material for mining technology from this computational universe that's a somewhat different story but then the the the obvious question is okay so what about fundamental physics and when my nks book came out uh i was kind of it was kind of an interesting phenomenon because in lots of fields of science people are like oh great new modeling methodology the one area where people were like pitchforks and uh and all this kind of thing was fundamental physics that was particularly ironic because essentially all of the pitchforkers were daily users of our technology and many of them were people that i'd known for many years so it was kind of interesting that there was this very kind of uh visceral oh my gosh this can't be relevant to fundamental physics kind of kind of belief and i think in uh you know it's it's like one well-known such person who said to me you know if you're right it's going to like explode everything we've done for the last 50 years and to which i my main response was i doubt that that will be the case because you know what's been done what we're talking about is something at a very different level from what has been been studied in physics to in the last 50 years but anyway um as it was it was kind of a a uh uh it didn't seem like a great market for developing fundamental physics when the people who were in its primary market uh were like please don't do this project type thing so i went and built wolverine language and built wolf and alfred and things like that and then um at our summer school actually in 2019 um uh max pisgahnofan and jonathan gorod were there and i had made a sort of minor technical um not really breakthrough but minor technical un uh uh un cobwebbing or something of the way that i was thinking about the underlying models and then telling them about this and they're like uh you know you've really got to do this it's got to you know got to work out work out does this work or not and so we got started on this project and remarkably quickly in the in the late 2019 early 2020 um made just amazing progress much greater progress than i ever thought was possible um i really thought that you know in my lifetime so to speak the best we will be able to do is start talking about the first 10 to the minus 100 seconds of the universe and we'd be arguing about kind of micro details of of you know does this really could anything emerge but instead we started to be able to see kind of the big picture of how physics as we know it emerges and and that sort of was very exciting of course we were about to announce the stuff when when a pandemic descended and so that um that led to other other issues but so let's talk a bit about the architecture of what we figured out and then i want to talk about some uh sort of i'm going to talk about kind of several levels here there's there's kind of what we figured i'm not going to talk about that in detail because you probably have read about it and lots of other people will talk about it um i'm going to talk about sort of the overall architecture and what i see is the as the current challenges then i'll talk about i'll talk first about physics then we will descend into the ruliad which is sort of the more general thing that underlies physics and we'll talk about the kind of philosophical character of the roulette and its relationship to physics and then we'll emerge again and talk about metamathematics um that's uh that's my plan here and maybe we'll talk a little bit about the general paradigm and multicomputation okay so let's start on on some physics here so um let me just uh bring up something here just to have something to talk about um so uh let's bring up this visual summary of um of the physics project and as i think you all know uh sort of a a key starting point for the physics project is this idea that space is not a continuous thing just like fluids are not continuous things they're made of molecules space is not continuous it's made of atoms of space that's an idea by the way that people thought would be uh you know they thought that was going to help it was was going to be how it was going to work right when quantum theory was was originating in fact people have sent me endless times now the quote from einstein from 1916 that says it was just after they mentioned general somebody you know says in the end space time will turn out to be discrete but we don't have the tools yet to be able to analyze how that will work and so then it took another hundred years and so now uh you know old albert was right um that seems to be how it actually works so the i think the um so that sort of the the starting point here is this notion that spaces is made of discrete things discrete elements of space atoms of space one might call them um the later on we'll talk about how that sort of a more general concept of atoms of existence and what we're calling eames emes um that correspond to those atoms of existence but in the way of talking about them in terms of physics we're interpreting them as atoms of space and the only thing we say about these atoms of space is how they're related to each other we make this hypograph that represents relations a collection of relations between atoms of space and then sort of the the big idea is that um we look at rewriting rules let's say local rewriting rules on uh little regions of that hypergraph we take the limit as we apply those rewriting rules many times we see what kind of structures we get okay so the and and the main point is that we imagine that those structures eventually limit to something like continuum space-time or continuum space at least but the spatial hypograph limits to something which is continuous space when we're looking at the the sort of the history of updates and so on then uh then we we are generating causal graphs and space time and so on but so first big question is okay so you've got this big hypograph and um it's kind of somehow limiting to something like space how does that limit actually work and we can compute things about it we can compute the effect of dimension we can compute all kinds of geometry gd6 all sorts of other things um but how does that really work well one thing that is true is that it's it's not necessarily interdimensional it's not like a manifold with interdimensional space that's locally euclidean and all those kinds of things it's some new and weird kind of object and essentially one of the projects that we're trying to do these days is this kind of thing i don't know what its final name will be but we've been sort of internally calling it infra calculus kind of what is the generalization of calculus that applies to things that don't end up being like manifolds that apply to things like these limiting hypographs how do we construct all the ideas of calculus and differential geometry and all those kinds of things you know what's a riemann tensor in 2.6 dimensions things like this so that's a kind of a big mathematical build out that we're in the process of doing and which i think is going to be very fertile and we've certainly done pieces of that but i want to see that be done more systematically as one does that so as we start looking at well what does this limiting structure look like it's kind of a thing that is qualitatively like the story if you take a bunch of molecules bouncing around in a fluid and you say what are the emergent equations of fluid dynamics i happen to work on that for cellular automata back in the mid 1980s and the derivation of the fluid equations from an underlying discrete dynamics this is a similar case and the question is what is that what are the emergent equations and basically with a whole bunch of footnotes and a whole bunch of untied down mathematics the answer is it's einstein's equations an important piece of that story is the identification of energy with activity in the network um and so that's that's kind of another part of the story there's also stories about causal and variance and so on there's there's lots of stuff that i'm not going to talk about here but basically the sort of architectural point is the continuum limit of this hypergolf rewriting process is the analog of the fluid equations is einstein's equations now let's you know we'd love to nail that down better you know jonathan has done some work trying to actually do kind of numerical relativity by using these kinds of techniques we'd love to see more of that done we'd love to actually see how uh how one can do practical numerical relativity this seems to be a good scheme for doing practical numerical relativity but more than that that kind of gives evidence that our model is really working because we can say look we can compute black hole mergers just like people who start from einstein's equations and discretize them we are starting from a discrete underlying dynamics and coming up to something which has kind of continuum-like behavior but what's more important about that is as we do that we can both validate that we're actually getting the same answer but we can also see how do we deviate from the answers that are gotten from using continuum equations how do we see for example the effects of you know shot noise on some rapidly rotating black hole how do we see how do we see down in sort of some gravitational microscope down to the underlying uh sort of atoms of space uh can we use some weird uh i guess one of the big questions there is sort of uh you know when you make up experiments you're kind of you're kind of poking at different parts of nature to try and see how it works so if you have fluid dynamics you don't self-evidently know that there are molecules in a fluid that's something that you have to come up with a clever experiment you know you have to invent brownian motion and realize you can look at pollen grains bouncing around and see that there's something molecular in in the fluid or you have to look at hypersonic flow or something and see that there is something sort of molecular about the fluid so similarly for space-time we'd really like to find are there effects where we can see down to this kind of uh level of atoms of space you know is there some weird gravitational lensing effect is there some effect of you know photons in orbit around a black hole these kinds of things where you can see that and we kind of expect that there will be for example dimension fluctuations in the early universe that because we think the the universe basically we can think of it as starting infinite dimensional and gradually sort of cooling down we don't know like the friedman robertson walker metric uh for for an evolving universe we'd love to have a version of that that accounts for dimension change um because that accounts for the evolution of dimension as well as the evolution of curvature um and we suspect that there will be curve dimension fluctuations left over from the early universe and those might be observable for example on the cosmic microwave background so we kind of expect that there'll be some absolutely bizarre effects that one never would have expected could possibly be there that will be suggested by our models but we need to actually nail down how do you actually do the astrophysics experiments and so on to see those things and i mean we've been uh it's very nice that um uh very unlike the the time of the pitchforking of uh from 20 years ago i would say that the response to to what we've been doing has been very positive and we've sort of been been inundated with experimentalists saying you know just tell us what to look for and we're like we don't know what to tell you yet because there's a bunch of actual physics that has to be done to go from this sort of underlying model to uh to the thing that you can actually point a telescope at and see what's going on so that's a that's the thing that i think is a very important thing to do i'm a little bit you know if i were to say what's ahead of schedule what's behind schedule in terms of what i might have expected from the development of the project i would say sort of the phenomenology of the connection between our models and what's observable is maybe a little bit behind schedule um at this point and something where i think there's a lot of fertile stuff to be done now i might also mention uh well let's let's uh go on and talk well okay there are no let me go on and talk about quantum mechanics sort of the big idea there is this idea of multi-way systems the idea that there can be many different rewrites that can happen and don't just say i want to pick this particular rewrite i want to be running a you know you know the universe is a monte carlo simulation or something but we're just going to pick one path no you're looking at all paths and those paths sometimes they branch sometimes they merge we generate this multi-way graph i just did an analysis recently about games like tic-tac-toe where you have multi-way graphs where the different possible moves you can make correspond to different branches you can follow and then if you get to the same board configuration that's emerge and you can study those in terms of multi-way graphs you can study many way many things in terms of multi-way graphs in fact this whole sort of pure multi-computation idea is itself a very fertile one for studying all kinds of systems so in any case the um the sort of the big idea there is inevitably you end up with getting a multi-way graph from these models of hours from the the microscopic updates and the in the in this hypergraph there are many possible uh sequences of updates that can occur many possible uh threads of history so to speak that can be produced and you get these complicated pictures well this is a multi-way causal graph here but the the skeleton of this is a multi-way graph and each each one of the nodes here is the state of the universe and there are different you can get branchings you can get mergings and so on so the story of quantum mechanics we think is so we're pretty sure is very sure actually i would say at this point is is the story of this whole branching of all these different possible histories for the universe now one of the features of that is okay so in in physical space we know how that sort of laid out in the hyper graph what is the thing that you get by slicing across this multi-weight graph it's another kind of space we call it branchial space it's the space of possible let's say quantum branches and you can say which branch is near what other branch well you can look in terms of common ancestry in the multi-way graph that gives you some some sense of distance in branch hill space so branch real space is sort of the space in which quantum states play out and we can start asking about all kinds of things about about branchial space well let me say something about us as observers of what's going on in physical space we're not sensitive to the individual atoms of space because we're really big compared to the atoms of space and so we sort of average out big chunks of of this and that's that's why we think of space as a continuum in the case of branchial space we're also kind of big compared to the individual threads here because just as the universe is branching all the time so too our brains are branching so in a sense the story of quantum mechanics becomes the story of how does a branching brain perceive a branching universe and that that's um and so it's all about sort of how big are we in branchial space what kind of uh what kind of um what kind of sampling of branchial space are we taking just like we might say about physical space what kind of sampling a physical space are we taking so there's there's a lot of uh that we there's a lot we don't understand about branchfield space for example we don't really understand how to coordinate eyes bronchial space we have indications in certain simple cases but even something like the double slit experiment we really want to nail down more precisely how does the coordination of branchial space relate to physical coordinates of of angles and things like that have some ideas but it doesn't i would say fully nailed down i think that the um uh the question you know our our rough interpretation is that position in branchial space corresponds to quantum phase and kind of the statement that uh i've been making is that just like in the case of space time i didn't really talk about this actually that the in space time people kind of got the idea as a result of things minkowski did in 1909 or so that space and time are the same kind of thing which not really in agreement with our intuitive expectations but in our model space and time are not the same kind of thing space is the extension of the spatial hypergraph time is the inexorable kind of uh computation of next states and so on so the bundling of space and time together may have been a mistake in physics and similarly we tend to think that the bundling of quantum phase and quantum magnitude is also a mistake that quantum phase is a different kind of thing it's position in branchial space quantum magnitude is more of counting of the number of paths in the multi-wave graph so okay so there's a uh there's this whole sort of story of brawnshield space there's the whole uh uh the whole thing we've been doing about looking at multi-way graphs as a an underlying way to look at quantum mechanics and that looks very promising through zx calculus and things like that it seems that we can pretty much compile sort of standard quantum mechanics and things like quantum information quantum circuits and so on into multi-way graphs and then compute there and then sort of translate back and so that's again that's a great way of validating our models is to say look they're actually just directly equivalent to perhaps in some limit to what you normally see in quantum information and actually we've been we have now have a quantum framework that's been rather nicely developed nick has been nick merzin has been very involved in that as mads has been recently um the uh uh jonathan's been involved in the past multiple people have been involved um but uh uh nick just won some nice quantum computing competition by using our framework which is nice using our quantum framework in morphing language um that's not yet as completely connected to the things that we're doing uh in multi-way graphs as it could be jonathan's done a bunch of work works he's had him i've also worked on that on that connection um so uh in any case the the um that that's quantum mechanics then there's quantum field theory and then this quantum gravity and all of those things are kind of part of the story for quantum field theory the big point is to sort of combine the spatial degrees of freedom with these branchial degrees of freedom and that's just a mathematical well i wouldn't say mess it's very interesting very rich but it's complicated and the because here every single node has the complete state of the universe which is incredibly wasteful because between this node and this node there's very little difference and but yet we're sort of representing our data structure is such that we're sort of copying the whole universe over here so there's also the notion of local multi-way systems and global multi-way systems which are ways to kind of refactor how you present the data of a multi-way system and that story of that refactoring i think ends up being the story of quantum field theory and just as we can talk about doing direct uh sort of uh simulations of of space time using our models we also will expect to be able to do direct simulation of quantum field theory using our models kind of giving one a a different kind of story from lattice gage theories and things like that a more direct way to just do kind of field theory computations um using uh using the sort of underlying discrete stuff so that's a that's another direction then comes quantum gravity uh we kind of think that uh the the the multi-way causal graph is the thing that sort of knits together spatial structure and branchial structure that's probably that knitting together is probably exactly the adscft correspondence that again needs to be nailed down better and the extent to which you can kind of project in the spatial direction or in the branchial direction and get kind of the the general relativity or the or the field theory side of things um but that's that seems to be a very much a concretization of the kinds of things people have thought about in in other areas of mathematical physics and so this is another theme is that what we have seems to be kind of the a an explicit machine code raw material for lots of different approaches to mathematical physics and that's something that again is a very nice feature of what's emerged from these models something i didn't expect at all is sort of everybody's right it's not the case that you know we're right and everybody else is wrong no actually everybody is right one's taking different kinds of limits and things like this so you know spin networks um you know causal set theory causal dynamical triangulations probably string theory these are all things which seem to emerge as various kinds of limits and specializations of our kind of model um now there's a lot more to be done in nailing all of that down i mean we have the qualitative picture i think and a number of people at previous summer schools and winter schools have worked on these things but there's more to be done to kind of have the definitive nailing down of those things and among other things i'm i'm really curious whether my kind of ridiculous pun of the fact that some string theory is the continuum limit of string rewriting systems turns out to be correct something like that is probably true whether that actual pun turns out to be exactly the right thing i don't know yet but anyway so so that's a another kind of a big direction is is sort of connect what we're doing to other approaches to mathematical physics um and i think there will be that's a very fertile thing both for for our project and for those uh those other approaches to mathematical physics okay so uh that's sort of a very rough outline of some of where we're at i mean when i talk about the physics project there there are pieces there are there are great big gaping holes um so as we do this infra calculus and we understand how to really how to really set up the structure of um uh of kind of uh the things that we're used to in in you know differential geometry and things like that um there are there are things that we can then start to see like what's a particle well we think a particle is a topological obstruction of some kind kind of like a vortex and a fluid type thing but really nail that down you know what does it mean when we have uh something like a a fiber bundle embedded in our emerging from these systems that we're looking at what what do what does local gauge invariance look like we've got very good ideas about this and and toy projects have been done but we need to continue that we need to even understand what does rotational invariance really correspond to in these hypergraph systems and so on how does the how does a lead group continuous group emerge we get some idea of how continuous space emerges how does something like a a lead group emerge and and one of the big questions is what is generic and what is not and what emerges so for example we don't think that three dimensions is a generic result that emerges but it's possible that the league group you know that everything you get has to be a subgroup of e8 or something we don't know that's possible that will be a very exciting thing to discover and it might be something that is a necessary feature of these models now there's another whole direction here which uh i would say jonathan xerxes etc have been working on which is kind of the the the kind of um connecting this to sort of mathematical structures like in category theory higher category theory and so on and understanding kind of the the uh the relations between sort of using that as a framework to think about what we're doing and using what we're doing as a framework to think about those kinds of things james boyd has also been involved in in this in this effort so the um uh and um actually most people have been involved i haven't been as involved in that effort as as uh in some of these other things um but any case the the the that's sort of a another direction um and uh again one of the things that's really great about our models is they're very concrete i mean they're they're you know it reminds me greatly sort of in terms of history of science with what happened with turing machines you know people had combinators they had lambda calculus they had um uh you know various kinds of rewrite systems and so on but it was touring machines that made people kind of understand concretely what was this thing that was abstractly being thought about not really in a coherent way that emerged as computation and i think what we have uh with our models is sort of a similar kind of concrete machine code from which a lot of things can be built so in any case the the that's sort of a rough outline of some things with the physics project let's now go deeper down the rabbit hole okay so one of the big questions that emerged with the physics project was let's say we're successful we managed to find a rule that can reproduce physics as we know it oh i i'm sorry i missed one thing i want to talk about i want to talk about particles so i talk about you know making local gauge invariants fiber bundles things like this and these topological obstructions i mean we want to find the electron you know we will be able to find something like that but we haven't done it yet and that's one of the things but i think we need a certain amount of more infrastructure development before we can get to the point of being able to do that and you know if you're a particle physicist the thing you always have to do is you know find the spectrum of particles that's kind of the name of the game in particle physics and i think there are all sorts of exotic possibilities in our models and to even know what's conceivable like very low mass particles that you know maybe dark matter like things or who knows or weird things that aren't really particles in the ordinary sense of localization and so on these are things we should understand because these are things one can go out and actually look for in the physical world and if you're in the business of making models of physics it's uh you know it's it's very very useful to make something like calculus which provides sort of a way of computing things but it's kind of a good wow factor if you can say go turn the telescope in that direction you'll see this weird thing and somebody does it and they actually see that weird thing that has a very high wow factor and it's something that we would uh would like to see happen um i think that uh uh you know and whether that is accessible in 2020 you know to 2023 uh 2057 i don't know but uh it's something which i i think it will be uh it will be frustrating if we just hadn't kicked those tires to find out if it's something accessible in 2022 or 2023. so in any case okay let's let's go further down the rabbit hole so question is you think you have uh you know you've got this rule and it reproduces physics as we know it weird situation because you say why was it that rule and not another rule and i was wondering about that for ages and ages and ages and what became clear from this whole multi-way way of thinking about things is actually just think about what would happen if if all possible rules were followed what would you get then and as an observer embedded in this thing where all possible rules are being followed where you also have all possible rules being followed inside you what do you observe and how does it relate to what we actually see and that led to this idea of the ruliad uh the ruliad is the entangled limit of all possible computational processes so you know you can look at it that's a very abstract concept but you can break it down you can sort of coordinate it in a variety of ways so for example one coordination would be take those computational processes to be turing machines and just say start off all possible touring machines with all possible initial conditions let them run what do you get well you might say you're not going to get anything with any structure at all but you'd be wrong because the equivalences between states of turing machines means that a turing machine you might start with one state there are two different turing machines they end up with two different states then you run them again and then they can merge to produce the same state again and and so i made lots of pictures of of what you get in the sort of rule limit of touring of multi-way touring machines and how you make sort of a coordination of the ruliad using turing machines for example but so the ruliad is the structure that is this entangled limit of all possible computations and then then what we are saying is that the ruliad is a necessary object it just is something that given the formal structure of given just the idea of formalism so to speak you necessarily get the ruliad it's just following all these rules and you don't have to say i'm going to pick this one i'm going to do this special thing i'm going to have this special kind of uh you know curation of the whole thing it just necessarily has this form so then the question is well what about our observation of so if that's what's going on uh and uh that that's sort of the universal possible universe universes all entangled together it's not kind of a multiverse type thing where it's just a bunch of separate branches this is something where everything is deeply entangled together and so uh oh you know i forgot to mention one other thing i'm just going to mention i'm just going to go back i'm sorry to do this we're talking about quantum mechanics in brownshield space i just want to mention that one of the areas that i'm also really interested in is is sort of what sort of experimentally accessible predictions might there be from things that we can say about essentially quantum money body systems from thinking about them in terms of branch hill space one of the things that emerges in our model is a maximum entanglement speed analogous to the speed of light in physical space there's a maximum speed of information propagation in branchial space and i'm kind of suspecting that there is a way in something close to chemistry quantum chemistry to observe the effects of the of the maximum entanglement speed maybe i mean who knows what the magnitudes are we really have only one parameter in our models which is equivalently the elementary time the elementary length the maximum entanglement speed these are all once you have any one of those you determine just using planck units and so on you just determine every all the other ones but we don't know that value of that parameter but so i'm sort of interested in in kind of working through how does one think about quantum many body systems in these terms and are there predictions that come out of that all right i'm going to come back now to the ruliad um and uh uh just as so okay the rulier is out there and how do we experience the ruliad we are taking a tiny sample we are part of the ruby ad but we're taking a tiny sample of the rouliad and that is our experience of what happens in the world okay so here's the important sort of philosophical point that we realized in the last year or so that's the following point the as observers of the ruliad we know we have certain characteristics two very important characteristics one is we're computationally bounded that is we don't get to actually figure out what's happening with all those different branches all those different computations happening in the ruby ad it's analogous to when we look at uh molecules bouncing around in a gas we don't get to trace every individual molecule we just have some aggregated uh sort of computationally bounded representation of what's going on in the gas so first point is it's a we have a computationally bounded view of the ruliad one more condition the other condition is we believe that we are persistent in time you might think that's kind of trivial but it isn't because at every moment in time we're made of different atoms of space yet we believe we have a consistent thread of experience and that fact kind of ends up driving the um uh a whole lot of things and so so i think the the um the point uh so those two conditions we're computationally bounded and we have a belief that we are persistent in time so it turns out that once you have those two conditions combined with the principle of computational equivalence combined with computational irreducibility plus uh sort of a a bunch of other machination it is essentially inevitable that you will derive general relativity and quantum mechanics or so it seems and in a sense this is similar to what we've seen before it's similar to fluid mechanics it's similar to thermodynamics you know it doesn't matter what the underlying molecules are you still always get the same continuum equations for an observer like us if we were little tiny molecular scale critters we would probably not believe in fluid mechanics we might not believe in thermodynamics either but because we are observers like us we come to the conclusion that the world works this way and so philosophically what we've realized is that from the ruliad the the sampling that we that observers like us take of the ruliad is one where where you um uh where necessarily we get the laws of physics as we the general laws of physics as we observe them um so in other words it is it is our nature as observers that essentially combined with the necessary character of the ruliad that gives us the physics that we we are familiar with now you can ask okay what about the aliens who operate differently and don't sort of have those conditions well they will observe a different physics and their physics will be perhaps bizarrely incoherent with ours now for example even in the evolution of our own sort of civilization so to speak we can expect changes like as we are able to be uh to have sensors that measure different things about the world we are essentially moving outward in rule space or even as we have different ways of thinking about our models of how the world works we're kind of moving in real space and we can expect some sort of gradual expansion in rural space as our technology and as our methods of thinking about things improve we are we're sort of gradually able to see further in rural space now i mean things get things out very weird here because because i mean sort of at a philosophical level the you know being at different places in rural space correspond to having different views of how the universe works and so every different mind has a slightly different view of how the universe works human minds have fairly consistent and coherent views about how things work in the world they are close together in rural space by the time you get i don't know a dog or something there are certain things you can see are pretty much the same you know wags its tail when it's excited and so on you can kind of interpret that as you get further away in rulial space to you know the weather which might have a mind of its own so to speak that mind is far away in rural space and and very hard to interconvert the with the kinds of concepts that we have but as we kind of in a sense develop further our own sort of technology and way of thinking about things we're kind of moving outwards in exploring real space one of the things that i sort of realized recently should have realized a long time ago is that one of the bizarre features of of nks and of what we now call ruleology just looking at uh different possible rules out in the computational universe is you get to do these kind of jumps through rule space while while many of the things we do are sort of gradual expansion where we're kind of connecting them back to typical human experience when we just say oh pick a random rule out in in in the ruliad or in in in the computational universe pick a random rule there's no human connection necessarily to that and so that's that's that's sort of that's sort of a a different experience of kind of going out and and doing that anyway there's there's lots of kind of philosophical and fairly fairly deep philosophical things about about uh kind of definitions of technology and consciousness and all sorts of things like that that um uh come out of this thinking about the ruling adam maybe i won't talk about those particularly here um but that's something which to me is as important as a way to sort of understand uh the kind of the bigger picture of what's going on some of some of the questions that one one ends up asking questions that were asked a thousand years ago but but kind of by the time mathematical science took off people said that's going great let's you know we don't have a way to ask those kinds of questions let's not worry about those for right now we're back to the after the right now so to speak so we get to think about some of those kinds of things and there are a lot of very weird ideas i mean for example uh the analog of particles in real space is kind of something which a particle is like something that maintains its coherence through time moving in space even the fact that pure emotion is possible even in physical space is not a trivial thing the fact that you can take a thing and move it without it changing is not obvious you know near a space-time singularity we even know from traditional general authority that you take a thing and it's going to get smooshed and it's not going to stay the same as it just sort of moves around in space well in our models we have to kind of derive the fact that there is pure motion the fact that you can take a thing and for an observer like us have it move without change because the thing is is changing in the sense it's made of different atoms of space but at least for an observer like us it seems oh that's just a particle that just moved around so anyway so you can ask questions like what's the analog of a particle in rule space and i think that is a kind of a thing that moves from one essentially mind to another without change and that kind of ends up being something like sort of a one can interpret as something like a robust concept because you know two minds have a completely different representation internally like two neural nets and there's a good thing somebody could study actually of looking at two different neural nets that got you know learnt things in two different ways and uh the question is is there some robust thing that can be can move from one to the other that becomes something that's sort of a concept that is moves without change in rural space okay so we're we're kind of descending into sort of uh deeper philosophical kinds of things and there are lots of things one could start to unravel i think and it's it's been lots of fun recently to thinking about some of those kinds of things but let's come back out in a different direction let's talk about mathematics so the question is uh from this point of view of uh of thinking about things and sort of figuring out what are the underlying structures in uh of things how do we think about mathematics so let me bring up something here um let's see oh where do i have it [Music] ah there we go okay let's see that's a good place to start all right um actually i'm going to go to a different place yeah this is that that first thing there that alien intelligence and the concept of technology was the thing i wrote last week or so it's kind of an interesting philosophical application of some of these kinds of things um talking about well what the title says uh um okay and talk about metamathematics so okay so let's talk about kind of um how we construct mathematics and one of the things we're going to say is that physics is ultimately our sampling of the rouliad physics is kind of the emergent thing for us as observers from this inevitable necessary object that is the ruliad the claim we're going to make is that mathematics is exactly the same kind of thing it too emerges from the ruliad but by a different kind of sampling that the mathematical observer is different from the physical observer both have certain characteristics in common that are a consequence of essentially the fact that they're both humans so to speak that humans are at the bottom of both of those kinds of observation but they have differences okay so let's talk about kind of what mathematics how we start constructing mathematics so we want to talk about what's also the meta model of mathematics so we can think about mathematics as defined you think about it as having some kind of we typically you know this is the hilbert russell etc kind of idea uh well it goes back to euclid in some sense and through piano and frager and people like that this idea of let's represent mathematics by having axioms and then let's derive theorems from those axioms and uh so we we're going to then we can just build up these networks which say given a particular this case an expression we have certain axioms that derives another expression a true theorem of mathematics will then be a path in this network of possible expressions where every edge here is uh corresponds to an axiom of our of our system of mathematics and then a proof is the steps on that path saying how do we transform from this expression to this expression now at a slightly more technical level it's convenient to think about things in terms of um let's see oh boy lots of technical stuff here um the the um to think about things where we where we have the nodes in our graph are our actual relations and what we're doing is we're applying laws of inference entailments to these relations to these theorems to derive other theorems and it's a story of multi-way graphs again because from one theorem you get to derive a whole bunch of other theorems from those you may end up well you end up getting this whole structure that is um a structure where you start off with certain uh uh propositions certain original axioms and you derive other theorems and you build this whole network and this whole network is a multi-way graph just like there's a uh just like the multi-way graph that we have in uh in physics and you can go and you can look um what is that one humph let's see you could look just to sort of connect this with things that one might know about um we can um uh we can look at sort of empirical meta mathematics i had looked um uh at well for example these are these are um results in boolean algebra um where you can say what is the the um the graph the the entailment graph um that connects these results in boolean algebra um and you will derive all kinds of results some of them may be results that are significant that humans talk about some of them may just be sort of intermediate results that one doesn't particularly talk about so this is an example this is the branchial graph of that comes from derivations in boolean algebra just like we can talk about light cones in physics where you have an event that occurs and it has consequences for other things in the future so similarly you can talk about an entailment cone in mathematics where you say here's a theorem what other theorems are entailed in the entailment cone of that theorem and so you can look at the kind of the transversal uh to that you can look at this branchial space that you get which asks the question how do you uh how what are the kind of ancestral relations between between uh results between theorems you get so here are some notable theorems of boolean algebra and this shows essentially their their branchial connections in the entailment cone well we can go through and look at this um this is uh euclid's elements this is euclid's um uh the entailments that euclid uh you know asserted about the theorem so these are the these are the up here are the propositions and common notions of euclid and down here is all the things that you could proved and right down at the bottom is the proof that there are five platonic solids and that proof in the end has a very complicated um uh well you can see the structure here of what leads to that proof well you can do the same kind of thing you can take um uh yeah these are this is this is the sort of fully unrolled euclid proof um you can you can take uh places where people have tried to make formalized versions of mathematics uh things like um lean or or a system called metamath which is a nice clean simple system and you can take they've they've encoded you know 40 000 theorems of mathematics together with their proofs and you can ask the question how are those things related and how what is the sort of entailment structure of all those things and this is the pythagorean theorem i think yes there's a pythagorean theorem in the matter math proof uh assistant system and uh you can see somewhere down here is the pythagorean theorem being proved somewhere up here a bunch of axioms of set theory and this is a complicated mess that goes through 6970 intermediate theorems to prove the pythagorean theorem and you can go through and you can look i mean sometimes this is kind of ridiculous uh you can unroll things and you can find out that modus ponens appears 10 to the 36 times in the proof of uh the pythagorean theorem this is sort of a a sense and one can even as i did here one can even start to compare the sort of number of uh the number of steps necessary to prove the pythagorean theorem with the number of elementary events that happen in physics to make something that we experience in the physical world but you can kind of go on and you can look at well all kinds of things you can look at sort of how uh well this is actually something slightly different but you can look at um things about how uh in metamathematical space in the sort of space of how of things entailing things how different fields of mathematics are related here for example are some famous theorems of mathematics and where they lie effectively in this network of of uh this sort of axiomatic structure network okay so one question is is this what mathematics is actually about this grinding down to these low-level axioms and looking at what happens there let me make two comments here um one comment is that uh well actually okay first comment is even the level of axioms is not the bottom level at the level of axioms as we normally formulate them with kind of variables and quantifiers and things like that there's already a lot of structure that we put in there we can actually go down below that level we can construct those things from something even lower level we could for example use well you could if we were writing this in in computer code we could write it in machine code we could uh if we're writing it in a slightly more formal way we can write um let's see i have an example here uh where is it um section 22. thank you this is what happens when one writes these things too quickly is that that um you know the nks book it took a decade to write this this particular piece only took about nine months to write and so i i don't uh uh i'm not as personally um um each individual page doesn't doesn't occupy such a large piece of my my personal experience um anyway this is sort of going below the level of um sort of named variables and things like that this is using combinators as a basis by the way in case people are curious combinators you know i i was it was the 100th anniversary of combinators in in 2020 in december of 2020 and so i thought i'd better go actually understand combinations i've been meaning to really understand this for ages and i ended up writing this whole book about combinators um but turns out that the effort that i spent on combinators which might have been thought to be sort of overkill at some level um in um just studying combinators combinators are a super useful toy model of lots of kinds of things and they're really worthwhile as a way to think about sort of the link between sort of meaningful computation and multi-way systems and things like that so and by the way that was the same reason that i i just studied i'll just show you that example for a second here um let's see if i can pull it up um just just for fun talking about multi-way systems i just wrote this thing about games and puzzles as multi-way systems and you know here's tic-tac-toe simplified version of tic-tac-toe as a multi-way graph and uh you have somewhere here i have actual tic-tac-toe um there we go there's the three by three board that's the game graph of tic-tac-toe which is a multi-weight graph and we can start doing the same kind of analysis of it talking about branchial graphs all that kind of thing as we've done with other uh with these other systems this is just a way to kind of humanize um uh the phenomenon of multi-computation humanized this idea of multiple threads of time and so on um and uh i looked at needless to say we looked at all kinds of others this is all kinds of things about winning conditions in tic-tac-toe and which side is winning at different places in the multi-way graph and so on and needless to say we looked at uh oh all kinds of other things let's see what's oh yeah that's that's a fun application of ramsey theory to to to tic-tac-toe but this is relating it to uh things like um uh random walks and then mazes and uh then to um various kinds of games that's a game invented by um uh william rowan hamilton uh where hamiltonian circuits came from that's different anyway lots of multi-way graphs but the point here is is sort of a a humanization of multi-way graphs um uh by by seeing them sort of in action there's a simplified rubik's cube there's the multi-way graph you get for that that's a two by two by two multi uh rubik's cube and that shows its kind of game graph another multi-way graph story um so in any case the um uh but we're now descending below uh traditional axiom systems we're looking at them as built from combinators um we can see that we can start making uh well this is arithmetic being built from combinators and we can cut start seeing all these different kinds of things built up from combinators now what we realize here is what we have descended to at this point is the ruliad basically we have descended below axiomatic mathematics and we're at the level of the raw ruliad and what we're doing in that raw ruling ad is we're fishing out things and saying that clump of things in the raw rouliad we can identify that with the integer one and then we can identify that other clump with this and then when we go further up we say well that corresponds to this particular piece of some axiom system there may be many ways to identify the number one in in the raw rouliad we think of it as being the the sort of the underlying data structure is a bunch of atoms of existence or eams which are knitted together by the ruliad and we're seeing sort of the emergence of the structure of things of even things like variables are emerging from these eames um in in at the level of the raw ruliad so that's kind of going down to the raw ruliad we've got this kind of uh this sort of undefined uh ocean of of computations which we're then sort of identifying pieces of fishing them out and saying that looks like a plus sign so to speak and we can explicitly see just for fun even in empirical matter mathematics we can start to see that phenomenon happen um see if i can get to that here um even in these formalized mathematics systems that is the breakdown of plus in terms of lower level operations of uh that ultimately are the things that appear in the axioms of set theory in this particular case logic and set theory so that's kind of that's a little bit of the way down it's not all the way down to the ruliad but it's a little bit it's part of the journey going down breaking down the definition of plus that's breaking down the definition of gcd in terms of lower level objects and what we're doing is we think about the ruliad is going all the way down even further much further down than this we're reaching the raw ruliad in the end and so what we perceive as mathematics is this thing that is some kind of emergent structure um in the in the raw rouliad okay let's talk about mathematics as mathematicians actually do it because most mathematicians do not operate at the level of individual axioms and and uh at the level of sort of the the grinding detail of for example that derivation of i mean the pythagorean theorem nobody gives a derivation of the pythagorean theorem that's based on this kind of stuff um it's uh with 10 to the 36 uh instances of modus ponens and things like this instead mathematics says done by humans is done in a much higher level it's kind of like our perception of fluid dynamics it's at the fluid level not at the molecular dynamics level it's like our perception of the physical universe is at the level of something close to continuum space not at the level of atoms of space so somehow when we think about physics we as observers of physics are operating at this higher level how does it work in mathematics well i think it's the same thing i think that there is a higher level mathematics where we just say it's the pythagorean theorem we don't think about oh it's based in this kind of representation of the real numbers it's got this kind of particular definition of this that and the other it's enough to just say it's the pythagorean theorem so then what's not obvious i think i have a picture here of of the um uh of the different derivations of the you can you can you can reseat the pythagorean theorem differently even in these axiomatic um even in these um uh proof assistant systems and so on um i think i had a picture of that where is it james where is our beautiful picture go down wrong way further down yeah ah there it is okay these are two different formulations of the pythagorean theorem derived from from the same axioms but they're sort of that these are but for a mathematician it's like it's just the pythagorean theorem yes there are differences if you grind down to the level of atoms of existence these can be very very very different but at the level of the way mathematicians think about it one thinks about them as the same thing okay so the question is does that work and it's like saying can you do fluid mechanics or do you have to go down to the level of molecular dynamics there are effects when you're studying hypersonic flow or something where you have no choice but to go down to the level of underlying uh of the sort of underlying molecules but most of the time it's enough to just think about things at the fluid dynamics level so the claim is that what we're making is that essentially the fact that higher level mathematics is possible the fact that you can consistently just think about the pythagorean theorem not this ground down thing in terms of atoms of existence that is the same basic phenomenon as the phenomenon that allows us to think about continuum space and it happens because it's observers like us that are doing the mathematics now it turns out that the way we think about how we sort of are extracting things from the ruliad is different the way we coordinate the ruliad for mathematics and for physics is different physics is a very time oriented coordination of the ruliad mathematics is much more of a of a kind of a transverse uh way of thinking about the rulia we think about entailment fabrics we think about the mathematical observer kind of building up this kind of zone of of the ruliad that is that sort of the the set of things that they think of as being the sort of the the the things they think are true in their mathematics so to speak and and we kind of think about uh moving from there and and deriving other kinds of things so for example here's an example of something if you think about the notion of falsity in mathematics remember the things we're doing here are deeply constructive we're saying you just derive this theorem from this theorem from this theorem so what happens what is falsity well falsity and this is something people have talked about since the middle ages the principle of explosion the idea that once you have something which is false you can derive everything and so in our picture what that is is something where you you have it's like a white hole in physics it's something where it's just sort of spewing out um results now for an observer the issue is that a computationally bounded observer can't deal with that a computationally bounded observer is trying to have this coherent view of what's going on this this bounded region of rule space that the observer kind of uh hasn't has in their mind as being what they think is true in mathematics and as soon as you introduce this exploding sort of uh false proposition that is no longer possible you can no longer maintain that kind of coherence so any case the the sort of the the idea here is to think about uh mathematics as something which emerges from the ruliad there's a notion of meta mathematical space which has all these theorems embedded in it and and connected in different distances in in meta-mathematical space and so on and there's this notion of the observer um who is uh essentially deciding you know to take a particular sampling of the ruliad now uh so what what are some predictions of this kind of approach so one important thing is one of the things about physics that is not obvious at all is that there are laws of physics that is it might be the case that you're stuck talking about individual things about individual atoms of space and so on and there won't be any zoomed out laws of physics there won't be anything like general relativity anything like that it's all details so to speak but we know there are actually laws of physics global laws of physics so the question is are there global laws of mathematics if we have this picture that mathematics also emerges from the ruliad what are the global laws of mathematics here's an example so imagine that uh in in physical space there's a certain homogeneity to the universe the universe here is pretty much the same as the universe somewhere else as far as many aspects of the structure of space time and other things are concerned what would that be like in mathematics that's like saying one place in meta mathematical space is much like another place in mathematical matter mathematical space what that means is if you take some structure that you've built in one part of mathematical space you can transport it to another part of mathematical space what that's like i mean you can think about it category theory with functors and things like that but you can also think about it as just sort of the the statement that you have one area of mathematics and there will be some duality to another area of mathematics it's kind of explaining to you the homogeneity of meta-mathematical space is kind of the statement that there will be sort of that that the the structure of one area of mathematics will have a correspondence after some translation that corresponds to meta-mathematical motion to another area of mathematics so that's kind of the the type of thing that you would conclude from thinking about things in terms of matter mathematical uh in terms of this kind of really added structure of mathematics another thing not yet fully worked out is the idea that what is the analog of a black hole for example what's the analog of energy in mathematical space presumably proof density what's the analog of a black hole presumably so you know one characteristic of black holes these space-like singularities is that time ends so in our models usually you're just getting to update the network but you can be in a situation where there is no update that applies so in a sense time has ended in that place and that's uh and that's the interpretation of a space like singularity in our models and so the question then is well what about um uh what about the um uh what's the analog of that in meta-mathematical space what's a black hole in meta-mathematical space well the picture of that is it's it's kind of like a decidable theory where essentially in in a in an ordinary undecidable theory uh where you know girdle's theorem applies and all this kind of thing there are proofs of arbitrarily long lengths and of unbounded lengths and that's like saying there are paths you can take and you just never time never stops time just keeps going you keep wandering around further and further but in the case where you're in a in a in in this black hole setting the you know you have these gd6 of limited length and the analog of that is you have proofs of limited length and the details aren't quite nailed down here but the basic picture is the sort of proofs of bounded length are the ones that show up in decidable theories and so what you might think is there's some something like singularity theorems of physics that say when there are enough proofs around you will inevitably end up with a decidable theory and so one question that you might ask is what's the long-term future of mathematics what what will it look like you know when we propagate mathematics forwards and so on what will the result be and kind of one amusing thing is that we think about you know in physics we might think about well you know all the matter is going to just aggregate into a bunch of black holes and so on in mathematics you're saying well all the mathematics is going to aggregate into a bunch of decidable theories and so that's kind of the sort of the common future of physics and mathematics now i mean there's there's lots of things to say slightly more philosophically about um about the structure of um uh what you get from thinking about kind of the mathematical observer the physical observer and the correspondence between those there are you know things about platonism and so on and the fact that if you believe that the universe exists the physical universe exists uh that is sort of our interpretation of our sampling of the ruliad the ruliad necessarily exists we are uh we uh that our our existence is the result of we think about that in terms of sampling the ruliad we can then come up with the conclusion that in the same sense that the physical universe exists so also there's ultimately something underneath mathematics that similarly exists lots of lots of kinds of consequences like that let me let me paint one more picture and then then we should uh wrap up for now which is okay so oh by the way i mean one of the things is in doing empirical matter mathematics there's a lot to be done in empirical matter mathematics one of the more bizarre things that may be true is that it may be easier to get evidence of our theories in mathematics than in physics that is by doing empirical meta mathematics you can discover features of the sort of the mathematical the structure of meta-mathematical space and so on you can discover things like oh there is this necessary uh you know you can observe these things like the homogeneity of mathematical space and so on you might expect to be able to do that empirically in in mathematics and i think the um so that uh uh i mean i should explain that empirical matter mathematics done from human mathematical databases is the same thing as doing geography rather than geodesy that is you know you got the surface of the earth and we know where all the cities are and that's the human geography of the earth that's where humans chose to populate on the earth and so similarly in mathematics the theorems that are in textbooks there are a few million of those um are the places that humans chose to put little pins in in all possible meta-mathematical space and and so what we might conclude if we know where all those pins are and we we knew that for geography for the earth then what we would conclude is that um uh you know by golly we notice that well all of these places where humans have populated on the earth they all lie on the surface of a sphere and and so that you know from the geography we can conclude something about the geodesy basically and so similarly i think that's the way that we imagine concluding things in empirical matter mathematics that's one approach to empirical mathematics is to be able to see from what has been populated what the underlying structure has to be we can also look at the at the pure underlying structure one of the challenges there is just that these entailment cones get very big and um they're sort of the just like in physics we have the same problem going from the atoms of space to something we can actually observe in a physical experiment of human scale is difficult similarly going from the the uh the sort of raw entailment cones to something we can observe in in sort of the mathematics that humans care about it's difficult it's not quite as difficult in mathematics because we have this intermediate assembly language level we're not quite at the machine code level because we have this axiomatic layer that isn't quite the same as something that we have in physics and so we're not going from the raw eames the raw rouliad up we've got a little bit of a um you know a little bit of a boost there okay so so let me just outline a couple of other things i i think um uh the um uh one of the points here is so there's this whole kind of way of thinking about physics way of thinking about mathematics in a sense one of the core ideas here is this idea of multi-computation there's several core ideas i mean there's a kind of a core idea of ruleology of studying the space of possible rules there's the core idea of metamodeling of being able to take kind of things that have been studied as models of things and ask what the essence of those things is but then there's this idea of multi-computation this idea of many threads of time this idea that kind of goes beyond the paradigm of computation i mean i mentioned these sort of three paradigms from the structural to the mathematical to the computational and in the computational paradigm it's just like you take these rules and then you evolve them forwards and you see you go from one state to the system to another to another to another and so on but you can always observe the what the the in that model it's just like there's a definite state of the system in the multi-computational setup you've got all these different threads of time and there's no way to to make any sense of that without sampling across many threads of time without having a notion of an observer and so this idea of multi-computation is something which necessarily involves the observer it has many threads of time and it has to be those those have to be uh that there has to be some coherence that's generated by the observer and and so one of the things we've been very interested in is this notion of observer theory the question of can we make a a sort of a a meta model of the observer a little bit like turing machines they're a meta model of computation um and so uh various kinds of thinking about that one of the projects i've been interested in recently is kind of inventorying all the different ways one measures things you know in world from alpha and welcome language we have about 10 000 kinds of units and we have about a thousand we've sort of curated about a thousand different kinds of measuring devices and so those are different ways you can measure things and the question is what's the bigger picture what's the what's the meta thing that's like computation that's what's the meta thing about observation it's kind of like you have a bunch of molecules that are bouncing around and they all hit some paddle that that they push a little bit and that's how you measure pressure how do we think about that in a general way and how do we characterize i i how do we characterize the observer or more to the point how do we characterize observers like us how do we characterize how do we make a meta model for those aspects of observation that are the ones that we humans in the current stage of technology have been involved with uh in a sense there is a more absolute notion of what is out there to measure it's the whole ruliad and and one of the more bizarre sort of philosophical points here is that we are now sort of we sample the ruling out in these tiny tiny little uh uh sort of slices of the ruling out of what we're sampling and we have we are able to we in our minds we can keep coherence in our minds because we're sampling these tiny slices of the rouliad as we kind of evolved to to sample bigger swaths of the roulead it becomes more and more difficult for us to have sort of a coherent notion of what's going on so our existence and our kind of experience and mind-likeness and so on is a consequence of the fact that we're actually sampling these tiny pieces of the ruliad in a sense as we think about well what's the future of you know technology and and uh and intellectual development and so on we say well we're going to expand out we're going to we're going to colonize the whole ruliad the problem with doing that is by the time you've spread out to the whole ruliad you kind of don't really exist as a coherent entity anymore um so it's kind of a a disappointing you know uh be careful what you wish for kind of thing but but any case the the um uh this this notion of multi-computation this notion where observers matter we have to try and develop an observer theory this idea of multi-computation applies to a whole bunch of other fields and we've started to look at them and i just list off a few of them i might talk about these a little bit more uh maybe next next week um when we have uh uh uh other people also also also here um the um uh but just to just to sort of rattle off a few of the different areas so you know after physics meta metamathematics was the first one we looked at uh another one uh um you know we're hot on the trail of is chemistry um when you think about a chemical reaction you're thinking about um the the you think about chemical synthesis for example it's another multi-way system story of you can make these different moves from this chemical you can you can do this you can do that but as you go down to the lowest level to the kind of eemish level you're looking at individual molecules and their interactions and so we kind of suspect that there are probably aspects of what is essentially the multi-weight graph of all interactions where when we do ordinary chemistry all we look at is the total concentration of each molecule but there's more in the multi-way graph than that um and uh ironically we're not really even talking about quantum chemistry here we're talking about ordinary chemistry just with with classical molecules but yet the formalism that we'll use is a formalism that is deeply coming from quantum mechanics even though there's nothing really quantum about it but then the the the idea has to do with looking at again i'll talk about this some more detail another time um is is to look at kind of the choreography of molecules that goes beyond just how much of each kind of chemical there is and it is my suspicion that uh that it is a key thing that's been missed in molecular biology is the importance of the choreography of molecules so to speak and that it's kind of like when when you know dna and the idea of digital genetic information was was discovered in the 1950s um that was kind of a thing that opened up a bunch of thinking about biology i kind of have a suspicion that there's the possibility of a similar kind of thing using this kind of multi-computational paradigm to think about sort of the full details of what happens with molecules beyond just the how much of this kind of chemical species is there well related to that is how you do molecular computing and one of the things that i think is true is and one of the general things we're looking at is what's what is multi-computation you know in ordinary computation you feed in an input it's like typical or from language workflow you feed in an input it goes crunch crunch crunch and you get an output now in multi-computation that's not what happens in multi-computation you get this whole multi-way graph now one of the things and i i really realized only very recently how exceptionally lucky i was back in 1979 when i was first developing this idea of transformation rules on symbolic expressions because i i realized at the time i knew perfectly well at the time that when you do transformations on symbolic expressions there are many possible transformations you can do but i thought well let me just do the simplest thing let me do the you know left most innermost transformations and let me just keep running them until i get to a fixed point okay and that's what wolfram language does that's what the evaluator of wolfman language that's what it does it it runs it does transformations it keeps going it chooses particular transformations to do and it keeps going until there are no more transformations to be done turns out doing that is really useful turns out that's what lets us do all the computation we do in wolfram language um and that was enough that's the humanly useful computation but in a sense there's a multi-computational version of language where instead of following just that one path you follow all possible paths i actually tried to do something a little bit like that parameterizing different paths back in 1980 or so 1981 and the problem was i didn't really understand it and nobody else understood it and so this is sort of a challenge for right now can we make a multi-computational version of awesome language but one of the big issues there is what's an answer in multi-computation because an answer depends on the observer and by the way logic programming probabilistic programming these are all different instances of different ways you sample the answer from a multi-way graph so in any case the the um so so another project we have uh nick is particularly concerned with this one um is practical multi-computation involvement language and and what are the analogs of functional programming for multi-computation but one of the reasons that's important is because i think for molecular computing whatever that whatever the answer to that is that's what we get to use in molecular computing because in molecular computing it's again we've got all these molecules going around and we we have to have an observer who is sampling them in a particular way and we have to know what how do you deduce the answer from that sampling so that's another case so back to biology uh recent thing i've been thinking about is biological evolution and um using well the question is how much of biological evolution is essentially kinematic that is does it not really depend on the on the detailed dynamics of things is it just a consequence of structure and so what you imagine is imagine the genealogical graph of all organisms that have ever lived so you know usually when people do biology they don't think about anything like that that's a way to small scale thing but just imagine i don't know i i i'm trying to remember i know i did an estimate of how many organisms i've ever lived and i thought it was in the ncaa's book and i can't find it but i maybe it's 10 to the 40th organisms i'm not sure but you can imagine a genealogical graph of all of those organisms and somehow that graph is colored by well this is the clump that corresponds to this species that's the clump that corresponds to that species and so on and you can imagine that just like this the branchial distance in that genealogical graph is the literally ancestral distance that's essentially how related you know is it the is it the the the 15th cousin seven times removed or something that's essentially distance in branchial space it's telling you that degree of genealogical relation and similarly one can think about in in biology one thinks about it was a spatial position and just like in the physics project there's a branchial position there's a spatial position there's also i don't quite know what the analog of this is in the physics project there's trophic levels like things eat things um and that has that also has some consequence for this graph so in any case that those are all uh sort of pieces of of this picture and the question is what can you deduce globally about biological evolution from thinking about that limit of that graph so there are just a whole bunch of other ones of these i mean there's economics there's another area we've been looking at that kind of started with trying to use the physics project to make a practical version of distributed blockchain sort of more distributed version of blockchain where there's not just a single ledger that you're building up and that kind of led to the question of well how does economics really work fundamentally what what is the emergence of value in economics you know the elementary events of these elementary transactions how do the elementary transactions get knitted together into the thing that's like kind of continuum space where there's a definite notion of price and so on and that's the thing james and i have been particularly james has been working on um and uh of how that how that might work and whether there is what we what we then wonder about is what is the economic observer like and one of the things that might be true about economics is that the things that might be the obvious things that a human observer economic actor is interested in turn out not to be the things about which there is a theory so you know there might be a generic general relativity-like theory for the underlying sort of really ad-like structure of economics but that might or might not be the thing that you can trade on with a hedge fund or something um it's uh so that that's that's another area there are also applications to uh machine learning and also applications to linguistics and what am i forgetting uh i don't know that that that seems like enough um the uh the basic point is just like the computational paradigm gave us this new raw material for making models of things so the multi-computational paradigm is again giving us new raw material for making models of things the big difference though is that while the computational paradigm computational irreducibility is its great big sort of phenomenon and computational irreducibility is a limitation on predictability in multi-computation because of the interaction with observers we again have the possibility of having things like the laws of physics that are kind of simplified narrative statements about what happens so anyway that that's um that's a little bit of outline of of of things and and one of the things that's been really really wonderful is that as we make progress even in thinking about thinking about metamathematics from the thinking about that we understand physics better from the thinking about uh like multi-computation as a practical matter we understand things in mathematics and physics better and vice versa you know we understand things about multi-computational programming better from physics and so on and it's really a lovely thing that this kind of one idea of multi-computation is knitting together all these different areas because there's this kind of common underlying formalism and that that's sort of a another wonderful feature of this anyway so that's a very very quick survey of of some of uh where we're at with some things and um i'm kind of looking forward to uh these summer schools you know that the whole physics project got launched at the 2019 summer school so i'm almost a little bit afraid of what could get launched because the physics project is already big enough um by the time it adds in metamathematics and all of multi-computation it's already pretty big so so uh i you know it's um it's it's maybe um uh but um uh it will be interesting to see what we um uh what we managed to launch um uh in in this in this year um and i think um as i say i i fully expect there's just an awful lot of low-hanging fruit and it's been you know there's a slow and steady process of beginning to pick some of it and uh there's a lot more to pick i think you know the time scales were we're sort of right you know we're in that five to ten year period when sort of the the most low-hanging fruit is that i have to say that these these areas this is deep and complicated stuff and it is also technically complicated and like if you look at the history of general authority it went from 1915 to the 1960s to untangle the technicalities of what had to be done um even before you started getting sort of more of the the real traction on on uh exciting effects and so on um and i think i i'm hoping that we can accelerate that uh i mean i think it really helps that in a sense the methodology we're using is this combination of in in training all of the methodology from mathematical from mathematical physics and things like that plus we have two other kinds of methodology one critical methodology is is practical computation the fact that you know the physics project would have been absolutely impossible if we hadn't been able to simulate things and see what was actually happening because all the time the things you actually see are not what you expect to see i mean my my statement about that is the computational animals are always smarter than we are so to speak it that you always see things when you actually do experiments that are things you didn't expect and wouldn't have predicted so that's very important the other thing that has increasingly emerged is kind of this philosophical approach to thinking about things is is very very powerful and useful and i think one of the things i've noticed in recent times okay that's a bit of a knock on on um uh you know there tends to be a among many scientists there's a thought that uh all thought is incremental and that that when when you do a project or something that or you when you are confronted with a new idea that somehow it can only be epsilon away from something you already know and so there has to be just this one step and i've noticed that people who are a good sort of professional philosophers for example they're kind of used to the fact that there can be multiple steps that you have to take to get to an idea and i think one of the things that's happened in this project is we're realizing there's actually some philosophically quite complicated ideas that you need to need to sort of wrap your brain around um and once you understand those things a lot of stuff becomes kind of intuitive and obvious but it's important to do that and it's not sort of a pure and the very fact that one even thinks about doing that is important for kind of really being able to make progress in the project so it's kind of a funny thing where we've got this kind of uh you know we've got the philosophy and we've got the computation and those are sort of at opposite ends the philosophy is kind of very in a sense qualitative and conceptual the computation is very much how do the bits actually work and somewhere in the middle is all of the stuff that comes from existing mathematical physics and so on and and from existing pure mathematics and so i think that's a it's a pretty powerful combination and it's one that's worth sort of bearing in mind as you try and learn about the stuff that there is it's it's a little different in texture in a sense from what one learns about in typical technical areas of science
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Channel: Wolfram
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Length: 111min 1sec (6661 seconds)
Published: Thu Jun 30 2022
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