INVITED KEYNOTE - Stephen Wolfram

um well our next speaker really does need no introduction at all uh he's the founder and chief executive officer of Wolfram search creator of Martin Massacre the absolutely legendary um black muscle software the inventor of the Wolfram language author of A New Kind of Science um this book here this is Spencer Brown's personal property of the new kind of Science and the originator of the Wolfram physics project will you please welcome as keynote speaker Stephen Wilbraham hello thanks boy I can't okay I can't hear you guys but that's probably all for the better because uh otherwise we'd have all kinds of feedback um great well very nice to be here and um I realized that uh uh I've had I had this book on my shelf for a very long time and um I uh ended up talking about it a bit in uh the book that was just shown which uh came out in 2002 and uh the an important part well one one piece one page of this book I talked about having found the minimal Axiom system for Boolean algebra and as part of the notes for that I talked about laws of form you know I was searching my archives to to uh see what kinds of interactions I'd had with George Spencer Brown and I have to just tell you uh of um uh uh the kind of an introduction that I had to him this was in 2003 December of 2003 uh a mail message to get sent to me from a customer service person at our UK uh office um which which reads as follows I've just been speaking to Professor Spencer Brown from Warminster he said he received a copy of nks from Mr Wolfram and wishes to speak to him regarding it whilst talking to him on the phone he informed me that he was dismissed from the RAF for flying his plane into the Black Bull Blackpool Tower during a fly pass for the people on Blackpool Beach I said that it's good he's still here to tell the tale and he told me that in fact he's written a book about flying and was only able to do this because he had so much free time after having been dismissed anyway it uh the message ends but there's a that's one of the I think it has to be up there in them uh just a an interesting personality um I I ended up talking to George Spencer Brown on the phone and we ended up having uh several interactions and uh getting him a I think a computer and a copy of Mathematica as I as I recall and as my email seems to suggest in any case what I want to talk about here was um about the getting to the foundations of the foundations of things which I think is a topic of uh of interest in the in the spirit of laws of form and of George Spencer Brown um and talking about sort of just how simple things can be underneath and what those simple things can produce I might mention uh since I did mention um Boolean Algebra I suppose I should should make the comment um the the one line in um a new kind of science that um uh is a reflection of my um uh my efforts in Boolean algebra um I can um um show you here um the this is a piece that I wrote to him more recently about it but um one of the question that I was interested in answering was if you just consider possible Axiom systems in mathematics represented equationally for example you just start enumerating them where do the ones that um show up see I am sharing the right thing yes I hope I'm sharing the right thing um you're seeing you're seeing a let me just check one thing here um oh yeah we were seeing it I'm sorry yeah great great great okay um the uh um let's see so so the question is if you just imagine that you're generating possible equational axioms uh at random for example how far do you have to go before you reach axioms that correspond to things that we've studied in mathematics and so the thing that I went to do was to find what's the very simplest possible Axiom system for Boolean algebra and that's the answer there where we can interpret dot here as a as a nand operation or nor operation um interestingly the the proof of of the correctness of that is something you can now generate in important language in a couple of seconds with find equational proof but an interesting feature of it is that in the in the 22 years since I first derived this um or at least my computer first arrived this for me um it's uh nobody has has really been able to say much that is of human value about what the proof of this means and it's sort of an interesting example of kind of uh raw computation meeting kind of things of mathematical interest all right well so what um uh maybe I should describe a little bit about how the the journey that I've been on of trying to understand sort of the foundations of the foundations of things um I was I was lucky enough when I was a kid in England in the 1970s to to do a bunch of work on particle physics and cosmology and things like that um and so I got involved in trying to understand kind of uh the the foundational questions about uh the universe in terms of of what one might call traditional physics um but then as as an effort to um be able to do the kinds of computations needed in physics I started developing computer systems and developed a thing called SMP symbolic manipulation program which was kind of a forerunner of mathematical language and when I started developing that in 1979 uh one of the questions that I wanted to answer was all these computations we want to do all these things in mathematics other kinds of computations what's a good underlying representation for the kinds of computations that as I would say now we humans want to do and you know we had all kinds of things where you would describe variables and make loops and and so on in the traditional programming languages that kind of hewed close to the actual Hardware construction of computers but I was interested in sort of the more theoretical question of what what would be a good model for the way that computation gets done and that we think about computation being done and what I ended up with was a approach based on transformation rules on symbolic expressions and so the basis of SMP was everything that you define was a transformation for a pattern for symbolic expression so you know a function would be well a modern notation of modern language would be f of x blank colon equals you know x squared or something and that means if you see F of blank anything named X then transform attack squared so the idea was to take kind of symbolic Expressions that were just sort of pre-structured uh things with with arbitrary symbols in them and to have the whole sort of process of computation be one that involves successive replacement of those those things with Transformations which have been defined for patterns on those symbolic Expressions okay the big surprises that works spectacularly well and now we're 40 something years later and that's the foundation for Orphan language and more from Alpha and everything we've done is this idea of represent everything in terms of symbolic expressions and have the actual sort of core computational operation be successive uh rewriting of those of those symbolic expressions using pattern patterns that represent classes of those expressions so in a sense it's a very simple idea about kind of how one should think about computation that we humans want to do and it's worked spectacularly well well so having built SNP I then got sort of interested in some basic science questions and the particular question I was interested in was we see all kinds of complicated stuff in the world where does it really come from what's the kind of uh what's the foundation for making complicated stuff and I tried using sort of methods from mathematical physics and and all those kinds of things try and answer that question and got basically nowhere with it then I started asking the question well if we just imagine that we are making models where we're taking uh sort of we're just thinking about what what is the meta model for how things work in the world and what I kind of started thinking about was well maybe the right meta model is more what we see in computation in programs than what we see in things like mathematics traditional mathematics so that got me interested in the question of if nature operates according to programs what kinds of programs might they be and in particular if we imagine that those programs are simple well then there's a very basic science question what do simple programs typically do and we didn't know the answer to that and so I started exploring this question of what do simple programs typically do and my kind of favorite class of such programs are cellular automata and so if I just make a an example here of a very um simple uh cellular automaton we just have a rule and again in a sense this is substitution rules all over again because What's Happening Here is we say we have a line of cells each either black or white and at each step the color of a cell is updated according to the color of the cell of that cell before and the color of its two nearest neighbors so there's a simple example of that and if we say um uh Let's uh let's actually run that rule and let's run it starting off from just uh a single white a single black cell um let's run it for 40 steps or something let's um make uh just show it with the mesh lines here that's the result we get so very simple rule very simple Behavior so let's change the rule a bit let's say we use um this rule here um well uh then we can run this again we'll see we get some more complicated Behavior but but if we run it for a while we'll discover that this is uh even though it's very intricate it's ultimately in a sense a very regular pattern it's just a a nested pattern here okay so now we can do now let's let's imagine that we we have this kind of computational Universe of possible programs out there let's kind of see what happens if we kind of turn our computational telescope out into that computational universe and just find out what what's out there what what possible things can happen with programs like this so it's these days it's really straightforward to do that so we can just say um just show us uh for all of these let's say um uh let's just do 40 steps of each one and let's um uh say here we want to show the first let's say 64 of those of those rules Okay so each one of these pictures is a different rule of this type up here um uh starting from just a single black cell and we see many of those rules give very simple Behavior sometimes we'll have a nested pattern here's one of those um we keep going we're sort of looking out into this computational universe without computational telescope we get to rule number 30 and we find something that to me at least was extremely surprising um in fact when I first saw this it took me a couple years before I was really uh uh realized what what I had seen but let's let's try running this again up here let's look at what rule 30 is it's just a uh a substitution rule essentially for these triples of cells let's run rule 30 for 400 steps let's say um here's the result okay so you know what I at first thought was there's such a simple rule when we run it the thing we must get out must show uh must be very must somehow be simple so you know you try and you apply various mathematical methods and statistical methods and cryptographic methods and so on and you try and crack this and what I found was that I couldn't crack it in fact a number of mathematicians rather distinguished mathematicians who happen to be sort of interested in what I was doing uh tried also to crack it and they didn't succeed and their main conclusion was well we'd only just have to give up what I realized and I only kind of realized this this sort of uh point of History recently the main thing that was probably my main achievement in those days this is the early 1980s um was realizing that the very fact that it was hard to crack what was going on in this case was itself a significant fact and that if we imagine if we ask the question what will this thing do after a very large number of steps well to to figure that out we can do a computation we can just do the computation that this system itself does but the question we might ask is is there some way to kind of jump ahead to have a smarter computation that in much more rapidly than this system itself can figure out what it's going to do it can just say and the answer is 42 or something and and jump to the end so let's say we want to find what this does after a trillion steps we want to find the center column of this uh of this um rule after a trillion steps um you know can we do that much more efficiently by just them just by running a trillion steps in this Evolution okay so the the surprising conclusion is we probably can't in fact what I think is that this is an example of computational irreducibility an example of a case where there's really no way to find out what it does much more efficiently than just by running it and see what happens seeing what happens that's kind of a corollary of a thing I call the principle of computational equivalence which is kind of the statement that if you think about a system like this computationally and you ask how sophisticated is the computation that it's doing as soon as you get above a very low threshold most systems that you just kind of pluck out of the computational universe will be equivalent in their computational sophistication none of them will be any more able to sort of jump ahead of another than than uh than anything else so it sort of means that if we are with our brains and so on we are trying to figure out what is the system going to do our brains are kind of stuck being at the same level of computational sophistication as this system and that's kind of why this looks complex to us and why we can't expect to kind of jump ahead of it okay so the um uh the the sort of the the big thing uh from this was realizing that yes you could even with these very simple rules you could get very complicated behavior um you could have certain hypotheses about um um uh things like the principle of computational equivalence um which uh gave one sort of predictions about what one should see in the world like that one should typically see that um as soon as one sees complex Behavior one will see things which are as computationally sophisticated as anything among other things things which are computation Universal this is one example of a a piece of evidence for this that was from 2007 um where uh I had found the simplest possible turing machine this is a turing machine with two states and three colors uh the simplest possible turing machine that does not have obviously simple behavior and it was uh uh a young chap um uh rather quickly managed to prove that this machine is actually Universal giving one a piece of sort of evidence for this principle of computational equivalence okay well so the uh one of the things that that happened after discovering kind of the phenomenon in rule 30 and computational reducibility I kind of went searching for just how General is this phenomenon and how relevant is it to sort of the secret that nature has by which it uses by which it sort of makes complexity all over the place and I kind of came to the conclusion that this really is the core secret that nature has that allows it to make complexity is this phenomenon that in the computational universe there is this computational equivalence and it's it's very easy to have systems that are uh capable of of uh of of complex Behavior and the only reason that we don't know that more obviously is that in doing engineering and so on we've always tended to avoid systems where we can't foresee what they're going to do so we've always tended to use computationally reducible systems uh so that we can foresee that they're going to do the things that we want to do from an engineering point of view okay well so one of the things so so my my big book new kind of science is is about kind of studying the computational universe and uh its implications for existing sciences and so on um one of the things that sort of came up in that book was a use case the use case is what about our whole universe what about the physical world what about fundamental physics is that something where these ideas of simple computational rules can be applied so I made some progress on that in in the book and then I left that for many years um and uh uh pursued kind of more the uh the the questions about um uh well Building Technology around ideas that came out of the science and uh things like wolf malfur and so on but uh about three years ago now um as a result of sort of a minor technical breakthrough I came back to this question of well what about physics is there a way to see whether physics is computational all the way down and the huge surprise is that well yes there is and we kind of figured out how it works um which I consider to be very exciting and maybe many of you have heard about this already but I'll I'll give you sort of an outline and then tell you some of the the newer things that we've figured out um this is kind of a visual summary of kind of our physics project and and sort of the real question is okay so what is the universe really made of and the uh the first thing that sort of we start talking about is well what is space made of um we kind of would normally think from Euclid on that space is just something where you put things in it it's not something that's made of anything it's you know we might it's just like we might say we've got a fluid like water and we pour it and it does all those kinds of continuous food like things and we just think of it as a fluid but actually in the case of the fluid we know it's made of molecules down there well the first sort of Point of Departure for the physics project is to realize that space is actually not just a thing you put stuff in it's something that itself has a structure and the structure that is the the easiest way to describe the structure is as a hypograph but fundamentally space we imagine consistent a bunch of discrete elements we can think of them as discrete points the only thing we know about those points is how they're related to other points we don't know for example how they're embedded in in three-dimensional space we don't know what coordinates they have they don't have any of those things all they are are a bunch of discrete elements and what we know about them is how they're related to other elements so that means that what we're building there is some kind of um uh is a is a let's say we have a a simple um uh graph here that's just showing sort of the atom atoms of space and showing the relation between atoms of space well now what we imagine is that a bit like an acellular automaton we imagine that there is a there is a an evolution rule for these hypergraphs this particular one is just a graph but we could imagine a hypergraph where um where instead of just an edge connecting two uh atoms of space it relates more than two atoms of space but we can just say let's let's um what what is the Dynamics of this well the idea is that there is uh you are updating this structure just like you're updating the the line of cells in a cellular automaton you're updating it by saying whenever we have a particular um piece of of network let's say a piece of network that looks like this replace it by a piece of network that looks like that okay so let's try doing that so in this particular case we start off in that initial condition and then after a few steps we're building up this kind of complicated complicated thing now remember this picture that I'm showing is just showing the particular embedding in in two Dimensions that Waltham language chose in displaying this graph what is fundamental about this is only the relations the connections between these atoms of space okay so the question one question is what's the limit of this if we keep running this for quite a while what kind of thing do we get as the limit so sometimes we can see these limits um let's let's try running something like this this is an example uh the limit of that is something looks like this we can recognize that as being something that's essentially a two-dimensional object or it kind of plays as a two-dimensional object a very simple structure of object here's another example that's a slightly more complicated kind of curved object a slightly more complicated example all kinds of examples here now one thing we can do is we can say well what's the effective dimension of the thing that we have that is the limit of this big uh big graph well the way we can estimate Dimension is we say start at a point and then just uh make a ball effectively by going to things that are one unit away on the graph two units three units Etc and simply ask the question how many distinct nodes of the graph do we reach after we go our steps and if the answer can be approximated by R to the D we say d is the effective dimension of the limiting object limiting space that emerges from this graph well things get a bit more complicated and just like if you were drawing a um uh you know a a circle on a sphere the the area of the circle and sphere is not exactly pi r squared it has a correction term that depends on the overall curvature of the sphere and so it is with these typographs you can compute their correction terms and so on and you can ask the question when you look at the limit of one of these things um when you've run it for a long time um what is the what is the overall structure of this and and how can you describe kind of is there a Continuum limit that you can describe for these for these types of things Well turns out that there is and it's a little bit like if you imagine a bunch of molecules bouncing around and they are colliding and they can serve momentum they they uh and so on the limit of a bunch of molecules bouncing around the Continuum limit is the equations of fluid dynamics so what's the Continuum limit of this update rule for these uh for uh kind of this these some uh atoms of space and this hypergraph turns out the Continuum limit is Einstein's equations for the structure of space-time and so what you can do here is you can you can derive uh you there's there's many more pieces to this whole story but but um basically the analog of the navi Stokes equations for fluid dynamics for an updating hypergraph that lives uh that just has this sort of connection information um is the Einstein equations uh with with certain conditions so a thing to realize is when we think about well what what is the universe made of then well first thing to say is the universe is just this big hypergraph so in a sense the universe is just made of space there's nothing in the universe except space but there are aspects of space that we interpret as things like electrons and quarks and things like that it's very much like if you have a fluid like water there's all these molecules bouncing around and there may be structures like vortices and the fluid that have a definite identity and persistence that we can identify as as sort of definite kinds of objects and it's the analog of of those kinds of vortices and things that we think of as being things like electrons and and so on in this evolving hypograph and we can and kind of see these analogies uh probably between black holes um and uh and things like electrons in the way that that those appear in the structure of this evolving hypergraph but kind of a little bit disappointingly it seems like the the fraction of the activity of the universe that is all of the things we care about like all those electrons and photons things like that is some tiny 10 to the minus 120 or less of all the activity in the universe the vast majority of the activity of the universe is devoted to the knitting together of the structure of space and to maintaining this kind of connection of this hypograph well okay so so one of the things you might wonder about in this whole description is uh you know I've described space as kind of the extent of this typograph time is something very different from space and this kind of picture it is the kind of computer inexorable computational process of rewriting this hypograph so the progress of time is a progress of computation and one of the things that you can then ask about is so how does relativity work and how does the sort of relationship between space and time and relativity work um basically the answer is what really matters to an observer embedded within this hypograph it's very important that we are we are if we believe this is a model for our whole universe we are embedded within that within that model to an observer embedded within that model the only thing you can tell is the causal relationships between the update events that happen in this hypograph and as soon as you start drawing these causal graphs and there's a property we call causal invariance um once once you have these causal graphs you will discover that it's inevitable that relativity Works um for an observer embedded within this hypergraph okay so the the next sort of big idea and I'm going super quickly here and I'm happy to try and answer questions and so on later on about this but the um the the next sort of really big idea here is um so how does something like Quantum Mechanics Work in um uh in these kinds of models well it turns out that quantum mechanics is not something where you have to say oh let's add quantum mechanics to the model uh quantum mechanics is an inevitable feature of this model and I'm I'm in talk in a moment about how this actually relates quite directly to laws of form and so on but let's um let's just talk about Quantum Mechanics for a second um so I mean the the big sort of distinction between classical mechanics classical physics and quantum mechanics is in classical physics we imagine that definite things happen throw a ball it goes in a definite trajectory and quantum mechanics sort of the big idea is no there isn't a definite trajectory there are many possible trajectories and we only get to see a sort of sampling of probabilities across those trajectories well in the way that these hypergraph rewrite systems work um the what we Define is there is this rule that says you if you have a little piece of hypergraph that looks like this you should rewrite it like that but the point is that we are not saying where exactly to apply that rule and there are many different places where that rule could be applied in fact there are many sequences of different rule applications that could be made and what we can do is we can make what we call a multi-way graph that represents the kind of separate all possible histories of this hypergraph corresponding to all possible ways that this rewriting can be done and so what happens is sometimes you'll have a particular state of this typograph and the rewriting could be done in two two three different ways whatever and you'll have a branch where you can end up with different effectively states of the universe sometimes two different states of the universe will end up up being updated and end up being the same state so you end up with this kind of branching merging structure that comes that represents these different states of the universe now you can you can uh that there are uh much simpler setups you can just do this with string rewriting systems for example um you can make a multi-way graph with just string rewriting where we just have rules for for replacing A's and B's and so on and we get this uh we get multiple possible results here and then they merge and so on Okay so uh what happens in uh so what we imagine in our model for physics is that this hypergraph can be written Rewritten in all these different ways and we end up with this multi-way graph of possible parts of history for the universe well then the next question the next question is well what do we actually observe in the universe we've got all these different possible parts of History what do we actually see happen well the first thing to realize is let's let's say what let's say we slice this to correspond to the possible states of the universe at a particular time what we get then is what we call we can we can we have all these different possible states of the universe but we can make what we call a branchial graph that represents the relationships between those states of the universe we can simply join two states of the universe if they had a common ancestor on the step before so in quantum mechanics this is essentially a map of quantum entanglements um this uh is is a way to represent a Quantum State um and it's what what you're seeing is at every moment in time and by the way the the the the choice of what constitutes a particular moment in time has a kind of reference frame issue just like it does in relativity um but anyway you you if you pick a particular what you consider a particular moment in time you you can produce this branchial graph and you can think of it as as a kind of branchial space in which the the uh coordinates in this bronchial space represent different possible histories for the universe okay so how come we think definite things happen well it's kind of interesting to see how this works because in the case of space we don't think there are little atoms of space it's new news that there are little atoms of space our experiences space seems like a Continuum the reason we have that experience is because we're really big compared to the atoms of space and we have this uh we are aggregating our experience as observers of of this spatial hypograph we're looking only at these very aggregated uh features of it in particular down underneath there's all this computational irreducibility all of this very complicated behavior in the detailed atoms of space but we as computationally bounded observers of that all we do is to look at these aggregate properties and that's why we conclude that space seems like a Continuum and so on well just to give you um a little warning of uh four minutes four minutes before the Q a is that all right uh just another one I'm not going to get to okay well I'll I'll give you I'll tell you what I'll just give you a menu of other things I could talk about okay the I could tell you how quantum mechanics works I could tell you um the relationship between um I could tell you about uh the relationship between this and laws of form I can tell you about combinators and um uh how the whole story of multi-way graphs and so on Works in terms of uh evaluation processes for things like combinators I could perhaps most interestingly tell you about this thing called the rouliad which is this kind of ultimate limit of um uh all possible sorry um we just want to make one more little interjection here real quick um Stephen how's your the last speaker of the day why don't we try and give you another few minutes so that you can put across what is what it is that you want to put across sure I'm happy to I mean I have I don't have my next meeting for a little while after this so so we're I can I can try and uh all right let me let me try and go really fast um yeah well let's see well let me let me just I okay I was I was just going to talk a little bit about okay let's um let's see what we can do here uh all right let me show you as an example just one fun thing here oh yeah um that on the on the the commentary on on our chat um and in the room in fact um a lot of people are very curious about the connections um with laws that form in particular um perhaps the theory okay here's a real simple thing this is a this is a multi-way graph made from uh I think I made this in as a result of uh an email with with Luke Huffman a few months ago um Lou had sent me this is this is kind of a representation of laws of form in terms of string writing and so you can uh you can start off with with nothing um and you can end up with a multi-way graph just like the kind of multi-way graph that we're making representing possible histories to the universe so this this is something where you're looking at all possible rewrites that can be done based on the laws of form rules for for a a null expression so I I don't know the um so this is uh so let's let's put that on one side just so you see that the um a little bit of connection there um the thing that um maybe I can show uh well let's let's show something about combinators here um uh let's see um so combinators um this is a typical evaluation of a combinator expression where you're just saying this is one particular path in the evaluation of a combinator expression in the case of combinators um it doesn't matter what path you follow your law is always eventually get to the same fixed Point um but it can work in different ways so for example if we look at and if we look at combinator Expressions oh gosh these are this is kind of the ruliology of combinator Expressions this is with a particular evaluation order just looking at the sizes of combinator Expressions as a function number of steps and you can see very wild things happen um but uh the the thing I really wanted to show was the multi-way graph for combinator evaluation um so here we have uh a combinator expression which can be um transformed in a variety of ways it will always end up with the same result at least it always end up with the same results so long as it terminates so here's another this is let me see I've got other combinator Expressions here um showing the different kinds of multi-weight graphs that you get from combinator Expressions um and you can end up with um uh sometimes you end up with combinatorial Expressions which terminate sometimes you have ones that do not term that have branches that do not terminate so I don't know which um let's see if I can find a good example of that um so this is this is kind of the the basic story of multi-way graphs is this idea that you're evaluating expressions there are different ways to evaluate them those different ways to evaluate expressions correspond to the different parts of quantum mechanics in our universe we are just as the the universe is branching and merging so are our brains branching and merging and so quantum mechanics becomes this kind of strange story of how does a branching brain perceive a branching Universe now the thing I I thought you guys might be particularly interested in is kind of understanding um what you can how you can go uh okay so so one one thing is you have this kind of rule and from this very simple rule you can generate all of this structure that reproduces all sorts of laws of physics and um seems to be a very promising way to understand that sort of physics really is computational all the way down and it connects with all sorts of other kinds of approaches to mathematical physics we can enumerate a bunch of those but one question is so let's say you've got this rule you think you have the rule and it gives you our physical Universe you then in this very strange position why did we get this Rule and not another Rule and so that seems very mysterious at first but then you realize that actually just as we said you're applying this particular rule in all possible ways we can imagine applying all possible rules in all possible ways you'd say what a huge mess that will make all possible computations done in all possible ways well what you realize is just like what happens with a particular rule you have something where there's branching and there's merging you have this kind of giant entangled object which is the entangled limit of all possible computations so for example one one example of that would be you could just take touring machines and you could take for example all possible um uh or all possible um Turing machines started off with all possible initial States and as they evolve uh many of the states they reach will be the same so you will have this complicated entangled object we call that object the ruly ad it's kind of the ultimate limit of sort of computational processes and what uh seems to be the case is that we can think of our physical universe as being kind of a particular sampling of the our perception of the physical universe is a particular sampling of this ruliad object and so the the um it's sort of a strange thing because what happens is uh the rouliad is this entangled limit of all possible computations um but the the question is what can we conclude from that well it turns out for observers like us it is inevitable that we will perceive certain aspects of how of that of from that ruling ad so just like if we have a bunch of molecules bouncing around and we're observers who are large compared to those molecules and we're observers who don't disentangle all the details of how those molecules are bouncing around we will conclude that the gas laws are valid so similarly as observers with certain characteristics in the rouliad embedded with as observers within this rouliad we will conclude that certain aspects certain things are true about about what happens in the rouliad so the conditions that we seem to need for observers like us are that we are computationally bounded and second that we believe we are persistent in time so you know at every moment we might be made from different atoms of space but we believe that we are persistent in time those two conditions turn out to be sufficient to give us General activity and quantum mechanics and not just roughly give us those things give us the exact mathematical formalism of general relativity and quantum mechanics so in a sense if we were sort of alien observers not like us we might conclude that the universe works very differently it's a very different slice of the rouliad but for observers like us we will conclude that General activity and quantum mechanics work okay so one feature of the rouliad um that has many implications but one of the ones that I thought I would mention here is uh where we're again going to get back to laws of form I think um is is the question of okay so the foundation of physics is this really ad which we are sampling as observers what about mathematics well we also imagine that the rouliad is the foundation of all mathematics and we are and the mathematics that we do and and and perceive the the three or four million theorems that human mathematicians have written down those come from a certain sampling of this rouliad the sampling done by a mathematical Observer is a bit different from the sampling done by a physical Observer a physical Observer is much more oriented towards Evolution and time a mathematical Observer is much more oriented towards sort of extension and metamatical space but it's the same idea and we can start looking at uh sort of um from let me show again sorry um the uh uh we can we can just just to show you very quickly here we can start looking at kind of meta mathematical space and we can kind of um uh start seeing uh we we can look at you know let's imagine we have some uh relation here we can start looking at a multi-way graph that represents the the consequences of that relation we can start thinking about sort of theorems here where we have the theorem that this thing is equal to this thing has a proof that corresponds to that path we can kind of unroll that and say that's the proof um we can go and do that um for lots of kinds of uh we can sort of understand how mathematical structures um you know this is I think one from I think this is group Theory and we can kind of see how how sort of the elements of groups we can sort of see underneath group Theory um how how things can be constructed from this we can construct kind of the analog of branchial space in matter mathematics which is this is kind of meta mathematical space it represents kind of the relation between different theorems and Mathematics we can oops that's not what I wanted um and uh we can go let's take a look here we can just go looking at um uh at things like we can we can sort of empirically look at um uh we can we okay we can look at things like um uh this is sort of the derivation from that relation we can derive these relations we can keep going doing that we can uh kind of for different kinds of um starting relations this is kind of like the big bang for meta mathematics we start from this initial condition and then we derive all these other theorems of mathematics um we can go and start looking at um how that relates to sort of theorems of present day mathematics so here's the semi group uh relation and we can go and say well what if how do we make this entailment graph that's a multi-way graph that gives us the theorems of pure semi-group Theory um we can go and do that um uh if we're interested in I think this is this is again semi-group Theory um if we're interested in particular interesting theorems here um we can oh maybe this isn't some group Theory maybe there's no this is this might be group Theory actually um we can we can we can start doing things these are Theory Improvement this is a theorem proven being run on this thing finding a particular path through this uh through this multi-way graph now somewhere here I have um I hope I have Boolean algebra um so this is um we're starting from uh some simple not not as simple as my accents for Boolean algebra but some sort of standard axioms for Boolean algebra and or not Boolean algebra and we're just generating the multi-way graph of all true theorems of Boolean algebra and so then we can start asking well where do the theorems that we humans care about whether they appear in the space of all possible theorems fully in algebra so here's a couple of couple of trivial theorems of Boolean algebra we can keep going we find a few more theorems of Boolean algebra as we keep going further so essentially we're fishing human interesting mathematics out of this kind of uh General multi-way graph in the end ultimately we can go all the way down so let's take a look here if we if we do this for um uh and see we can we can do this for arbit reaction systems but we can also start doing this um for uh um actual um well this is this is the the um uh the kind of dependency graph of the theorems of Euclid um but we can we can ask if we really try and drill down to the foundations of things um what um uh where do where does sort of existing mathematics lie so this is based on set theory this is the derivation of the Pythagorean theorem based on set theory uh from a proof assistant um uh from from things built up in a proof assistance system we can ask questions like uh for different kinds of uh famous theorems of mathematics how many ultimate instantiations of the axioms and Mathematics are needed to get those theorems it's kind of like building up physical reality from the underlying atoms of space how do we build up mathematical reality from sort of the low-level structures of mathematics and we can ask questions this is perhaps uh you know things like a plus sign get broken down into much smaller kinds of objects um and and we can ask questions like if you look in the space of all possible mathematical and sort of in this multi-way graph of a possible mathematical theorems where the different kinds of theories of mathematics lie where do different famous Axiom famous results in mathematics lie but we can also actually go down if we're looking at kind of the the very low level of um uh of sort of we're trying to see to the very lowest level in mathematics we can actually go below the level of of uh of of um uh of sort of standard um symbolic mathematics below the level of variables and operators and things like this and we can ask the question uh in in the rouliad none of those things exist the rouliad is just sort of this pure pure computation and when we uh we're we're finding structures that we interpret as things like variables and operators and so on just like if we were looking at a fluid we might find structures that we interpret as things like vortices and such like and I had some examples here in terms of combinators um of kind of what you might how you might um fish kind of um like like here's how you fish Boolean algebra one way to fish blue in algebra out of combinators this is not the only way to fish bullying algebra out of combinators but in a sense what we're what we're interested in here is is when we go to something even below the level of kind of symbolic mathematics um what how do we build up what we interpret as symbolic mathematics for something even lower level but I suppose the main main conclusion here is that um we can think of this rouliad object this entangled limit of all possible computations as the foundation of both mathematics and physics um and we can ask questions about well for example somewhere fished out of the raw rouliad is the laws of form and we could certainly derive that if we wanted to we can we could derive it by going through combinators and things like that but it's interesting to kind of see how something like that sits in this ultimate limit of all possible computations and this ultimate kind of view of of uh of things as sort of computational all the way down all right I'm sorry I went to rather long and I went way too fast there um but uh I'm happy I'm Keen to chat about whatever you would like to um uh to chat about object Okay so the comments here about the halting problem um I mean just to to be clear this whole idea of computational irreducibility is kind of a uh a generalization of that sort of notion but the halting problem is just asking what will happen in the end after an infinite time computational irreducibility is saying even for any given time it is irreducibly difficult to find out what will happen so if you want to ask about infinite time that will be something you can't answer on the basis of of the computations that you can do so I'll start the Q a and I'm gonna um go ahead and start with graham who um has a question I I think um part of it is also um included um visually on on this this thing we're about to share here so here's Graham hi Stephen great presentation thank you very much indeed um what you've done for um um minimalization in um the actions of spreading sequential um Boolean uh algebras I've done for Spencer Brian's primary algebra I don't know whether you saw my um no presentation a couple of days ago um but I in the 1980s I went into this very very deeply and with a bit of luck I can pull up maybe you can do that sir uh Florian it just he keeps scrolling down if you scroll down to the first uh um keep going keep going keep going sorry about this stuff uh yes that one that one there I think yeah okay it turns out that the primary algebra is completely demonstrable from undivided extension shown here cool all the all the food in algebra is uh demonstrable from that from that simple expression in fact I've investigate it a lot more thoroughly it turns out there are four other uh simple Expressions non-trivial simple Expressions but in fact they're also an infinite number of uh single initials which perform the complete primary algebra how difficult is the proof uh not not very difficult I mean I did it by hand not by machine um it was fairly the first the first two which I call undivided extension and uh divided extension actually I didn't find um divided extension that was the first time by Dr Rodney Johnson the student of Spencer browns with Naval research laboratory in Washington DC in 1980 that I didn't know about it at that time I only knew about it but I sent this result to Spencer Brown in 1983. but should we remark should be remarked that these initials don't uh already assume commutativity associativity that's right that makes it much easier yes yeah of course if you're curious Beyond in my talk I quote version Russell saying that the perfect notation um would include um um commutation you you wouldn't have you wouldn't actually have to um um state it explicitly and the reason why it's not necessary in Spencer Brown's notation is that Spencer Brown's notation is planar not linear not sequential like like conventional string notations I mean I think I think the interesting question so by the way if you're curious um let's see if I can if I could share the screen again for a second I can show you one thing um uh let's see if you're curious in the um oops if you assume commutativity so that that's that's the simplest Axiom uh assuming nothing but if you assume commutativity um uh oops Theory if you assume commutativity then then that is what what I found to be the minimal Axiom system actually it's an interesting question the minimum Maxim is uh if you assume associativity as well um and uh I could run that we could probably even run that right now and tell you the answer I don't know the answer either because brackets are in the the expensive line boundary performs the function in Brackets this is what I understand this point I'm I'm trying to understand in terms of in terms of minimizing things you know it's the thing I was sort of trying to talk about at the end there is you know in this kind of view of matter mathematics as emerging from this rouliad of pure computation um all of these things that we talk about whether they're you know uh you know a hook Opera hook objects or or you know even variables or you know kind of concatenation of variables things like this all of those things are things that have to emerge from this underlying computational structure and so you can ask the question how difficult is it to get those things I mean it's uh and it's sort of interesting you you it looks like you have a nice example that would be interesting to look at you know how difficult is it you know what how does that emerge how can you see the emergence of something like that from this underlying sort of uh all possible computation structure yes that's it it looks looks nice I I didn't know that result I I know in the in the notes to the nks book I talk about some laws of form and I I don't think I I knew that result at that time and um my presentation was recorded I think you might find it very interesting because I also um just discuss the the core components of um the primary algebra um I I factorized the fundamental operations of the algebra I think you may find that uh very interesting but I mean the thing that just just to make a comment about notation and things like this the you know I see my main life work as being trying to take what is sort of computationally possible and figure out how to bridge that with what humans think about and care about because there is much more that is computationally possible than what we care about so to speak many things in nature for example show lots of elaborate computation but we have no way to connect those things to our technology or our ways of thinking about things and so you know my big effort in building wolfen language over over the course of all these years is really an effort to Define what is the notation for the computations that we care about and so I think it's the the thing to realize sort of more generally about notation is notation is is kind of a story of how do we how do we kind of pluck out of what is computationally possible those things that are worth parameterizing for us humans to think about and uh you know I think that the um the the way that um so I mean to my mind what's interesting there for example with laws of form is well with with nand and so on in general um is it is a strange fact that in human languages I don't think maybe somebody knows that knows differently I don't think there's any human language that has a has a an everyday word for nand um that seems to not be a thing that we humans mostly tend to think about now I don't know think in terms of so it's an interesting question whether you know as you if you have some notation that makes it really you know really clear what's going on can one start thinking in those terms and come on start thinking about the things that one sort of everyday cares about in those terms and that would be uh as I say it's a it's a it's a you know I kind of feel like um as the way I would interpret it in modern times is in this sort of rural space different different minds are different positions in rural space just as they might be if they're embedded in brains they might be different places in physical space so these different Minds sort of interpret the universe differently and are in effect at different places in rural space and so the the sort of the the question uh you know when we understand more that's like saying we are more and more extended in real space and I guess the the um uh the sort of questions about about where do we how do we what what happens when we extend in real space what happens when we start being able to think in in in terms of you know as I say my effort as a language designer is mostly about taking what we humans think about and trying to connect it to what is computationally possible and so I I you know I struggle a bit with nand and general laws of form as an example to you know to see how do I connect that I'm curious for you guys do you think in terms of these laws of form I mean if I ask you some everyday question okay we've got we've got one nodding head that's that's a good sign you know if I ask you a a um you know an everyday question that would be what Aristotle would have you know when Aristotle originally invented logic you know his his approach what his idea was so far as we know it to sort of take typical forms of human argumentation and find templates for those and you know what emerged from that was things like ants and oars um so the question is um the uh uh so I'm sort of curious I mean if I were to ask you you know I don't know what's an everyday question um like uh um if you haven't eaten dinner yet um and uh and something else which I might talk about in terms of hands and ores would you guys be able to fluently think about that in terms of kind of laws of form constructs but we can translate it into that symbolism but the question that we're really focusing on is in a given situation is what are the distinctions that are involved what distinctions are being made and what concepts are coming from them so the question is how do you think about that in terms of the rouliad and I think you were answering it because you say a larger grasp of the roulade corresponds to understanding yes I mean okay so that of the question is how do you see that kind of phenomenon arising from the computation okay phenomenon of the Observer having a wider and wider grasp of the roulette okay so you know one of the things that is I think philosophically complicated is what we're imagining is what the way that the sort of structure of the of the kind of approach to science here is to say let's let's think about what an observer with these characteristics would conclude about what's happening in the rouliad there's a separate question how does that Observer come to be constructed in the rouyad but yes it is not difficult for the Observer to be constructed in the roulia the question is where where are we where is our experience you know the rouliad has all this stuff going on but somehow we have a certain each of us in a sense has a certain threat of experience and that threat of experience represents some sampling of the rouliad just like in physical space each of us has some particular position in physical space there isn't a theory for what those positions in physical space are really I mean yes we could have a theory that you know I walked this place in that place and so that's why I'm there but that would be a that to ask the question why is the over where the Observer is in rural space is like asking why are you where you are right now in physical space it's something which yes you could break it down into sort of the physical steps that got you to where you are but for for the purposes of of understanding what you are going to perceive it is reasonable to just take it as a given that you are where you are and and then you can make conclusions about what the universe will seem like to you from where you are and so I think the the step that we're taking right now is to say for observers who are like us who are computationally bounded believe we're persistent in time we can make certain conclusions about what we will see about the rouliad now an interesting question would be if you have two different observers and that are essentially in different places in rural space they might also be at different places in physical space they might be right next to each other they might be in different galaxies but they are can also be at different places in rural space and the question is what is the translation between those those n entities and and you know in terms of universal computation we would just imagine you know let's say these entities were represented by Turing machines we can imagine some interpreter that goes from one touring machine to another but in this picture of rural space we get a somewhat more geometrical somewhat more uh somewhat richer version of that question of how translation works and one of the things I I currently suspect is the following bizarre claim so in in physical space you know particles are presumably something like topologically stable things that persist through time and can be identified as the same thing moving through space that's kind of what a particle is a is a persistent lump in this uh you know in this evolving structure that is space so now the question is in rural space what is the analog of a particle and my current guess is that the analogo particle is essentially a robust concept because you have two two minds you have two different entities in real space place and the question is what can they exchange that is you know that the details of how each one of them operates are different but nevertheless they can communicate in a sense by exchanging something they exchange a thing that is sort of a robust concept that is a robust sort of topologically stable thing in real space now we haven't filled in the details of that um I think it may be possible to fill that in using we've studied kind of Turing machines in real space and it may be possible to fill in those details a bit more with Turing machines but at a qualitative level kind of the one thing to realize is as you go further and further away in real space it becomes more and more difficult to communicate you know I don't know it's not quite an inverse Square law but it'll be some some way of estimating sort of the the difficulty of communicating as you're more distant in real space so for example between different humans well there's often reasonable efforts to be able to communicate humans to cats well we can identify some pieces of of kind of you know emotional connection or something uh humans to the weather for example much harder you know the weather we might claim has sort of a mind of its own but it's a mind that we are not that is far enough away in real space that it's really difficult for us to kind of uh translate our way of thinking about things to its way of thinking about things and as we kind of go uh you know this is kind of the the you know we think about sort of the the aliens so to speak it's kind of we can ask both how far away are they in physical space and how far away are they in rural space and to what extent is our way of thinking about things sort of uh uh you know and their view our view of of of things like the universe coherent with a view that some entity in a different place in Royal space would have I mean one of the one of the things that that we imagine is that the growth of technology is a growth of of being able to understand more about the universe and being able to have sensors that detect more things about the universe the growth of kind of paradigmatic thinking is about being able to understand more about what we might think of more about rule space so as we as we learn more we're essentially extending our reach in rural space one of the things that is a little bit of a downer I think and this is kind of takes one to the real limit of this stuff is if you imagine taking the you know the the infinite future where we have just expanded our reach in rural space we have lots more technology we have lots more science we've understood more and more and more in Royal space one of the downers of that situation is in the end I don't think there's any meaningful sense in which we exist anymore because in a sense our existence is a feature of our cohere appearance in rural space we have a coherent view we have a coherence to to what we are and that coherence in a sense forces us to have a limited extent in rural space just like in physical space if we were diffused throughout the Universe um we wouldn't feel that we had the same kind of coherent existence and so I think it's it's sort of a a be careful what you wish for you know if in the ultimate limit of sort of infinite knowledge about uh about things the the the the limit of that if you if you try and sort of wrap all of that together in one mind then you end up with something where you do not have a coherent sense of existence so that's a that's a very a very limiting case of this kind of thing but I think the um um the uh uh let's see I'm I'm just looking at comments on the on the chat here um uh let's see and if you could just um find your favorite or something because um we are we are a bit over now sure let's see yeah but by the way people are interested in in like the things I was talking about about mathematics you can if you in that writings.civamorphan.com they'll find all of these all of these things um position and physical space hold on yeah um let's see Lori asks does the rouliad produce a kind of hyperboltzmann blame which is not computationally bound oh okay that's that's an intense one I I think that we do need to wrap it up um Can Can we clap for you again I just want to thank you one more time um thank you great all right nice to see you all right

Channel: Kunstforum Den Haag

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Length: 66min 46sec (4006 seconds)

Published: Mon Jan 30 2023