History of Science and Technology Q&A (March 8, 2023)

One from Matt here asking this. Can you discuss the history of graph theory and network theory? What was the role of computation? The idea of a graph is a pretty natural idea, but I don't think it occurred until the 1700s as a sort of a formal thing. And it's usually attributed to Leonard Euler, who kind of captured a pastime of the time in the city of Konigsberg, where he lived, which I think is now in Poland. There was an assembly of bridges that went to an island, to the mainland, across river, et cetera, et cetera, et cetera. And the question was, on the Sunday afternoon stroll, could you go over. Could you wander around these bridges, going across every bridge just once? And Euler figured out the answer to that and figured out the answer, I think, was no. And that was kind of. And that's what's led to what are now called Eulerian circuits of graphs. But that was kind of the first example, I think, that sort of in recorded mathematical history of thinking about the abstract graph. Just the connectivity of. Where could you go on the bridges? Not the geometry of where the bridges were laid out. I think. Well, let's see. Probably by the 1800s, there were a bunch of other kind of similar things. I think Hamilton, William Rowan Hamilton, inventor of quaternions, probably in the 1830s or so, was interested in analyzing this thing. I think he maybe even invented this thing called the Icosian game. I happened to look at this recently. It's a thing where you're interested and you kind of lay out the net of an icosahedron and you're interested in can you make a path that visits every node on that laid out net. And I guess that that led to an alternative, another sort of examination of graphs and the question of whether you could take a graph and wander around it visiting the nodes rather than visiting the edges. And I think there's another sort of small appearance of graphs and networks. So I wonder what other appearances there were in the 1800s. Probably another case will have been when people started looking at chemistry and started having the idea of chemical bonds. Although that didn't really get. I mean, it was all rather vague until probably the. Until quantum mechanics came along in the 1920s and so on. But there was this kind of notion of, oh, there's an atom and it's sort of connected by a bond to this other atom and so on. And I think that led to a collection of. To thinking about graphs that were these sort of abstract graphs of collections of atoms. And for example, I think these Questions like how many configurations of atoms can there be? There's a Cayley's theorem that says the number of different possible assemblies of atoms that are captured in this graph is something like N to the N or some. Some such other number. And that was again, something we' have happened in the 1800s as a way of thinking about abstract graphs representing molecules. Of course, people didn't know that molecules were real things at that time, but people still were imagining sort of atoms linked by edges, so to speak, to other atoms. That was another kind of use of. Of graphs. And I'm trying to remember when the benzene molecule, when that was figured out. And I'm thinking very early 1900s, maybe another case where sort of there was a graph representing the molecule, in that case a hexagonal ring. But, you know, I have to say, even when I started doing Science in the 1970s, graph theory was at best an obscure kind of thing. It was not a mainstream story at all. It was something where turning things into a graph was. A thing that was sort of a layer of abstraction, a layer of kind of, oh, there's this field of graph theory. It's sort of an area of mathematics, maybe, or maybe it's an area of applied mathematics, but it's not something that was part of the mainstream story of things. Now, I think there were other types of graphs that have been invented. So, for example, in the probably 1920s, there was this big enthusiasm for, you know, factories existed and people wanted to make factories more efficient. And there was this kind of idea of time and motion study of people going to a factory and saying, you know, on the production line, which thing has to be put there first, which thing comes second. There was a husband and wife team. Oh, the Gilbraiths, I forget. The Lillian Gilbreath was the. Was the wife's name. They were sort of big on the time and motion study business. And they were, among other things, invented flowcharts, which were an example of a kind of a graphical way to indicate kind of the sort of the. The flow of actions and the dependencies of one action on another kind of the causal dependencies of things. Notably, that couple were famous for being very concerned with efficiency, which led them to have some number 12 or something children, which led to the book by one of their children called Cheaper by the Dozen. It was kind of a kind of a practical use of sort of their efficiency studies. So that was another case. Then, for example, there were Feynman diagrams that Dick Feynman introduced in the 1940s. I guess, late 1940s. A representation of the progression of interactions between elementary particles. You have a line representing sort of the space time line of an electron, and that would emit a photon which could be absorbed by another electron and so on. You would have a graph representing that collection of interactions. I mean, I have to say that even in earlier times, when relativity had been being invented, I don't know whether Einstein was the one who started off talking about world lines, but this notion of sort of there's a line in space time that represents the progression of a thing, both its motion in space and its progression through time. And then when you start thinking about photons being exchanged between things, light pulses being exchanged between things, you're sort of building up a graph there. I'm not sure that that really arose as something where people thought a lot about graphs in modern times. In our physics project, for example, down at the lowest level of things, there are giant causal graphs that represent all of these elementary events and their relations with each other, and I suppose very junior versions of those. I'm trying to remember whether I don't think that was in Einstein's sort of, for example, popular books in the 1920s about relativity and things like that. So I think Feynman diagrams were sort of. These sort of space time diagrams are very abstract interactions between elementary particles. You could think of these Feynman diagrams in some sense as just being diagrammatic representations of mathematical formulas. But that was another use of diagrams, and again, one that felt a bit exotic. So the kind of, the sort of. The serious use of graphs for sort of production purposes and big graphs and things like this, I think that was a phenomenon of the 1970s and 1980s. I mean, there have been things that were, for example, shortest path algorithms. I think Dijkstra's algorithm maybe is from the 1960s of a way to given a graph, given two points on the graph, find the shortest path between those. Now, clearly, by the time people were interested in doing, for example, having maps of cities and road networks and things, and finding what was the shortest path from one place to another on roads, you had to start thinking in terms of graphs. But I certainly remember when I was a consultant at Thinking Machines Corporation, which was a parallel computing company in the early 1980s, the. There was interest in using this parallel computer to do pathfinding in graphs and things like this. But that was sort of an exotic, abstract kind of thing to do. One use case at that time was a very early direction assistance system. Maybe I've told this story before, but it's a fun story. So I can, I'll tell it anyway. This was at a time before there were cell phones, before there was GPS and so on. And the question was, could you take a map? And I don't know how they had an electronic version of the map, but somehow they had an electronic version of a map of Boston that had kind of the network connectivity of the roads in Boston in it. And so using that network connectivity, you could in principle work out directions from here to there, so to speak, on this map. The question was, if you were actually going to use those directions and you were kind of driving around, how would it know where you were and how would it tell you those directions? And so the scheme used was there were payphones in those days and payphones had numbers and it was possible to get apparently for Boston, something which gave information on this payphone with this number is at this location. Well, I, being an out of towner at that time at least said, I'll volunteer to test this direction assistance system and I'll start driving around Boston and see if I can use it and give comments on it. It was a total disaster. Now I should have had a backup paper map to use. I didn't. And so there I was driving around and stopping at a pay phone, calling this automated, this voice response system and trying to get it to tell me, you know, which roads to go on next. And the whole thing was very, I would say the user experience was exceptionally poor. I mean, it probably didn't help that, you know, I hadn't really thought through. I've got to take notes from what I hear about, you know, turn left, turn right, et cetera, et cetera, et cetera. And there wasn't a good way to get that list of, oh, it's, you know, such and such street. How do I have to write that down in my notes, things like this. But I got absolutely irreducibly lost and it took, oh, I don't know, probably four or five hours for me to get back to where I needed to go. From my experience in this direction. Assistance System, probably the very first of its kind and probably this must have been 1984, 1985 type time frame. But anyway, that gives some sense of the fact that graph theory was just not really a well explored thing. I think it really became popular coming into the 1980s and by the 1990s it was very well known. I mean, by the time Mathematica came out in 1988, it was certainly a thing one knew was of interest, although it wasn't something that was so core that in the very first version of Mathematica, it was a thing that was a built in feature. The Combinatorica package by Steve Skeena. Notably that name really emphasized combinatorics rather than specifically graph theory, although in fact there was a lot of graph theory and what was done there, you know, there have been a few specific things where sort of graph theory had kind of showed up. Like the four color problem. The question of whether, given a map of the world, let's say with countries on it, is there a way to color the countries so that no two countries of the same color are next to each other? How many colors of countries do you need to color the map that way? And the claim was that on a flat Earth you needed or even a spherical earth, but not a donut shaped earth, you needed four colors to be able to always successfully not have two neighboring countries have the same color. And that theorem was proved, actually it was a computer proof by Appel and harken in the 1970s, maybe 78 or something. And that four color problem quickly turns into a graph problem because you think about what country is adjacent to what other country and you make a graph of the adjacencies of those countries. But I don't think people thought about it particularly as a graph theory problem until after the fact, so to speak. So yes, I think that graph theory really came into its own probably in the 1990s, probably, and became something where people were sort of abstracting from all kinds of things as graphs. It certainly helped by the time the web came into existence and everybody was thinking about hyperlinks and so on, that became sort of an intuitive version of a graph. Clearly graphs have been used in other places I mentioned flowcharts. Another sort of use case for graphs was the sort of the logical layout of things like, I don't know, the London Underground train system, for example. The maps of that were not sort of geometrical geographical maps, they were sort of connectivity maps and so on. And I suppose also another use case will have been electric circuits where one was just drawing the connectivity of things, not specifying where things were. And I guess that was. Yes, that will have been another thing. The theorems that let you, like the star delta transformation that lets you say this collection of resistors in this shape is equivalent in resistance to this collection of resistors. I think those theorems, gosh, when were those from? I'm kind of thinking the 1930s, but I'm not sure. But you can tell from my rather rambling Description here. Graph theory was not really a very coherent subject until quite late on. And I would say that sort of it was the existence of computers and the possibility of studying graphs with computers that really made it much more feasible to deal with graphs. I will mention another thing about graphs, the question of laying out graphs, that was really hard to do. And for example, yeah, I used graphs, a whole bunch in the 1980s and the mid-1980s to study finite state machines. So finite state machines had been represented by graphs since they were introduced in the 1940s, 1950s, but they were not usually very big graphs. In the 1980s, when I started using finite state machines to represent these sort of discrete dynamical systems and cellular automata and things like this, those ended up being big kind of computer generated graphs. And I had a very difficult time laying out those graphs. I was just recently found actually a bunch of the pieces of paper where I had tried to lay out 100 node graph and figure out where to put the nodes so that you didn't have a lot of lines crossing over. I had various efforts with pieces of string and things like this, but in the end I think mostly did it by just sort of drawing it out and then redrawing it and so on. It was a very tricky thing to do. Automated graph layout was not something that became seriously easy to do until the 2000s at that time. And it was an important thing in our sort of standardizing graph theory in Wolfram language, which happened maybe 20 years ago now. It was an important point was you make a graph, how does it actually appear? You want something where a human can readily tell sort of what's going on in the graph. And for that you need automatic, fairly rapid graph layout. And the sort of most standard method that became sort of was this kind of a spring electrical kind of setup where you imagine that there are nodes of the graph and they are being pulled together by springs, but they're repelling electrically, so to speak. And that's kind of the way you lay the graph out. But so I think it probably will be fair to say that the existence of computers and the fact that graphs could be represented in computers in the 1990s probably was the thing that really made kind of graph theory take off. And I think the kind of statistical study of graphs also kind of originated around that time. People studying all sorts of networks and asking statistical properties of those networks. That was a thing of that time. And clearly by the time there was, by the time the web was getting established Also in the 1990s, there were started to be questions about sort of visiting places on the web. The kind of the, you know, the PageRank algorithm for deciding what's an important, what's an important website based on sort of random walking around the graph. In that way. You know, I'm thinking, as I'm trying to explain this to you, I'm thinking of all these different very fragmentary places where graphs had showed up. Always a little bit exotic. There was a thing called the beta lattice. Hans Bethe had invented that. It's a tree like lattice where instead of. And that was a thing which had been introduced and was relevant for lots of sort of easy to prove theorems and statistical mechanics. I'm kind of thinking that came from the 1940s or 1950s. So just an incredible number of kind of fragmentary appearances of graphs and networks that really became kind of a coherent field I think pretty much in the 1990s. Let's see. Tori is commenting. So graph theory evolved as a theory after practice, like thermodynamics and steam engine. There was theoretical graph theory, but it was a very minority field. I mean, I think that things like the spectra of graphs, taking the adjacency matrix and finding its determinant. I think that might have been sort of known in principle in the late 1800s, but it was a really obscure thing in the. I mean I remember I made use of spectra of graphs in the early 1980s and it was like, yeah, you could find a book on it, but it was not a mainstream kind of thing. And that. Yeah, actually Peter here is commenting. Graphs as knowledge representation were popular in AI in the late 1960s and more formally in theoretical computer science a decade later. Yes, that's right. That's another use was people trying to represent. Well, let's see, I mean the, This kind of the, the idea of a knowledge graph, so to speak. I'm trying to think about. Oh yeah, I'm going to make another, another example, another fragmentary appearance. This one. I know the date 1912 was when game graphs first appeared. They were introduced by Ernst Zermelo who was most famous for his role in set theory. People had been studying what were possible chess endings and things like this. And he said with a lot of reference to set theory, oh, there's a graph you could make that represents all the possible states of the game. And I think when people started talking about automated theorem proving, which happened very early in the history of computers, I think there was some awareness that that was kind of a graph related thing. Although I don't think People had drawn. I mean, even when I was first using automated theorem provers in the late 1990s, people were not drawing the graphs that got made by those things. It was still very much of a sort of an algebraic thing behind the scene. I think people. Yeah, I think even with relational databases, other kinds of things. Yes, I certainly saw graphs being drawn in to represent kind of the data structures, the connections of pieces of knowledge and things like that. In fact, the very name I mentioned, the connection machine, its very name came from the idea that there was sort of a representation of knowledge in terms of connections between things as a graph. I had forgotten that. I don't think the connection machine really sort of the graph aspect of all these connections was something that came to the fore very much once the machine had been built as sort of this massively parallel machine. But yes, that's a. That's a good point. And I would say there are other examples. Gosh, this is a fragmentary area. It's terrible. Another example is Petri nets, which were kind of a representation of originally of chemical reactions as kind of a graph theoretic representation of kind of what's going on in a chemical reaction and what's going on when multiple chemicals are involved. I could say Also in the 1960s, there was starting to be metabolic pathway charts that showed sort of the collections of chemical reactions that happen in a biological cell, for example, again, could be thought of as examples of graphs, but pulling them out and saying, this is a graph, let's analyze this as a graph. I don't think that happened until the 1980s. I think the idea that it was sort of abstractly, oh, there's this chemical here and it's connected to this chemical here that existed certainly in the 1960s, possibly even a little earlier than that. But pulling it out as sort of a data structure on its own, I think happened later than that. So, yeah, messy history. Gosh, you know, I'm reminded of these things where, for example, you know, Euler's sort of introduction of that idea based on this sort of, you know, pastime in the city of Konigsberg. It's always fun when these things that are sort of very kind of past time, ish, everyday kinds of questions end up throwing up what ends up being deeper mathematics. I just noticed recently James Clerk Maxwell, most famous for electromagnetic theory, for things in Statistical mechanics and so on, he wrote in maybe the 1860s, a paper called On Hills and Dales. He had this estate in Scotland, which he walked around all the time and so on. And it had these hills and it had these dales, valley like things. And he got curious about what amounts to the graph of where can you have a peak, where can you have a valley? How are these all connected? Later, that whole question about what maxima, what minima could exist and so on got called Morse theory in the probably 1940s or something. But Maxwell had already sort of developed that just on the basis of, oh, I'm curious as I wander around, you know, just how many times am I going to go up, how many times do I have to go down? And so on the, you know, I'm thinking of yet more examples of graphs. Another one, which I suppose was one that I sort of mentioned a bit earlier is nets of polyhedra, where you. And things related to that, where you're asking, you know, where you have a lattice, for example, you know, asking what are the nearest neighbors, what's the connectivity of the neighbors? And so on. Those are things. Certainly nets of polyhedra were pretty well studied by the late 1800s and people were the kind of. I'm not sure if Kepler, who studied polyhedra a lot, I'm not sure if he drew nets of polyhedra. I think he was more thinking about, you know, polyhedra in 3D, so to speak. It's, let's see, you know, I will say in terms of graphs, You know, I have become a huge user of graphs. I myself, as I say, used them sort of in the wild a bunch in the 1980s as a way to represent the kind of the states of these cellular automaton systems that I was studying. State transition graphs. I use them in several different ways in cellular automata. Actually this was probably early 1980s state transition graphs. I was generating large state transition graphs. I don't think that had been done much with computers. I'm trying to think about that. For shift register sequences, people were most concerned with maximal length shift registers, which have a very simple state transition graph that's just a great big circle with 10 state separated from that. But I myself became a huge user of graphs starting well in the early 1990s. I got curious kind of what the underlying structure of space time might be if there were discrete elements of which space was made. And I sort of knew that it couldn't be some very regular lattice like a cellular automaton. It had to be something much more flexible. And the thing that I started thinking about was, well, it might be a graph. Later on I realized it was better represented as a hypergraph. That's Very recent realization. But back in, starting in the early 1990s, I kind of had this idea, if space is going to be discrete, it must be a giant graph. And so I started needing to be sort of a much more industrial user of graphs. And when I wrote my book New Kind of Science, I had all sorts of code in Wolfram language for doing kind of graph theory sorts of things. But those was not a standardized part of Wolfram language until quite a few years later. And this question of was it difficult, someone asked her, was it difficult to integrate graphs into orphan language? The answer is, I would say that the real sort of important technological innovation was being able to do fast automated layout of graphs. Because, you know, to just represent a graph and say, blob, here's a graph and you've got to go look at its adjacency matrix. Got to look at this properties. But you can't really see the graph is much less useful than you make a graph and oh, it's right there on the screen and you can actually manipulate it by saying, I've got a function, you know, I say vertex count, open bracket and I put the graph right in there. That becomes a much better way to do things. Now, this whole question about programming with graphs, thinking in terms of graphs, it's an interesting issue. It's something people have tried many visual programming languages where one says, let's represent things a bit like having flowcharts as the way of representing the actual program. It tends to work well when there's a single chain of things you have to do. Do this, then this, then this, then this. Sure, you can represent that very easily by a graph. It's a rather trivial graph. Sometimes when there's a single branch or two, you can also represent it. But by the time you're doing more sophisticated kinds of looping, functional programming, things like that, in my experience, it becomes very impractical. I mean, it was something that was, I'm trying to think in the early versions of Logo, the turtle motion language invented by Seymour Pappert and so on in the 1960s, that was originally a textual language where you'd say, make turtle go forwards this amount, make it turn, make it go forwards and so on. But by the time it emerged into mindstorms, that must have been in the 80s, and then scratch, it had become sort of a visual sort of click these pieces together and see the flow type type language. And there were a variety of other such languages. A thing I've often been interested in is whether there is a way to truly represent, sort of truly write code that is a graph. And what do I mean by that? I mean that normally sort of programming languages and our computational language are very much based on sort of the metaphor of human language, which is essentially a one dimensional string of tokens. It might represent, it might be something where you have sort of a hierarchical tree like thing that you're representing, but in the end you're writing it out as, you know, F of G of X and so on, all on a line. In Wolfram language back in the, in the mid-1990s we kind of expanded that to sort of this two dimensional format for math notation where you could have sort of built up pieces where you're not just going all in one line, but you can have something where the actual language has, you know, the, an integral sign and it knows that the limits are above and below. And just like in, in a standard programming language you might talk about having a prefix operator, like I don't know, a, a. Well, let's say like a not symbol, an X clam or something would be a prefix operator or you might have an infix operator like a plus sign x plus Y. And so we have this idea of over script and under script operators, things like that, where instead of just going sort of linearly, you're taking the language being in two dimensions. So I got interested in the 1980s actually in whether you could make a graph fix language, a language in which kind of the relations between entities were sort of fundamentally represented by something that was a graph rather than something that was a pure kind of one dimensional string or even a one dimensional string with some sort of up and down pieces to it. I never managed to figure out how to make that work at that time. In fact, I concluded at that time that we humans were not well built for thinking in terms of kind of graph type representations of things that we have good built in probably even hardware for dealing with sort of what language which goes sequentially and for dealing with things in the world that is sort of a two dimensional array of things in an image. But this notion of a graph was something for which we don't have good sort of intrinsic built in hardware or something. And so I sort of thought that was not such a good way to think about programming, even though it had many interesting advantages. In more recent times I've sort of come back to this question because when you are thinking about, well, in our physics project we're thinking about these giant graphs that represent the causal relationships between all events that happen in the Universe. And one of the things that goes on there is this notion of multi way systems where you can have multiple threads of history where different chains of events are happening and one can think of some of those events as being simultaneous, some of them as not being simultaneous, and so on. And this is important when one's thinking about parallel computing because that's something where again one can have different chains of events happening and they're not happening in any sort of. There can be multiple parallel chains happening. And so I've sort of gotten reinterested in this question of how can you make sort of programming with, in, in the presence of these kind of graphs of connections between things and have a bunch of new ideas about how to do that that are very much based on the fact that we now have a pretty good idea that physics works as sort of a thing that emerges from all these underlying graphs. And so we can start using ideas from physics, like ideas like reference frames and things to think about how to sort of make these graphs comprehensible to us. You know, there's questions about graphs. It is interesting that particularly with hypergraphs, Hypergraph is so a graph. You're just taking the nodes of the graph, the, the, the vertices in the graph and you're saying every edge connects two vertices in. And in a hypergraph you can have kind of triple hyper edges and things like that, where instead of connecting two nodes together, making a connection between two nodes, you make a sort of simultaneous connection between three nodes and so on. And hypergraphs have shown up in all sorts of places. They're still not very kind of standardized. We still haven't put them as built in things in Wolfram language because there are a bunch of different ways you can think about hypergraphs and it's not clear what the best one is and whether it matters, for example, that the hyper edges are directed ordered. These kinds of things. In our physics project, we think about those hyper edges as just defining relations. That's something very similar to what happens in things like relational databases where you are dealing with things in terms of hypergraphs and so on, or whether representations in terms of that. But hypergraphs are still. Hypergraphs are probably about at the stage today that graphs were sometime in the early 1980s. Let's say they're out there, but they're a little bit exotic and not yet sort of mainstreamed and standardized. I think the. But hypergraphs are a very wonderful flexible representation of lots of kinds of things, including I think the space, time in our universe and things like this. But you can someone was asking here about compiling things to graphs. Absolutely. You can compile things to hypergraphs. And in fact, in one thing we've been looking at recently about multi computation, the idea is precisely to compile everything to hypergraphs because hypergraphs have very convenient properties in terms of the sort of the causal relations between events that update. Hypergraphs are very cleanly representable. It's very easy to understand what's a change, what affects what, and so on when you're dealing with hypergraphs. So that's relevant for those purposes. As a question here, has anyone formulated Zemelofenkel set theory in terms of graph theory? Well, okay, so in my fairly recent work on metamathematics, it's a giant graph theory story where the nodes of the graph are theorems and then you're asking for sort of the derivations of things are connections in this graph. And so, for example, a proof from one theorem derive another theorem is a path through the graph. But in terms of the raw material for set theory being graphs, I don't know of such a, such a formulation, although I think it should be possible to make such a thing. All right, another question here from Bennis. At what point in history did mathematics reach a level where a single individual could no longer learn all known mathematics within their lifetime? You know, I have the impression, let's take some data points. I think Euler probably knew all the math that was sort of going on at his in his time, and that was in the mid-1700s and so on. I think Gauss in the early 1800s. I don't know. I would think probably, mostly I would suspect that the time when things got to the point where there were sufficiently many specialized areas that you couldn't really, you know, get your arms around all of math. I think that probably happened in the first few decades of the 20th century. I think that in, you know, I would even think people like David Hilbert around the end of the 1800s had a pretty good global view of what was going on in math. And certainly when Hilbert presented his 23 problems at the International Congress of Mathematicians in 1900, that was sort of an intended way of kind of sort of touching all these different areas of mathematics. So I would tend to think that sometime in the 1920s, 1930s was when that stopped being possible. One could sort of pick on particular people and say, did they know all of math at their time? Whether it's GH Hardy or whether it's John von Neumann, these kinds of people? My feeling is that by their time it was not so easy. And I think that there started to be areas of math, I mean math had been sort of a. There had been corners of math, let's say number theory that really hadn't developed much. Number theory had been almost a recreational area with a few bits and pieces that had been done with, that were sort of follow ons from the Pythagorean, from Pythagorean triples and things like this and diphentine equations. But number theory as sort of a major area didn't really come into its own until probably maybe even the mid 20th century. I mean there have been sort of contributions to it. Riemann had made connections between number theory and analytic kinds and analysis, calculus and so on. That was in the 1860s or 1870s. And then, well, Ramanujan in 1910s was sort of introduced all these kind of, look at this amazing number theoretic fact about this weird formula for this weird property of numbers, this weird thing about the number of partitions of an integer, things like this. But at that time that was still sort of a corner of mathematics, an almost recreational corner of mathematics. It didn't become of really mainstream I would say maybe until the 1950s or even 1960s. I noticed this once because I was actually studying Ramanujan and his kind of practice of sort of doing experimental mathematics which confused people like J.H. hardy who were very much into let's make proofs of everything. Ramanujan was just like, I'm going to see whether this relation looks like it holds. I'm going to try a bunch of examples and I'm going to have good enough intuition to know whether it's likely to be sort of globally true. But I was surprised that his work hadn't really gotten much attention around the time it was done in the 1910s and so on. And it had taken quite a few decades before it really, really started to get serious attention. And that I think was a result of the fact that number theory had not been a widely thought about area of mathematics. And I think as these different sort of areas of mathematics, whether it's started to sort of come online, I think that fragmented the field to the point where it was no longer possible to have this sort of universal view of everything that's going on now. You know, one can have a pretty good sense of sort of the core, the core of mathematics. You know, many good mathematicians have that, but I think that there will always be sort of, at this point there are, there are enough sort of vertical towers of mathematical development that, that are quite separated from other ones. I would say that one of the things that's sort of a trend in modern mathematics with things like the Langlands program, for example, is can you pull it all together again? You know, a big discovery from Descartes in the 1600s was the fact that geometry is representable algebraically. The fact that geometry and algebra have a certain equivalence between them. And that was something that really informed a lot of mathematics in the particularly 1800s, I would say. And now there's this question of are there much bigger correspondences between different areas of mathematics that might in the end, allow one to have a much more kind of universal picture of mathematics by learning only one thing? It's a common feature of the development of science and other things that there'll be sort of something that develops and there'll be lots and lots of cases that get studied. And it might take a really long time to get educated about all those cases, but then eventually those cases all get sort of rolled up together in some abstraction, and then you just learn that abstraction. You don't have to learn all the cases. And that keeps the sort of frontier of education finite. If it wasn't for that, it will be truly hopeless because there will be sort of all this different knowledge that's been accumulated. And, you know, if it wasn't getting abstracted into bigger buckets, so to speak, it would be very hard to learn that. And it's the same type of thing with mathematics. I think there's a certain effort in modern times to see whether one could sort of pull together a lot of these different areas in these kind of mega theories. I have to say, I think that our analysis of kind of the physicalization of metamathematics tends to suggest that that must be work. And it sort of must work because there's a certain sort of uniformity in metamathematical space. Different places in metamathematical space correspond to different kinds of fields of mathematics. But just like space and physical. In the physical universe is quite homogeneous. Things are pretty much the same in terms of, for example, laws of physics at this place in the universe and at some other place in the universe. I kind of think that the same is true in mathematical space, but. And that in the end there are correspondences, there must be correspondences between these different mathematical areas. And there's a certain sense in which things like category theory capture some of those correspondences, but that in the end it must be possible to kind of talk about mathematics in a much more unified way. You know, category theory is a. Is A modern, last probably 20 years or so effort to have a formulation of a bunch of areas of mathematics that has that sort of commonality between different areas. I tend to think that category theory has certain specific sort of make it tractable assumptions which will sort of lock it out of some kinds of discussions of mathematics. But in the places where it works, it begins to give you this kind of universal view of what's going on. Memes is commenting here. It seems too often amazing discoveries go years without being picked up by a particular community. Yeah, absolutely. I mean, the. There are, you know, it's complicated how, you know, in order for a discovery to get picked up, there has to be sort of a conceptual framework and to some extent a kind of social framework that that discovery can fit into. So there are things that people notice, things I've discovered in my time, for example, where I realized there were sort of precursors of those things from 100, 200 years earlier, but there was no framework into which to fit those things. Yes, the digits of PI look kind of random, but so what? Well, eventually you get this whole framework where you think about the generation of complexity by computational processes, and then that starts to fit in and, and be something that's interesting. But earlier than that, it was just sort of a random fact that didn't seem to fit in anywhere. You need these kind of frameworks to kind of support having sort of things that were, as a point, thing discovered much earlier. You know, I have to say it's, it's. There are, there are different dynamics of sort of things that got discovered and then didn't really get picked up. I mean, neural nets are a great example, invented in 1943 or so, but finally sort of really, really picked up in the last decade or so. That was an unusual case because I think that was a case where sort of the conceptual framework kind of existed, but one needed practical technology tools to really make something out of this idea. It's sort of unusual because most, the most common situation tends to be you don't even know that it's a thing until you have some of the tools necessary to study it. And by the time you have those tools, you can build a framework and you can start to sort of integrate things. But yeah, there are lots of examples of where things were sort of discovered ahead of their time, often as isolated facts and not really picked up. Now, it's confusing when one studies the history of science because there's a certain tendency, once one understands something in today's terms in the framework that's been built up. You go back and look and you say, wait a minute, those squiggles that somebody wrote down a thousand years ago, they remind me of what I now know and what I now understand in this framework. Really, this was discovered a thousand years ago because look at that squiggle. And you'd probably be wrong. And probably if you went back and talked to the person who made that squiggle and said to them, hey, we've got this modern thing and it works this way, they'd look blankly at you because they wouldn't know that modern framework. But even if you go back and sort of probe what they were thinking when they wrote that squiggle, it will be something pretty far distant from what one has in sort of modern times. I mean, this question of sort of communicating ideas across history is a difficult one. And there are ideas that are rather straightforward to communicate. They're things where we've built a tower where we go back and we look, and the tower we built really had the things that were there, let's say axioms from Euclid or something, as bricks in that tower. And there are also places where one looks back and the things one can see are features of human nature where we can understand them, because human nature back in ancient Greek times is no different from human nature today, so far as we can tell. And so that's another place where you can have that kind of understanding. But when it comes to, I don't know, some thing about computation where you might say, well, there's sort of an ancient Greek analog of this. And let's try to go back and explain that. I mean, for example, I don't know, the Heraclitus was big on the sort of everything is flow. There is nothing unless there is change, so to speak. I realized recently our modern theory of how space works, physical space works, is just like that. There are all these sort of. We call them eames, these atoms of space. And the way in which there are relations between those atoms of space, the way in which those relations sort of come to exist, is by virtue of events that change these atoms of space in ways that are correlated with other atoms of space. So, in other words, space wouldn't be a thing, it wouldn't be knitted together as a thing, were it not for this sort of continual process of change. But I sort of imagine going back to old Heraclitus and explaining to him the kind of theory that we have now about causal graphs and these slices of causal graphs that make hypergraphs that limit to the Structure of space and so on. And the question is, you know, his statement, everything is change. There is nothing without change, so to speak. You know, could one really have made the. Made the connection between these modern ideas and what Heraclitus was talking about, or is there really sort of an irreducible distance between those kinds of sets of ideas? I'm not sure. I kind of tend to think that there's a fairly irreducible distance and that it isn't that. You know, you're sort of. You can take the modern idea and you can back project and say it's cool, that there's something that sort of was a shadow of that idea that existed long ago, but in some sense it wasn't really the same idea. When you ask of a modern idea, can one build a tower on top of that idea? And if the answer is yes for that idea, but then you go back and say, well, could Heraclitus have built that tower on the basis of what he was talking about? And the answer is no, then you have to kind of think, well, it isn't really the same idea in some sense. Let's see, Questions here. Oh, actually, there's a question here very relevant to this from Luis. If Aristotle were alive today, how might he describe modern technology? How would one explain modern technology to someone from Aristotle's time? Interesting question. You know, I thought about this. I thought a few decades ago, actually I was thinking of writing some kind of book, maybe I'll eventually get it done, of sort of fictionalized interviews with scientists and other thinkers of long ago. And I kind of had this kind of setup which was you're there with your laptop, and you've got your laptop and you're opening it and you're showing Pythagoras how things work on your laptop. And so then I've sort of wondered, well, what would Pythagoras think the laptop is? And, you know, I think one imagines that would see what it was doing and would think it was some kind of form of disembodied soul that existed in the laptop. And you kind of wonder about the conversation which goes like, well, you know, these actions that are happening in this laptop, this sort of spontaneous action that we can see on its screen, that's kind of like a disembodied soul doing things. You might say, well, you know, how did it get to be able to do those things? Oh, one might say, people programmed it to do those things. Oh, so one might say, well, is that, you know, is it the case that they sort of put, you know, that that act of programming is kind of like transferring sort of the disembodied soul to the laptop, so to speak. And one can imagine, you know, I kind of think that's probably in the Pythagorean period. This notion of sort of, when you see this, this, you know, computation is something a bit like their notion of a soul, a bit like a sort of disembodied process that is not. That is sort of can be physicalized, but can be physicalized in different ways. That. That's the kind of way that that would be, that that would play. I think if one thinks about more, more modern times, people like Newton, for example. I'm kind of guessing that Newton would imagine that it was some form of alchemy, that it was some sort of collection of complicated chemical processes that were happening that were causing the screen to do all the things it was doing. That that would be the picture there, maybe. But for Aristotle, I'm kind of thinking that the. Some of modern technology will be sort of thought about in terms of disembodied souls. That will be kind of the computation idea, the notion that you can. Well, let's see. I mean, there are many, many pieces of technology sort of had precursors. I mean, something like a, you know, a camera and the ability to take the image from the camera and put it somewhere else. I mean, there were pieces of glass and sort of the beginnings of lenses, although not really image forming, I think, back in those days. So that wouldn't be so shocking, I think, when it comes to, oh, I don't know, what's a piece of modern technology? I don't know. An airplane, for example. I kind of think that, you know, I wonder to what extent that would be a thing that would be thought about in terms of its sort of. How would it be implemented by having, you know, a flock of geese? I mean, Lucian back in antiquity had already described you a trip to the moon, you know, on a. On a sleigh, sort of flown by geese, if I remember correctly. And so, you know, seeing an airplane. I mean, the thing that I suppose would be notable is the fact that so many things that were at that time purely biological kinds of actions, like being able to fly, things like that. The fact that they could turn into things that could be done with machines is surely notable. Although it's not like there wasn't the idea that there could be a machine coming from the sky. I mean, that's what the deus ex machina of Greek plays was kind of the elevator, like Descent of some. Some. Some character into the middle of the play. So I think the. The idea had certainly arisen that you could, you know, have a thing that. But, but how that. How that kind of, you know, how that would be achieved technologically, that was. That was not known. I mean, I'm trying to wonder what would be some of the things that would be the most surprising. I mean, I don't think that in modern medicine, for example, the fact that, oh, you take this, you know, this drug and it. It cures you of this or that thing. You know, there were plenty of Hubble sort of medical ideas that existed at that time. And the fact that these things were, you know, were packaged up and put in these little pill, you know, pill containers, I don't think would be that notable. I'm trying to think about other kinds of technology. I mean, I think that a car, for example, an internal combustion engine or an electric car or whatever, The fact that it's sort of possible to have a thing that sort of spontaneously can drive itself around surely is notable. Although this question of, well, it could be a little donkey inside there, that's, you know, that's. That's turning those wheels. I'm not sure that the, the lack of a donkey in connection with the car, you know, how that would be. How that would sort of be read. But I, Yeah, it is a very interesting question what, how one would describe things. You know, I've sort of thought, and somebody should do this, of training an image identification system with only those kinds of objects that you find in an ancient Greek dictionary. And so imagine that the world only has the kinds of things that exist in antiquity. And now you point the image identifier at some modern kinds of things. What would it say that telephone is, given that it only has in its kind of visual lexicon, the things that existed in antiquity? I think, yeah, I suppose that's another. That's another big area is radio and all those kinds of things. That's a, you know, I don't know to what extent. I mean, in, in. In a lot of traditions, this notion of kind of the spirit world or something where things happen but we can't see what's going on is pretty common. Not sure whether Aristotle was so bought into those kinds of things, but that's certainly been a common tradition. And so the fact that, you know, there's radio and it's a thing we can't see, but it nevertheless communicates from here to there. Might not have, might not really seem that notable. I mean, it's a question, you know, at different times in history, this was particularly big, I suppose in the things like the 1940s, 1950s was the kind of whole extrasensory perception. And you know, can you have direct brain to brain communication through some mechanism that might still. Might still exist? Who knows? But we don't know it yet. But the fact that things like that might be possible I think was not inconceivable in antiquity. It just the actuality of oh yeah, you can have a cell phone and you can be talking to somebody else there that obviously wasn't there. But that, that would be inconceivable, I think is probably, probably not, not the case. It's an interesting question. What the most surprising pieces of modern technology would actually be. The most kind of. I mean, I think computers would be high on the list of most surprising kind of kind of technology. Yeah, it's spare parts is commenting. Would you say the word soul or spirit was used in the past in much the same way we use the term software today? In some ways, I mean, I think computation is kind of the, the abstract thing, which is the abstract process that goes on in things like brains. And that is somewhat similar to the abstract thing that a soul was supposed to represent when that was first being talked about. That there was something about brains that had an eternal character to it that was abstract. Not a thing that was a property of the physicality of a brain, but something that had some existence independent of that physicality that that was. So I think, yes, I think that's a good correspondence between those things. And it's quite interesting to me at least that so much thinking was done about the character of souls and so on in a time before sort of modern science. And many of the sort of logical deductions that were made about things like souls probably apply today to things like computation. And it's an interesting kind of circulation of we talked about sort of things that showed up before their time. The idea of sort of the disembodied thought is something that surely showed up long before it could be embedded in the framework that we have today for computation. I should probably wrap up soon here. Let me just see. Well, Look, who says why would cell phones be inconceivable? They would work the same way speech does. The only difference is that the ancients didn't know about larger magnetic field. I think that the concept that you could talk here and somebody could hear you a long way away is a little weird. It's not Incredibly weird. It's a little weird. Again, I think that it's an interesting question. What, for example? Well, our own efforts, like with Wolfram language and so on, the ability to compute things like mathematical kinds of things. How surprising would that have been to a Pythagoras or whoever else? You know, again, in the more is discovered earlier than you think. The Antikythera device, the kind of clockwork computer that we know we have one example of from antiquity is a case where there was sort of a mechanical mechanism for computing something, in that particular case, various astronomical phenomena. But the. The idea that this laptop box or whatever could compute, you know, Pythagorean triples and just print them out on the screen. Yeah, we can even. We even have a, you know, for compatibility with ancient times, we even have a Greek numeral form that you can use where Pythagoras would have been right at home in terms of the way the numbers were represented. It's like, okay, here are a whole bunch of Pythagorean triples. You know, roll them out quickly. Would that have been notable? I don't know. As I say, I think that the best model that a Pythagoras would have had is this kind of disembodied human soul type model for what was going on in a computer. Callie is asking here, my experience, have there been major changes to a field of study that changed the way one would view a certain topic? I remember being in school studying astronomy when Pluto was declared to no longer be a planet and the professor had to adapt their lesson plan an instant. Yes, there are absolutely things that change how one use a topic. I think, for example, our physics project in time will completely change the way that one talks about space, time and quantum mechanics and so on. Even the things I've done recently on the second order of thermodynamics should be things that are used to explain the second law of thermodynamics from the beginning. And it's interesting in all three of those cases, actually, that the way that one explains these things today is much more abstract than what ends up being possible with the kinds of things that have come out of our physics project. It's something where, in a sense, one has to, in that particular area, one has to sort of tiptoe around what's going on, rather than just saying here, step by step is what's going on. I've seen that actually a bunch of times that when the foundations of something get better understood, one can start just talking in terms of those Much more solid foundations. And it's actually easier to understand what's going on than when one is merely describing kind of this thing floating in the clouds very abstractly without sort of bedrock type foundations. I think that's a pretty common phenomenon that leads to sort of a change in the way that things are presented now. They're also just straight sort of new facts that are discovered. And you know, when people were teaching about, oh, I don't know, genetics before the discovery of DNA, they will have described, it's a little bit like what I was just talking about. They will have described all these phenomena and all these different details about, oh, we've noticed that when you cross this with that, you get this thing and there's this rule and there's that rule and all these kinds of things and then new bedrock. You know, we know genetic information is stored in this molecule and it's stored as digital information. And then you have kind of a completely different way of explaining what's going on. I think that that's, that's another example of where kind of the, the, the, the things that, how that works now. You know, I'm always amused because biology is a field that's advanced a lot over the last sort of 50ish years since I kind of learned it in school. And I'm always amused every time I see mention of the Golgi apparatus. I remember that when I was studying, you know, cells when I was, I don't know, eight years, nine years old, whatever it was, you know, memorizing the different organelles in the cell and memorizing the fact that the Golgi apparatus was an organelle of unknown function and the endoplasmic reticulum likewise. And so I find it very amusing that that's now, you know, you look in a modern biology textbook and there's a whole chapter about how those things work and what they are and how they're involved in protein synthesis and so on. But for me back, you know, at that time it was memorizing. You know, this is an organelle of unknown function. So things like that happen. And I think there are now, in terms of things that happen very suddenly. That's an interesting question. What has been discovered suddenly that led people to change how they describe things? Possibly the Michelson morley experiment around 1900, discovering that there wasn't an ether drift that made one not want to describe there being a luminiferous ethereum. But I think for a while people don't really know what to substitute for the luminiferous ether. And my guess is, if you looked at the textbook, it would be an interesting question to ask. How long did it take before mentions of the luminiferous ether died out in textbooks? My guess is quite a long time. I mean, my guess is that one will be shocked how long sort of things survive in textbooks after one really knows much better ways to talk about them. And a very common phenomenon is this. So, for example, let's say there is a modern discovery in physics, like some of the things we've done. Well, people who are like teaching physics might say to themselves, wait a minute, first we teach classical physics of Newton, then we get to modern physics of Einstein, relativity, quantum mechanics, and then, okay, the very advanced thing that's very much recent is, oh, this modern stuff about discrete computational models of physics and so on. But that's very modern. That's, you know, the end of the chain of sort of the historical progression of physics. But that's completely the wrong way to teach it because, in fact, that sort of the things we've done in recent times are much easier to understand than for the typical person when you start from that, than when you say, you sort of wave your hands around and say, oh, well, there's this equation that you have to memorize that works this way. Nobody knows why it works that way, but you just have to memorize it. The fact that one sort of has a fighting chance to understand the bedrock of why things are built the way they're built makes it much easier, I think, for people to understand. I found it notable when we built Mathematica and we have this notion of how functions work and how F of X blank means you'd have an F. And then there's anything blank and you name it X and you transform it to something on the right hand side. Well, that's a very much more mechanical view of what a function is than what you think about typically in pure mathematics. And what I found really interesting was that people who were teaching calculus in the early days with mathematica back in 1988 and so on, were telling me, you know, we always used to have trouble explaining what a function was abstractly, mathematically. But now that we have Mathematica and it has this concrete, mechanical representation of what does a function do? It takes a blob of stuff that sort of. You can put it in this slot and it transforms it in this way. It became a lot easier to explain the notion of functions in calculus because you were not floating up in these abstract clouds. You were kind of Doing things from kind of mechanical bedrock, so to speak. So I think that's a thing which again, people often get wrong in thinking about teaching. They think the newly discovered knowledge must be taught last because it is the thing, because there must have been a progression of knowledge that led to it. And you have to build that tower to get to the modern thing. But often, if the modern thing has really managed to sort of reset and solidify the foundations of the field, that may be the thing to start from. But that's something that is often got wrong. As I said, Memes is commenting. I wonder what Professor Einstein would have been able to do with WolframAlpha. I think when Einstein came to the Institute for Advanced Study in Princeton, he was in 1933 ish. I think he was already both very famous and sort of a celebrity scientist and general celebrity. And so he had, you know, a rare scientist who had sort of the paparazzi type folks buzzing around him. But somebody asked him, professor Einstein, what equipment will you have at the Institute for Advanced Study to help you with your research? And I think he responded, this might be an apocryphal story, I don't know. I didn't really check this. But he apparently responded, the main thing I will have is a garbage can to throw away the paper of all my wrong theories and so on. But Einstein was a person of, I would say, very penetrating sort of philosophical style intuition about physics. It was interesting to me to study recently his early work, particularly in thermodynamics. And his efforts to do things like derive the second order of thermodynamics didn't work. 1903, 1904, was working on the second law of thermodynamics. Didn't work out. 1905. He takes some of the same kind of, in a sense, sort of philosophical intuition that he and other people had put into thinking about the second law. And he applies it to sort of the idea of photon, the discreteness of the electromagnetic field, the kind of mechanics and its relationship to electron dynamics and relativity and Brownian motion, all in 1905. And it just works in all those cases. And I think that Einstein was someone who had just this very sort of take it to the bare principles kind of intuition about physics and so on. And I think he could see in many cases there was a kind of mathematical structure that those bare principles revealed. That happened with Brownian motion. It happened to some extent with relativity and things like four vectors. It later happened in quantum mechanics, although he wasn't as involved in that. And it Happened very, in a very large scale with general relativity in 1915. That kind of, see, there's a mathematical structure here and there's got to, you know, and okay, mathematicians, do you have something that fits into the mathematical structure? And in the case of relativity, you know, Herman Mikovsky showed up and had, knew about quadratic forms and started talking about four vectors and so on. In the case of general relativity, I think Besso was the person maybe who introduced Einstein to tensor calculus and the whole Riemann's work on differential geometry and things like this. And Einstein was fond of saying that he really wasn't mathematically very sophisticated. And I think he was part of a long tradition of physicists where you can see you need the math, but you've got to kind of drag it in from what mathematicians have done and then potentially reform it for yourself and do computations with it. And I think, I have no doubt that this sort of ability to compute would have been something that would have been super useful to Einstein. I'm not sure that maybe Einstein was more of a sort of think about the principle of the thing more than a computing kind of person. I mean, people like Gauss and Riemann were very computing oriented and some of these people were more kind of tech savvy than others. Like Gauss, for example, was tech savvy enough to try and hook up kind of an early proto fax machine based on a telegraph system for his observatory. And you know, he did actual astronomical observations, discovered the asteroid Ceres for example. But. It was something. So he was, Gauss was very tech savvy. I think Hilbert was amusingly somewhat tech savvy. I know Hilbert people found it entertaining that, you know, the esteemed professor really liked his gramophone. His. Oh gosh, what is a modern translation of gramophone? Oh, turntable audio thing for people who are too young to know about these things. Vinyl records, you know, little needle runs through the grooves and plays sounds and so on. That was high tech of its time. And actually I should have. No, I'm being unfair to Einstein perhaps because Einstein, you know, when he got his PhD, you know, he went to work at these sort of the then analog of Silicon Valley, which was the, the patent office at Bern where he was very much dealing with inventions and particular refrigerators and things like that, which were the, you know, refrigerators were probably the, the large language models of their time, so to speak. But yeah, the, that was so, yeah. So I think, I think Einstein would have, would he have used a computer? Did Einstein use. Einstein could have used a mechanical calculator. And I wonder if he did. I don't know. Certainly von Neumann did. I think I know that Gauss did a lot of hand calculation which he didn't publish in his papers. He just calculated things on the side like the number of primes and so on in a particular interval, things like that. It is an interesting question which of those folk would have really embraced using tools and how quickly they would have done so? I mean, my observation in modern times, and I should close with this, my observation is from just looking at people who use the tools that I've been involved in building, it is a, a very nice stratum of the kind of high end of the research community. It is a really notable thing that, you know, it'll turn out there's somebody who's a big user of our technology and you know, that's sort of all I know about them. And it will turn out, oh, actually they're a leading person in their field and maybe the not so leading people don't use our technology. And you know, I can, I can take. I think it's correlation, not causation. Perhaps it is the case that people who end up being able to make. There's a certain sort of tendency to use, kind of use tools and use the best tools and learn tools and things like that that tends to come with the package of being really effective in doing kind of the high end of research in particular areas. And I've just noticed that correlation really quite dramatically. It's really quite almost amusing to me because literally I'll know people just because they happen to be big users of our technology and will really not happen to know anything about the field that they work in. And then I'll discover, well, actually they're a leading person in that field and it's sort of a remarkable correlation. Anyway, I should wrap up there and I think go back to my day job and I think I have another live stream actually coming up in just a couple of minutes here. But thank you for a bunch of interesting questions and comments today and look forward to seeing you another time. Bye for now.