History of Science and Technology Q&A (September 18, 2024)

Sam. Hello. Welcome to another episode of Q and A about history of science and technology. You know, it's this whole process of trying to understand history is always interesting. I'm in the middle of a. A little minor history project that's part of a piece of work I've been doing about nestedly recursive functions. I suppose one thing to say is why do history? There's a certain set of things that I've discovered. There's a certain set of things where I'd like to understand. I'm curious why people didn't discover these things before. I also want to both give respect to and sort of understand what was done in the past of this particular field. But this particular case is a very fragmented, sparse area. It's not something where there's a huge amount of work. Sometimes you'll find an area where there are 100,000 academic papers about that particular kind of thing. In this case, there are perhaps 20 spread over a long period of time. And it's something where sometimes there'll be an area where a particular thing gets a certain name, and then you just can go look, and everybody refers to it by that same name. And sometimes people will have discovered the same thing completely independently and given it completely different names. For example, cellular automata, something I've studied a lot in my life, were given many different names because they were invented many different times, including by me, independent of knowing that they already existed for other purposes. And in fact, back in 1983, I had the fun of organizing a conference at Los Alamos, actually, where I invited basically all the people who had worked on things that were equivalent to solar automata, though often under completely different names. And almost none of them had known that anybody else worked on the same kind of thing. So there were names that were things like tessellation, automata, homogeneous structures. Then there were things like forests of stunted trees and all sorts of other kinds of names. And at that time it was, well, even now to know that there are different things, there's the same thing. Being referred to by different names isn't so easy. It's a place where perhaps LLMs could actually be helpful to say, what are some other places? Given that you're reading some vector database of literature and so on, and you can kind of find out what things have similar meaning vectors, what can you disentangle about what are other possible names for this thing? But in any case, the challenge is often, first you have to figure out where are these different pieces? Where are the different things that might have contributed to whatever it is you're studying. When the thing was being invented, it didn't yet have a name. And so if it's invented many times, it typically will be invented without any particular name at all, or with different names in different cases. And then there's always the challenge of understanding people and what they did. And the one principle that I have in sort of history of science is in the end, it all makes sense. In the end, there's a thread. You can connect this and that and the other two. And there are no jumps. There are no things where something totally mysterious happened where somebody figured something out. There was a big thing that they suddenly figured out, and there was no precursor to it. There was no kind of chain of getting closer and closer to it. I mean, I think the thing that is often the most challenging is even when you have something where you did the experiment and the thing you observed, such and such a thing, did you understand what you saw? And this also relates to things like, well, who should get credit for that discovery? They saw it, but they didn't understand what they'd seen. Does that really count, or do you have to understand what you've seen for that really to be a discovery? Now, often in the chain of what happens with discoveries, you really have to have understood it for it to have any possibility of having kind of a chain going beyond it. And sometimes the. So what then happens is you start reading documents, you start talking to people and so on. It's sometimes even confusing to talk to people because people will say, oh, yes, this is what happened. And actually it won't be. And it's not because they're intentionally misleading you. Sometimes they are intentionally misleading you, just to be clear. But most of the time, that's not what's happening. Most of the time, people 40 years later have sort of a recollection of the high points of what happened, and they don't really remember how did they get to those high points. They just remember this is what, you know, got done at this time, but they don't remember, oh, there was this chain of things that led up to that. And sometimes with enough prompting, it's like, oh, yeah, I was thinking about that, because this had just come up because this was a topical story, because I had worked on this or that or the other before the case. That's most embarrassing to me because I've done a fair amount of this history is when I'm looking at my own past efforts and I'm realizing I didn't know. It took me decades afterwards to realize that's why I did such and such a thing, because I had been led to it, because this was something people were talking about, that was something I'd studied, et cetera, et cetera, et cetera. So in the particular case I'm looking at right now, I'm really wondering about that because I know I knew about. I thought I knew about these things in 1993, but actually I realized that I knew about them by 1981 and possibly by 1980. And then why didn't I do anything about it? Well, well, because I didn't really understand the broader context. I didn't really, really understand the broader context. Well, I understood the broader context by about 1985, but by then I was working on slightly different kinds of things and I sort of maybe didn't get back to it until 1993. But anyway, this is the kind of thing one has to untangle a bit of methodology, I suppose. But it is always. It's really a very. It's a bit like doing science. You have to look for all these clues and then put them all together. What can be kind of interesting about history of science is that there's all possible forms of evidence that you might end up getting. I mean, some is very straightforward. Oh, the person published a book and there are gazillions of copies of that book out there. But sometimes it's much less clear than that. Sometimes you're going through boxes in attics, sometimes you're contacting archivists, sometimes you're talking to people and saying, do you remember this? And then it's, no, I don't remember that. And with enough prompting, it's like, oh, actually, yes, there's this thing that happened and then you're off and running. Sometimes you wind up talking to the descendants of people. And that's its own challenge because they'll know certain kinds of pieces of lore. And I know that about my own past, where when you actually check up on it, it's like this story was told and then retold and retold and it didn't really. What actually happened was not the third level retelling of the story. The third level retelling of the story had some relationship to what happened, but it gave you a clue that something happened. But going to find what actually happened is its own separate challenge. In any case, the comments, history is very good at preventing humanity from making the same mistakes. Would that that were true. I find for my own personal experiences, I do try and make sure that that's the case that the things that I do in my life, you know, when I realized that was a mistake, I try to remember that's what happened, that was a mistake. And I'm kind of pattern matching something that's going on now and trying to make sure that I don't make the same mistake again. And I. I'm actually reasonably pleased with myself for not having made the same mistake multiple times and across a. A wide range of different kinds of things. Well, let's see. Lowen is asking, how would you explain the history of PI? Not a huge expert on this, but let's, let's talk about a few things. I know the name PI. Who invented that? I think it was a chap called Jones in England. Mathematician, 1700s, early 1700s. And he used it to represent the perimeter of a circle, PI being the Greek letter for p. And that was sort of perimeter. Jones wasn't particularly notable. And that really didn't get a lot of pickup. But when Leonhard Euler started using PI to represent the circumference of a circle, then it really took off. And then people started using that PI as a concept. Had existed in Babylonian times, for example. There were approximations to PI that were known at that time. I think maybe 22 over 7 was the best that people came up with. I'm not sure exactly, but the. That was. But so PI was known at that time, and PI steadily got slightly better computations done on it. I think Archimedes had a way of computing PI based on taking polygon, a regular polygon with n sides, and figuring out the perimeter of that regular polygon. If each side is length one, then if it has n sides, length N. But then, then figuring out the radius of the polygon. And I think he found some. Actually, I think he found like three digits divided by three digits. Ish. Approximation to PI back in 150 BC. Right. That's right. Right. Or is he. No, he's. Later than that. I'm not sure of that order. In any case, the. Then I think by the 1600s, if I'm remembering correctly, there was a chap called Ludolph somewhere in the Netherlands, maybe type area in Europe, and he computed quite a few more digits of PI. I know he had them put on his gravestone. And I think for a while in continental Europe, PI was actually called Ludov's constant. And then it became called PI. As I say, Euler was the main one who sort of really promoted the name PI. And then there's a question of the computation of PI. Well, there are some funny incidents in the history of PI. One of them was the funny one often quoted the bill in the Indiana state legislature to legislate that PI equals, I think, 22 over seven. Obviously that's a crazy thing to do because PI is just a mathematical construct and it has whatever value it has. You can't just say we describe by edict that PI is equal to this or that. Of course, if you are doing that to say it's a little bit complicated because if you're saying we're going to use that as a way of converting from something which is measured linearly to something which is measured in circular area or something, then that might define other units and it might be a slightly complicated chain of whether you're actually really redefining units of area or something. But it was a bit confused. But I think it was sort of an amateur mathematician, I think, who was somehow connected with the state legislature who put this up. And I think everybody kind of knew it wasn't totally sensible, but maybe some point was being made, probably some political point. Those often end up leading to craziness. And so, you know, it was a bill that made it through at least some stages of the state legislature. This must be in the nineteen, was it the nineteen hundred, early nineteen hundreds? It was before, I think it was well before the time of the Scopes trial, which is almost exactly 100 years ago now, was 1925. I think that was one of these kind of, you know, do you believe in science? Type questions. Although I'm not sure that was really what was being debated there. But any case, the, when it came to the computation of PI, the, the need for kind of very many digits of PI wasn't particularly clear. But then when people started wondering in the 1800s, people were already wondering if, if you were to compute PI to a lot of digits, we know it's 3.14159, et cetera, et cetera, et cetera. How random are those digits? So one thing that back in antiquity had been figured out is that there are numbers which are not rational. There are numbers you can specify, but they cannot be given exactly as a ratio of two integers. So that's the thing that Pythagoras reputedly was very sort of freaked out about because he was kind of had this belief that there were sort of harmonies in nature, just like, you know, a perfect, an octave is a factor of two in frequency or in string length for a stringed instrument. And a perfect fifth is a three halves ratio of Lengths for a stringed instrument or now as we know in frequency. And he kind of assumed that, or so it said, that sort of everything reasonable in the world would, would be some rational ratio, it would be some ratio of integers. And then he realized square root of 2 cannot be expressed as a ratio of integers. And that was kind of a defining moment for kind of the growth of mathematics and abstraction in mathematics. I suppose that that wasn't something where you could just say it's a ratio of integers. So that was a thing you. Then later the idea of transcendental numbers arose. So there were algebraic numbers that would be the solution to an equation, a polynomial equation like x cubed plus x plus 1 equals 0 or something solve for x. The solution of any such polynomial equation is always an algebraic number. Doesn't necessarily mean it can be expressed in terms of square roots and cube roots and fourth roots and so on. But it has certain characteristics that one can describe it as being an algebraic number. A big question. So by the 1500s people had find formulas for the solution of cubic equations and quartic equations, quadratic equations. That was known in ancient times certainly to the Egyptians, I guess also to the Babylonians. What the formula? Let's see, let me embarrass myself. Minus B over 2a plus or minus square root of B squared minus 4ac over 2a. I think that's right. But gosh, it's a long time since I remembered that formula. In any case, It's kind of, that's a formula that's somewhere deep inside Wolfram language. And whenever I actually need to solve a quadratic equation, which does happen from time to time, it's just computer, no human remembering formulas type thing. But any case. So the, the cubic and quintic were solved in the 1500s. And then came the question of the, sorry, the cubic and quartic fourth degree equation. So then came the question of the quintic. And that was a big question in mathematics for a few hundred years. Was there a closed form solution in terms of fifth roots and things like this of the quintic equation. Finally, young avarice Galois in 1830, something proved using, well, actually what began group theory proved that there couldn't be an expression in terms of radicals for the solution in general to a quintic equation. There's an important sort of early impossibility proof that got produced there. So then, okay, so we know those are kind of not, those are algebraic numbers still, but they can't be expressed in terms of fifth roots and so on. But then the notion arose of a so called transcendental number, a number that couldn't be expressed as an algebraic number. So what were examples of those? So PI was a quintessential example of a transcendental number. I think that was proved in the 1860s, maybe that PI is transcendental, E is also transcendental. And various other kinds of numbers were proved to be transcendental. I think square root of 2 to the power square root of 3 is also transcendental, if I remember correctly. And there are a bunch of things like that that will prove to be transcendental numbers. Proving numbers are transcendental, that is not algebraic is not easy. And for example, there are some that, oh, I think Zeta of three, I think is one that was done fairly recently, by recently, maybe 1950s or something. But there's sort of a gradual effort to prove that numbers are transcendental. Okay, so then the next question is, well, if you were to write the number out in terms of digits, how random are those digits? And so the idea arose that the digits might be so called normal, which meant that there would be a. That if you look for like in PI, you know, every one of the digits 0 through 9 occurs, do they occur with equal frequency? If you look at blocks of digits 0, 0, 0, 1, 1, 1, et cetera, et cetera, et cetera, do all of those blocks occur with equal frequency? A number is said to be normal if in, let's say the decimal expansion or binary expansion or whatever, every block of numbers, every block of digits occurs with equal frequency. Every block of digits of a given length occurs with equal frequency. Okay? People have been trying to prove that numbers like PI are normal for 150 years. I don't think there's any progress at all that's been made on that subject. Interestingly, the only thing that has been done is numbers which are kind of born normal have been produced. So a chap called David Champanone, who was a friend of Alan Turing's in college, and in fact they were involved in sort of thinking about building chess computers and things like that. Anyway, David Champanone in the 1930 ish timeframe, came up with a number called Champanone's number. And Champanone's number is you write down 1.0.12, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 3, 1, 4. But those are the 10 that I'm saying there. Those are just two digits put down in the number. Anyway, what he showed is that a number written down by concatenating all these digit sequences is a normal number. But that number, is there an expression for that number in terms of, oh, it's the limit of this mathematical function and it's this thing like that? No, there isn't. So there isn't really a connection made between numbers that are known to be normal and numbers which are expressible in some kind of concise form, symbolically, so to speak. Anyway, that was that. David Champernon went on to be a. A rather distinguished economist in England. But any case, the thing that. So that was kind of on trying to prove things about things like digits of PI. Next question is, how do you actually compute PI? Well, One of the big breaks in the computation of PI came from Ramanujan. This was around 1910 or so. Ramanujan noticed that E to the PI square root of 173, I think, is very close to an integer. Many people would have just said, so what? It's just a numerical coincidence. If you try enough, E to the PI square root of something, well, 163, that's the one. E to the PI squared of 163, if I remember correctly, is very close to an integer. And so the question was, so what? And Ramanujan was very big on doing experimental mathematics. That was his big trick for discovering things, was just calculating a lot of stuff and noticing patterns, calculating things not only numerically but also algebraically and noticing sort of a pattern in the series and things like this. Anyway, so he noticed this coincidence and he built that into a whole theory. And that theory told one that this was the theory of sort of almost integers, and it turned into a whole theory of elliptic functions and so on, quite elaborate kind of set of constructs. But any case, that factor that was very close to an integer allow Ramanujan to produce a series approximation to PI. That for a long time was kind of the best way of actually computing PI on a computer. There's another method, which is the arithmetic geometric mean method, where if you take two numbers and you work out their arithmetic mean, which means you just take the two numbers, add them together, divide by two, and you work out their geometric mean, which is multiply the two numbers together and take the square root of the result. And you keep doing that repeatedly, that will converge. And the thing it converges to is an elliptic function. And that elliptic function, I'm forgetting all the relations here, is, can connect that to PI. And so that's another method of computing PI, which is also quite good. But anyway, the ram nujam Method was very long used. I'm not sure what we use right now in Wolfram language. I think it might be. That's right. I think it's one due to a chap called Gene Salomon, who was a sometime laser physicist and sometime part of the MIT kind of hacker group that in the 1960s, who produced a document called Hackman that's quite famous, which is sort of a list of interesting factoids. And one of those interesting factoids was a. A series. The PI that I believe is. Is. Is maybe the best one that's known right now. I'm not sure about that. The other thing that happened in the. In the Story of PI is, well, you would think you're computing PI, you're going to get it more fairly accurate, and then you add more terms to the series, you get it more accurate and so on, but you're just finding numerical values. So if you want to find the hundredth digit, you're certainly going to have to know the first 99 digits, and you're just making a small correction to what you've got, and that gives you the hundredth digit and so on. That was what people assumed. And then along came an amateur mathematician named Simon Plouffe, who came from a family of weavers in Montreal or in Quebec, I think, and very kind of not sort of educated in fancy mathematics, but somebody who really played around with numbers in a slightly Ramanujan kind of way. I think Simon Plouffe was. I think he's still around, but. So one of the things he did was there was a Handbook of Integer Sequences that had been compiled by a chap called Neil Sloan at Bell labs starting in 1964 or something. Neil Sloan had kept on sort of index cards. Oh, this is a cool series, you know, 112358, et cetera, et cetera, et Cetera, or 1122233. Whatever it was. He'd kept all these things on index cards. Eventually they'd been published in a book in 1973 or so that was. And now that's turned into an online thing, the Online Encyclopedia of Integer Sequences, which has many more sequences. But somewhere in the middle of that, Simon Plouffe was very enamored of the Handbook of Integer Sequences and decided he would actually type in all of those sequences to create some kind of online database of these things. Subsequently, he collaborated with Neil Sloan to make another edition of the book and so on. But the very act of taking this thing, which obviously had existed computerized form and typing it in was a kind of a matter of devotion to sequences. That was quite something. But anyway, Simon Plouffe, sort of lover of sequences, noticed a totally bizarre fact that in base 16 there was a way of kind of parachuting in to the billionth digit of PI and figuring out what it was without figuring out earlier digits. And actually the reason this works well, you can do it also for log 2. I forget in which base. Maybe it's also base, maybe base two, I'm not sure. But it turns out that in the right base kind of the terms in these series align in such a way that there typically isn't sort of crosstalk between this term. Oh, it's really only this set of terms that contribute to the such and such digit because the others are sort of all zero by that point or something. So to everybody's surprise, there was this kind of formula for the nth digit of PI, more or less it's a little bit of a squiggle because, and this is something that is not, I think, so widely known, but it sort of assumes that there isn't a huge collection of carries that build up as you're working out certain computations. It's something where assume there aren't a huge number of carries. If there were a huge number of carries, you might be able to flip over that bit, but absent that, you're going to be able to parachute into the nth bit and find out what it is. So that was sort of another innovation in the computation of PI. Now in terms of sort of records, Bill Gosper, long time well person who so always using Wolfen language and Mathematica and sending out the most bizarre kinds of results and so on. And it's been doing it sort of, I, I've. It's been sending out sort of strange mathematical results in the kind of style of Ramanujan, certainly since I've known him, which is 1979 or so at quite a rate. But in any case, the Bill Gosper for a while had the record for the maximum number of digits of PI computed. Then I think it went to a Japanese group. I'm not quite sure who owns that record now. I think it's reached a trillion digits by now. And one of the things that's interesting to look at is, okay, we believe PI is a number that's normal in the sense that it has this property of random digit sequences. But is it really normal? Well, you might be able to tell maybe at the trillionth digit something bizarre happens. I'm led to tell a kind of A weird story. So there's a movie about extraterrestrial first contact called Contact that was based on a book. Was it which came first, the book or the movie? I'm not sure. Anyway, Carl Sagan was involved in the whole thing. And it's a movie where sort of a radio astronomer, played by Jodie Foster, discovers the signal, and then eventually they make a sort of a spacecraft like thing. And it's all kind of interesting. I won't spoil the movie. I haven't seen the movie in years, but I thought it was a good movie when I saw it. When it came out, it had some interesting kind of ideas and commentary on the world and so on. But any case, the book version, so in the movie and so on, eventually it's sort of. They're extraterrestrials. They've given these sort of elaborate. They tell the humans a bunch of elaborate sort of future technology. In the book version, the Jodie Foster character kind of meets the aliens and has a big discussion with them and sort of says, well, to us humans, all of this kind of interstellar transport pathways and so on is very remarkable. And that character says to the aliens, so what's remarkable to you guys that's remarkable to us, but for you, there must be something that's remarkable about the universe. And in the book version of this, what the aliens say is, well, somewhere at the trillion, trillion, whatever digit of PI, we found a picture of a circle. In other words, sort of the creator of the universe put a picture of a circle that was the implication into the digits of PI as something to find. Now it's interesting to untangle that idea because it's philosophically just nonsense because, you know, PI is what it is, the universe is what it is. The question of circles and so on is a human interpretation of things. One of the things that's sort of philosophically interesting is the if you imagine that the universe is fundamentally computational, then its character is like the digits of PI. In other words, everything that's happened in the universe is determined by some rule, just like the digits of PI are determined by some rule. So in a sense, everything that we look at in the universe and say, gosh, it's amazing that this or that is true. It's the same type of thing as saying, gosh, it's amazing in PI there's this collection of digits because both are kind of determined by rules. And to say that it's remarkable, so to speak, is in a sense to say that the observer of that thing considers it remarkable. It is what it is. There's nothing about it that was sort of tweaked in a particular way. It just is the way it is. But it's a question of whether the observer finds it remarkable. Anyway, just a fun sort of feature of a little PI philosophy and so on. I am reminded of one last thing about PI. There's this whole kind of notion of PI day with the American date system. It's 3 14, March 14th is 314 sort of PI day. And in fact, we have a website called My PI Day, which we built a number of years ago, which allows you to actually enter your full birthday. And then it will search the digits of PI to find where your birthday occurs in the digits of PI. And remember, if PI is a normal number, which so far as we can tell, it is, your birthday better appear somewhere in the digits of PI. Some people, a person who was, you know, some birthdays occur very early in the digits of PI. Some occur, you know, many digits into the digits of PI. You can kind of estimate, since if a birthday is a six digit thing, you can kind of estimate it's going to be in the millions to tens of millions on average that the birthday will appear, although some can appear much earlier. Anyway, that whole sort of pieology thing led to the idea that March 14, 2015 was PI Day of the Century. 31415. So that's sort of a funny thing. I have to tell this story. We were. There was a conference trade show called south by Southwest in Austin, Texas, still goes on, that was kind of really in its heyday, actually at that moment. And we were going there and we were having some presence there. And we knew it was going to be PI day of the century right in the middle of that event. And so we were like, let's celebrate PI day of the century. And famously, the person who was handling events and now handles other things at our company as well, I said to her in some way, let's get a big pie for the sort of event we're going to do at this pie day of the century. And so then it's like, okay, next meeting comes around and she's saying, I called all these bakeries in Austin. And I'm like, why are you calling bakeries? Oh my gosh, you thought I meant pie P I E? No, I meant an object that looks like a Greek pie that we can put up as a thing. So anyway, we ended up getting both kinds of pies for that event. And it was kind of a fun thing because we also were able to sort of count down to the pie moment of the century, the 3.11514159. See, I'm not using Ploufe's algorithm. I have to remember the digits of PI in sequence, so to speak. Okay, Let's see. Oh, my gosh. Various comments on what I was just talking about, WW comments on Brahma Gupta and rules for arithmetic and algebra with zero and negative quantities and so on. It's always very difficult to trace early history of mathematics, particularly in India, because little was written down. Mostly things were oral traditions and they stayed as oral traditions, often until even the 1600s and so on. And so when you say so and so discovered this thing and one has some markers of when that person was alive. But it's not so easy to have these markers given that it's an oral tradition. And it's just like, you know, it's harder to know when Troy was. When the fall of Troy happened. It wasn't even clear that the works of Homer, you know, that the Iliad and so on, was a sort of a historically based document until in the 1800s when the actual archaeological remains of Troy were found. Although famously, you know, like many archaeological sites, there were many layers of Troy. And Schliemann, the person who had pushed for many years as sort of an amateur archaeologist to find Troy, kind of had the theory about where the historical Troy, the Homeric Troy was and blasted with gunpowder or something through, or maybe it was even dynamite by that point, through the layers of Troy, which includes probably the one that was Homeric Troy. Big shame. Kind of a lesson in the complicated issue of, of when do you excavate things in archaeology? Because if you leave it for longer, there'll probably be better methods and you'll be able to preserve more of the history. But in any case, it's hard to know when all these things came from because things were an oral tradition for a while. And I think with Indian mathematics, ancient Indian mathematics, a lot of that was very much an oral tradition and sometimes very tied into things that were part of the kind of, I don't know whether you call it religious tradition, but the cultural tradition, so to speak. I mean, I know that things like the Fibonacci sequence, which sort of to the west was introduced by Leonardo Pisano, the Fibonacci in 1205 AD or something in his book Liber Abaci. But that idea, the Fibonacci Sequence, was known presumably in ancient India. And the reason, I think, is that in the Vedas kind of Religious, I suppose, texts. There are kind of, I'm told, sequences of words. And it was. It's kind of. They're short sequences of words, but it was something where it was common to kind of speak as a kind of. To chant incantation, you know, incant, these sequences of words, and to do it in all different orders. So, you know, it's like we're saying the cat on the mat, the mat, cat on mat, cat on sat mat, et cetera, et cetera, et cetera, in all these different orders. And it was that kind of activity, I think, that led people to wonder, well, how many different orders are there? And turns out when they're repeated words, you're in the Fibonacci number sort of domain. And so that's an example of kind of how the math got in there. Now, in terms of. I know there are claims of sort of the first zero in India and I think in Cambodia also. And it's not completely clear what that was, how that worked and so on, certainly not to me. I mean, the very idea that there would be negative numbers, for example, or, for example, zero, why would you have the number zero? I mean, if there's none of it, why are you even talking about it? And negative numbers, why would you talk about that? Actually, negative numbers really became cemented in mathematics around 1500, I think, with Cardano, who was really involved in trying to solve cubic equations. And to solve cubic equations, you actually need two things. You need complex numbers with imaginary square root of minus one, and you also need negative numbers. And the casus irreducible is. The irreducible case of the cubicle was one that seemed to need both of those. And that was kind of. You really need negative numbers. You can't really express these things without negative numbers. And also, by the way, complex numbers, which today are a much more elaborate concept. So that was sort of the place where negative numbers were forced to come into mathematics. Now, they had implicitly come in through accounting, and particularly Luca Pacioli's development. That must have been mid-1400s, maybe, am I right? Not sure of double entry bookkeeping. But of course, double entry bookkeeping specifically doesn't. You know, you have credits, you have debits, you don't really have to talk about negative numbers. And even in accounting today, negative numbers are represented with parentheses around them. They don't have a minus sign in front usually, or sometimes in. In. In some cases, they're represented in red ink, so to speak, you know, as in, in the red. But that was literally red ink. When people like when Ramanujan was an accounting clerk, I'm sure he was putting negative things in red ink. The. I think that it's kind of a. He had kind of a. A kind of a. A job that was probably rather easy for him. I think he worked at the port of Madras, which is now. Oh, gosh, I'm going to embarrass myself with Indian geography. I think, isn't that Chennai now? I think. I'm not sure. I think I'm wrong about that. Anyway, never mind. It was in those days it was called Madras and Ramanujam was a clerk. And in those days, before there were computers and all those other kinds of things, clerks, the main activity was adding up columns of numbers and so on, which obviously Ramanujan could do very quickly. And so I think he got to do a day's work in the first 15 minutes or something because he was just really, really fast at doing those kinds of calculations. Then he probably got to spend the rest of the day playing with numbers in ways that were kind of more important for the history of mathematics, so to speak. But in any case, the. So I don't think that, you know, I think accounting specifically avoided having to introduce the idea of negative numbers. And in the end, negative numbers. It's sort of interesting because negative numbers are not that conceptually complicated. They're certainly much simpler conceptually than complex numbers. Yet the invention of those things took a really long time in kind of mathematical history. It's kind of like the invention of. Of universal computation and the idea of programmable things that wasn't really invented with any degree of clarity until the 1930s. There's a sort of precursor in 1920, and there were pre precursors in the 1600s. But that concept, once you, you know, if you explain that concept to young kids these days, they understand it. Now, it probably helps that they've seen computers all over the place, but I think they'd even understand it without that. It's something where it's like an idea which once you have it, you can't kind of go back. You can really understand it quite easily. But it took a long time in intellectual history for that idea to arise. And I think the same was true of negative numbers that once you kind of have the idea, it's not so difficult to understand. But getting the idea is not trivial. And these things where it's like, why would you make that abstraction? That's often the thing that people ask, as if one looks at the history of mathematics, for example, that's often the thing. Why would you make that abstraction? Why would you care about something that's like arithmetic but doesn't actually have the rules of arithmetic? Why would you. Et cetera, et cetera, et cetera? That's been a continuing story and certainly in the sort of learning from history. In my own efforts in mathematics and metamathematics and so on, I certainly am keenly aware of the fact that there are things which haven't been abstracted yet, but where the abstraction will be important, but where right now it's like, why would you abstract that? Why would that disembodied thing actually mean anything? I think one area where I've been really pushing, but I haven't figured it out yet, is I'm pretty sure there's a theory of what one can call bulk orchestration. So when you have a microprocessor and it's doing all these different computations, there's something you can say about the thing in bulk that has been set up for all these individual purposes. And I think that thing you can say is. Is highly related to what you can say about biology. And I think that at sort of a conceptual level, it's a little bit like inventing the idea of information theory, where you have to invent the notion that there's something worth saying about sort of messages in general, independent of the actual content of an individual message. Let's see. So many interesting questions here. Calvin asks, do you know if there was any physical reason that the Greek elements were associated with particular geometrical shapes? I think they had arguments for that. I mean, in other words, the, you know, water. Oh, gosh, I forgot what was what fire was. Maybe the octahedron, because it was all pointy and it sort of poked at you and so you felt hot. And then the dodecahedron, I think was the most mysterious. The sort of quintessence type thing. They had arguments for these things. I mean, I know that pointy shapes were associated with both fire and. And the soul and the fact that organisms did things, the fact that sort of the initiator of motion and so on. But I don't know beyond that. I think they were just like. Well, we have these shapes that we know, the Platonic solids that we know exist, and they seem very fundamental and so everything must be made of those. And let's just guess which one corresponds to what. I think that's as far as one could get. Let's see. I see a comment here from F and H. The PI day thing is great. I Think I might get a shirt. That's cool. Yes, there's this website, My PI day. And yes, you can order a T shirt from there. I believe that is your PI day in a nice swirly pattern. Let's see. Oh, yeah. I'm being reminded that is it next week? I think it's next week. Wednesday Next week turns out to be math storytelling day. I don't know who makes up all of these such and such days, but last year I had some fun doing a part 1 math storytelling about arithmetic and I think I got to algebra and I think I promised to go on. I never did. So now we get this year I'll go on. I'll try and get to calculus and things. So coming next week. All right, let me see. All right, Very different question here from Elsie. To what extent did your own path and work intersect with the heyday of Bell Labs and notable people therefrom? Okay, so I was a consultant at bell Labs in 1983, 84 type timeframe, which was a really, I would say a wonderful time at Bell Labs. It was the time when the UNIX operating system was really taking off and there was a lot of activity around that. And I was interacting both with the physics group and with the. What was it called in those days? Computer. It wasn't called computer science. I don't think this was at Murray Hill, New Jersey. This was the headquarters of Bell Labs. I was always very impressed with that organization as a generator of research. I thought that it was a place where their whole process for having bright people who would work on research but in a fairly managed way, and they would promote people who kind of showed sort of management potential, but who were also good researchers to be the managers of groups. It was very regimented in many ways. Like I remember the generic office had a linoleum floor, but if you were a manager, you got a carpet on your floor. And if you became a manager, they would come in overnight and fit a carpet in your office. And if the office was no longer used as a manager office, the carpet would magically disappear. I think the Bell Labs also had these very, very, very long corridors. And it had this whole elaborate coordinate system for numbering offices, which actually I copied for our company, although our office space is much smaller, vastly, vastly smaller. But it was always, people were like, where is such and such an office? And it's like, well, let's have a coordinate system for this building. Which we did. And I think it's helped over the. We've been there 30 years now. So I hope it's helped. But in any case, Bell Labs had this whole numbering scheme for offices and I always thought that to some extent it was like, okay, you want to come and compete with the Bell system? We've got researchers and we've got a mile of corridors or something and it's researchers all the way down, so to speak. It's just the mass of research that was possible there was really huge. And I thought it had a lot of very talented people, I would say in the UNIX group. Well, all sorts of people were there. I think the guy who ran it was a chap called Doug McElroy who I think is still with us, who sort of was interested, actually curiously enough had been interested in some math kinds of things to do with number theory and so on, which intersected some of my interests. Oh gosh, where can I start here? Let's see. I mean, I remember, I think before I hung out there I gave a talk at Bell Labs and I was talking about cellular automata and it's sort of a funny story. I was going to talk there and I was showing some plot and talking about some formula for the period of repetition of some cellular automaton thing. Somebody puts up their hand and says, you know, I say, I think the formula is blah blah, blah based on this numerical data. Somebody puts up their hand and says, did you try the case? I forget what it was. 157 or something. No, it's probably not a prime. But is that even a prime? Any case the, the. No, that one's not a prime, is it? No. Might be, I'm not sure. This is terrible. I, I should know this. It's, it's like there's a, there's a number. What is the number? The 90 something or other which is often known as Grothen Dieck's prime. Grothendique was a very talented, although slightly strange. I could tell you some story of Grothendeeck, but that's a different story. In any case, he was a very creative and successful very high end pure mathematician in France. And in some seminar he's like, what about a prime number? Like, and then he quotes some number which turns out not to be a prime. But of course he wouldn't know. He didn't know about the specifics of individual prime numbers. He was thinking eight levels of abstraction above that and in fact ended up inventing some mathematical constructs like the Infinity Groupoid that are very closely related to this ruliad construct of mine. But that's a completely different story. In any case, not embarrassing myself with what numbers are prime. Any case, I was giving this talk and this person puts up their hand and says, you know, did you try this case? And I'm like, no, I didn't actually get that far. So then I, I go and I try it and yeah, the formula that I had didn't work in that case. Person who asked that question was a chap called Andrew Odlisco who was a number theorist at Bell Labs and who became a friend of mine and we even wrote a paper together and so on. And these days he's, he's retired and actually just sent me some stuff. He's been working on bubbles, and by bubbles I mean financial bubbles. And I think most recently he was studying the railway bubbles in the 1860s and 1870s and so on. And I was about to go look at some of what he's written because I'm curious about some questions about how bubbles end in connection with modern tech issues. But any case, that was. Was one of my early introductions to Bell Labs was Andrew sort of saying, have you tried this number? Well, it turned out he knew that that number was special because of some things to do with, well, various kind of number theoretic properties which were in turn related to things called shift registers. Whole collection of stories about that. There were other people around. Ron Graham was a. I think he was the head of the math department at Bell Labs at that time. Ron Graham was a sort of discrete number theory is more discrete mathematics person who was quite, quite a playful character. He was a kind of a friend of a chap called Paul Erdos who was not a Bell Labs person, but was a big fixture in the number theory world. Paul Erdos was kind of a person who was a Hungarian chap who had kind of. I don't know, I think he existed. I'm not sure he ever had a place where he lived. He would always just go visit one mathematician and then leave that one and go to another mathematician and keep going. And he was sort of traveling around the world as a traveling mathematician, so to speak. And everywhere he went he would kind of state problems and like, do you know whether the sum of this number of cubes are this and that and the other? And for each person he would kind of say, I've got a problem for you type thing and come up with it. I met Aros a few times and I have to say it was always a little frustrating for me because he's a person very. Oh, I don't know. I think he was A very kind of wired kind of person, possibly from. From taking all kinds of wiring things, but. But anyway, very, very wired older person by the time I met him. And, you know, sort of everything that came up, it's like, I know a problem about this. The problem is, blah, blah, blah. And it's like, let's talk about the bigger picture and let's talk about what do you learn from the fact that there are all these hard problems in number theory? Do you learn something about the sort of metamathematics of number theory? Do you learn things about decidability? How many of these problems that you are actually, even in principle, doable? I couldn't get him to engage on those topics multiple times I tried. But Paul Erds was a big fixture of the sort of number theory world. And people often quote Erds numbers. Like, how many. If you've written. If you wrote a paper with Paul Erds, and he wrote many papers with many people, you have an Erds number of one. Erds himself, of course, had an Erds number of zero. And if you'd written a paper with somebody who'd written a paper with Paul Erdos, you have an Erdos number of two. So I think maybe I haven't written an academic paper myself in a long, long time, but I think that paper that I wrote with Andrew Orlisko got me an Erds number of two, at least right there. In any case, when it came to the computer side of things, oh, gosh, all these people, and I haven't thought about this in so long. There was one young chap called Rob pike, who'd been a student at Caltech when I was there, and who was sort of a physics student but was really interested in Unix. In fact, Rob pike was a person who probably 1978, 1979, told me, C is the language of the future. At that time, people were like, fortran. Fortran is the language of today. He's like, c is the language of the future. And kind of he was the reason I learned C. But anyway, Rob then sort of, he was like, well, should I go work at Bell Labs and do all this UNIX stuff and so on, or should I stay as a physics graduate student? And I was like, just go work at Bell Labs. It's really an exciting place. And my colleagues in the physics department at Caltech were kind of horrified by that advice. And I got in some trouble for that advice. But so it goes. But anyway, Rob went to Bell Labs and had a very successful career at Bell Labs and later at Google and Later developed this language called Go and also the Google file system and a whole bunch of other things. At Bell Labs he was working on the thing called the blit terminal, which was a. This was 1980. Well, certainly by 83, this was a bitmap terminal. And kind of the idea was that it's kind of like a thin client terminal where you would sort of interact with this terminal. It had a bitmap nice display was, you know, it was. What was it? It was green on something or something white on green, I forget. It was a green display, I think. And it. And the idea was that the computer was connected by a phone line because after all, the Bell system was about phones and was connected by a phone line to your terminal will be connected by a phone line to the computer that was doing all the actual computing. And so that was Rob and I think a person called Bart Lecanthe's sort of thing was the blit terminal blt, but BLT stood for that was to do with bit blitting, which was taking bitmaps and copying them to other parts of the screen and so on. In any case, at the time there was sort of the big kind of big competition was between that idea of the local terminal and the remote computer and the fully local computer. Personal computers existed, but they were starting to be workstation computers and particularly the sun workstation, Apollo and other companies. And I remember one time, it must have been right around then, having dinner with Rob pike and Bill Joy. Bill Joy was one of the founders of Sun Microsystems and listening to all the arguments going back and forth between the two of them about is it going to be these thin client type. They weren't called thin clients in those days, but the terminal just on your local terminal with the remote computer, like cloud computing today, I suppose. Or is it going to be the. All the computer is right there and this whole thing about is it remote, is it local for computation has gone back and forth so many times and been determined by many different trade offs about compute speed versus communication speed versus memory needs versus whatever else. But in any case, listening to that argument, I was like, Rob should win. But actually Sun Microsystems won. And part of the reason for that was that Bell Labs and what was then Western Electric were really not set up to do something like take a, you know, take sort of develop a computer and distribute it to the public. Just wasn't what they did. You know, they were providing phone service and they were maintaining the long lines, you know, phone network and things like this. That wasn't and so they really were. I think, Robin, people were pretty frustrated in the end because just wasn't a thing you could do from there. There's a similar type of problem to the problem that happened with Xerox and Xerox parc, which had been the Xerox, you know, photocopier company and office automation company later on. But, you know, they had this research center in Palo Alto that was, I think, if I remember right, 3333 Coyote Hill Road, I think was it. I think that's right. It later became taken over by. By next Steve Jobs's company. And I don't know what it is now. Maybe it's a museum now. I have no idea. In any case, the. The it was. Xerox PARC had many innovations, including the very first bitmap terminals, the small talk language, this that lots of other things were created at Xerox parc. But Xerox got very little benefit from that. In the end, a lot of those innovations the about graphical user interfaces and so on. Steve Jobs and Apple were really the place where that took off and they had known about what was done at Xerox parc. And ironically, as I said, I'm pretty sure if my memory is right, that Steve Jobs moved into the very building that park had been in 15 years later or something. In any case, the thing that. So that was another case where there was a company and it sold photocopiers and these innovations just didn't really fit. And it wasn't really possible for that company to take those innovations and put them out in the world. Now, interestingly, Bell Labs, when it came to Unix, Bell Labs did something very similar to what companies often do today, which is they say, we don't know what to do with this. Let's just throw it out in the public. Let's make it open source, basically. And that's what they did with Unix. Unix was originally developed as an operating system to run the phone switches. And I think there have been, if I remember right, the electronic switching systems. There were a series of ones. The number five ess, the number six, the number seven. I think it had gotten seven by the time I was there. And I think those represented, you know, 10 to the five phone lines coming in, 10 to the sixth, 10 to the seven phone lines. And they were all getting computerized. And the Bell system had actually made their own computers. Three B2s maybe I've forgotten. It was a type of computer that was used in the phone switching systems. And there was a version of Unix I Think Unix rt, that was a real time version which had actual commands in it, system call commands for things like initiate a trunk call and so on. It was intended for phone switching. But the rest of Unix as an operating system in general, they didn't know what to do with it and so they pretty much gave it to the world. And that was so. But there were a lot of people around who had been involved in doing that. I mean, I remember Ken Thompson and Dennis Ritchie were the two sort of key people, both of whom I knew, I will say the one piece of kind of bad advice those two gave me, or not really bad advice, but something or other. I had developed SMP, the first version of that came out in 1981. And SMP had the feature that it had lots of commands, lots of functions, and they all had short names. So it would be array, would be AR and things like, oh, I don't know. Well, the key function of the system was in fact called smp, which stood for simplify. And there were all kinds of other short cryptic names. And when it came to mathematical functions, they would all have cryptic, you know, two, three letter names. Actually, the king of weird names was the function Rex. And Rex was a random expression generator, which I suppose would rightly be called the king of functions. But in any case, I remember talking to Dennis Ritchie, I think I was having dinner with them, Dennis Ritchie and Ken Thompson, and they were telling me the thing they really liked about SMP was short command names, just like unix. That was a really bad feature of SMP at the time. One was somewhat forced into that because the typical person who was a potential consumer didn't know how to type and they were pecking away with one or two fingers and there wasn't anything like autocomplete. So it really did make a difference if you had to type an eight character thing. That was a very time consuming thing. So it made some ergonomic sense at the time, although it made the code unreadable and the whole thing much more obscure than it needed to be. But anyway, those two really like that idea. I remember Ken Thompson was something of a hacker and was proud of this black box that he had, which was some kind of phone system, I don't know, something where you could make a call into this black box and you could call out to anywhere. And they didn't really quite understand what the whole story was. There's another chapter named Joe Condon, who'd been involved in the invention of touch tones. The tones, the two frequency tones that phones make and he was somehow involved with that. I forget all the details. Actually, with Dennis Ritchie, I actually only learned very recently or fairly recently his background in theoretical computer science. I mean, he'd gone to Harvard and done theoretical computer science. And in particular he had studied a lot the notion of recursive functions and different kinds of recursive functions and the difference between essentially for loops and while loops in a while loop, at least in the sort of classic version, in a for loop, in the sort of classic version of this, you're saying, I know what the range of variables is supposed to be. Just go range over that range of variables. And nothing that happens while the for loop is being executed can change what range of variables you're going over. And that led to this notion of primitive recursive functions. A while loop classically had been, well, you're doing this while some condition is true, and the thing that you're doing in the actual computation can affect that condition. So you don't know in advance how many steps you're going to have to go and so on. And he wrote, I think, his thesis about that, that very topic. And then it's sort of amusing. Two things are amusing. One, that he was really involved with such theoretical pure ideas about computation, which I think led to some of the simplicity of the C language which he developed. But also it's kind of ironic that the very thing he wrote his thesis about, the sort of distinction between for loops and while loops, he broke that distinction in C because you can make these conditions in a for loop in C can be affected by the actual behavior of things during the execution of the for loop. So I'm trying to remember some of the other things. I mean, Brian Kernighan was another person who was there a guy called Bjorn Stroustrup who would later go on to do C. There were people, I'm trying to remember the trough typesetting system that was done. Who did that was one of those guys. I mean, sometimes it was easy to remember who did what because like orc, the utility, the search utility, I think that was Al Aho, Peter Weinberger and Brian Kernighan. And the letters of their names spelled out the function. So one kind of knows. I don't know if anybody knows that anymore. Probably they think it comes from the name of a bird. I don't know. Perhaps it did come from the name of a bird at that time. Although GREP is presumably not the name of a bird. GREP is another related function in Unix, which stands for generate report. What else can I tell you about those guys? Gosh, I'm. It was, yeah. I mean there were a lot of different things that were going on. I mean Unix had sort of was splitting into these various different forks and well, there's many stories about that, but that's as much as I can immediately think of there. Let's see. Oh boy. All kinds of people with comments about. Okay, VP is asking, did I ever use an Amiga computer was made by a company called Commodore. Commodore was a. Was an odd company. It was. Headquarters was somewhere in Pennsylvania, somewhere slightly obscure in Pennsylvania. I never visited there. But when Mathematica was young there was a Amiga. Commodore was sort of developing a higher end computer and there was certainly talk of whether we would get Mathematica to run on that higher end computer. The Amiga was primarily a gaming computer, is that right? Yeah, there was some. Commodore had a computer called the pet, which was kind of a desk calculator type computer. And actually that was the computer. Dick Feynman got a Commodore PET back in this must have been 1978ish timeframe, maybe, maybe a little later than that. And actually that computer was the computer that he used to try and when I had invented the Rule 30 cellular automaton, that was the computer he tried to use to sort of break. He and his son Carl tried to use to kind of sort of break the randomness of Rule 30. But any case, the Commodore had had that history and then they. It was a funny company. I mean I certainly heard, I don't know if it was really true that the CEO of Commodore was somebody who had. Oh gosh, this sounds bad at the time, but then it sounds like my life too. But anyway, had basically almost never visited the plant, so to speak, the actual place where the work got done at Commodore. And I say that sounds bad because I've been a remote CEO now for 33 years and I don't go to our headquarters location very often at all. So I sound as shocking as the chapel was running Commodore at that time. I certainly know there were at that time it was always a slightly sort of odd computer company that was off from the mainstream. I do remember one story, I think perhaps the names won't be included but it was kind of a funny story of Commodore had wanted a new operating system and it was going to be a Windows system. And there was this young chap, a couple of young chaps who were. Who were going to build this Windows system and they said we can build your window system and we'll build it in a weekend. And it's like really, really, really? Is that possible? Well, they did. They built a window system in a weekend, but it took like, I don't know, five minutes for a new window to open. And in the following, how many years they were supposed to actually make it real and it never really happened. So it's one of these cases where if you have a specification and you say, make me a Windows system and somebody says, yes, I can build it in a weekend. Be careful what you wish for because you have to be careful that the specs really were what you thought they were going to be. In any case. That's a little bit. Let's see, we have. D0. Let's see, it's commenting. Okay. Dellamort has commented with mobile devices we're basically going back to terminals. Yeah, I mean, I don't know. And then there's more that gets done on the mobile devices. It goes back and forth. You know. I know that when I learned to type when I was, I'd used a typewriter from when I was, oh gosh, probably seven years old, something like that. I still have that typewriter, but the seven year old me did not have particularly strong fingers and it was a, you know, old manual typewriter. And so it was like, how do you actually get that key to press? And so I was like a one fingered typist and I was pretty fast. I got really fast. I wrote whole treatises when I was like 12, 13 years old which were all done with one finger typing. And at some point when I was probably, I don't know, 14, 15 years old, I thought I'm going to learn to type with 10 fingers. And so I did. And it was actually surprisingly quick. It only took like an hour or something to be decently proficient. I haven't looked back, so to speak. Although I think I use weird fingers for keys because I taught myself that, as I've taught myself most things. But in any case. But anyway, so I had always thought being good at one finger typing useless skill until cell phones came back and now of course I haven't properly modernized for thumb typing and I'm doing my 1960s style one finger typing. But so it goes. Let's see. Doug, comments. I used to have a Silicon Graphics Indigo 2 sitting on my desk for AutoCAD and 3D modeling. Great machines. Fun times. Yeah, the, the Silicon Graphics Indigo, I think it was actually an indigo color. It was a dark purple machine. And Silicon Graphics was a company that when we first built Mathematica was one of the companies I always am grateful to these companies was one of the companies that signed up to distribute Mathematica for their computers. And they signed up before we released the product. And so a chap called Forrest Basket, who was, I guess the CTO of Silicon Graphics was at our original product announcement on June 23, 1988, because we were running Mathematica version 1 on the silicon Graphics machine. Actually, a good friend of mine, John Mazuris, who was founder of mips, was the company that made the, the processors that were used in Silicon Graphics machines. MIPS was sort of the first RISC architecture computer company. It kind of. Well, John had been a physicist originally, but then had been at IBM briefly and then at Stanford and then MIPS spun out of there. I do have to tell one story. John's been a good friend of mine for many, many years. But when Mathematica was young and we were running things on the MIPS computers and there was some bizarre bug with some, you know, all the tests passed except for some bizarre bug to do with the values of the zeta function were wrong on this MIPS machine. And so I was telling this to John and his immediate response was, oh, it must be a problem with the hardware. And he was like, oh, I wonder whether it's some floating point, this, that and the other, the other. And I always viewed that as a nice example of if you've designed a particular component of some chain of systems for many people, many people's personalities, they always assume it was their part of the chain that had a problem. I have to tell you, in terms of porting to different machines, the all time favorite was the Cray 2 story. So crays were these supercomputer type machines. I never met Seymour Cray, one of the designer of Cray computers. I just one of those people. I never had a chance, never had a reason to meet him, unfortunately. But any case, the Cray 2 computer, these were like five $10 million computers and we were doing this port of Mathematica to run on this computer through Cray the company. And what we got it up, it was running, it was going to put its first input prompt up, type two two and I kid you not, came back with the answer five. So this giant multimillion dollar machine, two plus two is five. How could something like that possibly happen? The reason was that the way that machine was architected, it was all about floating point. It was all about doing sort of calculations for partial differential equations and physics and so on. It was really optimized for floating Points. So integer stuff was kind of an add on. And so that meant that somehow the pathways and the instructions and so on for dealing with integers was sort of something that you did on top of floating point. And so somehow something got done wrong by us. Presumably that led to the two plus two equals five incident. But it was certainly, for me, it's kind of one of the more dramatic porting problems that I've seen. Let's see. Okay, I should go in a moment here. Well, Elsie is asking, speaking of McCarthy, John McCarthy, I assume in those days, do you think that sticking to S expressions as opposed to M expressions and Wolfram language style ones, impeded Lisp's adoption historically? I would assume the answer is yes. I mean, what happened? And I didn't know this history until very recently. I mean, John McCarthy always had kind of a grudge against me because he had developed Lisp starting in the late 1950s as sort of the language of artificial intelligence. John McCarthy was sort of the originator of the term artificial intelligence back in 1956. And he'd kind of, you know, Lisp was intended to be sort of the language of AI and then. But it wasn't a terribly practical language because the machines of the time were just not big enough, not enough memory, the implementations of Lisp weren't efficient enough and so on. So around comes 1979 and I'm starting to build SMP. And although there was one person who was a bit involved who was a Lisp enthusiast, it just was completely impractical to build the kind of system I wanted to build in Lisp. And so I built it in C. And I think John McCarthy was never very happy with that. And he was probably wrong at the time. And I think he was definitely wrong at the time. I see people now building systems in low level languages when they should be using Wolfram language. And I always think that's kind of bad. I think this might have been the very first experience that John McCarthy had of seeing somebody build something not in the language that he thought was optimized for this. I would like to think the Wolfram language is better optimized for lots of things, particularly nowadays, certainly vastly, vastly, unbelievably better optimized for things than lisp was in 1979. But any case. So John McCarthy always had a bit of a grudge and I really regret the fact that I never really broke through his sort of curmudgeonly and actually frankly pretty jerkish behavior towards me. And I never really pushed hard enough to have A real conversation with him. And I regret that because the more I learn about him, the more I learned that he was actually pretty rambunctious with many people and that there was an other side to the whole thing. And I really should have pushed harder to break through that. And I think the thing that. But, you know, I hadn't really known about M expressions until quite recently. So in Lisp, sort of the original thing was don't have syntax, just have everything with parentheses and have, you know, open paren function space, arguments of function, very clean, very, very, very straightforward. But then there was the idea of M expressions. And even in M expressions it will be function, open, square bracket and then arguments just like in Wolfram language. I didn't know that until probably four or five years ago, amazingly. And that had been a thing that had been talked about probably around 1960 for LISP. I don't think it was ever implemented in an actual Lisp system, but it was talked about. Maybe. I think it must have been implemented maybe in Lisp 1 or Lisp 1.5, but in any case, it didn't catch on on Lisp. I don't really know why. Yeah, it would have been a good thing. I mean, Lisp had such a long trajectory to go before it had even the possibility of being sort of more practical that I think it was, you know, that it would have been a hard road in any case. And I think it suffered from. Well, it suffered from a bunch of things. It suffered from some not great design decisions at a practical level, like, can say that, from having done language design for a long time. We didn't make some mistakes that Lisp made, which were early versions of Lisp. And then Lisp became very, very complicated and it probably became sort of too complicated for its own good without having a clear sort of use case path. It was just like we're adding all this complexity, but it wasn't oriented at a particular kind of, kind of use. And I think so I think that was sort of an issue. But it's, you know, people say any sufficiently complicated programming language will eventually become like Lisp. Well, you know, that's the story of universal computation, the story of my principle of computational equivalence. You know, any system very quickly becomes a universal computational system. And all universal computational systems are ultimately equivalent. The real question is, when you go from the human thinking about something to the actual computer computing things, kind of, what's that path like? And what waypoints do you go through? And the idea of symbolic expressions, which is an idea, a fundamental idea. In Wolfram language, it's also, in effect, an idea in Lisp. I mean, Lisp is sort of, in a sense, symbolic, but in a sense, not so symbolic. It involves explicitly having to sort of evaluate things, it involves explicitly quoting things and so on in ways that. So it doesn't really have quite the same. Doesn't have quite the same character of symbolic expression usage that Wolfram language has. And really the core for Wolfram language is this transformations for symbolic expressions, which are certainly possible in Lisp as they would be possible in many other systems, but it's not sort of a core idea of Lisp in the way that it is in Wolfram language. In Lisp, the core idea is much more function evaluation. In Wolfram language, the core idea is much more transformations for expressions, which can also be used to do function evaluation, just as function evaluation can also be used to do transformations for symbolic expressions. But the emphasis is different. And that emphasis is part of why it's been so easy in Wolfram language. And I didn't see this particularly coming when I started developing transformations for symbolic expressions back in 1979, that it's been so easy to represent all these very different kinds of things in the language. And I think it's been a sort of irony of my life that I worked on physics very early on and came back to physics in a big way, like four or five years ago now, and ended up with this fundamental theory of physics that turns out to be, at some level a theory that is about physics being transformations for symbolic expressions, or at least for hypergraph expressions. So it's kind of an ultimate irony that I've been interested in physics from a long time ago, although I was not optimistic of making progress in this most foundational level. But then in the end, when progress is made, it really is made by leveraging an idea that. That was an idea I had for completely practical reasons decades earlier for the purpose of building. Building computational language. All right. By the way, I will say about John McCarthy, I learned recently that his whole file system from Stanford is actually available on the Web. And like, there's a letter he wrote me in 1984 that you can find on the Web in this big file system of all of these things. It makes for an interesting. I've sampled a few pieces of it. It makes for interesting read, I think I wish I'd had access to that archive when I knew John McCarthy, because I would have learned more about him and I would have pressed through the rambunctiousness, so to speak. I Will say, to give you an example, the very first time I met John McCarthy was probably 1980, 1981. I was visiting the Stanford Computer Science department and was at a lunch there and there was a whole bunch of people there and this white haired chap who was obviously looked very distinguished and so on. You go around the table, everybody's introducing themselves and that person says, my name is John Smith. And I'm like, I don't know who the heck this is. The web didn't exist. There were no, you know, there was no bulletin board with photographs of people and names and so on. So I had no idea who this was. And so I, you know, that was John McCarthy. But he chose to introduce himself to me as John Smith, which meant I had no idea who he was. And you know, I probably would have talked to him that day had I been able to connect the name which I'd certainly heard of his work. But that was, that was an early John McCarthy encounter. So, and it went on from there. But anyway, so, so it goes one of those I, I have to say and if I am to look at errors in my life in terms of, of interacting with people, sometimes one is put off by things like that. Well that was not the only thing but that sometimes one is put off by things and I think sometimes when the value is high enough one should press forward, so to speak. Despite those things, I will say one more thing about people and just another one of my foolish things I did. This must have been 1975, maybe 1974. I was a mid teenager in England and I was doing physics stuff and somebody told me, oh, Werner Heisenberg is going to give this talk in Cambridge. And that was a train ride away for me. And I was like, eh, I don't care, he's just an old guy, I don't think he's doing anything interesting. So I didn't go. And actually the person even told me and he likes to meet young people who had, you know, doing physics stuff. So terrible lost opportunity. Not least because I learned in later years that Heisenberg had been really interested in, well, in fluid dynamics and in these nonlinear field theories and the idea that you could sort of derive physics in a sense from purely kind of abstract principles and then even much more Recently I learned that Heisenberg in the 1930s had believed that space was discrete and so on. Of course I wasn't thinking about space being discrete back in those days yet, but it's a terrible lost opportunity. And when I try to give advice to people who are interested particularly in the sort of subjects and their origins and so on. It's like if you can go meet the people who originated these subjects because it gives you a lot more perspective about what was actually going on. And that was one of the cases where I goofed and didn't meet somebody who and subsequently it would have really helped me to understand. Something that I have found is when you're trying to understand people's work, it's like if you have met them or even if you see a video of them, it gives you quite a lot more perspective on the kinds of things that you expect that person will have emphasized in their work. So I find that really useful. So anyway, a lost opportunity for me. All right. Well, thank you for joining me and, and thank you for asking all kinds of interesting questions, making all kinds of interesting comments and look forward to seeing you another time. Bye for now.