History of Science and Technology Q&A (April 3, 2024)

Welcome to another episode of Q and A about history of science and technology. And I see that we have a bunch of questions here. I see one from Joy asking, is there a directionality to science and technology? I feel like the answer to that is straightforward. I feel like it's one of the few things about human history that does not tend to be cyclic. We build ideas in science, we build inventions in technology and they kind of progressively build up as a tower. We sort of couldn't have the modern things without the older things. Not always true. There are cases where things go in cycles. Like for example, in technology. Famous one in the computer industry is, is the computer power right with you or is it on some remote server? There were the mainframe computers and then personal computers and then there's the cloud and then there's this and mobile and this and that and the other. And that's an example of something where the pendulum has swung multiple times. I would say in the progress of science, there are certain cases where there are theories that people had that were just a bad idea and that led things astray for 50 years, 100 years, maybe more than that. But those are, I think, sort of exceptions to the general flow of just the more, you know, the taller the tower gets and the more you can build on it. So I tend to feel like that's unlike many other aspects of human history and human attitudes and things. You know, there's a fashion for this and then there's a retro fashion for that and it's things go back and forth, so to speak. That feels like a thing where there's sort of definite progress that gets made. Let's see. Oh gosh. Des asks, has anyone applied sort of hacker mentality to the Antikythera mechanism to figure out what else you could use it for? Like, like what kinds of uses could a time traveling von Neumann figure out? So the Antikythera device is this thing with I think 37 cogs that was in a shipwreck from about 2000 years ago, around either 1st century BC or 1st century AD. It's not clear exactly which was discovered originally in 1900 from this shipwreck and then wasn't really recognized what it might be until the 1950s. Then it was only in the last, I don't know, 15 years or so that it sort of got CT scanned and it got figured out, oh, here are all these cogs and there's sort of an interpretation that it's a thing for computing astronomical. It's kind of an orrery like device that, among other things, allows you to compute potential times of eclipses and so on. Now, you know, could you have used it for something else? I don't think so. I think it really is kind of an, a model, sort of epicyclic gear type, little solar system type thing. I don't think you could have taken those gears and oh, use it as a arithmetic computation device. I don't think. Not sure. Certainly having seen that device, one should surely have the idea of constructing what got done eventually in the, in the 1640s, 1630s and 1640s, which was starting to make kind of arithmetic computation devices. So I, I think the key idea which I think had existed in some form in, in antiquity as well is the idea of sort of an odometer like thing where you have cogs that you turn and then let's say they're in sort of digits in base 10, you turn the first cog and, and you turn it all the way to 10 and then it has another little tooth that knocks one unit up in the next cog and so on. And that's how you sort of add carries and you're essentially, you're saying, let me do addition by just turning these cogs a certain distance and so on. That idea got finally sort of implemented. Presumably the first time in modern times was a guy called Chicard who was a friend of Kepler's. And the sort of first surviving example I think comes from Blaise Pascal, I think from 1642. Ish. It's a brass thing about, oh, I don't know, it's about size of two hands or something that has little, has, has cogs effectively and does arithmetic. I, I have to believe that that idea probably existed in antiquity. We just don't have samples of it being used. But had. If you'd seen the Antikythera device, you surely would have had the idea of doing something like that. So let's see. Paula asks, what's the likelihood that ancient tech we've discovered had vastly different uses than what we believe? We don't know a huge amount of ancient technology. I mean, we see the, you know, we see buildings that have been constructed, we see, you know, aqueducts, things like this. And we see, yeah, I mean there's certainly you can have pots and things and you can have pots of different shapes and it's like, what was this pot for? Like, for example, one time when I was a kid, when the world was a little different, I was visiting Cyprus and you could just find like on the beach there, you could Literally, you were crunching, you know, pots from antiquity on as. As you. As you walked along the beach. And I found an oil lamp there, or at least I believe it was an oil lamp. That's a fired clay object. That is what I would interpret as the shape of an oil lamp. And it kind of had carbonization on the place where the wick would go. So it's like. I'm pretty sure that was an oil lamp, but I can certainly imagine many sort of strangely shaped pots and things where it's like, I don't know what this was. You know, maybe it was for cooking some particular kind of thing which we don't cook anymore, or maybe we don't eat anymore, or maybe we cook it a different way now. And it's hard to work out. I mean, I think one of the things about archaeology that is perhaps counterintuitive is that you might think the past is kind of whatever we discovered from the past, we've now discovered, and we won't know in the future more about the past. But that's just not true. I mean, what's happening increasingly is the ability to go back and look at little molecular residues. And you've got this pot. You say, what was cooked in this pot? Oh, actually, we can get little fragments of DNA here, and we know it was this particular kind of plant that grew in this particular place and so on. And we can deduce that in the future. I mean, sort of some of the notable examples of. There's certainly architectural forms where we say we don't really know what that was. There also are plenty of examples of kind of ornamental art or what appears to us to be ornamental art. But for all we know, it had some definite quotes. Purpose, I suppose. There's an example I know of in Aleppo, Syria, that's kind of this. This wall that has black and red squares. I think it's from maybe 11,000 years ago, something like that, a really long time ago, maybe a little less than that. But in any case, it's, you know, it's black and black and red. Well, actually rectangles in a particular pattern, presumably. I mean, it was clearly sort of done on purpose in some way. But what was it for? What did it mean? We have no idea. So I think there are definitely things like that. I mean, I suppose there are other. Yeah, I mean, this problem of kind of interpreting what things were for long after the time is pretty hard. And it's. Even when it comes to, I don't know, let's say people describing, you Know, birds and fishes or something. In ancient Greek times, it was a non trivial effort to match up what was this fish that was talked about in such and such a place, in such and such a book. What is that fish called today? How do we figure out the connection between those things? Let's see. Yeah, Des is commenting that some parts of the world have it better than others in terms of survival. For archaeology, people who made things out of stone or sort of baked clay and so on have done much better than people who made things out of wood. And so there's, you know, we know lots about the ancient Egyptians because not only did they make stuff out of stone, but they also had this idea that they would bury kind of dioramas of their life in the pyramids and things for the sake of the afterlife. And we sort of get to be that afterlife because we can get those dioramas and see things about how they lived. But for example, we know little about, let's say, the Phoenicians, I suppose, to know things about some civilization. It's good if there's writing, it's good if they wrote things down. It's good if they wrote things in a form that survived, like they chiseled them in stone somewhere. And sort of worst cases, the things that they had were so culturally significant that people copied them for centuries, possibly translating them through multiple languages and so on. And that there was sort of a continual chain. That's what happened, for example, with Aristotle, that his works. There's certainly no original piece of Aristotle's work that survived, at least so far as we know, I think. But through, you know, translations into Arabic, into other languages, you know, his works kind of survived in that way. Right. So Elsie is commenting, what does that tell us about how we should preserve things today? You know, my own personal experience with kind of archiving is that which is not on a sort of current storage medium, like a disk you can access immediately with your computer. It's hard to make it survive. If it's a thing where, oh yeah, you put it on a tape and then you read the tape many years later and maybe the magnetic bits decayed, maybe it's really hard to find a nine track tape drive, all those kinds of things, that's bad. But if you have it just in the form of, in a sense, abstract bits on your computer, then whatever your new storage medium is, oh, I'm going to copy it onto a, you know, 100 terabyte drive or something like this. You do the copy and the bits are there, sort of abstractly. And you know, if you keep them as sort of online storage, my experience is that tends to work okay when it comes to things like data formats. The more elaborate the data format, the more difficult it is to recover it. I think for myself, most of the things that I produced, I would say have been recoverable. Even things like, I don't know, faxes that were in some kind of fax run length encoding, pre TIFF type format, those have been pretty recoverable. I mean, an example that I have not recovered is paper tapes that I have from the very first computer I used around 1972, that in principle one might be able to buy an optical paper tape reader. I haven't done that. My assumption is that the right way to actually read those paper tapes is lay them out in strips, take pictures of them and do kind of image processing to figure out where the holes are and then just do, you know, write a simple program to figure out what the data meant. Now then one has the question of what was the encoding. And actually for those paper tapes, one of the. You hang yourself on your own path, so to speak, is that I wrote the loader that read those paper tapes onto the. At least for the ones that I created. I wrote a loader that would load those, the data from those paper tapes into the computer's memory. And it had some very clever, I thought at the time, error correction mechanism, which I have no idea anymore how it worked. And that would be sort of a piece of almost cryptanalysis to go and unravel how that works. Actually I have to say about something to do with preserving one's data back. The first big computer system I built a thing called S and P, built between 1979, 1981. I'd had the bright idea at the time that we should encrypt our source code. And I sort of as a, as a, some kind of security measure of the time. And of course, oh, and I also thought, let me not use the standard UNIX crypt program because if a crypto system is going to be broken, it will be the most standard system that everybody uses, will be the one people break. So I took the source of the UNIX crypt program and I modified it and then used that as the encryption system for SMP and for the source code of smp. And needless to say, in the intervening decades, I forgot the password and everybody who worked on SMP forgot the password. And so we had for years this encrypted source code. Not only that, we had the encrypted source code of the encryption program as well. And for a long time it was like, what are we going to do with this? And eventually a young chap, much younger than who was, you know, wasn't. Was born many years after smp, so to speak, managed to break that encryption not by finding the password, but by directly doing sort of crypto analysis of the statistics of the. Of the bits, so to speak, to find the plaintext without ever finding the password. So we managed to recover SMP that way. But that's another kind of gotcha of things. You say, I'm going to save this, I'm going to encrypt it. Oh, whoops, where am I going to keep the password, so to speak? Let's see. Des is asking about thoughts on ancient dodecahedrons. Yeah. From Roman times, there are these brass dodecahedra that are hollow and have little knobs on the sides of them. That's a good example of something where I have no idea what those were used for. And this is one of the cases where maybe they were ornamental. Seems unlikely because those knobs seem like they were functional in some way. But when in doubt it might be ornamental. I mean, what is a Rubik's Cube used for? Is that, you know, how would you classify that use type, so to speak, or some paperweight that one has, or something of this kind? I mean, I have to say, in my own experience, one of my own personal sort of classic experiences of this, of the what's it used for? Question. When I was, oh, gosh, probably six years old, maybe visited Sicily and picked up this thing on a beach that is kind of a sandstone colored thing. And it's more or less, oh, I don't know, it's kind of circular, but has an indentation on one side. And the bottom of the indentation is an elaborate pattern, but not an elaborate pattern, which is hard to interpret. So when I was a kid, I wondered what this thing was and lived in Oxford at the time and was able to take it into a. Well, I forget which one was first, whether it was the Archaeology Museum or the Natural History Museum. I took it into one of these places and they're like, the Archaeology Museum was like, no, it's not our thing. You know, take it to the Natural History Museum and vice versa. So to this day I don't know whether this object is some kind of natural thing or whether it's a. Whether it's something that was a thing that was part of a seal or something like this. A seal? Not in the biological sense, a Seal in the sealing wax type of sense. And I'm sure today there are sort of more molecular level tests that can be done. And I've sort of been meaning to do that for years. But that's an example of something where it's really hard to tell whether it's even a natural thing or an artifact. Let's see, questions here. You guys are probing my memory here. Maybe asks who started research on the periodic table? Can you discuss a bit about its development? I don't know that history terribly well. I mean, Mandela in the second half of the 1800s, I think, was. Was the person who sort of put that together. But what had happened was that the idea of different chemical elements had originated back in. I mean, people knew there were different materials. The idea that all materials might be made of a small number of different kinds of atoms that came from antiquity. And people would discuss the question of whether the kinds of atoms might correspond to the five Platonic solids. And maybe the pointy atoms that were the ones associated with the octahedron would be associated with fire and things like this. And it was a very kind of a notion of a sort of very kind of mechanical way of thinking about the atoms that things might be made of. And people had thought, you know, air and water, those are fundamental things, and so on. It was really by the time of. Well, certainly by people understood more about metals and ores, of metals, and being able to create the pure metal from the ore and so on. And then I guess the big thing in the Middle Ages and so on was the idea of alchemy and the idea that you could do things where you kind of transmute materials, one material into another, by doing what we would now call chemistry. And people didn't know what you could transmute into what. And that was kind of a big issue. So, for example, one of the things people really want to do is transmute lead into gold, because lead was cheap and gold was expensive. And whoever figured out how to do that could make a lot of money, so to speak. And there was this sort of notion of the philosopher's stone, which was sort of associated with kind of the essence of things and spirits. And how would you extract spirits, which might be alcohol and might be kind of spirit in the sense of a. A mind like spirit that was sort of somehow confused and conflated and sort of. The philosopher's stone was something which variously would be associated with transmuting lead into gold, with eternal life, and with a variety of other sort of good features. But this notion of sort of transmuting things into things, changing from one material to another, heat it, combine it with this, do that. That was a thing that in the time of the Middle ages and beyond, was the thing people were starting to figure out. I'm not sure at what point people really got the idea that there were sort of basic elements, so to speak, that were sort of fundamental and which could be combined in different ways. I mean, certainly Galton in the early 1800s had sort of noticed that when you saw certain chemical reactions happened and you would get this amount of this material, that amount of that material kind of had this argument based on the idea of atoms, and of course atoms, nobody knew that matter was discrete. That was a thing that had been hypothesized by in antiquity, but it hadn't really been established. But nevertheless, Dalton kind of figured out that you could sort of count things and see, oh yes, if water is H2O, so to speak, then that accounts for why twice as much hydrogen is produced here than there and so on. Very much a kind of count things and see what happens. Very similar, by the way, to what happened with the quark model in the beginning of the 1960s with people like Marie Gel Mann sort of saying, well, you know, we could account for the particles we see by a proton as upped up, up, up, up, down. You know, a lambda particle is up, down, strange, what, whatever else. It's basically doing this, in a sense, combinatorial argument for what's there without knowing. And in fact, Marie Galman didn't really believe that quarks were real things as opposed to mere mathematical modeling mechanisms. That became clear only in the, in the early 1970s when people were able to sort of bash protons hard enough to see that there were sort of hard bits inside which were then interpreted as quarks. But the original kind of argument for the existence of quarks had to do with sort of, oh, you can account for protons and neutrons and lambda particles and sigma particles, and you could predict the omega minus particle just by saying, well, we've got these entities that are combined in different ways. And I think that's really the way that atomic theory was thought about in the time of people like Dalton. I think the kind of gradually different sort of basic sort of atomic types got filled in. And what Mandelaev then did was to notice that there were certain regularities, periods of sort of related elements, and then arranged them in what we now call the periodic table. The interpretation of the periodic table came considerably later. I mean, the, the Idea, So, so a key observation, okay. To, to interpret the periodic table, you have to kind of know what atoms are made of. And the idea that there's a definite sort of sequence of elements that I think was, I'm not sure how that was figured out. The fact that what we would now know as you're counting the number of protons in the nucleus of each of each kind of atom, but I'm not quite sure how that was figured out. I mean, I think that there's a certain amount, yes, the atomic mass of the element. Not sure when that was figured out. But the fact that the, are these definite sort of regularities of elements that are similar to each other that required knowledge of sort of the electron shell structure of elements. And I think that came quite rapidly after Bohr's model of the atom, which was kind of a, a sort of pseudo pre quantum mechanics. And, and then this idea, well, you got to have these different electrons and with the exclusion principle knowing that you could only have sort of one electron in each particular state. So you'd end up with these different cases where you would have electrons in different orbital angular momentum configurations where you could have more electrons as you get to the outer shells and so on. And that that would account for the periods table. I'm kind of thinking that was a thing that happened in the, maybe the 1910s, 1920s type timeframe. I don't know that history terribly well. I'm sorry. Let's see. Well, Felipe asks what motivated the other to the phosph Fourier transformation? And who was its creator? What was its creator wanting to solve? And I'm, I'm going to embarrass myself by not, I think Hartley maybe was the inventor. Was it? I'm not, I'm not certain. But let me say what I, what I do know about. I mean, there was when computers first sort of came on the scene, okay, the question of algorithms, how do you work stuff out? Was something that people were certainly concerned about when it came to hand calculation. So in the 1200s on, people were trying to come up with different schemes for calculating things. Like Fibonacci wrote this famous book, Liber Abaci, the Book of the Abacus, which was a kind of description of how to calculate stuff. Most famously, the very end is this mention of the rabbit's problem and the now called Fibonacci sequence, which had by the way appeared many, many years earlier in Indian writings and other things like that. But in any case, there was this sort of idea, 1200s to 1600s and beyond. That people would write books about how to calculate stuff. And I think to some extent that was how to organize your calculation, how to say the ratio of this to that corresponds to this. And remember the rule of threes, which I now forget, but that was a big thing in sort of education. It's a way of doing proportions of things. I don't remember what it is. And there's another one called the rule of sixes, and there's another one called the rule of Z or Z's. That is some kind of calculational device for doing some kind of multiplication. I'm not exactly sure. That was, I think, developed by a guy called Richard record in the mid-1600s or so. But people had had the notion that there were sort of better and worse ways to calculate things that had been around for a while. And it was relevant for merchants who were doing, oh, yes, you can multiply by nine by doing this. And there's this trick. And so on and so on and so on. When computers came on the scene in the, well, basically Beginning of the 1950s, the first assumption was, well, we'll just do this. It was a big premium on writing programs. So it was expensive to write programs. It was expensive to store those programs on the computer. So if you want to do multiplication, you were going to do it in a very direct way. You weren't going to try and use any kind of trickery and so on to do it. And that was sort of the first step. But then people started realizing, oh, things like sorting, you want to sort things in order. What's the best way to do that? Well, a human does some very simple sort of transposing things around. But then, for example, I think that was Tony Hoare, who invented this thing called quicksort, which is kind of a recursive sorting algorithm. And that was invented as, I think, in the 1950s, probably early 1950s, as kind of an example of a surprising algorithm that actually wasn't too hard to implement, that a computer could readily do, but a human really wouldn't choose to do it. And that was kind of an early idea. There were a collection of different kinds of things. I mean, I don't know when, for example, hashing was invented. I mean, I know that sort of. I personally reinvented hashing. Oh, probably 1972ish. And, you know, I'm sure hashing was invented many times, but I don't know at what point it became called hashing and really took off as a thing people thought to do probably when databases were developed, which was probably mid-1950s. But in any case, these different algorithms, even for something like multiplication, there's a question of how should you multiply two numbers? And the most obvious way to multiply 2N digit numbers involves doing N squared operations. That's the standard schoolbook long multiplication operation. But then people started figuring out, oh, actually there are faster ways to do it. But that happened, I think it was 1970s before that really took off. There was thing called Karatsuba multiplication in which the main issue is you don't reuse in that N squared set of operations, you're actually reusing a lot of things. And so that method uses essentially some nested fractal like pattern of multiplications, which is sort of all you need because you can kind of reuse the pieces. And I think that gets you to, I don't know, n to the 1.55 or something like that rate. And then much later there were methods using fast Fourier transforms that were N log N type computation methods. And for example, we implemented those things in Wolfram language. At the time when we implemented this stuff, that will have been 1987 or so. I believe that Caratsuba multiplication was still fairly new. And the phosph Fourier transform method for multiplication was, if it was known, maybe it was known, but we didn't implement it at that time. Now, in terms of the fast Fourier transform itself, it's again this story of, well, a Fourier transform. The most obvious thing that you can do is an N squared operation. But again, it's a question of how you organize. You're doing a bunch of multiplications and so on, and you're essentially redoing things that you kind of did before. And if you organize the calculation better, you can get away with less operations and you can end up with N log n instead of N squared operations. And I think, I think I used to know why that was invented, but I don't now remember. I think it was certainly by the phosphorier transform was one of those things that sort of a calculationally organizing idea of which there were a whole family of algorithms that were like organize the calculation better and then you could do it more efficiently. There's one which I'm pretty sure I think I suggested this became called the Barnes Hut method for doing N body simulations, which is just again, it's a turn N squared calculation of working out sort of this gravitating body has this gravitational effect on this body and that on that body and so on. And the naive way of doing this is an N squared type method. And I think, I think it was one of those. Somebody tells me about a random problem at lunch one day and I say an obvious thing. That is a possible way to approach it. And that must have been 1983 or so. And at that time somebody like me knew all about phosph Fourier transforms, but they were not widely known. That was not a widely known idea that you could kind of take a calculation and reorganize it and end up with sort of fewer operations. And so it was sort of obvious to me, at least you could take a calculation like that and there should be a way of reorganizing it like you could with a fast Fourier transform. Actually, the method, the fast Fourier transform idea can also be used for pure integer operations leads to so called fast Walsh transform. And I know that was not understood. Gosh, that must have been the 1990s or 2000 or beyond. When that got understood. It was pretty obvious that you should be able to. For integer multiplication that you should be able to see how does this work? The. Oh, gosh, this is. Yeah, that you should be able to do something. Oh, how does it work? I don't remember. I wrote it down. Something about it in, in New Kind of Science. And we've certainly made use of that, those ideas in Wolfram language. But. Okay, so I'm afraid I'm, I'm not, not doing well on, on answering this. I will say that one of the things that people were very sort of keen on the sort of fast furry transform idea when parallel computers started to be kind of a popular thing. This is now in the 1980s and there were a series of. So there had been parallel computers where the computational elements were just in an array. So for example, the Illiac 4 I think was an example of that from the 1960s where you just have a sort of an array processor where there are, let's say in a two dimensional or three dimensional array, you just have different processing elements and they communicate only with their neighbors on that grid. But by the 1980s, people were starting to think about more elaborate ways of connecting multiple processes together. I mean, I worked a bit on this thing called the connection machine computer built by a company called Thinking Machines Corporation. I was just a consultant for them. But that was a machine that was intended to have the capability to take sort of. It had two to the 16 processors, 64,000 processors. And the idea was that it could sort of take data and route it from any one processor to any other processor. At some level. The processes were arranged in a Boolean hypercube, so a 16 dimensional kind of 2x2x2x2x2 array. But the sort of, the big idea was that you would be able to take data from any processor to any other processor. So for example, I developed this thing I called C, which was an extension of the C programming language in which you could have. When you could have a pointer and you could say, take this piece of data and when you follow the pointer, it's moving that data to another processor. All rather lovely, except that as a practical matter, there was a nasty habit of these different pieces of data that were getting routed around all these processors getting stuck in a queue. And there were various issues. There's a whole long story about what happened with that. But in any case, on that machine, one of the things that people were keen on was, oh, can we make this kind of butterfly, so called butterfly type connection, which is a particular graph of connectivity between the nodes achieved in software, in effect there, because that was the sort of connectivity that you need to implement a phosph Fourier transform. And that was a thing that was kind of imagined. In that case. It's worth realizing that when it comes to. Yeah, I think I didn't even explain what Fourier transforms are. But basically, if you have some quantity which is some Y as a function of X, you've got some curve, it's wiggling around and so on. There's a question of how do you describe that curve? One way you can describe the curve is just saying, oh, I know the value For x equals 0, y of 0, I know the value for x equals 0.1, y of 0.1 and so on. You just fill in a bunch of points on this curve and that's the way you describe the curve. That's one way to do it. And for some purposes that's very convenient. And if you want to know what's the value at 0.07 or something, you might just be able to, to fill it in by looking for a linear kind of just say, oh, there must be a straight line joining the point on the curve in this place and that place. That's approach number one. Approach number two would be, let's describe this curve by saying it's a sum of known curves. So for example, let's say we say this curve is we pick sine waves. So we look at something which is just wiggling up and down and we say, you know, we look at sin x, it wiggles up and down with a certain certain wavelength sin2x. It wiggles up and down at half the wavelength. So it's got more wiggles in a given space. And we just say, let's try to make our curve by taking a sum of contributions from a sine curve, a sin 2x curve, a sin 3x curve, and so on. So the Fourier transformation is to say, let's take our curve, our signal, let's say our sort of, let's say intensity as a function of time and let's represent it not by just saying the value at this place, this place, this place is this, but rather let's represent it by saying it's 0.1 times sin x plus 0.03 times sin 2x minus 0.07 times sin 3x and so on. So it's a different representation. Why is that useful? It's useful because, for example, the sine curves are, you can think of them as kind of pure tone single frequency signals. And for example, when it comes to sound and hearing, we can distinguish those definite frequencies, those things that are just wiggling at a certain fixed frequency. So the Fourier transformation is about transforming the data that you have about just the points on a curve to a representation of that curve in terms of a sum of sine curves and so on. And it's something where you're kind of taking that signal in the so called time domain where it's just we know what it is at each particular time to the so called frequency domain where we're saying we know what the contribution is from this frequency, that frequency, that frequency. So that's kind of the idea. And the. So Fourier transforms are used all over the place, not least because if you've got a sound, speech sound for example, there might be some hum or hiss that's very high frequency and you want to get rid of that. So you say let's transform it to the frequency domain and then let's just cut off the things that are at high frequency or maybe at low frequency, if there's some low frequency hum or something like that. And that's a convenient thing to do. In addition, when you're trying to transmit data, a typical situation is, oh, you've got all these details and all this wiggling around and it's got all these high frequency wiggles. But actually we don't notice that it's too high for us to hear or if it's in an image, it's too much sort of individual pixel by pixel variation. We can just cut that off. And so that allows us to have less data that we have to Transmit, because we're not including those high frequency pieces. So phosph. Fourier transforms occur all over the place. Fourier transforms appear all over the place in JPEGs and in all sorts of different. In all the sound formats, MP4, things like that. They're sort of a basic tool for. For manipulating signals of all kinds. The idea of doing it sort of with algorithms is one idea. You can also actually use light, for example, to implement transforms. In a sense, when you make a hologram, you are effectively doing the hologram. The content of the hologram is effectively a Fourier transform of the original sort of image that you're trying to encode or the original object you're trying to encode. And roughly that happens because the thing that matters is kind of the angle that the light was going at and sort of at the source, you're kind of. The different positions correspond, give you things that are at different angles. But by the time you're far away, what matters is kind of the phase. This is. Okay, too complicated. Let me not go down that path. Let's see. Greg asks, how advanced did analog computers get before we moved to digital computers? Was there any debate on whether we shouldn't move, should move to digital at the time? I think I did talk about this a bit before. I mean, I. Analog computers were still sort of a thing by the beginning of the 1980s. Like, for example, I worked for a little while at Los Alamos National Lab, and they were most proud of their digital computers. They had very fancy supercomputers and so on, but there were a few analog computers still knocking around that had mostly dated from the 1960s. And the problem was it was just a big pain to use them. I mean, I tried to use them for some things, for some solving some differential equations and so on, which is something for which they should be very suitable. And, you know, it's kind of cool because you can easily get an audio output and all these kinds of things. But it's just a mess to have things where you're plugging wires into this and that. Much easier to just say, I'm going to write my program, I'm going to put it on. Well, by those points, it was a timeshared system where it was just digital files. But even when it was cards, it was easier to sort of have a stack of cards that were a uniform stack of cards for any program you wanted to have, rather than analog computer where you had to actually be sort of plugging things together. As I recall, the one that existed at Los Alamos that I played around with, had sort of the beginnings of digital control of the analog computer. People have tried to bring back analog computing multiple times for various kinds of image processing for various other purposes. In a sense, some of the work that's being done on quantum computing is, is an attempt to bring back various types of analog computing. I think what tends to happen is analog computing is nuisance to set up. It's not as you don't know what you have, it's not as repeatable as digital computing. And in the end, it's only certain kinds of things for which I don't know voltages and currents and so on work in the particular way they work, end up being solvable with analog computers. Digital computers are much more general. Let's see. Cass asks, why did modern formal logic take so long to develop historically compared to other branches of mathematics or physical sciences? What explains the delay until the 19th century? Well, interesting question. I do know that history reasonably well. Let me tell you what happened and then we'll talk about why it happened. Back in antiquity, math was developed in two real branches. One was arithmetic for commerce, basically, and a little bit of kind of calculational geometry for the sake of land surveying and for the sake of saying, you know, I've got such and such a number of what we'd now call maybe acres or something of land. I forget what the ancient unit of that was. Cubits was length, but I don't remember what the unit of area was in any case. So there was a sort of calculational mathematics branch that existed in ancient Babylon and beyond, where mostly arithmetic, some calculational geometry. Then there was kind of the Greek pure thought tradition that led through Plato to Euclid and so on, where it's like, let's just sort of reason about things and think about geometry as something where you make proofs and so on. I mean, the person who was trying to compute the area of a field didn't care about proofs. They cared about. You can multiply this number by that number and then you've got the area. So the philosophical tradition of Plato, Euclid, et cetera, was sort of a different tradition from this calculational tradition. Now, meanwhile, in the middle of that was Aristotle, who was basically an organizer of knowledge. Lists of mollusks, lists of ways of making arguments. The lists of ways of making arguments turned into syllogistic logic. And that was sort of this pattern for how human arguments would be put together. And in the Middle Ages, people would have these mnemonics for, oh, it's a Barbara Syllogism or a cellarant syllogism. Those letters stand for. Well, there's some filler letters, but the main letters stand for the different elements of an argument. You know, this implies that and not that and so on. So that had been sort of the tradition, that tradition of kind of syllogisms. Syllogistic logic, I believe was pretty disjoint from everything that happened in both calculational and kind of philosophical mathematics. The calculational side of mathematics didn't. I mean, it developed the sort of. In this rather strange direction of who can solve the cubic equation. You know, let me make a public demonstration of how clever I am because I can solve a cubic equation and so on. That was a thing from the 1500s and such like. But I think that was pretty disjoint from. I mean, I think the people who will have known about that certainly knew syllogistic logic, but I don't think they made any connection between those things. There was no sort of mathematicization of logic at that time. It was like logic is about arguments and then there's math and it's about what was by that point algebra. And then there was sort of the proof based math which, yes, in fact, if you really take it apart and think about it, it involves something much like logic to deduce one step from the next. I don't think that's how that was thought about. I think that was just thought about as a. It's a matter of deduction, so to speak. And the connection with the sort of deduction for human arguments was separate from the thinking about deduction for abstract mathematical kinds of things. It came together with George Boole in the 1830s, I guess. And you know, George Boole had been mostly a schoolteacher and. But in those days people did lots of things and he was kind of. He was a little bit out in the boonies. I think he was in Lincoln in England, if I remember correctly. But there was a chap called Augustus de Morgan who was in London who was sort of a popularizer of math and kind of a, I think a little bit of professor, but mostly kind of a math tutor of people like. Like he was Ada Lovelace's kind of math tutor among other people. But Augustus de Morgan sort of wrote these books about logic in the kind of largely Aristotelian tradition. Meanwhile, George Boole and de Morgan invented things like de Morgan's laws for logic, which are kind of combinations of sort of things you could get out of syllogistic logic as sort of another level of kind of description, but still very non mathematicized. And I suppose, now that I think about it, De Morgan was a single person who sort of brought together the logic tradition and the mathematical tradition, but I'm not sure didn't really relate them. I think they were two different branches of his thinking. But then George Boole was kind of a said, well, let's see if we can just take algebra and sort of apply it to ands and ors and nots and so on. And he did that as a pretty adventurous thing. It was kind of like, well, we've got this kind of framework for how to do things in math with algebra and can we apply that to logic? I think that was really his idea. Now it happened that at the same time there were people sort of mathematicizing other kinds of things and creating kinds of versions of algebra that didn't work like with numbers. Like Galois, right around the same time was inventing group theory and sort of in a different branch, Lobachevsky was inventing the idea of curved space. Even though it seems like the space we have or it seemed like the space we have is just Euclidean space, you can still imagine these other kinds of things. So you can still imagine things where you do multiplication, but the multiplication doesn't work the way it works for numbers. Well, Boole kind of had that idea and sort of did violence to the standard laws of arithmetic to put in his Boolean algebra. Most importantly, that sort of if 0 is false and 1 is true, then true or true is just true, but one plus one in normal arithmetic equals two. They said, well, let's just set it up so that it equals one instead. And so in a rather rickety way with some inconsistencies, he created this kind of notion of Boolean algebra and that idea. And he wrote this book called Laws of Thought, which was pretty popular in its time. And people were then off to the races, kind of making little simple logic machines and so on and really systematizing the way of thinking about logic. Now I suppose also at the time, I mean, Lewis Carroll, Charles Dodson was sort of an expert on symbolic logic. And I guess the point there was, and there was sort of a precursor to Boole's work was this notion that you could take kind of syllogistic logic, Aristotelian logic, and you could fill in symbolic variables like P is the proposition, you know, Alice has a pet dodo and you know, Q is the proposition, a dodo is a bird or something. And then you could use that kind of symbolic idea. P and Q implies, I don't know, Alice has a bird or something like this, that you could fill in symbolic variables for the terms in a syllogism. And I don't think that had originated before that time. And that was sort of a necessary precursor, I think, for Boole's work. But people would make kind of mechanical devices that would implement logic of various kinds. And that sort of went on for a long time. I think the next kind of really big piece of this was probably Shannon's Masters, Claude Shannon's master's thesis in the 1930s, I guess, which was about switching networks and Boolean algebra. So people in the early part of the 20th century and the very end of the 1800s were actually. No, there's another piece I should mention here. But the part where this started engaging with technology was people had made all sorts of elaborate electromechanical machines. And the question was, how could you think about those things in a sort of systematic way? And Claude Shannon pointed out, oh, these switching networks, you can think about them in terms of Boolean Algebra and in terms of kind of Boole's mathematicization of logic. And that's how you could reason about these things. Big piece that I left out there was kind of 1870s, I guess, people trying to understand what really are the foundations of mathematics. Particularly people like Gottlob Frege who wanted to say, what really is math? You know, all these things we come up with of multiplications and numbers and all this kind of thing. And maybe it's group theory, maybe it's other sort of abstract things. What's this all made of? What's underneath that? And so there was kind of this idea that maybe what was foundational was logic. It's kind of a weird idea, actually. Not an. Obviously, I mean, in the end, what I've come to believe is that sort of. Ultimately it's all computation, which we can think of as sort of sets of rules. The particular rules of logic are particular. They are specific things about ands and ors and so on, just as the laws of arithmetic are particular to pluses and timess of numbers and so on. But anyway, that was a kind of a thought from the 1870s. And then that that sort of fed into Peano's work and then Whitehead and Russell's kind of Principia Mathematica effort to sort of see whether you could base all of mathematics, all the deductions of mathematics, on just thinking about things in terms of numbers and by Then sort of sets and so on. And you could sort of define a number to, to be the number one is like the number of sets of all sets which are not members of themselves. And that ends up being just the set with no elements and so on. And that's like a way of defining the number one. It's kind of a very early version of what we might do now in sort of basing sort of computational systems on different kinds of underlying elements. And you could say you can, you know, from this very abstract structure you can find a way to build a representation of numbers. I mean, in the end, you know, you're making a quantum computer or something and you're finding a way to represent bits in terms of configurations of ions or something like this. It's always like, how can you represent these things that you want, for example, in mathematics in terms of some lower level constructs. But I guess that the sort of, the big sort of theoretical flowering of logic which turned into mathematical logic, that happened in the late 1800s and that kind of flowed into later Godel's theorem and Turing's work. And then I think got a, you know, it came out of logic. But what one realized in the end was that just arbitrary systems of rules were the real story. And that as far as I'm concerned, is the kind of the, the ultimate story of computation and so on. Yes, Bool had a, I, I wrote a thing about Bool a number of years ago and bu married quite late in life to a somewhat younger woman and one of his daughters, Alicia Stot, Boole I think was her name, was a. Led a long life and was a important sort of amateur mathematician who worked on particularly visualizing things like four dimensional polytopes and had had a bunch of contributions to that, I think I seem to remember. Yeah, as I say, he sort of started out as a schoolteacher. In the end he ended up being professor in Dublin, if I recall, and sort of a respectable university professor type thing. And I think he had several daughters. But one of them was notable as a mathematician. Let's see. Nan asks, is there any knowledge in physics today that has been influenced by ancient texts like the Vedas and so on? Yeah, that's always so hard to tell. You know, the one thing we know sort of for certain maybe is that, you know, Hindu Arabic numerals, the notion of clear positional digital notation for numbers sort of was a, an Indian ish tradition that came to the west around 1200. In fact, Fibonacci was a big person who sort of brought it to the West. And I suppose that was what, at the time of. That was the time when there was trade going on across, you know, along the Silk Road and all this kind of thing. And I think that's sort of how that. How that knowledge got spread. But I think the. So, you know, for our positional notation numbers, we definitely have kind of the Eastern Indian ish tradition to thank for that. And without that, it's pretty hard to invent things like algebra because if you're doing Roman numerals, it's hard to even recognize what's the difference between a letter and a variable. Things like this were sort of very confusing to people. In terms of kind of thinking about physics. I would say that the Western tradition has had a pretty straight kind of shot. Terms of the development of physics, I would say there has not been to my knowledge, a lot of sort of direct causal connection between Eastern traditions and physics that we have today. Now, that is not to say that there weren't things that were figured out in the Eastern tradition which now have deep resonance and in fact possibly really align with things we're figuring out really right now about things about the ruliad and observers and our kind of view of what the universe is and our place in it and so on. Those things seem to have really quite remarkable resonance in some sense to things, at least that I can understand of kind of Eastern philosophical traditions. But I think in terms of the core developments of physics, the things that it is, I think there's. It really was sort of a straight shot from Babylon through Greek, Roman, then into with some Arab kind of world pieces there, some things in astronomy, particularly from that. But then I think it's been sort of a development, I think a pretty Western tradition development after that. So I don't know of direct connections there. I mean, certainly people are fond of saying that sort of mysterious things in physics, whether it's in quantum mechanics or whatever else, are somehow related to kind of Eastern ideas. And I think they're not wrong about that. But I think that the sort of formal development of the physics was. Was sort of a Western story. Now, interesting question to what extent the. The very idea of formalism exists in some of, for example, Indian traditions, I'm not sure. I think, you know, I've made attempts to kind of read some of the source material there. One example is people wondering about kind of the formalization of grammar and language. We know that we can make porous trees and diagram out sentences and so on. That's something that became sort of popular in the Western tradition in the 1950s, there's this question of sort of when can you make sort of systematic construction kit type grammars for things. And it's often said that a chap called Panini in well, it's hard to date these Endian things, but some number of centuries BC had created this kind of construction kit type grammar for Sanskrit. Well, certainly looking at sort of the works that seem to have survived, it doesn't read like a generative grammar would read today. It's more. Here's one example of something you can do in Sanskrit. Here's another example. Here's another example. Now, you can interpret that looking backwards from the formalization that we now have to. Oh, that's something like that formalization. But I'm not sure that one should think forwards from that. And I don't know to what extent kind of the kind of, well, things like notation that allow one to formalize things I don't think existed in those traditions. Now that isn't to say that people knew methods of calculation there sometimes. Well, I'm trying to think there were. Well, certainly, for example, the Fibonacci numbers were certainly known in ancient Indian traditions. They arose in trying to count the number of different ways. So I guess in those traditions the kind of way in which your sort of parse, in which the rhythm of language of spoken text and so on is important. And just as you know, when you have. Well, I only know it for Latin and Greek actually, I'm not sure I know it for English. But things like pentameters and hexameters, particular forms of poetic kind of construction where you would have short, short, long, short, short, long for the different syllables. That was something that I think existed in an ancient Indian tradition, probably in Sanskrit. And there was sort of a question that somehow arose what was the enumeration of possible forms of prosody? Actually, one thing that's sort of interesting is I don't think in the Western tradition that there was a lot of enumerate all possibilities of things, but that did exist in the Eastern tradition, the I Ching does that. The hexagrams of the I Ching are sort of the enumeration of possible forms. I think this Indian work on the enumeration of possible forms of forms of prosody is another example of sort of enumerate all the possibilities. I don't think that arises in the Western tradition until much later. I think ramon lull, for example, Catalan chap in the 1200s, had this idea of sort of the Combinatorial way of all possibilities. And for example, he had this sort of funny idea. Oh, what was it called? Some kind of. I think he had a book called the Great Art or some such other thing in which he's talking about how you can sort of enumerate all possible things that can happen and you make connections to figure out what will happen in some somewhat mystical type way that is probably not that different from casting sort of fortunes with things like the I Ching and so on. But that was where the 1200s, that's, I think, the first example that I know of of kind of the combinatorial enumeration of things being something in the Western tradition. And I think, well, Leibniz was quite influenced by Lowell, that's in the 1600s. But I don't think he talks. Oh, yeah, actually, yes. Okay. So this is a sort of connection point there. So binary numbers were almost certainly known in China probably in antiquity. They're somewhat related to these Indian tradition type things. There's sort of a big question of how did binary numbers come to the West? And Leibniz was very proud of having realized that you could make binary numbers. He had this kind of medal that he really wanted to have struck that says, oh, what is it? X nihil. From nothing you just have a one. And from that you can derive everything. So it means just with 0 and 1. And he writes out the numbers in binary. You can make any number just with 0 and 1. He thought that was very significant. But that's sort of an enumeration, a combinatorial enumeration type thing. There are all these combinations of zeros and ones that. That correspond to numbers. And I think that was a thing that sort of people. It's not clear whether Leibniz learnt that from missionaries to China and so on, or what. Probably he did. And that was a case where. So that's sort of an interesting kind of approach to thinking about things, of enumerate all the possibilities. That, as I say, I think was much more of an Eastern tradition and possibly, one would say, didn't really enter kind of the Western. I mean, the combinatorics of sort of counting things showed up a bit in studying games in the 1600s and inventing ideas about probability. The Bernoullis were involved in that. But kind of combinatorics as a significant field is pretty much a 20th century story. Before the 20th century, those kinds of. Oh, you enumerate all the possibilities of this and that that was a thing that would show up in recreational mathematics and in sort of puzzle Problems that were published in newspapers and things. But I don't think it showed up much in sort of professional, in the. In the professional mathematics, which was much more in the algebra, calculus and kind of analytic geometry type tradition. And so that was really a thing that emerged from me, for example, when I was a kid. Binary, the idea that you could write out numbers in binary was one of those typical sort of math enrichment type facts. It wasn't something you would be sort of taught in school in any mainline way. But if you looked at the, hey, there's cool things to say about math, those books would always talk about numbers in different bases and, you know, how do octopuses count? And things like this. And it really was, I would say, sort of the common knowledge of binary as a thing that would be part of the toolkit of everybody. It's probably a 1980s type phenomenon. Certainly, I know even the very idea that the digits of numbers might be significant. When we were first building Mathematica in 1986, 1987, some of the mathematical folk who worked on it were when I was like, yep, we're going to have a function that finds the digits of a number. Like, why do you want to do that? Who cares about the digits of a number? A number is just a single integrated thing that represents a value. Its digits are irrelevant. Well, of course, there are many algorithms where particularly the binary digits of a number become very important. But that was not sort of an immediately known thing at that time. All right, I think, I think we should wrap up here. So thanks for a bunch of interesting and unusual questions. And I'm sorry, a few of them I didn't have in immediate working memory the answers to those things. But thanks for asking interesting questions and see you another time. Bye for now. I should mention, by the way, that for those interested in the upcoming eclipse, if you're in the US Area, I want to advertise a little book that I wrote about the eclipse, which you can also find pieces of online and that I have a whole long description that I wrote and that I also did a live stream about, about the history of predicting eclipses. And we get to sample the success of that on what is it next, Next Monday. What is less certain is to predict the weather. And I know I've been planning to go to Austin, Texas for this eclipse and that may not be a good bet at this point. It's hard to predict the weather. I think people have wondered about predicting the weather for a long time and that will be another topic someday. I think I may have talked about that here before about sort of the history of weather prediction, but I'm about to find out how how good or bad it is. Anyway, thanks for joining me here and see you another time. Bye for now.