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A Story For Humanity — Minhyong Kim on Why Maths Will Never End
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Gauss famously described mathematics as the queen of the sciences. But how should we think of this discipline? Is it an aloof ruler, mysteriously governing the laws of nature? A vital tool for bettering our lot? Or a story that helps us understand nature?
Professor Minhyong Kim directs the International Centre for Mathematical Sciences and the Centre for Mathematics for Humanity in Edinburgh. He is an accomplished number theorist — particularly within the field of diophantine equations (the Kim in Chabauty-Kim). And a best-selling author on popular mathematics in Korea.
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Transcript
Mathematics and its Ivory Tower Perception
00:00:00
Speaker
Part of the mystique of mathematics is that it's very abstract. It doesn't seem concerned with the world, with what's going on. But this can be double-edged sword. My guest this week, Min Hong Kim, has said that maths can suffer from being an ivory tower within the ivory tower that is academia.
Mathematics and Interdisciplinary Connections
00:00:23
Speaker
Professor Kim sees mathematics as part of a continuum of disciplines of physics, chemistry, even logic, philosophy and literature, all of which share a goal of ultimately trying to understand the world. Mathematics is no difference. It is connected to the world in that same way.
Kim's Roles and Contributions
00:00:44
Speaker
Min Hong Kim is professor here at Edinburgh and also here at Watt University. He's the Edmund Whittaker professor. He's the director of the International Centre for Mathematical Sciences. And we'll also talk about the Centre of Mathematics for Humanity, which he also directs. More than this, he is a very accomplished mathematician.
00:01:04
Speaker
I am just someone who makes... semi-popular videos about mathematics and philosophy and other things. I do not contribute anything to this field, but he is someone who has made genuine contributions, ah particularly to the the area of diophantine equations. And if you work in that field, you've probably come across him and Shaboti Kim.
00:01:23
Speaker
If you're listening in South Korea, you will probably know him as well, as he's the author of numerous, very popular books in on mathematics in South Korea. And we talk a little bit about that.
00:01:34
Speaker
And I caught up with him just after our conversation and he is maybe about to pull the trigger on getting a translation out here in English in the UK and elsewhere. So hopefully that happens. um But this is someone who you should listen to. He's a very accomplished mathematician,
Endurance of Mathematics
00:01:51
Speaker
as I said. So We talk about many things. We talk about how mathematics should connect to the world. We talk about how many people see physics perhaps as drawing on mathematics, but it can go the other way. And not just in the famous example of string theory, but again in in Professor King's own work, he has drawn inspiration and and ideas from physics and used them in number theory.
00:02:16
Speaker
At the very end of conversation, we end up discussing whether mathematics will ever end. And I want to add just a little note on that here, because I reflected on this a bit kind of further after the conversation. So I hope this isn't a spoiler, but I feel like as long as there are problems to be solved, as long as we're inquiring about the world, math will be part of that.
00:02:42
Speaker
I'm James Robinson. This is Multiverses. I hope you enjoy this conversation.
00:02:53
Speaker
Minhyun Kim, welcome to Multiverses and thank you for hosting me here at the International Centre for Mathematical Sciences on this beautiful unusually sunny day.
00:03:03
Speaker
Well, le thank you for your interest. i I'm very happy to finally be interviewed and to have a chance to talk with you. I'm hoping today that we'll talk about different ways that mathematics intersects with other subjects and with human life more generally.
Defining Mathematics: A Philosophical Inquiry
00:03:19
Speaker
right But perhaps we can start just by saying, what is this thing, mathematics? You have a few lovely quotes on your website, but do you have a favourite perspective on what mathematics is?
00:03:32
Speaker
No, in some sense. that's that's maybe the That was part of the point of the quotes I put on my website. many Many people say many different things that cast light on different aspects of mathematics. But the point is that it's hard to say what anything is.
00:03:51
Speaker
right yeah So if you ask me what mathematics is, you might expect some kind of a concise answer. But suppose I ask you what ah even a cat is. That's very hard to answer, right? This is an old philosophical problem to say exactly what we mean by a cat. But if you move into the realm of, say, scholarly matters or culture or something like that, suppose I ask what literature is.
Mathematics as Literature: A Creative Perspective
00:04:18
Speaker
Yes. Can you give a concise answer to that question? not Not easily, not not a kind of analytic answer as the philosophers would would like. I guess it's, you know it when you see it, maybe? You know you when you see it, then you answer by some complicated procedure of delineating boundaries in some way, but none of the boundaries are even very clear cut, right? So if somebody asks, well, is...
00:04:46
Speaker
ah is mathematics literature. After all, it's written text and you you do a lot of things. And some of it has something of a literary flavor, other things seem more dry and so forth, right? So what mathematics counts as literature and what doesn't, it it's a pretty hard question to answer, right?
00:05:05
Speaker
So, I think it's the same with mathematics. What exactly counts as mathematics is very difficult. But I do like to point out, or at least call attention to some misleading characterizations of mathematics that exist out there, here here and there, especially among people who regard themselves as intellectual in some way.
Beyond Logic: The Explorative Nature of Mathematics
00:05:26
Speaker
So, one idea that was very popular through through a fair amount of the 20th century that's still with us is that mathematics is logic.
00:05:36
Speaker
It's just logic. The process of deducing B from A, right? can start from some assumption A and deduce some assumptions B. And I do like to point out that this is a very, very misleading characterization of mathematics for a number of different reason reasons. I mean, the first obvious reason is that A lot of people use logic, not just mathematicians. right of Anybody who does anything scholarly certainly needs to use logic to to carry out their argumentation.
00:06:06
Speaker
Physicists certainly use logic all at the time. They use mathematics all the time as well. And even in ordinary life, we talk about certain modes of deduction being reasonable and what and things not. We think that some argumentation is okay, some some some modes of arguing that that doesn't seem correct and so forth. So it's part of everyday life, the the usage of logic. And the way mathematicians use it is really no different from that.
00:06:35
Speaker
it's ah the The only difference is kind of a quantitative difference because we tend to be more fussy about it and we tend to be making more explicit. We tend to create more noise about something being correct or not, right?
00:06:50
Speaker
The way mathematicians employ logic is the way anybody else does. It's just creating convincing arguments. So that's one way in which mathematics is not logic. Everybody uses logic, not just mathematicians.
00:07:03
Speaker
And the other, of course, is that this mode of convincing arguments, convincing argumentation is just a part of mathematics. There's a lot of other parts of it that involve discovery, experimentation, calculations, philosophical speculations, building of theories and so forth. And there's so much of mathematics that doesn't involve exactly this process of correct argumentation. and what is In fact, there's enormous amount of preparation before people arrive at a logical argument to convince somebody else that some theorem, let's say, is true. So that's all being left out with the characterization that mathematics is logic. So those kind of things I like to point out. that Mathematics is certainly not logic.
Mathematics vs Logic: A Closer Look at Physics
00:07:53
Speaker
Now, am somewhat more favorable to the idea that mathematics is a kind of physics, in the sense that it seems to try to figure out, it seems to study properties of some very fundamental aspects of the world, and some of it which directly overlaps with physics, like if somebody tells you the volume of a sphere or something, right which is supposed to have been figured out by Archimedes, that's of course statement of physics as well. It tells you how much water can fit inside this ball and so on. So there are direct overlaps, but even where the overlaps are not so direct, it does seem like it' it's investigating features of the world that are very, very basic and fundamental to the way reality is made up. So there is some way in which it is closer to physics probably than logic.
00:08:47
Speaker
And then you probably know you know this cosmologist Max Stegmark. And he he has this idea that the universe is nothing but mathematics. right And i kind of I don't think I quite agree, but I kind of see where where he um where he comes from. It's because there are lots of physicists believe that the whole world is made out of elementary particles, for example. And it's hard to say what elementary particles are, except as a mathematical object.
00:09:17
Speaker
yes yeah And the way you build more complicated things up out of these very elementary things is also complete a very mathematical construction. So that's, I think, the inspiration for this idea that the universe is just mathematics. So from that point of view, maybe that's what mathematics is. It's just the structure of the universe. So
Defining Mathematics: Impact of Education
00:09:38
Speaker
all these different perspectives are possible. But as I said at the beginning,
00:09:42
Speaker
Saying what anything is is a very difficult task. It's a dangerous task. I'm not going to do it about mathematics. But by the way, it's sort of related, you know, in a vein that somewhat related to this idea that mathematics is logic. right I think there are people who expect there to be a clear-cut answer to the question, what is mathematics? And I do wonder where this expectation comes from, comes from right?
00:10:11
Speaker
But maybe it comes from the fact that in school mathematics, mathematics tends to have clear-cut answers. you You pose problems to people and there's supposed to be some correct answer, right? And if you're used to that mode of working with mathematics, maybe you expect even without mathematics itself to get a clear cadence. Many thoughts strike me. As you say, maths is introduced as this very certain subject at school. But I do sometimes wonder if in the long run it will be the undecidability part of maths that will sustain it for for longest, in that it will never be completed perhaps.
Mathematics for Humanity: Bridging Gaps
00:10:52
Speaker
but and on
00:10:54
Speaker
I was also struck by the first quote that you have on your website is the Vladimir Arnold. but Yeah, you know, bath math is just a part of physics where experiments are are cheap. way But I do wonder about that if if the hierarchy is correct there, or or indeed is the relationship not so much a hierarchy as some kind of symbiosis, and perhaps, yeah, there's a continuum between the concerns, I think, of of maths and and physics rather than one being subsumed by the other.
00:11:28
Speaker
i um I think there's a continuum between the
00:11:34
Speaker
interpolating all aspects of human knowledge, yeah it's not just mathematics and physics. so that's one thing But also, i think we need to distinguish a little bit here, mathematics or physics as subjects. or so so sort of um disciplines or areas of study and the objects that they study.
00:11:58
Speaker
yes And sometimes these two questions tend to get conflated when when when we discuss matters of this sort. So as subjects of study, there is certainly a continuum and there's a reliance on each other.
00:12:13
Speaker
But that's it's probably also true about the objects of study. Yes. Another thing I've seen you point out about the way that maths is represented, which is perhaps problematic, is that it is described as this very detached thing, almost underworldly, I think you've noted sometimes. um Perhaps partly philosophers are to blame for that. um But one of your roles is the director for the Centre for Mathematics for Humanity. literally um And so it seems clear that you're you're trying to ensure that mathematics is more connected to broader human concerns. yeah Perhaps you can speak a little bit about the way that you think maths is represented or about the the centre's work.
00:13:06
Speaker
Right. umm Well, of course, as a mathematician, and part of me wants to believe that all of mathematics benefits humanity. yes ah ah But ah it's it's still possible to talk about some benefits that are direct and some benefits that are much more circuitous. and So we do believe that there's a kind of grand and network of knowledge in mathematics, which is coherently connected to each other. And some of it
00:13:38
Speaker
directly touches on matters of human concern.
Mathematics in Global Challenges
00:13:41
Speaker
and So our Mathematics for Humanity project ah tries to support these kinds of things rather directly again by providing funding to mathematicians who work on problems like climate change, problems like sustainable energy,
00:14:00
Speaker
sustainable economics, healthcare. care So we've provided funding to mathematicians who work on these things and especially money for funding and other kinds of resources for people to gather here to have meetings on this subject to share ideas and try to bring up about progress on these very important challenges that face humanity. So that's largely what Mathematics for Humanity is about. There's also um part that has to do with sharing mathematical resources with the developing world, which is of course very, very important in in ah in the current situation. i mean I was just discussing something this just now with a mathematician is at our workshop this week. that it's probably It doesn't seem to be very well known that according to various estimates, for example but from the World Economic Forum, um within probably 10 years or so,
00:14:59
Speaker
more than 50% of the young people in the world are going to be living in Africa. So to think that we it's not our concern is a grand, grand illusion. right if you're now you You might also be one of these people who think that Africans can be confined to Africa. This is a ridiculous idea. I think everybody knows. Most people acknowledge this. But also,
00:15:23
Speaker
ah building up educational infrastructure in different parts of Africa is humanity's concern. So that's another part of our Mathematics for Humanity program. For Africa or other parts of the world where somehow we need more collaboration with parts of the world where we have more mathematical resources. So we are supporting meetings and training sessions and things of this sort in many parts of the world through the Mathematics for Humanity program. So this is a kind of a brief view of what we are doing, but as I said,
00:15:57
Speaker
It's probably trying to call attention to those parts of mathematics that touch on human welfare in a much more direct way than, say, number theory or particle physics. Right, right, yeah. I mean, I think it's fair to say that some branches of or some disciplines have really co-opted mathematics, physics we mentioned, but obviously economics as well. But perhaps then there are other places, as you mentioned, where there ought to be more mathematics, or mathematics can be a very valuable tool.
00:16:36
Speaker
um Yeah. i Is there any field in particular? I mean, I feel that climate science is right becoming or is pretty mathematical already. But are there certain places where you feel this this needs more?
00:16:52
Speaker
um Yeah. I'm not well positioned to really judge where the priorities are at the moment, but I can tell you there are lot of mathematicians already working on a whole range of things that are of the benefit human health. Among the projects that we supported was ah so somebody at the University of Edinburgh who is developing efficient batteries, for example, which is a very important problem. um We've supported a project called Rewilding Mathematics.
00:17:21
Speaker
They where you try to bring together people in mathematics and ecology to figure out the optimal ways of rewilding parts of the country where you know there's been a lot of environmental degradation. So things of these ah all of these are important. so i'm I'm loath to say that this is more important than another. right But there's lots of it. a lot of it is already going on.
00:17:43
Speaker
But I feel like the the announcement of our project, especially with a slightly pretentious name like Mathematics for Humanity, did create some energy. So for one thing, it it not it provided not just funding but also attention to people who are working on these vital problems.
00:18:03
Speaker
and also gave a certain sense of synergy and coherence to the different kinds of people who are doing such things. So um even without me saying that we we are going to support X, Y, and z as high priority, I think that can be decided by the community. And our job here is to respond to the community needs.
00:18:26
Speaker
Right. Yeah. Yeah. I think, as you say, just raising the point for making it more salient that mathematics should be serving human concerns ah is is a valuable endeavor in itself.
00:18:42
Speaker
I think the other piece that you mentioned about education is is interesting too. I'm thinking there's a quote from one of you know, Harari's book. I think something like the algorithm is the most important concept in the world right now.
Mathematical Literacy in the Algorithmic Age
00:19:02
Speaker
And it strikes me that ah mathematics is, in a sense, becoming more and more important within society in a way that perhaps physics is not. Our physical world is not really changing. OK, we have faster cars, but we're not living at relativistic speeds.
00:19:20
Speaker
But to a greater and greater extent, our lives are being intermediated by algorithms. um and i I wonder, I don't know, but I wonder is it important for that reason that people have greater mathematical literacy or do the details of algorithms perhaps not not matter?
00:19:44
Speaker
It's very difficult question and it's an interesting question. It is a question I think about quite a bit. right um It's sort of like a driving a car. right Many people know how to drive a car.
00:19:58
Speaker
Very few people know how to fix a car yeah when it breaks down. And it's probably not that important for most people to know how to fix a car. um On the other hand, if you do know how to fix it, it's it's a great sense of empowerment and you get compared to just feeling helpless when your car breaks down. So that's a kind of analogy one can draw. But I don't think that's quite the right analogy, exactly because of what you said. um Mathematics does tend to be more abstract than physics. That's a sense that many people have. and The abstraction is for a reason, of course, and a people mean this is well understood by many people. It's because sometimes when you abstract certain structures out of a specific problem, it turns out that what you study or the theory you develop, is much more broad more broadly applicable than the problem to the problem that you started out with. So that's where abstraction comes about. But as a result, it does seem to, even though i also pointed to the close relation between mathematics and physics, it does eventually seem to become more broadly applicable than ideas of physics, especially with the information revolution. So much of information is concerned with mathematical
00:21:10
Speaker
tools and concepts, and as as as we know, there's advanced number theorists in a number theory contained in each of our ah smartphones. right So all of these things are um astounding somewhat, and it happened in my lifetime, of course, this incorporation with high-depth mathematics into technology.
00:21:29
Speaker
and um so as to today Returning to the education question, I'm tending to digress, but um Of course, anybody can decide that they only need X amount of mathematics, right? So you learn the mathematics in primary school, secondary school, and so on, and more people wouldn't go on to study it at at two university. So you do have to decide how much you need, like in the case of a car, right? How much do I really need to know about a car in order for it to help my life, right?
00:22:02
Speaker
On the other hand, you also, real really strictly speaking, don't know how to, you don't have to know how to read. forget by in life. You can probably figure out various survival strategies to know how to read. On and the other hand, in in our modern world right now, if you didn't know how to read, your life would be very, very restricted by this inability. So that's the decision one needs to make.
00:22:25
Speaker
So...
00:22:28
Speaker
Sorry about that. Periodic, higher alarm test. I guess 11.30 on the dot. That's right, yeah. Yeah, so try to to try to answer your question. um
00:22:41
Speaker
it's I can't really say for any individual how much much mathematics state that they will need, right? But we do seem to be living in an environment, the more you know, the more empowered you become, the more options are open to you. And to some extent everybody knows this, almost any kind of workplace you go to these days, if you know math,
00:23:04
Speaker
It does seem to give you an advantage in some way. in Anything in the technology direction, for example, or information direction. or Even if you're a journalist, because of all the science news that's coming out, if you have some understanding of math, it seems to help ah you broaden your options, the the kind of things you can do. So ah the the more math you learn, the more empowerment it provides. I think this is a reasonable statement.
00:23:31
Speaker
but how much a given individual needs they have to decide. So as somebody working in an educational institution, i think the best we can do is as accurately as we can, give people equip people with an understanding so they can make good decisions.
00:23:52
Speaker
and I don't know if this is an answer to the question you were asking. No, I think that's a good answer. um I think that's very clear. I wonder also whether the arguments that we'll need to give for studying mathematics or or any subject going have to evolve.
00:24:10
Speaker
we We may be, again it's very unsaid, but we may be on the doorstep of a future where the economic value of studying something is is is very marginal. So it may come down more to the things you alluded to in terms of how empowered you are, you know what is the how does learning this help you flourish?
00:24:33
Speaker
And I wonder if in fact maybe maths may be in a stronger position in such a future or a weaker one. you know I think there are certainly some subjects which people just won't make much sense to study and the arguments will be around passion and human flourish. and yeah i'm I'm curious, do you think mathematics will could become more prominent if if we do move into a kind of um future of abundance Yeah, no, it's a first very hard to predict. i um
00:25:06
Speaker
Compared to many people, by the way, I'm i'm quite a fan of the new technologies that they are coming about. and It's very exciting. So, for example, I learn about lots of things using Chachi PT these days, especially because I work at the interface of physics quite a bit. And physics language is sometimes hard to understand from mathematicians. Sometimes I ask Chachi PT, what is this person talking about? It's often very helpful. so I'm a great fan of using whatever you can to learn new things, whatever is available. um Whether mathematics will become more important or redundant as a result of this is very hard to predict. But one thing that did happen in my lifetime,
00:25:51
Speaker
I can share, which is ah when calculators were becoming very widely distributed, right? Similar questions came up at that time, not exactly the same questions, right? But people did ask, do we really need to learn, say, mathematics in school when when everything can be done with a calculator?
00:26:10
Speaker
And as far as far as I can see, so so this discussion was happening say already in the 1980s, about 40 years ago. And there was a lot of concern.
00:26:22
Speaker
Now what has happened in the meanwhile is calculators or more generally computational tools have been incorporated to mathematics education. Nowadays you get the computational tools make it so much more interesting to teach different kinds of mathematics. you can see it on the internet, the various YouTube instructional videos and so forth. People use technology to enhance the learning experience incredibly. right And using some kind of scientific computation in mathematical research is also very common. not just in education. And during the the last 40 years since these probably these issues were first discussed, the need for mathematics in the meanwhile has become more intense, not less.
00:27:09
Speaker
So I don't know if that's considerable that helps us predict the future or not. But that is something that I've experienced in my lifetime. Yeah, yeah, it's interesting. I think we can probably see calculators have been some kind of cognitive prosthetic that has allowed us to I mean, possibly it means that less time needs to be spent on, you know, how do you add and zirk clearly and and you can actually get to the more abstract areas or explore more mathematics in the same amount of time.
00:27:39
Speaker
A large amount of the, by the way, within what people sometimes think of as pure mathematics, a lot of the interesting developments is computation with very abstract
Computation in Pure Mathematics
00:27:48
Speaker
objects. If you have abstract input and abstract output, how do you compute such things? that' That's been an enormous industry that my research itself has benefited from, for example. And I think what you're alluding to is this phenomenon that when a new tool shows up,
00:28:07
Speaker
people have a tendency to try to do not much, much more than they used to once you have this tool that you at your disposal, right? So from that point of view, maybe mathematics will become more intense because simply mathematicians will be trying to do much more than before.
00:28:23
Speaker
yeah The other thing I'm reminded of, i don't know if you've read the story, that <unk>s i think it's called That Feeling of Power by Isaac Asimov. and it's about this He must have written it around the time when calculators were... I don't think I've read it. It's a wonderful story. It's just a short one of his short stories. um and but it's when yeah yeah I think he wrote it when calculators were becoming more common and it projects kind of a far off future where no one knows how to add or, you know, do algebra themselves, essentially, or arithmetic themselves. um
00:28:58
Speaker
But there's this one guy who kind of reverse engineers it, he figures it out. And it's like I can't remember why, but he ends up being incredibly necessary for the survival of their civilization. But the the I think the title of the book or the story is very interesting, just that it's called That Feeling of Power. And it you know it says, well, even if, you know, suggests, even if we don't need this,
00:29:23
Speaker
um although i think the twist is they end up needing it. It's just that sense that I'm able to understand what's going on here and I can prove it because I can i can do the calculations. Sure.
00:29:37
Speaker
Yes, so and the need is itself a very complicated thing, right? what What do we mean by we need X? um So, well i can again, maybe just to mention some of my experience. I told you a moment ago that I learned things using Chart-GPT, for example, right?
00:29:56
Speaker
But I do feel like if I use it too much, it feels a little bit uneasy. So then, but after after getting general guidance, I do need to go back and do my own work to try to figure out, give some more substance to what the what the machine told me.
00:30:10
Speaker
So that might be related to the kind of experience you're describing in in thats that, that Azimov described in the story. I think that's right. I don't think there's any,
00:30:22
Speaker
there's no sort of matrix scenario where you can just wire knowledge into your brain. I think it is just intrinsically hard learning something new and it's sort of,
00:30:34
Speaker
to use that word again, perhaps needs to be hard. Yeah, well, that's related to this kind of software hardware problem, right? That even when when you think that you're going to just plug knowledge into your brain, that's a kind of pure information viewpoint on what knowledge constitte knowledge what constitutes knowledge, right? Whereas um in any of our human experience of it, there seems to be a very important hardware component. that Somehow the process of acquiring it in a certain way seems to be part of the knowledge we have.
00:31:11
Speaker
That can't be just reduced to the information that was ah injected. Yes, yeah. i'm also
00:31:21
Speaker
again the word need is is probably in my mind, but I should mention that i mean many people will will will know this, but others may not realise that you're a best-selling author in career and I think your breakthrough book was the moment that you need mathematics. yeah And as I understand it, it not only was you know a huge best seller, but it almost introduced popular at mathematics as a sort of subject in the book scene and in in
Mathematics in Korean Culture
00:31:48
Speaker
career. So, um yeah, I wonder if you could just say a little bit about that journey. What made you want to popularise mathematics? and And maybe you can give like a very brief um a description of that book. What was the the hook, I guess?
00:32:03
Speaker
Right. It was kind of an accident, actually, the book itself. Communicating mathematics is something I've always liked to do, as long as I can remember anyway. And then I started doing it rather um intensively around 15 years ago, actually. It's not that long that where I started seriously engaging with the public through public lectures, appearance on various YouTube channels or TV programs and so on. But then at some point a publisher contacted me with this idea for the book.
00:32:39
Speaker
And what they suggested is that I should give to their editors a series of lectures. And then the editors would ask questions, I would respond to that question, and we would discuss it, and let's turn that discussion into a book. That was their suggestion. And it worked quite well. so um the The editors put in a lot of effort, of course, to clarify, what they to to make clear for themselves and therefore for the reader.
00:33:07
Speaker
what I was saying. And so I owe a lot to them. But somehow, this book seems to have struck a chord. And this is tooting my own horn, but you I have been told what you said a moment ago, that somehow it started a boom for popular mathematics books in Korea. And it did it was within the, I think at its highest rating, it it reached the top 10 in the general books bestseller list for a period. so But exactly how it came about um and what what made it into such a popular publication, I really don't know. it was ah These things seem to just happen on their own. But the nature of the book, by the way, is something i don't think it would be very popular in the English language. that that it I did realize after writing this book and after various translations were attempted and so forth that the popularization of mathematics is it does also carry a lot of cultural baggage. You do tend to rely on things that
00:34:14
Speaker
a specific readership you have in mind is likely to respond to and so forth. so I have this on English version which in fact right now I'm discussing with a publisher. with it we should be we are done put it out right now. Well, actually I have to decide at the moment because it will involve quite a bit of work to make it relatable to the English reading public, I think. so But part of the reason is exactly this. It does take work. I realized that I need to tailor it somewhat more to the readership here or in America in order to make it into a successful book. But it was an interesting experience realizing these things. How much I had written it for the Korean readership.
00:34:57
Speaker
it's I mean, again, like as we said, math math math is obviously or so often portrayed as very detached.
Cultural Influence on Mathematical Reception
00:35:05
Speaker
You would think, well, you can just package this up and it will sell to everyone. It will make sense. so I guess it's is it is the difficulty the way that you connect it to.
00:35:16
Speaker
things that people know about and and understand. Things that people can relate to. That's one way of putting it. and so I think there were many such things, but two ingredients that that stood out as I was considering this challenge and problem was um Firstly, well yeah it's true that the Korean readership has ah has more tolerance for equations.
00:35:39
Speaker
So you probably know this ah as Stephen Hawking quote where he says that with his famous bestseller, each equation was supposed to cut the sails by...
00:35:52
Speaker
half or tenfold or whatever, right? So his publisher warned him about that and so eventually he only put in E equals mc squared or something. so um The Korean public does have more tolerance for equations, for mathematical formulas appearing inside books. So that was one sort of obvious thing.
00:36:15
Speaker
But another aspect which might have been harder to predict is that they do have more... they I found that they have more tolerance for philosophical digressions.
00:36:26
Speaker
So when you sort of try to get at philosophical motivations for thinking about some aspect of mathematics, Korean readers and audiences during lectures seem rather like this.
00:36:40
Speaker
what i found yeah giving sorry What I found in the UK is that during lectures when you are face to face with people, people like philosophical insights or or motivations.
00:36:53
Speaker
But when it's written down, it doesn't go over so well. That's my impression here. so somehow when you Maybe part of the reason is that people find mathematics so painful anyways. that Going into the philosophy of it makes it even heavier, to experience. So I don't know what it is, but these were the kind of standout features that seemed to be somewhat culture specific when I considered this problem of translation. Yeah, that's interesting.
00:37:20
Speaker
I wonder about that. I've heard that equation quote before. couldn't remember it myself precisely. But I just wonder how well tested that is. No, no. I know. One needs to challenge you first. But the publishers are wary. That's basically very wary. Yeah, exactly. That's the gatekeeper effect.
00:37:37
Speaker
and It does surprise me as well that you in physics, I think, has become very popular.
Popularity of Math vs Physics in Public Discourse
00:37:45
Speaker
and yeah so there are popular maths podcasts and there are popular maths writers in in English, but I just don't think they command the same sort of known share of audiences as popular physics. And yet the popular physics does touch a lot on philosophical issues. and In fact, I think that's why people buy popular physics books. That's a very good point. Yeah, and I've i exactly thought about this question as well.
00:38:09
Speaker
But so so, you know, reasons again for anything are very hard to understand, but that's why i um I'm somewhat leaning towards the explanation I gave, the second explanation in some sense that when it comes to mathematics, people would like a light touch because it's regarded as such a painful subject anyway. whereas physics that of course Of course, if you get to the inside of physics, as you know, it just involves enormous amount of mathematics, right?
00:38:44
Speaker
But that's not the conception people have from the outside. In fact, many people are highly surprised very very surprised when you show people how much mathematics there is in the actual research papers. they They don't quite realize that that's the way research in physics is done.
00:39:00
Speaker
So, somehow, physicists the public image of physics is very different, although on the inside, maybe it's not so different from mathematics. yeah yeah This might be a good moment to bring up your your own work. We've mentioned number of theory a couple of times just as an example of something that's completely disconnected perhaps the most disconnected part of mathematics from from from the world.
Number Theory's Real-World Applications
00:39:22
Speaker
um And yet, you know, Gauss famously said, maths is the queen of the sciences and and number theory is the queen of maths. um And you are a number theorist. And yet your work, as I understand it, does...
00:39:35
Speaker
perhaps unusually draw on physics in a sense, whereas often we see the reverse, we see physics drawing on maths. you You've sort of put your head up off the parapet and said, oh actually, math mathematicians can take something from from physics at some some level.
00:39:54
Speaker
Yeah, perhaps you could tell us a little bit about ah what your work is and and how it's drawn on physics. Well, maybe ah let me comment on the first part of your statement briefly first. At the moment, number theory is probably the most applicable part of mathematics.
00:40:11
Speaker
it's hard to It's hard to quantify such things, so I'm just making an intuitive statement here. But this, this again, is something that happened in my lifetime. And it's mostly because of two things. a Public key cryptography, that's one thing. And the other is information theory, mostly. coding theory.
00:40:29
Speaker
yeah so In order to make efficient codes that can correct correct errors automatically, you use quite a lot of number theory. And the cryptography part is also important. where you You can exchange information securely even with public publicly known codes. This is kind of the tricky part of public key cryptography. And as I understand it, each of our smartphones contain something called an elliptic curve yeah code, which is a very very number theoretic object.
00:41:04
Speaker
and you do it or do all of these things. these you You work with all of these things in involved you over something called a finite number system or finite field. That's what it's called, a number system with only finally many numbers. in it So these are all very highly highly abstract things, but from a kind of technological viewpoint, it's probably among the most applicable areas of mathematics at the moment. It's quite surprising. So when I was working in the US,
00:41:31
Speaker
I used to teach a postgraduate course on algebra at regular intervals. And i was i was amazed that over the years there were more and more engineers in my course.
00:41:43
Speaker
And basically they wanted to learn about these kind of structures. Yeah, I think it's i mean, it's I guess it's also not something that's just been left alone because they're still trying to prepare for a quantum future where they factorize things much quicker. Exactly. And you probably heard about this, but the early proposals for post quantum crypto systems also involve elliptic curves. Yeah. Yeah. And yeah, I was surprised that essentially we can use quite you know we can rely on the same kind of concepts. We just need to go a bit a bit further. Although there's there's other ah approaches as well. that' That's right. I mean, there are number of different approaches, but I was quite fascinated at the time that you took it again up to a slightly higher level of abstraction. consider not just single elliptic curves, but something called
00:42:31
Speaker
the isogeny graph of elliptic curves, a whole collection of such things that they contributed. It was hoped that they would provide a crypto system that was secure against quantum attack. So, number theory is very much in the thick of things right now when it comes to how to use mathematics to create new tools or make the world secure or things like that. So that's one thing.
00:42:55
Speaker
And it might not be unrelated to the second part of your question, the relation with physics. um So firstly, my in my view, good this ah interaction has been two ways for quite a long time. It's not it's not nothing new. that It's just that the degree of consciousness of how much traffic is going one way or the other, it might be the only thing that really changes as far as people's practice is concerned. So much of mathematics is developed in relation to physics. um
00:43:33
Speaker
This is a bit of a digression, but it's ah what as' a digression I find a bit interesting. so me and i hope i think it does provide some illumination on this question.
00:43:44
Speaker
um so People often talk about this unreasonable effectiveness of mathematics in the natural sciences, right? That's this famous paper of Wigner.
00:43:57
Speaker
and There's also a famous quote by a mathematician named Gelfand on the unreasonable ineffectiveness of mathematics in biology. that So that's been quoted often as well and that as a kind of strange phenomenon. Whenever people try to use mathematics in biology, it doesn't go very far. Now, of course, this is an exaggeration because it all depends on what you mean by mathematics and what you mean by biology. So, for example, everybody uses data science and information theory and statistics in biology. And this is actually very advanced mathematics by the standards of, say, even 100 years ago. so If you regard the that, but of course it's so well understood that it's regarded as kind of elementary mathematics right now, but within the scheme of history, it's very advanced mathematics. So from that point of view, already you're using a lot of advanced mathematics for biology.
00:44:52
Speaker
But what else? What Gelfand meant is that you should take some of the current tools of mathematics, like algebraic geometry, topology, things of that sort. They seem to have surprising applications to say string theory or quantum field theory, things like that. But they you can't get this kind of very structurally sophisticated, what's now regarded as structurally sophisticated in mathematics to feed into biology.
00:45:18
Speaker
Yeah, I had, um I don't know if you know his work, but Geoffrey West, who was a physicist, and he was actually a director at the Santa Fe Institute of Complexity for many years. But he he sort of later in his career, he turned to biology and looked at scaling laws and managed apply, know, lots of mathematics around fractals and anthropology to figure out why it is that um you know our blood systems and trees all kind of branch in the same way and the implications that that has on metabolism so it's really interesting and he sort of managed to derive particular scaling laws for example that as an animal doubles in size it's
00:46:01
Speaker
metabolic input only needs to increase by 75%. The same is true of trees and the same is true of pretty much any the animal which is more, um sorry, as you go from an organizer an organism one size to another that and that that happens.
00:46:15
Speaker
um But yeah, it's it's sort of, his work is kind of the exception. um um I don't think it's quite an exception. So let me give you a slightly different perspective on on that.
00:46:29
Speaker
which is that I find it very, very interesting, those kind of things. and That's not the only thing. Of course, there are people who find scaling laws in economics, organization of society, all those things that they do at Santa Fe are concerned with such things, right?
00:46:41
Speaker
But to some extent, Gelfand was really pointing to the fact that for most biologists, maybe even 99% of biologists, this is irrelevant. yeah I think that's a fair assessment. that that People may say, oh, that's kind of interesting, but whether it really does something for their research, of it's ah it's a very, very small portion of the population who would agree to such a thing.
00:47:08
Speaker
Whereas in physics, Almost all people working in say quantum field theory, which everybody regards as an important area of physics, think they should know more about topology, geometry, algebra, and so on. So it is a striking difference, even if you take that kind of research into account, the way it's incorporated into the community consciousness.
Precision in Mathematics and Science
00:47:31
Speaker
Yeah, yeah. i I do sometimes wonder with Figner's quote is if actually it shouldn't be so surprising that mathematics is so effective because it at least
00:47:45
Speaker
you know, physics presumes a fairly well-ordered world, one where you can write down some laws which always hold and describe things with precision. And I almost can't imagine any kind of physics, anything that conforms to that very loose description, that in a sense would not be mathematical, in that maths would not be your best tool for describing it. And so...
00:48:14
Speaker
Yeah, to me it perhaps doesn't seem so surprising that if you're studying something that's ordered, just by focusing on the ways that things can be ordered and the structure of things in the abstract, you can find concepts which actually do occur in in the real world.
00:48:32
Speaker
Right. so um as As I mentioned earlier, and as you know, it it largely does have to do with what we mean by mathematics and among other things. So that's why I started out saying that even in biology, of course, there's a lot of mathematics that's useful, right?
00:48:48
Speaker
And that's related to another quote I put on my webpage. You remember that there's this quote from David Gale and Lloyd Shapley, the economists, right? And their view was that any kind of thinking, once you make it precise enough,
00:49:03
Speaker
becomes mathematical. That's kind of their characterization of what mathematical methodology is like. And I tend to sympathize with that view as well. So also from that point view, then of course physics is going to be mathematical. Of course biology is also going to be mathematical. The more people try to make it precise, right?
00:49:20
Speaker
So, when Wigner was saying things like that, I think he was thinking, actually I think he was, okay, it's been a long time since I read that essay, essay so I'm a little bit reluctant to talk to try to guess at his intention. But my suspicion is he was thinking about specific kinds of mathematics. yeah So, for example, group theory.
00:49:41
Speaker
Group theory was an enormous thing in in quantum mechanics during Wigner's lifetime, to which there was a lot of resistance, as you recall, among physicists. right People spoke about the disease of group theory and didn't like to admit that this would be relevant for physics.
00:49:58
Speaker
But somehow it became a very important thing, not just in quantum mechanics, also in in special relativity. that People soon realized that the study of the Lorentz group was essential to understanding how relativity works, and especially essential to understanding how it combines with quantum mechanics to give you quantum field theory. So group theory started making its way into physics in a very, very serious way in the course of Wigner's life, in fact, and I have a feeling he was thinking about this kind of mathematics. it It doesn't seem to be the kind of mathematics that just comes out of precise thinking in the way, say, data science or or statistics or probability theory even might come out of trying to make your ideas precise, but still, somehow it's enormously powerful. so My suspicion is these are the ideas that are going through his mind.
00:50:48
Speaker
hat yeah So it's not as though, so in other words, if you talk about the ineffectiveness or effectiveness of mathematics, mathematics involves so many different things, it's hard to know exactly what we mean.
00:51:00
Speaker
yes but So we will probably do need to focus on specific conceptual tools. Like you might say, why is it why is group theory so so useful? Why is ah group representations or why is functional analysis useful? there and Then the questions become more precise and more of the mystery that he was struggling with probably does emerge when you ask these more precise questions.
00:51:27
Speaker
Yes, that makes sense. yeah yeah I don't think I answered the original question. No, yeah we we went on a really interesting tangent. Yeah, I think my original question was, yeah, tell us about your own work and how... Yeah, well, you see, I don't like talking about my own work.
00:51:47
Speaker
It doesn't seem so interesting. so That's fine, we can we can refer people to you. But maybe I will tell you one thing in relation to quantum mechanics, right? Because that is what I'm trying to use in a lot of the number theory that I'm doing.
00:52:02
Speaker
um And it might be of a number number of reasons. Again, it could just be that the ideas of quantum mechanics are also just basic things. It ah it already arises in the context of quantum mechanics. But it could be that the structures are just but as basic as addition and multiplication. So you expect to see the similar kinds of ideas and concepts arising in other contexts. So that's one possibility.
00:52:27
Speaker
But my more speculative hope or pipe dream, you might say, is that number theory tends to, especially what I do, something called arithmetic geometry, that's the study of geometry over small number systems like the integers or integers with a few more numbers thrown in or even finite number systems, right? This is quite different from what the number systems people usually use to describe geometry, which is like real numbers or complex numbers.
00:52:59
Speaker
with our Arithmetic geometry is geometry over a small number system. ah But the number of systems that you study in arithmetic ch that you use in arithmetic geometry are number of systems that have an intrinsic discreteness.
00:53:12
Speaker
so So in particular, the geometric geometric objects of study, the functions defined on these geometric objects are also discrete, intrinsically. So I feel like this is something that should be useful for quantum mechanics. So everybody knows that there should be some, I mean, in this idea of, say,
00:53:32
Speaker
The hope to get some some kind of a quantum gravity, you need to describe ah you need to combine this geometric nature of gravity with the discrete nature of quantum systems in some way.
00:53:43
Speaker
And you can't imagine a coherent geometry that combines with discreteness. but a version of it already exists. how Whether it's relevant to physics or not, of course, as I said, that's just a pipe dream.
00:53:55
Speaker
But a version of a coherent geometry using discrete number system already exists. This is arithmetic geometry. That's the research that I started from. So from that point of view, maybe connections are also natural.
00:54:07
Speaker
Yeah. Yes, that makes sense. I think, i mean, as you say, if if if we have... one way of describing a one system for describing discrete systems then it might just apply in all sorts of places. I'm reminded of there's this quote from Poincaré which is you know maths is the art of giving the same name to different things. I think physics is very similar like they they're both say looking for that same kind of unity where you might you know
00:54:39
Speaker
In physics, you can call it a gravitational field or you can say that actually you're just accelerating in space
Significance of Fields in Physics and Mathematics
00:54:44
Speaker
right there. Sure. um Well, the idea of a field, by the way, is a very interesting example. Of course, I think it arose in physics. It's hard to say about any of these things. But nonetheless, of course, that that turned out to be just a general purpose tool.
00:55:01
Speaker
Yeah. I do have to say, I'm not sure if the jury is out on whether the world is... discrete or continuous in that you know that quantumman you know quantum field theory is is a theory on on on fields and very continuous things but then you you get this quantizations of the the field so yeah maybe there's a kind of discreteness that um I know it emerges from the continuum as it were. but Or the other way around. the the
00:55:33
Speaker
Starting from the classical fields could be misleading. That's a viewpoint that many people have as well. In my view, that's a more consistent. I don't know what the reality is, but that is a more consistent view, I think, than starting from the continuous and moving to the discrete. Yeah, I think many people have speculated on this. Leibniz said nature does not make leaps. I don't if you know that one. You probably know the David Mumford quote where he said minds are discrete but nature is continuous. I don't know. No one really knows. I mean, it could be that minds are continuous and nature is discrete if you're a dualist, but that's probably unlikely.
00:56:08
Speaker
Well, all of these are difficult concepts, of course. We've talked a lot about how useful mathematics can be. I do wonder if though there are places where we over rely on mathematics. I'm thinking again of, um I'm thinking in particular of, I think it was Krugman who said, um what did he say? Something like, you know, we sometimes mistake truth, or sorry, we sometimes mistake beauty for truth when it's, you know, clad in impressive robes of mathematics. And then, you
00:56:45
Speaker
On the physics side, Sabine Hossenfelder has been arguing, physicists have been seduced by mathematics, you know to the extent which say they're relying on aesthetics too much.
00:56:59
Speaker
Do any of these resonate or do you think um we can never go too far with mathematics?
00:57:06
Speaker
um they well ah
00:57:10
Speaker
I fear that I'm not going to be answering your question again, but ah ah let let me pick on both of those quotes maybe simultaneously. right um ask Maybe I'll pick on Krugman first because that that somehow is easier to pick. I sympathize with what he's saying to some extent, except um To think that mathematics is about beauty is is a very narrow view.
00:57:42
Speaker
I guess this applies to Hossenfelder as well. The idea that you might be this yeah misled by the beauty of a model, I'm sure this can happen, right? But it's not the same thing as being misled by mathematics.
00:57:57
Speaker
Some beautiful ah model may be couched in mathematical terms, right? But for a mathematician as well, when you're trying to develop mathematics to understand the phenomenon, right, lot of it is often very, very messy and you have to live with that messiness. Now, sometimes after a few hundred years or sometimes a thousand years, the messy mathematics gets cleaned up to something more elegant and beautiful.
00:58:22
Speaker
But to think that mathematical models themselves are supposed to be beautiful, This is ah itself an extremely narrow view. So from that point of view, i somewhat reject the beginning of their of their of their sentiment. or Even while accepting aspect of it, the the idea that people can be misled by the beauty of a model, of course it can happen. But that's not being the same as being misled by mathematics. Returning to a subject that we've already touched on, if you take something like quantum field theory,
00:58:52
Speaker
Most of it is not beautiful at all at the moment.
Aesthetic vs Practical Models in Mathematics
00:58:55
Speaker
It is a major major major challenge how to make it properly consistent mathematically. and so
00:59:08
Speaker
I think you can take aesthetic sensibilities too far. I agree with this, but I don't think this is the same as taking mathematics too far. This isn't answering the question of whether you should what mathematics can take be taken too far or not, but that's a kind of an answer to your question. So maybe I will answer this question of whether you can take mathematics too far. So the question, in some there are again many different aspects to mathematics, right? Suppose we take Go back to this characterization of Gale and Shapley that mathematics is just precise thinking, right? Can we take precise thinking too far?
00:59:46
Speaker
That's another version of the question, right? Well, I think sometimes we attempted to make things more exact than we can. i mean, this is Aristotle's thing, like, you do not admit into your science more exactitude than than than it truly permits. um And I think certainly in economics, there's this perhaps desire to cut through all the noise and get to an equation. um But actually, you know, that's, we we don't have the
01:00:21
Speaker
I don't know, we don't have the quantity of data or we don't have the, or even just as humans, we don't have the predictability that you need in order to, you know, completely apply mathematics. There has to be an element of, I don't know, vagueness.
01:00:36
Speaker
um Yeah, the economics, of course, is a very complicated case in some way. um So I've had a lot of conversations with with economists about their mathematical models. So for example, economics has many famous theorems, like like even the existence of an Ash equilibrium, this Gail Shapley algorithm that I referred to a moment ago,
01:01:01
Speaker
um
01:01:03
Speaker
the Arrow's impossibility theorem and so forth. So I have asked economists, are these really useful? no and the The subject is economics is also changing very rapidly. of course they Many people are conscious of the idea that to get really useful economics you have to deal with much more messy collections of data. and This is being taken on board i think by the more serious economists right now. But even the older must very mathematical economics that you look at, the theorems I was referring to,
01:01:41
Speaker
I think their view is that they may not be directly useful, but they do give some insight on real phenomena. And that's why they like them. so um I forgot what your question was. Your questions aren't really so important. as If the answers are interesting, which they are, it doesn't doesn't really matter if they are. Yeah, so one economist I know relatively well is somebody named Ariel Rubinstein. And he likes to say economics is a story.
01:02:16
Speaker
and I think he means two things by this. and I've checked this with him and he he neither agrees nor denies, but my impression is that he agrees. When you say economics is a story, right on the one hand, he is acknowledging that you can't make precise predictions. It's not about making most of economics anyways, even though again that's changing, but most of classical economics you can't make precise predictions. It's not like ah experimental science or something.
01:02:45
Speaker
Even though even won experimental science is not like experimental science, but anyways, that's a different story. but anyways so So he said economics is a story in this in to express a certain degree of humility about what it's able to do.
01:03:02
Speaker
But I think he also means that it is like a story in the insight it gives. So when we think about great literature, and he said Chekhov is among his favorite favorite authors, when we think about great literature, again, it's not experimental science, but it gives us ah we can't avoid the feeling that it gives us a deep understanding about the way the word world is put together.
01:03:24
Speaker
That is what great literature does for us. And I think he's saying that economists can do this as well. in a different mode from literature, but still in the same sense of providing insight.
01:03:37
Speaker
So I tend to agree with him about economics in this way, and about a lot of usage of mathematics in different realms of human knowledge. That there are least these two different aspects, making precise predictions, making making things very very precise. The other is giving a global insight of some sort.
01:03:58
Speaker
so So maths is a story as well, a way of... So I gave a lecture called is What Kind of Story is Mathematics? I don't know if you've seen this, but I can send you a link. I have, so I've cheated. I knew your answer to that. Right, right.
01:04:15
Speaker
I think one of the really interesting points you made in that is that maths is a kind of continuous story. that's right it's and And it's actually very collective. It's a lot of people writing it together and and ah and reading it together. I hadn't realized how collective the reading of maths was, but you you you point out that. One thing I... the question that i was left with after watching that was,
01:04:39
Speaker
Is it a story that will ever end? In the moment, it just seems like it's you know one chapter after another. and dis get it but will do you think there will be a place where we I feel with physics, we assume that it is, you know, we'll we'll get to some final equation and that will be the end of what there is to be said. I may be wrong here, but I i feel like there's an assumption on many people's parts that once you get to the theory of everything, you're just going be doing, you know, figuring out how everything else derives from that.
01:05:17
Speaker
But is maths more open than that, do you think? or Or will we sort of get to a point where we've said everything novel that can be said? I don't see how it could end, let's put it that way. yeah that It seems to go on and on. By the way, even in the case of physics, people talk about a theory of everything as a kind of propaganda device. Yes. But I don't think anybody really believes. Yeah, i don't think they'll say, okay, well, we're out of the job now. I mean, yeah. If you ask them privately, nobody will express a very strong guilt belief that that it will end at some point. For example, and this is maybe a pedantic point, right?
01:05:57
Speaker
Suppose you are trying to understand the structure of an atom. So there are various rules for how the shells of an atom fill up, right? There is something called Hund's rule, there is something called the modeling rule and so forth, right?
01:06:10
Speaker
None of these are mathematically understood. none of them can be derived from Schrodinger's equation at the moment. So that's the level at which we don't have a precise understanding when you try to apply the fundamental theory of physics to actual phenomena. And you might have QCD, which is the theory quarks and so forth, right? But ah nobody uses QCD to understand nuclear physics. You have a separate model in order to do that, an approximate model, and that's what most of actual physics is like. So to hope that just because we understand something about so-called four fundamental forces, everything else will be understood, as it is is a grand illusion. Yeah, but I do wonder if there will then be a lot of filling in of holes and gaps and and so forth, but you can't envisage a point in which
01:07:02
Speaker
we know how it all hangs together up to some level. And then of course you can take to the level up beyond that and look at quantum chemistry and then sort of, you know, how does, how do cells form um and so forth. But it becomes, it kind of,
01:07:17
Speaker
starts to become other subjects, I guess. and um I think that, yeah, so I don't really know, of course, but my feeling is that in there is a kind of arbitrary division, meaning division between something you regarded as fundamental and some other things you regard as just gaps and holes.
01:07:35
Speaker
And I don't think ah the structure of our knowledge really divides up in that way. It's not at all. ah I mean, this is true even in mathematics, right? it Suddenly this came to mind. i hope it doesn't seem too irrelevant to you. But ah you you probably know any mathematics paper is full of mistakes.
01:07:57
Speaker
any Any mathematics paper in current journals, other than on the most elementary subjects, are full of errors. okay and now most of the Most mathematicians regard most of the errors as being kind of inessential. There are things that can be fixed easily.
01:08:17
Speaker
um But which things are inessential and which things are not is also also sort of an arbitrary choice that you make. and In order to justify your sense that something is essentially correct, there's nothing you can do but rely on a kind of community consensus. There are no objective criteria for doing this. right And um so I don't think there's any way to escape the idea that even in mathematics most of our knowledge is approximate.
01:08:48
Speaker
So some notion of of approximate truth, for example, is unavoidable when you think realistically about the way mathematicians practice and the way the subject evolves.
01:08:59
Speaker
But interestingly, in my conversation with philosophers, they are remarkably resistant to to this idea. They are much more attached to the idea that mathematical truths are about certainty than a practicing mathematician. So that's a kind of interesting thing. but So I think in in practice,
01:09:21
Speaker
it's hard to imagine any kind of completeness of understanding being achieved, even with existing mathematics, not just mathematics that's going to be developed, because we'll keep on recasting the same objects and improve our understanding and then they'll be connecting to some other things and so on.
01:09:40
Speaker
So this is one of the many senses in which I think the story is continuing. up You probably know this idea. This is kind of a very stupid example of such a thing. But you know this idea that the theorem of Pythagoras, for example, is wrong.
01:09:55
Speaker
No, I get it. This is ah a completely obvious thing, if you just look at it from a certain point of view. It's because according to relativity, the assumptions just don't hold, right?
01:10:07
Speaker
Yeah. if you if you If you created an actual triangle, the right angle triangle, right, and measured a B, and C, right, and the more precisely you measure it, A squared plus B squared will not be equal to C squared.
01:10:21
Speaker
Yeah, yeah, okay. Yeah. So from that point of view, it's wrong. For our, for the actual space that we live in. If you take the theorem to be one about actual triangles, right? Yeah, yeah. Even in an idealized sense that we can construct in our actual world, right? Yeah.
01:10:39
Speaker
On the other hand, if you take it to be just a implies B, is probably correct, ah although again, one never knows how examination of the assumptions will evolve over time.
01:10:52
Speaker
So ah any given mathematical result again has many different interpretations when you try try to start discussing the correctness or not. right And this is just, it just suddenly only came to mind as one aspect of the fluctuating nature of of the the achievement or knowledge that you get in mathematics and how it changes with time.
01:11:14
Speaker
Yeah. I think it's nice to imagine that maybe mathematics is a subject where you can make progress, and yet the progress will never be exhausted. It's also hard to know what exactly
The Infinite Progression of Mathematics
01:11:25
Speaker
what you mean by progress. Yeah. So somehow these are all things about which you make human judgment.
01:11:31
Speaker
Yeah. And there's not an objective criterion for it. Well, this has been such a wonderful discussion. No, no. Thank you so much for your question.
01:11:41
Speaker
Thank you.
01:11:56
Speaker
I'm going to go to the next video.