The Beauty and Unity of Mathematics

00:00:00 the question was enough mathematics has confronted I think every mathematician at one time or another and at those moments one can only feel that if I if I could feel if I were a biologist I would have an easy answer there's a single word life that would encompass protein

00:00:36 studies medical applications as well life it could be exchanged with mathematics it's very hard to see what unifying anchor it has on the other hand if you're a practicing mathematician and I can be contradicted which would also be interesting if people think otherwise

00:01:06 that there really is a unifying anchor to mathematics that is defies vocabulary this description but a one simply has a feeling what is it and there are some there's a certain sociology to the people who offer a responses to this

00:01:40 question for example the mathematician Galvan once said let me see if I can remember it that the thing that produces the unity of mathematics and therefore the notion that this one subject is the intense and this is what he says simplicity precision beauty and crazy

00:02:12 ideas and says that that identifies it with music well that's his his claim on there are other let's say sociological schools of thought about the unity of mathematics and one of them can be encompassed by language some people say well it's the language it's the well define admit

00:02:46 of our utter insists the there's some aspect of language but usually well-defined itness in fact the logical manner of the way in which our sentences succeed one after the other in what purports to be a demonstration of something anyway it occurred to me that it might be great to have a general jam session this discussion about what what

00:03:23 constitutes mathematics and I thought that if unity is one of the starter words not necessarily the finishing one beauty might be another and for that reason I called it unity and beauty but with beauty it's that opens up a new and

00:03:56 further topic what is beauty now there is someone who one simple question after another and there is someone who wrote the the blurb for this meeting mentioned and he put in the blurb I don't know who it was who did it for this meeting it was my comments but there is an

00:04:28 addition which I thought was really good probably better than my comments did you write it okay very good it's there's a mention of this 18th century philosopher philosopher commentator evil Eva he on play who quotes what I understand to be a lost treatise of st. Augustine who claims and

00:04:58 I think in none of his collected works that I find this claim that that beauty and unity are unified let us say the the the source and form of beauty has to be found in aspects of unity anyway so that's the reason for the subject and I this is supposed to be an unprepared thing and this is what I've just given a

00:05:29 semi prepared thing I'll never do that again just let let it go well my first thought when I saw the title yeah was that beauty and unity are have relate to mathematics in two very different ways so I think I do think mathematics is unified in some way that we can explore and but when you when we

00:06:00 call mathematics beautiful I think that's a that's in amphibole it's it's using a term from one domain and and just making an analogy I've always had a sort of pet peeve about people who said oh that mathematics is so beautiful because it seems to me different from the beauty of art for me so how is it different well for one thing it's not it's so this made me be my lack of imagination but I for

00:06:31 me mathematics is not sensuous you know it doesn't it it's it's ideas no their ideas or their or their the things I write down on the paper yeah but I don't have a maybe that's my problem I thought I'm Stein almost said that that when he had a really good idea that often when he was shaving he would have to watch out because we would cut himself the hairs would stand up there'd be a kind of a physical thrill so in the

00:07:03 sensuous its Houseman too with import poet you know that there is a kind of goofy side the physical side to it so you I said not saying and we Jung uses the word beauty about mathematics but but but it's a bit it's an analogy as opposed to unity which I find is a direct direct description of the of the

00:07:34 subject I don't know what is unified about it yet we do man to learn that from which is why I latched on to your comment about Augustine's attempt to connect the two your comments about mathematics how may be difficult to define mathematics one simple stroke that it reminds you what you said could also be said about beauty in a sense right so it's question head of how far

00:08:07 can you extend it how far can you extend mathematics under way how far can you extend it to concept of beauty on the other we could make a sociological experiment to see whether people actually just use the word beautiful for mathematics sorry I know that in academic English the community of literary scholarship at one point some

00:08:40 post Ruskin people decided never to use the word beauty when they talked about an object of literature and it's now there's now a revisiting of that that it's very difficult to talk about a domain in which people think it's beautiful without being able to talk about it isn't there also the the

00:09:13 problem that Proust talks about in his book that he argues that each true artist introduces not just beauty but a new kind of beauty and enables people to see in fact things as beautiful that they had not before regarded as beautiful but even on the contrary as ugly or devoid of any kind of anything like that so that it's not merely that there's a single kind of beauty but there might be a changing that the change of beauty or the opening up of a new domain as having a kind of beauty

00:09:45 that was never before recognized itself is important let me ask you a question have you seen something in mathematics that you thought was ugly oh sure well in fact there are in the and you know it students actually ser has something that he called the ugly Limor it was truly ugly for various reasons but the thing is one can't give to objective reasons

00:10:16 beauty if one has read the critique of judgment I mean it's not a rejected issue it's a it's a subjective console wanted to sort of elevate it to a universally subjective status let us say if you feel that it's beautiful that that really suggests that you also think all of humanity will feel that it's beautiful because you have some sort of

00:10:46 model of humanity in your in your mental faculties and you can test test this model does my model of humanity feel that this is beautiful not just me and I think that's the that the ugly lemma it's not it shouldn't be called the ugly lemma it should be said that the people who read that lemma feel in their universally subjective ways that it's ugly

00:11:17 which might start the question start the change the question - what is ugly in what its beauty and the story of the ugly duckling suggests that the ugliness itself could become dear and beautiful in a certain way which itself would be a new kind of beauty because it wouldn't be a meditation anymore on the ugliness of the duckling or a dilemma but how it was rejected how it had a difficult origin how it was hated by so many people and then it was embraced I want

00:11:57 to pick up from your suggestion never do a sociological experiment which we're not going to do and I'm not in a position to do it but I did since the rule was not to prepare of course I know I our parents where the marks are the marks of a mathematician is not to follow the rules also to follow the rules of one's choosing but the that's the no I prepared by it because I was under the impression that this this

00:12:28 connection of beauty and mathematics may be a preoccupation of Western in the Western tradition of mathematics and I looked into it and I inquired and I the conclusion although I'm certainly not I don't have an exhaustive knowledge of the literature's that in the Arabic tradition there was no association of beauty and mathematics or only or rather the the Association went in the other direction that is the same beauty was characterized in mathematical terms which also which is also seen in

00:13:00 Aristotle but some sense taken from that then in the Hindu tradition there was there are words that there were the closest word of that box space to the notion of beauty is Ananda which means joy the word beautiful never appears in the in the listener in the classic literature and then I asked one of my colleagues who was a Chinese mathematician whether when the Chinese mathematicians are speaking

00:13:33 to each other do they ever say this is beautiful and what word does he use well there are in fact they there are two words for beautiful as a matter of fact the English language is particularly poor in vocabulary for beauty it turns out the Arabic is as many different synonyms but there's the word May which means handsome or good-looking or good for an attractive face and then for a poof when they want to say that the proof is beautiful they use may meow and

00:14:03 Yao but she's mine which and now is a word that means clever so it's you would not refer to a mathematical proof as beautiful unless you also say clever what about what about the Grange and Greeks though where we're getting home yeah well you know the answer to that yeah well I but I thought maybe you had written about but it would have to be that what you think that I mean in that I think this goes to the question of the

00:14:33 unity of mathematics because for the Greeks mathematics was not a single thing in the quadrivium do you have a Resnick geometry music astronomy and they're sisters now how could you t not be an issue for them because music was there not speak of astronomy but the fact that they were not separate subjects of mathematics as such did not exist and it had two sisters and they were sisters but not

00:15:04 the same the same and not the same and so and at least for them it was not unified but there was something beautiful and having these two sisters there so that it seems to me if we're going to think about it this this is an immortal family so that it's still here these these goddesses are still present even though there may not be much talked about and they've had children which are the modern sciences and maybe modern mathematics itself but the family almost

00:15:35 always speaks it almost tells in this family lineage of mathematics I think is still present there is still a certain tension between arithmetic and geometry even to this to the present day because it's so deeply embedded in the fabric of number itself that all the attempts that we've made to unify the concept of number certainly for the Greeks would have been outrageous against a difference that they felt probably was even a beautiful difference and a

00:16:06 difference that was being respected not between arithmetic and geometry I mean for very deep and important reasons that so I I mean it seems to me that that family lineage is always is haunts us even today and so I just want to put it forward as something to talk about when we speak about mathematics I think the intent being must mathematics now also but it seems to me maybe it will be helpful even if this is all merely in the past which I don't believe but I

00:16:37 think what you say is really interesting I've never never really heard them but Euclid's elements of let's call the elements of geometry yes but book 6 - book 9 it's straight prime numbers and what sense is that geometry but it's all expressed geometrically it's expressed geometric as opposed to say diophantus who doesn't use the others yeah it wasn't it could just called stoichea elements I loved

00:17:08 sure that that it was of geometry and with their ability but I think that it was meant to be both the elements of both and I'm not sure that the word of geometry as if as if the word geometry were hanging over the whole I'm not sure that that goes back to the ancient texts in which this is the elements of the two sisters together in separate books for a very good reason and kept separate and to avoid the problem of the incommensurable the problem of irrational numbers which I

00:17:39 don't know whether that's beautiful or not and I don't know whether the response and this is another is the response to beauty us is it simply a sense a kind of an aesthetic glow or is it a kind of shock what about the joy of the Ananda that yes Michael refers to as Indian well I only have a few sentences and I they're under the chair but I think they won't

00:18:12 add anything to the discussion let's just kill it does it's clear just from the preparation that I wasn't supposed to do that there's a vast a vast domain of inquiry that a comparative of comparative study of mathematical aesthetics that no but that nobody has undertaken this goes back to this well you know I wanted to follow up the idea earlier that uh that the concept of beauty of you know in the post Ruskin phase was removed from a central position in aesthetic thinking and I and

00:18:44 I think that the sense at that time was if there was a platonic recent presupposition about the nature of category it's about essentially and that in fact to give him something oh and then a given subject could be defined by the essence that every member of human affairs and so that form of blatant ism was really kind of resident in sonnet aesthetic thinking until the fourth truck Ruskin phase when Beauty got removed and the reason it got removed is that there was a new awareness that there's no single

00:19:15 criterion for the adjudication of work of art the carries across all cases and of course if there were in essence then there would be such a criterion there isn't and so in fact in fact Vik and Stein opened his lectures on aesthetics with the idea that you know people think an aesthetic to the word central or the work you promote simple it is look at our actual aesthetic vocabulary which is vast and extensive pay attention to that and you learn much more about the concept of late then you will by trying to arrive at a platonic essence so so I think that's

00:19:45 part of what part of what sort of move Beauty out of the central point but then then that to connect that to mathematics that then gives us the question what is the vocabulary apart from beautiful or apart from the ugly but could there be for mathematics that's inept or that's clumsy or that's you know objectionable in various ways as subcategories of the ugly I don't I asked you I don't I don't know but I think that that's why beauty got removed and you're right there's a resurgence

00:20:15 beauty is back viki's but in a non-essential istic way maybe here music is of help because I mean the concept of dissonance I mean one thinks about the new kinds of beauty in the ever more modern forms of art as embracing greater more and more dissonant states and allowing them not merely allowing them but embracing them as having forms of the beautiful that previously were not allowed when those kinds of distances were were forbidden by certain rules and

00:20:47 it seems to me also in in mathematics it seems to me the proof of the that there's a rational number is a kind of dissonant proof in which there's um has shown to be a dissonance between the possibility of there being ratios and the attempt to find such a ratio that we described a diagonal of a swore a square the fact that it can't be done or that many mathematical proofs are reductio Zoar assertions that something cannot be

00:21:19 or that it's even better yet that it's non-existence cannot be which defines a kind of existence in mathematics each of them operates through a kind of dissonance or surprise which may be a kind of joy I mean that to go back to that term but it often involves assimilating a contradiction or a certain newness or something that has the shock of the new but which at a certain point is not rejected but embraced in some way

00:21:52 and I think that some of our ideas about beauty on the one hand and mathematics or the other are sort of static and it's interesting to go back to the idea of sensual in sensuality of mathematics if you can apply the word joy which has a little bit more of an explosive dynamic quality to it maybe it says something about what really may be going on in mathematics it's not a static thing it does evolve and grow and similarly beauty and our notions of beauty can you do grow and and that links nicely to

00:22:23 what is a fundamental question that is yeah is you know mathematics the kind of thing that is discovered we speak of Russell discovering Russell's paradox and new when was that 1900 or something like that and you know the set of all sets not members themselves if they are they learned if they aren't they are and so you kind of bump your head against the the wall if you will other of a paradox is that a source of beauty or is that a source of the doesn't work it gives Beauty that that is you know it's the pure thing that's full and complete the more beautiful thing or is the

00:22:55 imperfection in it that is after all a kind of mankind of memetic depiction of humanity is that the more beautiful thing you know so how does paradox work as a determinant or a distraction to beauty what would you think of the Russell conundrum that's kind of an agony it's a different dimension than Beauty it's more mental yeah I mean after all the 16th century Italian Alfa

00:23:26 Brea someone who make use of this the square root of -1 said this missing mental tortures because it was agony do it because it's going to solve your problem yeah so it could be it could it be in that direction that it's a question of joy from Michel Harris's ananda to discomfort well you know Russell but that's not a different dimension yet there's go tools proof which is pretty much dealing with the same sort of topic

00:23:59 right and isn't the people not think that's beautiful or because of the the wave to have dances you would have to learn yeah I would have to you couldn't imagine a person that was agonized and at a certain point might realize that their agony was actually a kind of super legit pleasure maybe the Joey is not simply not simply or special you know also be the sense when you watch a child taystee a new food for the first time usually the first expression is and then the expression may change to like at

00:24:29 first it's like well this is a new thing I haven't tasted this before and then suddenly it's like oh it's it's you or it's that and so it seems to me there's some kind of transformation so that the irrational could be some kind of horror or some kind of dark mystery that you would kill people over and at another point it might be something that would be embraced as being the most wonderful thing is it's really both it's like the Conti and mathematic goals its cover

00:25:01 that he calls it the mathematical sublime yeah which was a kind of a little three-act mini drama where you were confronted with something just larger than your intellect the infinite so to speak and that type form actu is you're you're in agony Act three is you come out of it saying oh my God my finite intellect can conceive the agony

00:25:35 of thinking about of this this larger thing exactly that's what he refers to the political sublime yeah exactly we kind of exemplify a contradiction in yourself that is you accont you you take into your mind something larger than your mind can accommodate and you realize you are accommodating that thing that reaches beyond you so it's it's a kind of step into a kind of transcendence there and and for him it for COD of course in the third critique he separated the sublime from the beautiful my question was always should that separation be as rigorous as as he suggested because it

00:26:06 seems to me that these kinds of experiences of the sublime which what you get in fact when you're a little kid you know and you learn numbers you try to come up with your sibling or something the highest number you know and you finally think of million alien and then you're you're smarter sibling says a million million plus one and then you think oh damn it you know well that moment right it's the moment where you actually realize to come there's you know you grasp the concept of infinity and and and at that at that moment you do kind of step over a platonic line from the world of the

00:26:37 sensory world to this higher higher world of relations and that that's a sublime experience it seems to me we can call that a beautiful form of beauty day I think bothered by feeling like maybe we're belong to some kind of high church you know some kind of yeah where these things we find sublime we find uplifting we find joyous I'm thinking while you were speaking I was thinking of my students my undergraduate

00:27:08 students for whom almost everything is an agony I guess it has to do with this subjective objective divine and I feel probably one of the things I believe most strongly is that mathematics is objective and that these feelings are objectively based but I keep coming back to these fears and I'm just fooling myself hmm like you know the guy at the

00:27:40 end of 1984 who believes that two plus two equals five and he's she's now happy with that result you know so that's I guess it's what how do we do the reality testing here to is mathematics the same experience for you as a mathematician and for your students maybe can you even talk about mathematics as being unified when so many different groups of people experience it different ways yeah what about engineer

00:28:11 a third way you know yeah I'm terrified before or the the sense of unity as a kind of struggle I mean we take unified as if it were a given thing but it seems to me that the from what I can know of the mathematical experience it's a struggle it's a struggle to unify things that seem to be while dissonant moving in different directions there are tensions between them in this musical sense and the attempt to grasp and put them together to form a unity but the

00:28:44 unity is always being searched for it's not a given thing from which things are simply unfolding it's experienced in some kind of dynamic way actually you know the 18th century Scottish philosopher Francis Hutcheson wrote a book in aesthetics and he he finally arrived after much struggle he arrived at the idea of uniformity amidst variety and he said that what is engaging what successful in the work of art is where you you hit this kind of point of equipoise between too much variety so you can't make sense of it or too much

00:29:15 too much unity so it becomes boring and kind of lifeless in a way and and I does that capture part of what is beautiful about mathematics if there is if there is if that's not just an analogy does that capture you know K mathematics can be tuned there there is a live tension between those two ways of viewing mathematic it's really live in the sense that there are people people I know you can sense who are making an effort to

00:29:46 hand over mathematics to machines is because they don't trust human beings with mathematics and they believe that the validation the only reliable validation can come from machine this goes back to Russell and and Sager the AI Russell who specifically referred to mathematics as a cold beauty which is quite the opposite is of sensual and that is for others for me particular

00:30:17 this would represent the death of math Magnus be superior in there didn't your textile it against the tooth that I have to say that the death wish is it is a feature of the mathematical practice because at a certain point you just want the thing to be proved you don't want to don't you hey you in some sense care whether it's you or or machine did it so what do you think about the two tenths the micro I'm very clearly in the in the

00:30:48 other cap but at the same time I understand most of the people who are actively promoting this have had a a crisis of one kind or another that is dismayed they found that there was the mistaken approved many years after they published their or very shortly before was said to be published or in the case of Tom Hales for example was unable to find anybody who proved Tom Hales and may famous for proving they kept their conjecture on the the packing of spheres and in three-dimensional space but there was nobody had to use a computer to

00:31:20 verify it and the code was so complicated that no human being who was able to to to vote forth and so deciding to he decided to develop the memory system automated system that would by reformulating it in a new language that only machines could read but could could could verify it I did but in a way that continues the same theme of reaching beyond I thought that she would that

00:31:50 that that the proof wants to reach beyond human capabilities and it then has to start to as you've written about it has to imagine the part that androids might play or Android mathematicians and computers that would carry out the steps and then the relationship between those androids and the human beings that would would or would not decide that they were going to accept what the androids told them about the proof proof being okay but yeah it seemed to me that that very that one of the things that you've introduced in your writings that has

00:32:21 helped me a lot to understand this is the the idea that to be a mathematician is to have a dream you know that at a certain level that the the a mathematician past a certain point is distinguished by there is a her dream and a dream seems to me a very interesting place to talk about the relationship of beauty and unity because it represents something which is consciously reaching beyond and yet has a very different form and is struggling

00:32:52 to encompass something which is the the shape of that mathematicians dream which may be different from mathematician to mathematician and may or may not involve Android participation in their dream why don't you tell about your essay about dreams in mathematics it's too it's too good you mean well right about the androids know about the circles disturb oh yes right yes right know there was a there's a story this is not tell the end

00:33:24 it's a sorority there's a this is it a book about mathematics and narratives way I recounted a story there was a mathematician named Robert Thomason published of paper and at the beginning and he published it jointly with somebody who was not only not a mathematician but who was deceased and in fact he committed suicide and his the participant II the contribution that years some time before thee the this work had been done and he'd been

00:33:56 thinking about this question for a long time and once he even day he woke up with a start he said because his friend had told him in a dream if you want to solve the problem you do stir you do this and we woke up he said well no that's wrong that's wrong I'll be I know that's wrong I know why it's wrong he said his friend was so insistent that we during the dream not afterwards that that he thought about it and he realized that this was the objection he had to

00:34:28 this this this this answer his friend gave was the only obstacle and once he over what once he accounted for that and he actually could finish the the proof and he and he the editors this of this of this volume agreed to include his his deceased friend as a co-author at the oh I think the only case it has but being wrong that was another form of being right exactly if someone says no no not an

00:35:00 absolute you know there's that wonderful essay or Freud in which he says if the patient says it's not your mother you know it's its mother there's no know at a certain level so or the word if something is really really wrong it's touching the truth or in some way or it's touching something that you absolutely have to look at if only to find out what the wrongness quote-unquote is all about I want to say

00:35:31 at least one word for unity yes I know that we're now in the completely justified and that is that what's shocking is when we work in mathematics is that there are analogies that unify different subjects I mean the most basic one is algebraic geometry or analytic geometry that it's say the algebra and

00:36:02 geometry have two totally different intuitions you you're in one mindset when you're doing algebraic equations you're in totally different mindset when you're trying to think of whether this triangle is isosceles or not and yet they can be joined by analogy they can be unified the intuitions themselves can be unified and what happens is you have a a synthesis you have some other thing

00:36:33 which you can't call algebra you can't call geometries so you use these combined phrases algebraic geometry now that's a new intuition which binds together two subjects and we in the midst of a life of different analogies in not only in pure mathematics but in also with the in

00:37:06 applied situations even more in a way and isn't that a unification I am sorry about that too good because our arithmetic and geometry for the ancient Greeks were separate because they had separate they separate domains and applied in several domains and they and the sciences were defined by the domains so they could not be the same and then when algebra was discovered by Elka

00:37:38 where is me the very first page he talked about how happy he was because he'd found something that could solve problems in arithmetic and geometry and then and on but then there was there was a trivial then the the afar avi a little bit later what decided to come up with a catalogue of the sciences and there were of five different kinds of Sciences one was mathematics but it couldn't figure out where to put algebra because algebra

00:38:09 was what they were this wasn't it supposed he was an Aristotelian but he couldn't he couldn't disagree but on the other hand algebra was there so he put it in with the science of tricks he called that he had a a a section in the in the chapter on on on mathematics for tricks that included mechanical devices in an algebra and and an engineer anything like that and in fact algebra started out as kind of a rough-and-tumble art of challenges

00:38:40 public challenges between or steal their answer I want to bring him so yes I disagreed on the beauty but entity and maybe one

00:39:13 way of viewing it is that there's a latent analogies among all yet but they they can all should be because that because in fact they do share a domain which is the domain of mathematics I mean I mean the Greeks Greeks ones where I might have limited you know in their vision that vision and but I mean that's it I struggle to

00:39:43 define you know what the donation mathematics is but I told my class yesterday the mathematics is the only subject we teach in the university right you know it's rude and that and that through history hasn't changed now I was lying I think a little bit with that but but if it's the feeling behind that statement that I think underlies my sense of community just one more thing it has something to do is the fact that we are country even

00:40:15 though I believe it's objective we're constructing our vision of this objective realm now that's I mean Heidegger Stressless that the mathematical is what's knowable to human beings because we are constructing it in a way that according that aligns with our own understanding of things and and it's that that domain I think that that is unifying it could I could I add to that though that unification costs something if you're going to take two

00:40:47 subjects that have a tension between them I don't think we can act as if unifying them is just something that you can do and it's all wonderful it may be wonderful but I think your students are will know that there's a price and that you have to stand in a different place the unity the part of the beauty of it I suppose in the is it is that you have to to think about algebra as every school child those you know there's there you have to put yourself in a different mind or you you can't approach it in the same way and

00:41:19 there's a kind of mental price that's exact that I think that I think that one of the things that bothers people when they're struggling with mathematics is that they feel that there's a price that has to be paid they don't know how much it is or what it is exactly but they feel their struggle to understand I never thought of it in that way that's really good and I think people wouldn't mind paying if they were told how much the trouble is that their professor says oh no no no it's all perfectly clear you should understand this it's all unified they know that

00:41:50 that's something about that bothers them they know that the mathematician is right what the mathematician is forgotten is how much he or she paid in a certain way for and at a certain point in their life which maybe is even forgotten to them but which now the student is being asked now you must pay in intuition in change your mind in I mean if if the Mathematica are the learner Bowl things I which I don't think this is what Heidegger is saying but I think it's the things that you can

00:42:21 learn but learning in each case means having to pay something having to to believe some of your blood behind in order to gain something because the more precious the thing is the more you would have to sacrifice and to give and not just of money but of something very very dear to yourself and people struggle with that their cart sort of was trying to protect himself I think um that by saying he said that his analytic geometry

00:42:55 corrected the defects of both subjects algebra and geometry so he felt that it was it was an issue that was there was going to be a change of viewpoint but he also felt that his union of algebra and geometry was a way of correcting but sure and after you pay the money you pay right but he begins by saying of common sense is the least common thing in the world his joke at everybody's expense

00:43:25 and common sense is going to be what what happens after you take his point of view and you see of course he's bright this is much better it's much more efficient but something had to be paid and it it seems to me that there's a kind of justice in that if you want to know more you really do have to leave some blood behind and it's true not just for the mathematician that's breaking it to new ground but it's true of the child that for whom geometry makes a certain kind of sense and then they're confronted with these symbols and it's someone acts as if oh of course you

00:43:57 should understand this right away and this is all kind of obvious and the textbooks treated as if it would be shameful not to understand it almost immediately but the child within themselves or if many adults do feel like no it's not clear to me no it doesn't make sense I I feel as if I have to view this in a different way but I'm not being allowed to do that and no place is given for the transition in which I would the the journey that I would have to take at the

00:44:27 end of which I would agree with Descartes say and now I see what you mean yes it is clear but the discussion would have to be what happened on route what were they what were the series of steps even emotional ones I've never had such emotional classes as those in mathematics it seemed to me in which ashes boiled over and feelings which were not given any kind of vent in the text of mathematics boiled over in the students that were struggling to understand things and and understanding that

00:44:58 somehow their frustrations were not inserted in the way of the sensible topic of the mathematics class that they were having but which they were feeling nonetheless there's some that's extremely interesting and maths on the my own experience a lots of ways another source of possible unity of the field it could be by following out a kind of ontological question about what kind of

00:45:28 thing is it and in our in our taunt here a number of us have used the word discovery I did in connection with Russell's paradox moment ago but Barry use the term intuition a moment ago and and finding algebra right and so in the logic of the concept discovery is of course the implication that the thing you discover exist prior to your getting there to see it right and so if we use that word then that platonic implication is in play if we use the word intuition that I think wasn't there a Royer it's

00:46:01 that sound right in the deserts of mathematician or a philosopher mathematician named Brower thank you thank you you know who had a different view entirely if I understand this and and and on that view it's not that we're discovering in mathematics we human beings are discovering something in mathematics that exists out there in this was not like well prior to arrival at it but rather were sort of like painting a painting I was a Picasso who said every single brush stroke in a painting changes the entire picture and of course in a way that's false but in a deeper sense it's true because what

00:46:32 happens is it with a single brush stroke the logic if you will of everything else going on now has to accommodate itself to that and so if I understand it there's this kind of the anti Platonic view of mathematics that suggests that we don't discover it that that's the wrong model we don't we don't find it there's something more creative at work but I would be the last person to be able to answer that I think well laughing you came close describing it but I think you you would fall short of intuition ISM as your favorite oh well you described yeah yeah

00:47:05 one of my problems is I see both sides of every picture all the time so I think in my heart of hearts I think mathematics is an objective domain that we are discovering but in you in the way we deal with it on a daily basis I think I have a lot of sympathy with the intuitionists and that we are constructing somehow maybe it's like a miracle we're constructing a body of knowledge or doctrine that corresponds

00:47:39 to the to this objective world well I want to go back to this comment about the paying paying for this and you didn't say though I think you made it very clear you pay in advance you're paying in advance yes yeah or the outcome and my question is so within what is the payoff and is that beauty or is that joy or in the particular instance in computers but it's elusive a synthetic new intuition which somehow

00:48:13 conjoins the algebraic intuition and the geometric so could you describe that as being a form of joy or an instance of joy or beauty yeah I think there's a certain shock that's associated with it and there's or even not even unify I mean I'm just thinking about the child that's that's that feels okay with geometry and is struggling with the idea algebra I mean you're paying kind of continuously there and yet this it seems to be often that there's a moment where

00:48:44 you start to see like oh I see if you stand and look at it this way the idea that you would have to take a different point of view already is a tremendous change for people because they have a certain based on what they think of their intuitions my intuitions say this and you're telling me that my intuitions that these are my intuitions that were speaking about and you very rightly say like but but look if you if you stand over here and you say over where okay well is there well and then you help me and I I struggle

00:49:15 with it and it finally maybe I get a little bit and I start to say like oh I could start to see this starts to make sense it did wouldn't have made sense I was right it wouldn't have made sense to my intuitions to the way I was but if I am able to shift myself into a different place it seems to me that's a large part of the beauty I mean I've if I don't know I would like to ask the real the mathematicians here I mean that sense of shifting and struggling and trying to find the new place to which which you're doing constantly mean and as much as the

00:49:47 child is doing and without yet knowing where it is that you have to stand or how much it's going to cost you but somehow at your point your truck you you have had experience such that it's going to be worth the struggle and the agony and the desire to have it all be over with when you reach that new point and can see things in an in a in a new way as happens in the perception of art all the time changing changing your stance metaphorically so that you see the object in question in a different set of relations there's no question in my mind

00:50:20 that payoff for the mathematician this is a very a very special population maybe maybe it could even be called a it's it's it's it's not in the direct that's the the it's not recognized it's not a recognized is not recognize this as a pathology but is precisely these

00:50:51 very rare moments which I've wouldn't hesitate to call joyous when suddenly everything makes sense in a way that they hadn't before attended and I there are many many many many mathematicians speak of this if you read what they say they talk about what but their motivation is and in fact there was a sociological study of mathematics compared in your book write effect comparing your and applied mathematicians the five mathematicians their their most their

00:51:22 most primary motivation is to do something useful and promote pure mathematicians primary motivation is pleasure and they say that that's also so pleasure achieved through pain so so they're heinous and masochist simultaneous they don't you know that the pain is easily in the same way that the I'm in an athlete will you know for whom you know are kind of trying to do even one you know a tiny amount of what they do for me might be an agony for them is their daily their joy in the

00:51:54 source of a kind of sustenance so that you but that's only achieved after a while the student sees it from the other end they see it as pure agony they don't they don't understand and I thought the part of our discussion was trying to understand how it was that this that the struggle to understand that what's that verse in the man stretch out that mind so that they have mice understand these things which seemed to imply from the point of view the speaker of the scriptures that that the the effort of stretching out your understanding was

00:52:25 was something that was suitable for human beings but also difficult for them and something that was a kind of divine divine call and says something what we call the patient has quality of pleasure

00:52:57 joy both for the patient even that aha may not be particularly may have painful qualities to it but there is something about that moment of understanding on the both sides that makes it joyous and special and maybe you could say beautiful somebody hearing that kind of interpretation is that you discover

00:53:33 something that is already there and as you keep discovering it seems to fit into place and fit into place and fit into place so from the moment you had the Big Bang from the moment of the Big Bang mathematic I think if I understand it correctly came about and that you keep discovering and there's something about this that as a specialness has a cohesion has a kind of beauty because it

00:54:07 falls so nicely into place that is what makes it particularly special and unified in unity that's what I the way I understood it in practice these these movements are movements Yuriko movements are those are the occasions for people to use the word beautiful and I think it may be almost set almost involuntary like a just an exclamation or ejaculation this is a this is it's for

00:54:41 lack of a better vocabulary of people in fact I even try to avoid it but once last spring I couldn't help myself I want to say something to the person with given a particularly inspiring lecture and I used that word and I felt terrible because it doesn't say anything it just is an expression of approval and well enjoy enjoy yes well what's wrong with joy it's conventional it's to

00:55:11 conventional yeah what Samantha that's not conventional but maybe all moments where the mind changes maybe that's the essence of beauty is that it's not merely the apprehension of kind something a static kind of proportion that is in some given way pleasing it seems as if the moments that we're describing are moments when there's been a deep change in the mind perhaps the analyst has maybe the patient himself

00:55:43 the analyst has set that same insight maybe more than once before maybe that it was the moment when the patient could understand where it where's is their insight not merely the sight because if it was just the psychoanalyst the person could say yes yes or no it wouldn't matter what they said it wasn't yet their insight and therefore it was merely empty because there had been no change of mine the moments that are both agonizing they're beautiful was the moment when a person changes their mind

00:56:15 and when you think about that's a very it's a difficult or perhaps even an impossible thing how am I to think differently than I think and to feel differently than I feel by definition it's impossible are there moments that that we that you recognize as a deep change of mind which produced joy or produce well I I think there's a

00:56:55 spectrum yeah yes I can remember a few real important moments where it's like almost like the duck rabbit where you know yeah you know your perspective shifts in something a different way and then that enables you to go go forward then those are those are very exciting what about the idea of turning point also involving sort of a condensation of sort that you you've simplified something that was unwieldy and chaotic isn't that a part of this feeling that we're touching on yeah

00:57:25 that's also very important historically the way the only way Mathematica can progress is this process of sedimentation where where ideas somehow gets summarized into probably a theorem and then into a definition and then into a word and then that gets laid then you can go on isn't that's part of the opposition to people who don't like computers doing math that you don't get there observe a collective aha moments is a total change of vocabulary when one

00:57:58 realizes as a community that you just just look random and take but symmetries are important you know that the idea of thinking of the group of symmetries of something rather than focusing specifically on the thing gives you an an inroad an insight know that it is certainly a change of viewpoint but it's not any one person something different goes in the history of it said it developed but it's a it is a kind of low

00:58:33 slow-moving aha moment of the community which I mean I am NOT reluctant to call it the sort of communal joy yeah as having a result of having it that's sort of narrative they work at that captures and I know you know this captures people's attention readers attentions if they're rendered in a chapter like describing even if it was very slow to develop its its captivating to read

00:59:04 about this so its development so my answer to Barry's question I see far-field but it actually isn't I'll be very brief about the far afield part it happened in my life when I was a very young person probably 14 years of age and I discovered the bossa nova on the guitar and and that was a revelation for me I'm still trying to recover and before that I just played block chords of there's a C chord of G chord F chord and so on and these are block harmonic

00:59:34 sort of entities that you that you move in a blunt way from from one of the other I learned the bossa nova on the fingerboard and suddenly all of these things that I thought were independent were interrelated through voice leading one figured by finger by finger so that you never moved in a block way at all it was always a subtle movement where you saw internal relations in one part to the next to the next to the next all of that music written on the guitar functioned the functions in that way and so I saw harmony in a completely

01:00:05 different way that's the far field part the closer in part is well that phenomenon of seeing relations between things in a much much more complicated way than you initially envisaged and it's the simple way has been in your practice you see this other way of seeing everything you still have C chords and G chords and of course but the end of relation between them are so complex they say spider webs of relations that you're working out on the fingerboard take you into a form of understanding you never would have had so that you end up really saying you know there really isn't anything such as

01:00:36 a C chord or an F chord it's all a matter of the relations into which things enter it and that's what makes them what they are and I'm as I say that that was a woman where I saw the world saw the musical world completely differently and it's certainly analogies all over you know aesthetics in the arts and I imagine there might be analogies in mathematics yeah that being able to B go together to things that once I'm thinking about something that I learned from berrywood and insight about the

01:01:07 numbers the natural numbers as being related to forces knots knots enough you know that's at an LED yeah and exactly and to me that was a moment there was that's why I asked about the shock because it was like I was shocked and then the next moment I said well it somehow seems familiar or it seemed right but I think part of it also has to do with two things coming together that there's a certain kind of tension it's a little bit like two voices in music or not two notes being heard at once and

01:01:39 that the unity is being grasped again through a kind of certain kind of tension between them I guess also in studying mathematics realizing that one was allowed to feel I remember struggling with it and that one was allowed to feel the sense of dissonance that one was allowed via privately if not publicly to struggle with things and that that was that even when you approach a dissonance in music a consonants and music are you tuning two strings as you approach very close to become more and more dissonant and you

01:02:10 know if you've been tuning it and they kind of jangle and then suddenly they fall into a kind of order my private theory about this is that this is probably completely illegitimate biologically is that we're descended from amphibians that those are our distant ancestors and that the beings that live both on the land and the water and that probably were laughed at by the other creatures that were not amphibians and thought that it was absurd that we would go from being in the womb and not

01:02:42 breathing and then breathing this is just silly but somehow our ancestors were able to survive when the land dried up and we they lay there gasping but somehow that there's something about the amphibian in us that it still remains so that when two things come together when there's an analogy it has that same force of we're beings that have to live both in the land and in the water and they have to go back back in the force between them at home on either or in both and so that

01:03:14 there's something strangely familiar to us about the situation of to the two parts of an analogy come together okay I'd open it up for questions if you're going to make a comment make it very short and then ask a question oh one word that keeps coming to my mind today is abstraction I'm not sure if you mentioned yet no and that

01:03:45 mathematics is always dealing with layers of abstraction maybe you start with something in the real world like five bottles on a table and then you geometry can tell you something about them and topology can tell you something think now it's your brother can tell you something and then you abstract those one layer or many many layers farther and you end up in unity and I was thinking about this problem with students in that it's really a lack of understanding or appreciation of this process of abstraction as beautiful in itself and I just think is was wondering

01:04:18 if you have thoughts about it I thought in particular the comment about dissonance as a as something that is kind of a commentary on what we already know and then we move past that dissonance and that kind of opens up a new world that in a way when you're wearing to the irrational sort of two girdles theory and stuff gosh I thought I had a question in there it's very good about about abstraction what well I mean

01:04:51 the thing is if you're doing mathematics abstract is a verb rather than you know because you're seeing something you do abstract it better thing produce a structure that's inherent in it but the minute you do it that verb it's no longer useful it's not an abstraction to you it says as concrete has Evan anything else that you're working with

01:05:22 it becomes as concrete as lines and triangles no matter how abstract it is but I think that fits in with what Peter was saying about the price you pay so this price to abstract the rides a certain energy you have to put in this energy to gain the abstraction and then be able to live with it and I'm going to go back to my classes with your thoughts in mind they're trying to write the last

01:05:53 about a day try and help them over this is so fun it's almost like an energy Oh to get the next level of abstraction so this is a lovely discussion I wondered how different it would have been if this caption happened 100 years ago when this society was formed or in fact a thousand years ago and I think many of us would agree that it wouldn't

01:06:23 be fundamentally different yeah I think there are there are certain things that we can offer these days so beauty is of course in the eye of the beholder but this is the wrong phrase it's in the brain of the beholder so we would like to know where in the brain and how in the brain beauty is represented so a good friend of mine Sam or Zaki has the same question in the University College London and has an answer so what you can but he has done is that she shows beautiful women with the foot man to people and a particular

01:06:56 structure called little clothes combats which is the site where dopamine is being released lights up then it shows to a lot of people professional musicians and people like me shown that let's say Bartok Antigone and surprisingly in the professional musician the same structure Liza then he recruited a bunch of mathematicians and they showed the

01:07:27 mathematicians beautiful equations and beautiful problems and also to people like me and the surprise was of course that do close up evidence if the fMRI lights up so if you are looking at from the point of view of the observer which is this in this case is the brain you have an answer that what beauty is real what beauty is is that what Peter said this is this energy barrier which is once it is sold or once you overcome it it releases dopamine and that's where beauty is your size of course as us is

01:07:59 also net if you are interested that's a wonderful description of interesting research but it also does I think misses the point about what's what beauty is this is the outcome this is this says that when people do find something beautiful that part of their brain lights up that's not saying what is it that makes it beautiful right because I guess you can infer from that study that different people with different perspectives or different funds of

01:08:30 knowledge might find something beautiful that someone else might not find beautiful right yeah but you shouldn't equate and not that you were saying this but shouldn't equate the nucleus accumbens as the the core of beauty like the pineal gland was where the soul entered the work of the body this is one question I agree with first part I think that the to say it's in the eye of the beholder I agree with you is entirely wrong but you said it it's in the brain of the holder I want to say non reductively anti physiologically it's in the mind of the beholder and so and from a

01:09:03 conversation we had earlier today I think that Edie will tell us that if there were a drug that could produce that particular neurological response that would be a response in the mind in the brain but it would not be an aesthetic experience it's cooking and and so vacuum that remainder really is what we want to focus on what is good enough and the violence the the other question URI would be about how the

01:09:33 transformation is made between the the student for whom the mathematical idea doesn't release the dopamine and the mathematician quote-unquote who is lets upon a time a student a child who didn't know five didn't know that there were five bottles on the table or anything like that something has changed in the brain so that the apprehension of a beautiful person seems to have a kind of immediacy and the difficulty what we're dealing with here is that mathematical beauty often is not immediate it requires a kind of struggle to reach it

01:10:05 that the mathematician is doing it that the student and the child is doing it this is the value over F that you remember namely the study is somewhere in between the totally predictable and totally unpredictable and neuroscience borrowed a term from physics and ugly ones called complexity but complexity is where the middle is which is sort of predictable but not quite so this is what makes music so beautiful that you can't predict what's coming next but not

01:10:36 quite this might fool is the same thing may things I don't know everything is this playful and you call it effort but the same thing and of course this is not the same in your brain and in my brain because it is a question of the tens of years of education that you have because in order to enjoy Bartok is not the same then enjoy a drum and so this level of complexity happens to be an interesting one because if you calculate the Fourier

01:11:08 transform of music it's the same as the equation of brainwaves or electrical activity of the brain so it's not a wonder that the brain resonates with particular type of music because it is the kind of dynamics it makes I just first I'm a mathematician and I

01:11:39 spent most of my career in an engineering school and and had to not focus on beauty another thing I want to say first is that I'm not nearly as well read as all of you about beauty and aesthetics so this is gonna be much more mundane and and you know I'm hoping you can that my question is you know whether you guys can tear it apart so when I talk to students I can't focus on beauty because these are engineers and I focus

01:12:11 on what I call the craft of mathematics where you're not trying to create new mathematics but you have to do mathematics and this is very important it's not just doing formulas or applying procedures and this is what I realized so I came up with the following rather mundane simplistic description of mathematics it's the use of abstraction and deductive logic to develop new knowledge from old know so did you had

01:12:41 to start somewhere but then you can build from there and so so that's my view of the craft of mathematics then um so then I also had to explain math to someone who knew nothing about mathematics um and trying to describe him this beauty or why there was this wonder when we do mathematics and the analogy I came up with is that mathematics so this very clayton assay but it's a you can visible universe it

01:13:13 only exists in our heads okay but it's a universe we all share okay there's a commonality and therefore unity because different mathematicians follow different paths and everything but then we always come to a common agreement of what's what's there in the universe I mean we may see different things but when you do it we see it in fact they remind me of the movie Inception if you know that one but anyway I mean that's kind of the way I see it so that

01:13:44 you know that's and and you know there's consistency in there so the beauty what's the beauty or the joy it's when you discover that these different parts of the universe that seem complicated or mysterious want to suddenly understand why it's there and when you've when you've explored completely different things suddenly you realize how they're really part of the same thing and and that to me is how I see math these days oh yeah

01:14:15 I just want to you said use the word Y which I think nobody around the table use but that in some sense what everybody all the mathematicians are striving will answer the why question rather really not they not the what question which is somehow beside the point and do your students understand why when you're when they're finished with it why because they can solve the problems or why what you told them allows them to solve the problems well

01:14:45 so when when we were working with weaker students they have been trained very badly where they thought they thought we were like the priests and we're supposed to give them the scriptures and then they would just follow the scriptures and solve the problems and we had to kind of point out to them that once they knew some basic things they themselves can put together things and and two things themselves and that's why I call it a craft and and and and there's that

01:15:15 too because the students I saw them with joy because we had a way of teaching where they would struggle with a problem and we would refuse to tell them how to do it we would simply teach them knowledge that was somehow we knew what was necessary and at some point the student would suddenly be able to put everything together and see how it all fit together and how it led to what they wanted and there was this moment of joy in the student that we saw and and in

01:15:46 fact the way they expressed it was they suddenly look at me and they say go away and and you know these were weak students it was a teeth it so for me that was a lot of joy to me that the that just to underline something you're saying that that that somehow the greatest praise of mathematics is it's the way in which you could change your mind in which you could find out what you don't know because it's building new knowledge we like myself or the

01:16:18 mathematicians that are the better students of mathematics want to know how to how to go from not not understanding to understanding it seems to me mathematics is the art of that above all the master art that maybe has taught Western philosophy that and maybe all of the world what it meant just the understanding understanding or solving solving well another thing is that you know we tend to talk about mathematics

01:16:49 as somehow objects and there was something some mention dramatic news but mathematics is really a process it's doing something and it's like you suddenly realize you can do something you couldn't do before and you could just you know and to me that's part of the power and therefore the pleasure of mathematics you you develop power yeah I think that's a fascinating discussion I

01:17:19 wanted to comment on the nucleus accumbens if I remember my neuroscience correctly it's an area that lights up when there is pleasure and would certainly one aspect of beauty is giving pleasure even ugly things can give pleasure but I would hardly equate it to beauty oh and I think that view comes from an anti preparing view of science you take things that you think are

01:17:49 beautiful show their nucleus accumbens lights up you don't attempt to break it so if you win in a roulette game you're gonna get a lot of action in the nucleus accumbens but I don't think we'd want to say does that rule that game was beautiful there's a distinction between simple fashion and the beauty so the management

01:18:21 said those things that equal areas so you say this is one thing in the brain because separate study we're a long way from finding aesthetics I would say that I know that Vidkun Stein was opposed to

01:18:51 this notion that questions about aesthetics have anything to do with neuroscience or psychology that they're it's a different sort of language game and not that this it's not that it's nothing or that it has nothing to say but it it there these are on different sort of dimensions of of discourse that's I think I mean would you agree with that I just want to make one comment on what you said about your students I mean that is the dream of any teacher of mathematics to turn students

01:19:23 into their only agent as being make students their own agents in creating the world of Mohammed I mean it's it's there's the craft which is essential and then that moment where the craftsman becomes a free agent you know I have a question that requires little interpretations but you can interpret it

01:19:55 my question if you allow me to I'm ready but it's not my my doing experience like yeah my experience during the years decades in both in music and sense this was I was lucky because of family constellations show me that some very great difference between the pure structures pure mathematical search actually every every search in order to the proof that every structure is pure you need to prove but

01:20:29 one of the very natural proof whether some structure is pure try to perform some mathematical structure by music or by poetry because any first lecture if you substitute by reaction any science with music sheets or something like that so express the harmony and the proportions and the beauty my question is until until pure structure of mathematics is not yet performed they express the sense of beauty but it's now

01:21:03 I speak about this grade difference but when the spirit of mathematics begin to perform they express not only sense of beauty but also sense of sublime all the great composers make this distinguishment between you know the what is the distance between ugliness and pity this is the distance between pity and sublime sublime ever included including within spell in oneself beauty but is there something more therefore

01:21:34 why so much not only passionate but people is so much impacted not by very pure mathematical starts but the performance of mathematics I can say that the mathematics is part of pure ideas in the mind of God and when God start to perform ideas pure ideas structures in his mind appears the universe when the man status men begin to perform the pure structure of the narrow mind they create art therefore

01:22:06 because the whole universe I think that the question is the mathematics is beauty or beauties mathematical performance of mathematical structure this is tautological because called universe at all universes consists from harmony proportions and symmetry everywhere the my question is because see the the the sense of beauty the numbers express the sense of beauty but the concept I mean conceptual rank the concept

01:22:39 doesn't doesn't express only the sense of beauty I mean the concept like pure philosophy or some pure classical poetry never the rest of the pure structure they expressed something different than the only the sense of beauty and what is the difference between sense of beauty essence of sublime sublime is I mean I finished and now to say that a correctly sublime is Beauty mingle with some bitterness and sufferings because if you remember the law of great genius of

01:23:10 music but confesses that the music is to transforms suffering of the content into joy by the beautifu static shape therefore aesthetic shape transform every negative emotion suffering bitterness into beauty which is the highest and pleasure because sublime is there is the gives change only one word beauty gives positive pleasure but sublime sublime expressed the sense of

01:23:41 sublime we feel as there is the pleasure different there is another kind of pleasure that is bound to a passion the stronger than satisfaction and this is the the some Milnes some synthesis some mixture between the pleasure and sorrow and dream and some bitterness because all the great genius of music they they nevertheless the takes they express the

01:24:13 performance of you stretch they abuse that super work already performed express the sense of sublime this is I think yeah I've made I think I was that's similar to something that I was trying to say that the there was brought up several times that they that there's a kind of dissonance that's experienced which would could be experienced is experienced by young people by learners of all ages as painful in which at a certain point might become pleasurable or might be

01:24:45 recognized as something that's beautiful I don't mean we haven't systematically tried to separate the beauty the beautiful and the sublime in mathematics so that we haven't tried to separate the moments where you would say that was a sublime as opposed to beautiful theorem the those they seem to go back and forth depending on your reaction or the amount of difficulty that you're having you might judge it to be supplying or to be this is other notion about aesthetics that it's engages this the free play of

01:25:15 the imagination hmm moving between as Hutchinson you referred to earlier between let's say chaos and order an order and it's that sort of free play of the imagination that sort of engenders a feeling of beauty or an appreciation of beauty and that touches a little bit on what what you're saying yeah and there might be there a nice analogy to psychoanalysis as ed mentioned earlier the process and the free play of the mansion as discussed in Contin third critique of the idea there is that you in fact individually in that moment make

01:25:46 sense of all of the parts of a work of art into a coherent whole that that takes place not in the work of art but in your mind and so that is the mind of the perceiver right working with the materials of the art form in a way that it's distinctive to that individual in that sensibility at that moment and I can imagine what it said moment ago about the about the progress of psychoanalysis there's a kind of taking the parts of a life and fitting them all together in a coherent whole that aha moment is where you where you say now I understand or now I've connected the dots in a way that makes a range of

01:26:17 experience coherent and makes it meaningful in the same way in there in the plane quite also the word craft is such a nice word because it gives us notion of being taking pride in what you're doing and in the process yeah that's something that was that it's related to that Freud praised the highest virtue courage that was more important than intelligence and it seems to me in this process courage is also required to have a free play of imagination especially for a child that's sort of facing a school and their uncertainties and the possibility of

01:26:48 failure takes a lot of courage and the ability to to to to a face the distances and the difficulties the struggle of imagination requires courage and the recognition of that may help the child to learn or the mathematician all understand that there is then hope because there is courage you know Michael and Barry might disagree with me here but mathematics seems to me precisely not a free play of imagination I mean it's as opposed to

01:27:20 art right the artist sits down with a blank sheet of paper and anything can come out but the mathematicians can't write down a straight one there's a lot of me injury so there's so there's there's a free play and perhaps in some parts of mathematics when you're starting to try to understand something but what I always feel is is that my up against the wall there are those walls okay and then you may get to it very quickly now to know because you're so

01:27:52 accomplished and you know how you're always at your wall I mean you're always at the wall at the rock face but ever no I want to go with Stravinsky just very quickly of you know he said give me a blank manuscript paper and an ink pen and I'm paralyzed I can't do a thing yeah I a single interval you know see up to a flat and he's not gonna fly and so there's a snake it back what I said about art yeah I was reflecting on the

01:28:24 Sanskrit language and because she brought up the word Ananda and in in the Sanskrit language there's the word suka and dukkha and their pleasure and pain ananda joy is as the other gentleman was suggesting this there's a big difference between suka and dukkha pleasure and pain and joy too something much beyond them now the word ananda is part of a longer

01:28:59 sanskrit word called sat-chit-ananda there's three three it's a it's a tripartite saath is consciousness chit is knowledge and Ananda is joy and satchitananda in sanskrit is said to be the nature of Brahman brockman is the unity it's the one it's it's the self it's yourself so there's as another yet another level jump up so

01:29:31 as Homo sapiens we're going through these experiences the students painfully having our su cos and our and all that I know occasionally a flash of anandam of joy or through consciousness through through knowledge through math or whatever and moving towards the

01:30:01 sublime unity and I just throw that out not so much as a question but if anything comes back from that and the the difficulty seems to be how to turn the pleasure and pain or mostly pain that people experience when they think of mathematics use the broad majority of people that would like to think it's beautiful buttocks whose experiences

01:30:31 mainly painful and how to show the path that's there I think that that we use the word lying I mean if someone once said that a poem is when words begin to fly that is that they leave the ground they leave the ground and you have this exhilarating sense that they're that they're moving in another two to whatever extent that seems to be part of the nature of the joy that that you were speaking about this but that's been experienced before which could be kind

01:31:01 of terrifying at first the idea of flying and not being on the ground but the moments where one hears words that are winged words that that that fly the idea that that could happen with mathematical concepts even the possibility even the thought that that might be possible is already raises one above the ground a little bit anyone who wants to say like okay show me how show me how to fly being raised above the ground changes the viewpoint and that's and that's so important in mathematics are usually

01:31:33 looking for some overview of things that would otherwise be considered separate and now you see them all is it as a territory and that's partly abstraction to go back to that word is that abstraction helps one to fly yeah yeah yeah I know what happens in the details though I've said No somebody comes up with an abstraction that in that makes unnecessary the

01:32:06 life's work of several generations we're all looking forward we can all know that that's going to happen to what we've done and that affection will be a concrete object which will be later abstracted he did its own right but the struggle of that seems important there's a once a man who loved to watch the butterflies hatch from cocoons but he was always very moved at the struggle

01:32:36 that some would die they were obviously in agony as they were going through it so he he had the idea of using a scalpel to make an incision to help the butterfly to emerge it worked very well the butterfly emerged from the cocoon it it flapped its wings there was only one problem it couldn't fly and so somehow that's part of the what did what it means that you would have to crash you would have to try to make it abstraction and it would be completely wrong and isn't that the nature what is your dear insisting on the pain okay

01:33:14 okay you know I have a comment that just seems to me to be crying out to be made especially in New York City among psychoanalyst and what we're talking about is the quest for beauty within mathematics and that reduces somewhat to a question of is math beautiful and what I'm drawn to as an answer is Woody Allen's comment about whether sex is

01:33:44 dirty and the answer is if it's done right that should be your next panel because mathematics dirty that's all I have a few thoughts about listening to

01:34:15 this wonderful discussion will lead to a question I think one thought is the question of a psychoanalyst of course of which I am one obsessed with the idea of meaning and I don't think there's been much reference in the discussion to whether mathematics has a meaning or whether particular mathematical ideas or facts have a meaning and the other thought I had was that from if I'm not mistaken I'm not also not a philosopher or a mathematician but if I'm not mistaken when you get to consecrate eke of

01:34:47 judgment and his discussion of aesthetics he not only says that there's some kind of inner aesthetic measuring stick sense but he also says that it that in order to find something beautiful it has to be Universal izybelle so you have to imagine at least that in talking with your fellow humans you could convince them if you have the right circumstances and

01:35:17 the third well really ever got just and parenthetically regarding the question of meaning I think there are certain mathematical objects which have meaning for all of us almost automatically natural numbers conic sections you know other like sphere has things that have holes in them and so forth but when you become more abstract and more technical and more specialized now I'm not sure

01:35:47 what the meaning is and then there is an anecdote I knew a very old mathematician who who was hoping to live to see the proof of the Riemann hypothesis and he never did live to see it he himself had had worked on a version of the ream I apologize for algebraic varieties but he never lived to see improved and it's a written Riemann's original form raises the question why is this a de function so beautiful if the mathematicians and

01:36:21 why are they so interested in where the zeros of the zeta function fall on the complex plane why is that more interesting than other questions well he did live this mathematician did live to see the solution the proof of the four-color theorem so I I made a point of asking him well you know what do you think of this new discovery he was also very already very old and not doing mathematics himself and he said oh it's not interesting that this proof is not interesting the fear the fruit the theorem is line usually it's not going to lead anywhere it just wasn't

01:36:53 interesting to him here's my question is it possible that the question of beauty and mathematics is a question of culture that there is a culture of mathematicians especially really good ones and great ones who have the right to define what is beautiful in mathematics among themselves and also educating the public define wouldn't be enough they would have to feel it to it whether you're a

01:37:27 Content and believe that beauty is a universally subjective notion but whatever it's whoever you are it's initially a feeling and yes if there is yes about the feeling the the other the other issue about the Riemann hypothesis which really what you I don't want to

01:37:58 get into that sorry the question technical but the culture question which has something to say about what's beautiful and not seen from another planet of what mathematicians are constantly doing there's professional mathematicians is defining the contours of the field by writing letters of recommendation by by by deciding to give lectures about one thing rather than another in it and now it's impossible

01:38:29 this is but you should try it it's hard enough to define mathematics I think it's even harder or to say what extent mathematics is is the creation of a particular stratum of society that is to say the mathematicians and to what extent it corresponds to something that would be there in any case and not just the mathematicians I mean you can have no canoes your hermeneutic of suspicion here and I had a friend who use this a

01:39:02 mathematician who would say the definition of beautiful mathematics is what the what the professors at Harvard and Stanford you know in Chicago when they say is beautiful and you know when I implement is different from that but that but that it was a feel I've been going back something very just today that was a feeling rather than a meaning and that you were asking about meaning and it seemed as it as with music I mean I've been going back to that it's a first of

01:39:34 all there's a feeling or the this is the this a strong feeling in front of nature is what Cezanne said was that what painting was for him the strong feeling and the feeling for mathematical objects I don't know if it's culture if at least a claim would be that this is the most available across cultures because it is having to do with objects that are apprehensive wise I guess even by extraterrestrial or Android type beings so it would be the least cultural them I'd like to know what the people who

01:40:06 prove the four-color theorem Oh thought about the four color theorem the reason why the four color theorem is the proof is let's say unsatisfying is one prove that there are a finite number of cases that you have to examine but that finite number of cases is very large and the computer examines them so that's why people are not happy about it on the other hand there two mathematicians

01:40:39 their differential geometry risen Colonel Homer and rosca have reconsidered the four color the four color problem is a is a is a simple single question which seems to have nothing to do with very much in the rest of mathematics but just recently people have replaced the four color problem with something in three-dimensional

01:41:09 geometry which is I mean not replaced it but if you could prove something in three-dimensional geometry you would have a really good proof of the four color problem so that become makes it that forces it part of a much larger let's say combined effort having to do with chern-simons invariants and things like that but it's now no longer an isolated particular thing and that's a funny thing that happens with a lot of these isolated marvelous problems for

01:41:41 example there's something called the kakeya problem which I can tell you you take a needle okay and you want to rotate it 360 degrees the cover on the on the table but cover the smallest amount of area on the table that you can cover right well the natural thing is just 20 and then you get a then area pi times the diameter of the needle the the length of the needle needle but Kukai a-- has suggested that

01:42:16 there might be sort of funnier ways instead of just rotating you might go in this direction and then go back in that direction and then we go in and and get some weird way of turning that needle 360 degrees and you would cover less of the table okay he conjectured this and it was finally proven that you can do this covering a small amount of the table as you want okay now that was the

01:42:48 standard regarded as one of these isolated single problems you know what can you do no it's and now it's now a piece of a really interesting very general theorem about the norms about the l2 norm so it's very hard to have an isolated independent problem in mathematic there's a on the question of meaning there's a I'll just mentioned in

01:43:20 one last time I promised something the Viggen Stein uncovered is actually I think quite a thing here we normally think of the primary repository of meaning as being in language and then we ask well how would mathematics have meaning from a linguistic point of view that 9 in the section of the philosophical investigations turned it around and he said imagine in a society that has only the numbers 1 through 5 and in that world you say 1 1 plus 1 that's ok foundational you know 2 plus 2 is 4 that's pushing the edges of what we you

01:43:50 know 3 plus 2 only the most in event you know some genius comes to how about 5 plus 4 and they say that's the incoherent question right and so so in that world right Vika Stein asked well when the number 5 have the same meaning in that culture as it does for us and the answer is obviously no what what he's doing is saying well words function like that too so we can learn something about word meanings by looking at number of meaning if you will and that is a matter of seeing a number in a very very

01:44:23 complex set of relations part of what you've been articulated any moment part what you can and and he was suggesting there that atomistic conceptions of word meaning or invariant word meaning the idea that words have meanings of themselves and they carry that meaning in parently across context is a misbegotten analogy is a slight kind of scientific picture that's out of control you saying words don't work that way because we know that numbers don't work that way so the meaning gets turned around an interesting way there by a

01:44:55 quick comment just a uncraft I'm a magician who has a lot of magician of mathematician friends and so I just wanted to speak up for craft and one time I was talking just Silvan Capel and he said I think you're right on that craft thing about math because one time Israel guffawed was with a student at Rutgers who was spinning his wheels and just worrying about a problem and not working on it and finally he yelled at him mathematics is not head work it's hand work and then with result talking

01:45:28 about your comment about roulette we shouldn't forget ed Thorpe's work with Claude Shannon on roulette that led to quantitative investment in Wall Street and has therefore blessed mathematics in ways that it couldn't imagine so in a beautiful way and and then I wanted to bring I've been a teacher's assistant to one of Barry's students module Raghava at Princeton the semester and he's a friend now for many years through John Conway and I want I and I

01:45:59 know John through the Friends of a Martin Gardner and Martin Gardner seemed to do all this work on the unity of mathematics by writing articles that kind of broke down gh Hardy's divisions between pure applied and recreational math that was written for political reasons and so I just wanted to it's kind of going jumping from beauty back to unity and to me it's just as amazing as a magician a guy who trained in philosophy at UFC and and then worked as a magician really use that to write

01:46:30 papers that helped unify math in a very real way it seems for many years in the last part of the last century thanks [Applause]