Permanence and Impermanence of Mathematical Concepts

00:00:01 yeah maybe later okay do you know count Ola good afternoon everybody Olaf good afternoon welcome to the Helix Center I'm Gerald herwitz I'm the associate director at Helix and I want to welcome all you all today to this very interesting uh Round Table we have with some very distinguished participants on the permanence and impermanence of mathematical Concepts uh let me introduce you to the uh speakers today and they will kindly

00:00:33 raise their hand when I mentioned their name first is Michael Harris who is Professor of mathematics at Columbia University and before that he held positions at brandise and the University of Paris Dido he obtained his PhD in 77 from Harvard University under the direction of Barry mour a name that should be familiar to you shortly he was organized he has organized or co-organized more than 20 conf conferences workshops and special programs in this field of number Theory

00:01:04 I'm going to go on because there's such a long list of accomplishments we'll move to our next participant Natalie Sinclair she is distinguished University professor at Simon Fraser University in the faculty of Education she is co-editor of mathematics and the aesthetic new approaches to an ancient affinity and what is a mathematical concept she has also led the development of two multi-touch apps for arithmetic learning called touch counts and touch

00:01:36 times Jared Weinstein is a professor in the Department of Mathematics and statistics at Boston University where he has worked since 2011 he studies number Theory which is ultimately the study of the whole numbers and their properties but which links promiscuously with practically every other mathematical subject um a New York native he received his PhD from UC Berkeley in 2007 Barry mazour is a mathematician at

00:02:08 Harvard University who has often taught courses in the history of Science and philosophy his books include imagining numbers particularly the square root of minus5 yes that's part of the title of the book U prime numbers and the reman hypothesis written with William Stein on in on Cambridge University press and he is edited with apost apostos Deus the book of essays circles disturb the interplay of mathematics and

00:02:41 narrative lastly we have Alma steinard who researches the interplay between politics and mathematical rationalities steinard second book manuscript accountable democracy mathematical reasoning and representative democracy in America 1920 to now examines how mathematical thought and Computing technologies have impacted electoral politics in the United States in the 20th century focusing on the census apportionment Congressional

00:03:11 redistricting ranked voting and election forecast she investigates how changing computational practices from statistical modeling to geometric geometrical analysis insinuated themselves into the most basic definitions definitions of fair representation of the American electorate in her previous book axiomatics mathematical thoughts and high modernism steinard excavates the influence of axiomatic reasoning on mid-century American intellectual thought so with

00:03:43 all that welcome everyone and Welcome to our panelist well I thought I would start by setting the tone and hoping that people will immediately uh change direction but this is uh this is uh at least how I was thinking about the the topic uh the title permanence and impermanence of mathematical Concepts uh developed

00:04:13 through a conversation among uh several of us and let me uh let me I'm going to quote uh from an essay by uh Kathleen Goldstein who has devoted much of her work to showing that m mathematical the the notion that mathematics deals with Timeless uh Concepts uh is limited that that that view he said uh mathematics is the art of giving the same name to different things wrote quank at the very beginning

00:04:45 of the 20th century but the view of mathematics encapsulated by this that it deals somehow with sameness has also found its way into the history of mathematics uh in the popular genres of the history is hidden the idea a that despite changes in symbolism despite the use or not of figures Etc despite the presence or not of proofs some mathematical thing is indeed the same and the rest of the essay uh she develops uh reasons to uh to to question

00:05:16 that that assumption and um that's one starting point and I I'm I should say maybe my motivation is uh comes from living through a period when the when certain Notions with which I've been working during much of my career are being uh are substituted by other Notions and the question is whether they are really the same whether the concept survives or whether somehow the way of thinking about them being so different has

00:05:46 changed the the the nature of the concept but let me just uh give you a few quotations because this this this this uh uh reactions to this are not always uh tranquil so here's a well-known uh quotation uh from Carl ludrick zel responding speaking for the past just now Lang has published another book on algebraic numbers which in Barry didn't want me to includ uh didn't want he

00:06:19 didn't want to include this in the uh in the uh in the blurb I think uh which in my opinion is still worse than the former one I see a pig broken into a beautiful garden and rooting up all flowers and trees so that's uh that's the past the voice of the past and on the other uh on the other hand uh voice for the future uh this is chevet speaking within borak Kei I'm going to translate it Loosely I uh wonder I asked myself

00:06:53 whether this mass of the most methods the most AC a which which I'll translate uh is actually not a practical joke by the other members of boraki this is a criticism of a of a draft of a book and P which I would like to translate as a flat asked and uh piss stained uh is that an appropriate translation translation is a is is is is

00:07:26 is a characterization of the old way of thinking and uh this is this appears apparently in some internal Baki documents it very very rarely got it made it into print although I found a few a few examples so with that in mind what's one more uh somebody who saw this this announcement the announcement of this uh wrote to me and said somebody who was just turned 6s said I'm often reminded of what another colleague who was in his 50s told me at

00:07:58 last year summer school on our on our on our uh on our branch of mathematics that he must be happy to be over 60 and not to have to worry about modulized stack of G bundles so so that that was in that was inspired by this by the title of this uh so I guess I just open it up anybody else I say that you just reminded me that I want to I have this pet pet idea of writing a something about you would

00:08:30 think that reviews mathematical papers will be boring but it's all full of gems like the one that you read just just sorry I just want to put it out there but yeah yeah I actually wrote a paper on some of reviews of mathematical thanks to Michael who shared them with me but was not supposed to share them with me so oh these these were secret internal reviews internal of a journal but yes with the names with the names removed yes so what could account for all this heat

00:09:00 you're the ones who would would answer that no really right you say folks think of mathematical reasoning and Mathematics itself is being dry and just one thing being named as being that's the same thing as this a equals B and yet people are getting angry and having potential fist fights over it how can that be well it might be one thing that the word the same doesn't mean anything that is to say if you uh as a old friend of mine

00:09:31 once said go to any important word in the dictionary and follow it through and you discover a circular Loop so so what does it mean for uh two mathematical Concepts be the same I mean let's scw it out what does it mean yeah you know my nephew asked me by text the other other day why does 09 repeating equal one and

00:10:02 it really brought me back to when I was his age he's 12 and I was having a really let's say spirited argument with another classmate like we were really into math so you say like why do why how could there possibly conflict when it comes to something like math but there was because we just couldn't agree on what these numbers meant and even what the equal sign in 0.9 repeating equals 1 meant and now as an adult mathematician I look back at that moment and this this very thing and lots of things like it

00:10:32 are invent invitations for someone to discover what uh the basis of the real number system is and you know what you can throw away some of those hypotheses and th so throw change some of those axioms and then maybe it's not true or maybe there's numbers less than one but greater than all rational numbers it's less than one and so forth um and you can discover new things this way but somehow I don't really know why people uh like me when I was 12 thought that

00:11:06 math was really just one thing math is math it does not change it is eternal and I think that most folk intuitions about what math is go along those lines and certainly if you get an education in math uh up through high school that that's that's kind of what you come away with but it's maybe not just a folk understanding of math I think in um I think to the Say by John Carlo Rota about um this desire for for for permanence for it it's much more

00:11:36 exciting if you think that what you've thought of or created is going to always be there and that's part of like given that we're here in the psychoanalytic Institute that feels like an important part of the experience of mathematics of having this feeling that you've created something that was always there that then it's maybe a little bit more villainous to insist that everybody else have exactly the same idea you you do through schooling but I think it's both the something that I again this is not I'm not a practicing mathematician I'm an historian so I I you know my work is

00:12:08 quite different but it's something that I noticed when I read a lot of mathematicians writing it's about this feeling it's not just that what you um what I kind of discovered it's not just about you what you created that has come of kind of permanence but what you with it what you have the concept that you have worked on is actually linked all the way down there's this feeling that the need to say that you one can if want to do the work um see your work as a continuation all the way you know all the way to ukl you you should be able to somehow you know that that this is this

00:12:40 is it's actually it's some of your language of kind of this idea of a continued conversation right that there's that math is this constant uh conversation uh that one can be you know again this feeling that you can trace it all the way back there as well so both future and kind of past looking as well MH I like the idea that mathematics is also a totally subjective and personal activity which is sort of what you were moving towards you know um I I know it because I began as a

00:13:13 topologist and I felt I had real training if you want in visualizing knots and pretty good but I had no idea that it's um it's a training and it's personal and it can come and it can go so as I moved from that activity to other mathematical activities I can see the

00:13:46 recession of the you know of the intensity of uh of intuition of knots in in my conception now I can't imagine that this doesn't happen to absolutely any mathematician and so to what extent do concepts remain the same well they can remain the same in the as things out there but they're not going to remain the same as inner felt

00:14:16 experiences as for knots there was a huge Revival or in not theory in the starting in the mid 80s and the lots of new techniques that had nobody had ever considered were related to knots uh were introduced and changed the so you think the knots are still the same knots uh the concept of not is the same having introduced these these relations to to other things that get knotted well since I'm pushing the

00:14:48 subjectivity of mathematical research of course I don't I think it's it um it depends on um one's inner inner experience should we explain to the listeners what a not is or what not theory is okay a not is you you're in this room and you imagine a piece of string that is knotted in the usual sense but the two ends of the string are

00:15:20 finally joined so if you're uh follow an ant following that knot you can go along a cular path forever um but you're allowed to take this piece of string and move it however you want and try to see if it looks different for example one possible knot is just a string going around in a simple Circle

00:15:50 that's usually called the unot because it's unnoted um now if you can move one of these strange tro structures um which intertwine with itself as you can move it continuously without tearing anything but move it so that it finally somehow gets unraveled and becomes uh looking like that Circle

00:16:22 it's called the unot and so not theory is a study of classification of the manner in which these curved um pieces of string join that their ends can be transformed can be moved continuously from one to the other and of course since the mathematicians they we mathematicians tend to think of the string as having

00:16:53 absolutely no uh cross-section uh as uid did uid is the definition two of book one is um a line is breadless length well yeah I think a lot of people would find it surprising that that such a thing is mathematics at all it doesn't have numbers in it it doesn't have functions in it but it absolutely is and people have some people are career not

00:17:23 theorists right right it's a good encapsulation of what mathematical quy can be uh Barry just laid out some definition what a not is and then the rules regarding how that definition Works you're allowed to deform knots continuously in space but not ever cut them and then one lays out the questions if I draw a knot is it possible to prove that the knot is or is not the unot yes there uh this is a good flavor for what

00:17:54 mathematicians do yeah so how does that matter to you as a mathematician ition whether mathematics is um permanent or or not in in in your description of mathematical practice just there oh well oh there's a lot to that question sometimes we are just stuck on a problem and new tools need to be developed to solve it and um then those new tools might be revolutionary um well so you're saying

00:18:27 that it's important to you that it's um that mathematics changes otherwise you wouldn't be able to hope that there would be a new tool that would um help help you solve your problem yeah perhaps um for instance I don't know about imaginary numbers I mean let's try to keep it a few centuries ago so that it doesn't get too technical right at one point they just didn't exist or people thought they were absurd impossible worthy of derision I mean we call them imaginary numbers after all it's like a vestage of

00:18:58 of the fact that someone thought they were absurd uh but they're incredibly useful I mean they're useful for the Sciences they're useful for physics like the laws of quantum mechanics governing How Stuff moves at a very small uh scale uh it's required to use complex numbers to study these things so I mean one simple answer to the question of why math has to change is so that developments like these can happen but that's about change and it's so people can say yeah okay new

00:19:30 Concepts come along but isn't it a different question about whether once they've been invented no matter what crazy word they were given do they change after that are they always the same I don't know our Revolutions in math really like Revolutions in the other Sciences like we used to think the sun revolved around the earth and now we know the Earth revolves around the Sun so the old theory is just dead but if if we take ukian geometry and we

00:20:02 relax one of its axioms so that you get nonukan Geometry which by the way kind of describes how SpaceTime works is ukian geometry gone no not at all it's still there so somehow math changes but it doesn't really like vary its old Concepts I think they're still around they get added on like a big tower I think that's a a nice example because uh just to take not the whole thing of non ucan geometry but a triangle which um we've known about for a long long long

00:20:33 time when nonukan geometry comes along we have a very different idea of triangle that emerges even just visually but also in terms of wait what does it mean to have three sides and then maybe it's that that changes backwards our original idea of triangle so our original idea of triangle gets um improved or multiplied by this new understanding of triangle that we have absolutely yeah you you come so far from your intuitions when you learn in school

00:21:03 that the sum of the angles of a triangle has to be 180° you get tested on it but what about a huge triangle imprinted on the surface of the Earth so that the curvature of the earth matters guess what you sum up the Angles and you get oh no is it [Laughter] more more more I'm going to be testing on this yeah typical yeah just to help you imagine it if you drew if you connect the North

00:21:34 Pole to two points on the equator which are 90° apart it could have a triangle that's got three 90° angles so but then you go back to your old geometry and you realize oh I've sort of been liberated from the old rules and that's a kind of a wonderful um liberating feeling that's I mean it's kind of the excitement of math is kind of progressing this way through ever um yeah ever more general

00:22:07 concepts Elma as a historian how does it matter to you whether Ma I think for me it's just a question of how I've mostly been thinking about how mathematicians are thinking about it like what what is both how they're thinking about it and how has that impacted mathematical practice over over time those was the kind of sort of question that I've been trying to understand there's there's a article that I really like that was published in uh in the 1960s by uh which to me kind of gets to exactly the the question of this spanel which is so it's an article that was published by a very

00:22:38 famous mathematician Richard Quant um and the title of the article is is called mathematics in the modern world the entire idea of this entire is the article is to it was in Scientific American it was public facing the idea was let's explain to the public to the kind of broader public what mathematics is today because mathematics is different now in the 19 60s and the and the article itself and all the article in that journal you know where they start with Egyptian mathematics so the title is mathematics in the modern world but everything starts in the Egyptian so

00:23:10 it's all a story of progression uh how things exactly what we're talking about right now everything changes but the underlying kind of sometimes spoken some sometimes unspoken assumption is that there is an Essence that there's an Essence that stays the same and that to meet to what is it for mathematician and why why is it important right why is it important for practicing mathematician to have this idea of that there is that there is some sort of essence um even if it's not well defined not something that anybody can

00:23:41 explain exactly what's the essence of a number is uh why is there still a a need maybe need is not the right word to hold on to this idea right why would you not start a mathematics in the modern world with here's what's happening right now and I think that's a common idea I mean I use this as example I really like it but I I think you can probably find Contemporary um accounts that will do the same thing that will start somewhere they will try to expand something that's happening today by going back you know X number of years and and

00:24:12 I I myself am struggling with this and mean you know it's a question that's like the driving question to try and understand why is that the case because you don't see it I think it's true that you see it less in in other in other fields in other fields of science as well yeah I mean you were asking about KN do you think that that's not so the same you were asking Barry so it's like we can throw the question back at you I'm not a a not theorist but I noticed that both Barry and Jared defined knots in 2023 the same way they would

00:24:42 have been defined in uh 60 years earlier so in that sense that definition so the definition has not changed but does that mean that the concept hasn't changed because it's not because the uh the the the uh without using word like Essence the uh the relation uh the Practical relation of mathematicians to not has changed immensely because they uh you start if you were to learn not Theory you know after the you go past the

00:25:13 definition then the first Notions uh that you study are not the same as the ones that were studied 60 years ago because of all these uh introductions of algebraic methods uh from in the 1980s starting in the 1980s and so I do not have uh I don't I don't see how I could make a uh give you a coherent answer to that question but uh but uh but alna you're

00:25:45 you're you're using the word essence as something that's in in itself ungraspable but what it's the center of gravity of this uh concept which may actually be changing in everybody's Viewpoint yeah yeah I mean but but it itself is ungraspable ungraspable undefinable UN um yeah I think that's the that's yeah that's part of the but the quotes you the things you quoted at

00:26:17 the beginning and then the rest of the discussion makes me wonder why is this permanence and impermanence in mathematics why don't we ask ourselves I never heard somebody say permanence and impermanence in genetic studies permanence in impermanence in Biochemistry in other words that feeling of uh almost finiteness or closeness

00:26:47 doesn't really occur to one in quite a lot of other fields so what is it about mathematics that makes those people you quot it react so strongly and why is the subject one of Interest well you know you would say of course mathematics is going to continue to develop forever so why is this a question in mathematics it's different from the other Sciences in that at least formally it's not connected to anything in the physical world that might be one easy

00:27:17 way of answering the question why don't we that was another St is correct of course not yeah so right so as you mentioned mathematics didn't used to be thought as something separated from the physical world I mean it was geometry which literally means measuring the Earth or right but the way we think of it now is well maybe that's why people defend its permanence just because it's supposed to be divorced from anything in the world it's supposed to be inherent in logic itself so it somehow can't

00:27:50 change um I think it's related to the fact too that um we keep we're still teaching um in schools mathematics that's been around for 2,000 years using technology that's been around for a thousand years and symbols it's just like in any other one of the school disciplines they've sort of caught up to to you know people read novels of today not just the novels of the past but isn't not a part of the initial appeal

00:28:20 to Future mathematicians you know Jared you were mentioning the debate over point9 repeating in one I think is it possible that initially this sort of platonic version of math that it is completely fixed and infinite and permanent that's part of its initial appeal and then as we go along as mathematicians you go oh it's not exactly that doesn't work out so well yeah I you know I think a mathematician working on

00:28:51 mathematics always I can't not always sometimes at least once wobbles between a totally platonic sense it's out there it's constant it's permanent and I'm going to understand it and that's the platonic View and then there's the Conan view that um I am sort of dealing with my own intuition space and time and I'm using

00:29:25 that to represent perhaps things out there but those things out there have no name they're sort of like the Essences of Alma yeah and then there's another view that refuses to dichotomize between those two oh yeah well that's why I said wobble okay I I think one wobbles and I I I I feel it too that hey anyway famous uh um philosopher uh hsh's uh uh phrase which is mathematicians are

00:29:56 plonis during the week and and formal during the weekend so but that still dichotomizes what's the middle position yeah I'm definitely a wobbler myself it's it's very appealing to think of a mathematical Essence and mathematical progress as just getting closer and closer to understanding what the true thing is like what a true triangle is let's say but also doing math can feel like writing poetry like there's a beauty to it it makes you feel good talking about it uh

00:30:27 builds community and is there like a is it was there an Essence underneath a Shakespeare sonnet or no it's an act of human creativity so I'm definitely wobbling between these two extremes all the time this you turn to the triangle and imagine the first person uh who realized that a triangle was not just uh an object by itself but was one of a whole family of polygons you know this was and

00:31:01 you know what and many of the things you could say about triangles you could also say about polygons like it has the same number of Corners as it has sides for example that was a that was and so then then comes I'm thinking about of course about about this this this this uh funny funny three mathematics but then then uh so then the triangle is it the same object as it was when it was just a triangle when now it's one of a of a

00:31:33 family of of similar objects and then you can say then somebody realizes that there are three dimensions and instead of just looking at two-dimensional figures you can look at three-dimensional figures and then much much later there are n-dimensional figures and you could you could do topology and and and uh and everything you were saying about the triangle well you can still learn about triangles in uh in in high school but maybe the

00:32:03 purpose of learning about triangles is to prepare you uh to do topology when you know when you grow up and this is the whole of mathematics is constantly growing up in the sense that as Jared says one is going to find that the essence of a triangle is not to be just something with three sides but to be one of un uh un unlimited range of figures in

00:32:36 geometry and all kinds of geometry and uh there is a drive which is not I don't know if this is one of the seven drives you were mentioning before to generality to to to to uh stop thinking as this is this quotation the French quotation uh was about think thinking about what this subject matter was in a limited way as opposed to in the most General way possible that's an expression that was

00:33:07 characteristic of the boraki group in particular taken to extremes but it's uh in asking about the permanence of the concept it when it changes in its level of uh of generality does that mean it was revealed to have been all along uh something that was only uh discovered Millennia after it was first considered or is that a different object at these two different uh ends of the of

00:33:39 the historical process one thing you you could even go further back from triangle to um the number five as in our little abstract where um I guess people may have started with uh the uh descriptive gesture the adjective five cows five stones and the

00:34:09 Great Discovery or creation or invention is to noun ify that uh adjective and all of a sudden there is something that can be studied in its own without the stones or the as a an artifactual aid that that even that's in the same uh spirit as your triangle but must have

00:34:41 occurred earlier I think it I think that that sort of thing must be the Breakthrough for mathematics some real discovery that you can noun ify that adjective well that was a breakthrough but in retrospect was it a good thing maybe it would have been better to have the five cows and the five rabbits and there well I think six cows is actually better what six cows are better than five oh oh but I'm wondering about

00:35:14 this just this sort of trend towards abstraction and increased generality in mathematics and that seems to be okay that's one of the things you might list under essence of mathematics if you're going to try to generate a list of possible um criteria um how could it be that with that movement which includes early the Early adoption of platonism within math because that seems what what is five if it's disconnected from

00:35:45 objects if not something Plato would say you know in heaven how did that lead to where we are today where people say oh wait wait wait wait we're not right about that does anyone care to respond respond to that one well just I think it depends a little bit on your theory of um learning and or knowing and uh I think you could say um there is an Essence uh to five but that's taking five to be like a thing that exists out there somewhere nobody has been able to say

00:36:16 where yet so there's that um but we could take a little bit more of a probabilistic view of what five is which is basically all of our experiences that we have each had had about of five and in all of those experiences which include this one here with all of you people here sitting down um that that is an instantiation of five that will be quite different than the one that I had when I turned 5 years old but that is also an instantiation of five and all of these multiple instantiations of five

00:36:46 that I have will have some commonalities and some repeated patterns that sort of get more and more engraved over time but that doesn't mean that the that I'm going to forget all of you here the next time I think of five cuz you're still there you just might not be part of like the most repeated pattern that I have and one of our urges I think in the seven urges is to be able to communicate with each other and so I'm I'm checking out with Barry like what are our common experiences of five we would like to be able to share something about five and

00:37:19 we'll have some not all of them will be the same but that's the basis on which we are able to talk about generality or or abstraction is on that sort of reduced set of overlapping and also um probabilistic and maybe even Quantum experiences of five geometric of course and geometric yeah but don't you think there's a tension that with for mathematician on the one hand um to think about Concepts exactly in the same way that you are just describing kind of experiential practice-based uh uh the

00:37:51 way you learn about it and on the other hand there is this need in mathematics to Define things right to come up up with a great mathematician need to Define their concept um and there it seems to me that those two um uh those two uh desires somewhat somewhat understanding or desires are are operating somewhat at odds with one another right and one to keep the concept uh open and and on the other hand mathematicians desire often to we need a definition right like you I mean

00:38:23 you will tell me but they the kind of defining concept is foundational to accepting it in same way if you can clearly Define it uh again it will be hard for you then to to uh get the rest of the commun community to accept it no I mean this is a question to the mathematicians I think you just gave a definition of what math is it's somehow the art of speaking clearly in a way that's unambiguous I think this is a very unique field in which we can at least attempt to do this when you say five uh

00:38:56 well I could could say it's a member of the integer ring which has such and such property and ab Bays these axioms and at least give a definition and you just gave a definition of knots earlier that to me kind of is what math is it's this art of definition making you're setting the rules of whatever game you're playing but the way you know so like the way you know the concept is not limited to the definition right so this is I think that's what I was trying to get there's the there's the there's some tension between the need to the it and

00:39:26 the other hand on the way that you actually come to grasp uh the way that one comes to grasp the concept has to do with learning it in all those different ways or you can think about a circle as an equation you can think about a circle as a projection you can think about a circle as me drawing this but you come to learn that you can define a circle but the way you learn a circle is actually now I understand yes these these two things really are and conflict especially when the in the intuitions perceive the formal definitions

00:39:57 which happens an awful lot like with well the definition of five has run through a lot of turmoil I mean yinian vulin who is an extreme constructivist mathematician um kept um trying to put some doubt as to what the initial definition actually of the number five where one thing it could be the the

00:40:27 quink clunks that is say the four vertices of a square and the central point and if you think of it that way well it's five objects five objects with a certain property and the minute you think of it that way uh every notion of five objects has some conditional property to it so that uh

00:40:57 David Yume I think in the tretis on human understanding said um we can um identify a number in that he had rather wonderful verbs but I'm not going to use his vocabulary uh um in if every unit of the one thing

00:41:28 that you think of as that number corresponds to a unit of the other thing that you think of as that number in fact is acquainted with is is one of his terms uh and if so then you can pronounce those two sets as equal and um equal to five if there are five of them right so there's as many people around

00:41:58 this table as there are bottles of water yeah there's a because there's a correspondence right one to one yeah yeah okay yeah okay that's a kind of the modern definition of number in a sense precedes all the modern definition preced it precedes that it's yeah see cuz maybe there's no progress maybe all of the ideas are there all of the time like a block Universe oh I love that concept did you mention the the term constructivist or right do you want to say a little more about that so people

00:42:29 yeah yeah uh well I I mean I can I don't want to okay uh here's how little this man y vulpin believed in infinite or even very large he was asked and this is uh you can YouTube this I think uh he was asked is two a number he would say yes is two squared a number he would say

00:42:59 yes is 2 to the 4 a number he would say yes when he gets to 2 to the E he would say yes all right so much for the permanence of the notion of number wait sorry why why so resistant to that number is not very large well maybe he got to two to the 10th

00:43:29 yeah it kept going it kept going if if if it's still on YouTube yeah I see it's worth uh trying to find it I see I mean this is one possible reaction to Notions of infinity which challenge people like if if someone really thinks that 09 repeating cannot be one one of their objections might be like well you just can't write infinitely many nines I just reject the whole premise in the first place that is one possible reaction that's not not a crazy reaction either I mean philosophers do

00:44:00 it this is particularizing certain mathematical uh Concepts or making them specific to a real experience or act right because these are steps algorithmic steps right and you say you can't do 0.9 repeating no one can do that hence they're not equivalent right I mean that's not the mainstream view within math not at all we use Infinity there's no branch of mathematics that doesn't engage with infinity in some way

00:44:32 yeah par was a little hesitant what was he yeah I mean he was he was not as as hesitant as uh just in inulin but there was a kind of attempt to to keep an attempt to keep to the finite you know I mean there's a moment historically there there's a there's a particular moment uh at the kind end of the 19th century in the beginning of 20th century that people refer to is the foundational crisis in mathematics which

00:45:03 is this moment that mathematicians and philosophers uh uh were rethinking some of the basic concepts really kind of asking what is a number how do we know what a number is and and you know many some of the most famous philosophers at the time were thinking about those questions um so I think that you know if we had this conversation you know if we can go back in a time machine to the 18th century this conversation would sound differently we are the conversation that we are having right now is a conversation that's happened in a way the kind of

00:45:34 post post foundational crisis uh so we think I think that the the way also we talk and just think about those questions changes over time um and the mathematicians that I think that the way mathematicians were trained at the time were not the same way the mathematicians are trained today um with regard to large numbers I was at a uh philosophy talked the day before yesterday and at the time of the foundational crisis um before the foundation crisis people would just just keep counting and they

00:46:06 wouldn't worry about it but at the time of the foundation crisis it uh the the consensus became that in order to know that you can keep counting you have to include this among your axioms and so yet at that at that uh philosophy talk it was uh raised I don't remember who who which philosophers question the the the validity of the successor Axiom which means to say that after each number there is another number and we were arguing whether can you can you

00:46:36 actually have mathematics if you uh don't uh if you if this is not admitted in which case is it really an axiom or is it just a part of what it means to be doing mathematics so this was so the number before this crisis was somehow innocent and then it lost its innocence because you have to uh you have to add an axium in order to keep to to to say make even the most Elementary claims about arithmetic and that's another

00:47:08 example of whether it's the same concept or not can I ask this where is the data coming where does the data have to come from in order for for mathematical Concepts to evolve where's the data come from uh well we gather data in a sort of funny way we stare at blank blackboards and at our computer screens well I I don't want to say only

00:47:40 I have gathered emperical data by writing a little computer program to like test a hypothesis but the funny thing is is that most of our theorems are true for all inputs and that could be infinitely many inputs and so you will never prove such a mathematical statement just by running an algorithm because you'd have to wait an infinite amount of time to check every case so instead our proofs are finite but they're abstract enough

00:48:10 to handle infinitely many cases and perhaps the question is not where the data where is the incentive comes from so we are like we're in a group of mostly pure mathematicians I mean there have been some mathematical development has come along because there were uh there are incentive outside incentive that somebody had to uh figure out some kind of new physical Theory or you know the that ideas in MA came into mathematics because of trying to solve ideas in the physical

00:48:40 world or trying to solve so it's those necessities of that the problem arises not in one's brain but arises at in the world that brings in trying to solve those problem that's a new problem maybe it's a new but I I think for Pure mathematicians one might argue differently I'll give a an example a historical simple historical example which is U roots of of polinomial so uh a very very long time ago uh I don't know how long ago

00:49:14 probably far back as the Babylonians it was their formulas for for the for the roots of a quadratic polinomial were already uh understood that you could WR down b u square of B b^ 2us 4 a c that formula the quadratic formula you can blame the Babylonians for inflicting memorization this form there's there was actually uh I think there was a an oped I I read once why why should uh high school students have to learn the

00:49:44 quadratic formula you know they've got their calculators and it's it's by somebody who was who had a was a professor somewhere and then uh but then came in the Renaissance quick succession formulas for uh cubic roots of cubic equations and then degree four equations and you know notice that going from cubes to degree four you've lost the connection to geometry and so it was a problem to

00:50:17 continue this you know why would you that you know where were the incentive to continue this come from but that's you know it's you could say it's a natural incentive where people people wanted to gain Glory they got glory for the that one cannot imagine now for the degree 3 and degree four and then it was turned it turned out to the early at the beginning of the 19th century that you couldn't find a formula and so at that point you've got two options you can just give up or you can say well that

00:50:50 was not really what we cared about we really didn't care about the roots of these equations after all all that time you know people were were fighting duels about them but they were they were mistaken in fact what we wanted to know is just that there are Roots you know and then you could say there are roots and then say what as much as you can about them and that's the beginning of uh you know that's the beginning of algebraic number theory in some sense that's part of what we what we do and and uh galwa galwa was the person who created the way of thinking about Roots

00:51:22 which is but although it's not the same galwa theory that people teach us not what gawa gawa nobody teaches what gawa actually wrote for example but building on that idea of what we care about what matters I I wonder whether you know um um uh what's his name uh oh my gosh I can't believe I keep thinking of P the um p no the philosopher the ancient Greek philosopher no uh Plato whether he sorry

00:51:55 whether he set us up for um focusing so much on um objects and existence and uniqueness and identity and what um and not focusing as much on how things are are different and sort of your work on sort of thinking of different ways in which things are the same and uh I'm thinking a bit about category Theory and how it is not um so much interested in just objects but in their transformations how objects are related

00:52:26 to other objects and if we if we take this category Theory idea in mathematics and apply it to our conversations it's not maybe the important thing isn't just what these ideas are but how they relate how they transform and that's part of the the package that's maybe more interesting in the end than is five the same maybe the more interesting question is what are all of the ways in which we now relate five to other things and if we follow um infinite category Theory

00:52:57 will eventually get from five to a Taurus you know somehow because there'll be all of these Transformations that will be there which reduces the um emphasis on identity you're you're moving towards the venin view right that um uh the meaning well for him it would be the language game the meaning of a word is really in some sense uh nothing

00:53:29 much or more than or at least extremely given by the network of relations that that word has to all the other words or the meaning of an object is the network of relations that that object has to all the other objects that are akin to it yeah in category Theory there is a um there is a statement that that says exactly that y's LMA MH um but uh uh I

00:54:03 so I said you're moving to are you you you moving to towards that or I don't think of my sign myself as Vicken Stein in I sort of more Whitehead in but so have to think about that a little bit I mean I think for Vicken Stein it is just a language game and I don't see it that way I see it as much more of a p language historical material sort of okay practices category Theory came up do someone like to take a crack at

00:54:33 explaining it okay oh me oh fine I'll try Okay so uh who am I I'm Jared Weinstein I'm a mathematician I'm from New York but that's one way of asking that answering that question but another way is uh I'm a brother I'm a son I'm an uncle and so forth so recognizing that objects don't come in isolation but rather studying their relations with all other um objects of a similar type the collection of all objects that have something in common

00:55:03 it's called a category like it's not used in its colloquial sense it has a specific meaning and I think lots of pure mathematicians maybe all of them agree this is a fantastic and Superior way of talking about things it's incredibly clarifying yeah the first time um someone tried to publish a paper on category Theory it was rejected by the Journal the um uh the editor who is a great number theorist

00:55:35 actually rejected it saying um uh that that is the most vacuous paper in mathematics I've ever read and the author said that's the point not car seag what not Carl lud seagull with a pig in it that was it no no no no you know who submitt this do you know who submitted ite

00:56:05 and and someone else and we always forget but isber in the clay it was called abstract nonsense that's how people that's right that's what we call it we call it abstract nonsense but it's the hood of so much of what we do the thing about it is that it produces a language and what's great about this language is that it has only two words well you might say three the two words

00:56:38 are noun verb and composition sure so I mean chumsky would love this by the way because he's uh has his notion of syntactic syntactical grammar but for him he thought it was syntax to know that um subject verb object was the initial Axiom for what a grammatical sentence consists of and

00:57:10 then you build on it by composition in some way right well the noun in category theory is called object the verb in category theory is called morphism let us say transformation for one object to another but uh transformation is a loaded word because you would think that you would see it it's just a label from the point of view of category Theory it's a labeled thing that you label

00:57:40 transformation and just with that and composition you can express uh as a language um essentially any mathematical Theory not only that you can make analogies between mathematical theories by taking this structure of for uh one Theory and seeing that it somehow is mirrored by the comp uh uh the uh

00:58:14 um the same structure for another theory and so that would identify two theories on the basis of this uh similarity in terms of these two simple words this tiny language anyway that's category Theory but the the movement I was asking previously about how abstraction which is one of the key points of mathematics has had its own Evolution over time and

00:58:45 yet now we've come to the point where people are questioning where does that lead mathematics in general is can you say anything about mathematics in general those sorts of question questions it leads us there so some of the solutions are these I would say more pragmatic Solutions I mean even including Vicken Stein's approach as a sort of pragmatism to it right and the solutions you're suggesting and the and this category Theory sounds like another instance of that so this is some sort of pragmatic approach um and is that the is that and then at the same time axioms

00:59:17 are involved in this form of pragmatism which one might think is different from just sheer abstraction you think it goes along with abstraction but in a sense it allows a form of pragmatism that may be the way math math is going these days let me take a stab at this this uh so on once the category theoretic language came in it became possible to read it into the previous literature or to R read the literature of the past as

00:59:49 if they were talking about uh categories in fact and then many of the a lot of the the the syntax the the the structure of arguments was common you know you could say that the uh that category Theory generalizes and uh uh formalizes a common pattern of mathematical argument and so that that you could say brings clarity now at the same time you can take this uh too far

01:00:20 you can nobody as far as I know is has gone so far as to say that uh uid was really doing category Theory but there was a controversy uh between historians of mathematics and mathematicians as to whether uclid did algebra so the the uh mathematicians wrote uh that clearly he was doing algebra because when we can when we read it we understand it using algebra and they star and say no that is an illegitimate uh uh projection onto

01:00:52 the past and I think even now now the majority of mathematicians would agree with the mathematicians of the time and the majority of historians and Mathematics would agree with historians so that's a another example of whether the concept remains uh REM is permanent or whether it is uh is evolved because if you reread the the concepts of antiquity in which in the only way you can because this is how you've been trained to to to read them uh you're uh you're no longer able to be in dialogue

01:01:23 with with ID or so so do you think that that's the only way you can that we can say it's the only way we can because you've been trained that way so the only way I can say I I mean I can also train myself to read uh uclid in a in a ra in in you know with the the gomans and everything but you know that would be that would be a a uh fairly painful uh experience and uh which is why I I

01:01:55 haven't done it right but what is invested in in so what is invested in as you said let's take either the example of category the or the example of the fight whether uid had algebra or not what is invested in arguing for of rereading the past with that language or what's invested in arguing for the fact that it was there is algebra why is I lot of people argue yeah I would think on the part of the uh the the mathematicians who were defending that

01:02:26 the it was there are two things first dogmatism because that's what they said and secondly they didn't want the historians to tell them what what what to think and that's also and that is a very present uh reaction I've I've heard mathematicians very good mathematicians even some who know the history saying well you know that you really have to be a mathematician in order to understand history I'm sure you've heard that yes God but there's politics invested as

01:02:56 well right I think that's what you're trying to get at go ahead no no was a question just I mean there there's the politics of who's who who invented the idea first was it us or was it them and there's the politics of assuming that we're always moving towards progress so that like whatever was before was obviously primitive and what's now is more evolved and that is a political position because it's about a story of post-enlightenment society and I I think that's very um relevant to me

01:03:28 why I'm interested in the as an educator in the question of permanence and impermanence is because I don't want there to be one um idea or concept of five because that really limits the um uh the the number of people the diversity of people and of cultures that have access to that as a part of their practice and so I'm interested in the plurality of and in what you said was really difficult you'd have to like relearn

01:03:59 something maybe that's what we have to be doing more of is engaging in this translation of you know different ways of thinking about five that are hard for us because they're not our natural ways of thinking about five anyway that's what I think teachers are are uh being asked to do as um we move towards more sort of Multicultural and culturally sensitive uh or appropriate education that so now that that can lead in one of two directions either the direction of

01:04:30 uh constant dialogue among the various uh interpretations of five or could lead in direction of of several mutually exclusive uh uh and uh incommensurable versions of five hard to hard to imagine but there are other kinds of mathematics where where uh each each uh culture would defend its own uh against against the Rival Notions you know and uh I

01:05:02 don't think that'll happen because we're way too mixed up already as cultures we don't I think we have the quintessential definition that was good there are interest in numeracy in different cultures you know and whether every every culture has a sense of number and and how high does it go and I think actually five may be around where some cultures cut off and beyond that that's considered many and I wonder if

01:05:33 you go back to what you were mentioning when you mentioned K before Barry whether this sort of synthetic quality of these numbers uh has something to do with how we think about them and how we might be creative about them one of the reasons to have people have a a diverse sense of what five means is that it may create the the basis for ination and for new ideas there is a five in K's literature uh and he uses it to a great

01:06:05 uh Advantage I think for his uh attitude towards what he called the the synthetic aiori so the the analytic a a priori for con is sort of topology definition so perhaps uh uh 7 + 1 is 8 wouldn't count as anything uh other than topology because that might be the definition of

01:06:36 eight but he has in the critique of pure reason this equation the only equation he has 7 + 5 = 12 yeah 7 + 5 = 12 you have to put the five on the other on the as the second as the second one yeah you know why yes okay that is um uh not a topology for him that is

01:07:06 what he would label as the synthetic a priori that is say it's it's something that you have to use your own intuitions uh in order to comprehend and of course I I think the reason I Ed five was this he would say there seven and then he uses five fingures yeah and just count but um that itself is a strategy and there are other strategies for getting this 7 + 5 = 12 equation but you

01:07:39 need a strategy and it's not a topology anyway well I'm wondering whether that's where arguments come in where we appeal to intuition and people have strongly different that there's room for wobble there could be well should we um open up the floor for questions if anyone has a question please come and step up to the

01:08:10 microphone would five of you please come up to the microphone well two and then three um as a former high school math teacher I'm really interested in thinking about how some of these ideas are being passed down to students and what might happen I must say that when I was a high when I

01:08:40 was a high school math teacher which was 40 years ago um I was very distressed that some of these abstract Concepts which I thought were fascinating stuff that I had learned in college colge um were absolutely invisible in the American High School math curriculum and that it was so algorithmically oriented that people had no idea of

01:09:13 these things even though there were very simple examples that could be taught and and followed through so you could get some sense of some of these abstractions like the idea of a group or a field and you could see oh yes there are all these different arithmetics if you because there are many different fields that have the kind of arithmetic structure that we expect of numbers

01:09:45 so where do we stand in that area that we all have a lot of opinions about how to change the high school math curriculum it's a very controversial topic right yeah yeah I mean so just to get us started I think one one uh thing we have the hangover of the new math right curriculum so it's going to be very difficult to introduce anything like group Theory or category theory in

01:10:16 high school can you remind us what was New Math so new math was uh I guess it started in the 60s um late 50s okay late 50s and um rethinking math um not so much as arithmetic but as um properties of of of groups and looking at uh um group Theory basically but using some numbers being a very technical thing is a very abstract notion yeah and sub

01:10:46 subsequent to development in in mathematics and to Baki and and that kind of thing so that was very difficult uh it was a very um great idea in some ways and but in terms of how it uh played out was disaster um in part because nobody thought to sort of inform the teachers of what that was going to involve and also um parents are actually one of the biggest determinant of what school math looks like cuz they for some bizarre reason feel like their kids

01:11:17 should go through exactly the same horror that they went through themselves so if the kids aren't learning multiplication tables there's probably something wrong with what's going on at school um so that that very arithmetic sort of focus as being like that that's what math is about is um what is prevalent I think in society and expect part of the expectation of what school math should be about and then there's like calculus like that's the where we're trying to get to all the time because that's the first class that

01:11:48 course that students have to take when they get to University so everything has to sort of vector to towards that which is a big algebraic apparatus then there's the depth of geometry died I was unaware do you have a geometer in your math department oh sure we've got plenty we have a whole research group like topologists or geometers a geometer sure well sure sure well um yeah there's plenty of you know geometers of

01:12:19 different Stripes represented absolutely yeah um but I just those are three I'm sure you have more you want to add to oh about the high school curriculum not I'm really I'm just not a high school educator I just know that and this mathematicians will relate to this I was out with friends and I made a new friend who asked me what I did and I said I was a mathematician can you guess the response oh I hated math with together with an I'm sorry as if I was expecting every

01:12:49 person I know to absolutely adore the subject oh I just hat it it sorry uh but then after then they asked well what does that entail and I said well I teach I teach calculus and I teach linear algebra and other things and then I also do research I make new math and the reaction to that was but hasn't everything already been discovered because what is the math education that you learn in high school it's um it's from a few centuries ago

01:13:21 generally and they kind of is oriented towards you the student solving the problem and getting the answer like I was looking at what my uh niece in n9th grade had a study for for her tests it kind of surprised me it was all the different centers that a triangle could have back to triangles a triangle I don't know if you knew this can has like seven different centers no like 3,000 oh 3,000 all right well my niece had to

01:13:52 know about seven of them those are the important ones and she had to memorize them and she had all these different pneumonics for like memorizing of how to name the different centers so that she could identify a picture of a triangle with that Center and pick out the name that's just not what math is I am not an expert on this subject I do you think yeah it's that way what

01:14:24 math day teaching school is the same as long time ago Bo is it that was I I guess these things are done by committee and they it converges on something that no one is happy with that's the only explanation most other subjects are probably done withes too what is special about math that's happened to I think math is also historically has been um associated with the way of reasoning that you'd learn math but you actually learn how to reason so it's not just about learning

01:14:55 the math that you're going to learn but somehow rationality itself is being developed as use study math it's that has historically always been the case people that's just there are many examples of that um I think that to some degree there is assumption I think that it's cumulative in a same way that I mean you can't I think M that's what happened with the new math which there's still the idea that you first need to teach students what are numbers how to add numbers how to you know there's certain kind of basic Ari arithmetical

01:15:27 properties that one needs to learn um and the one of the famous critique of the new math was why it's a book by mathematician um Martin Gardner has the why Johnny why Johnny can't add yeah that's the one I was thinking about why Johnny can't add which is idea you learn the idea of the critique was okay you taught my son my child all of this High you know High abstract concept of mathematics but then my child cannot add you like get like my child still like he understands all this number like all this concept supposedly

01:15:58 the mathematical concept but cannot just add two numbers um so I think we also have societal expectation of what we assume uh what we expect rather that a child should be able to develop uh sort of kind of mathematical concept mathematical ideas uh that we kind of expect a child going through let's say elementary school should be able to to do um choose the right cell phone plan but more cynically I mean math operates in our society as a gatekeeper so it's

01:16:30 it's uh the best tool we have so far for deciding who gets to go on to higher education in what field and how much money they get to make after that so it can cause more anxiety in in the students is that what you're well that definitely yeah goes back to the idea that it's associated with reason right like that somehow it's it's reflect that that's the way that there is no other ways of reasoning or that that if that's somehow the only way that one can uh be a a rational adult is my briers can show

01:17:02 that but if it was just reasoning then Jared could decide the curriculum and we would all think it was really beautiful but the fact is reasoning is hard to assess so it's much easier to just decide whether the students got the algebraic or the equation right or whether you know you can just check check check check it's it's a lot more difficult to assess people's reasoning so I think that combination of uh then it becomes not about reasoning actually and about a lot of multi um memorization you could

01:17:32 also say to be even more cynical because it's possible that it's a training in uh not questioning Authority if you if uh the authority is is vested in the people who control whatever the mathematics or or or or uh the internet then uh then if you have uh been trained that uh the ideal subject the ideal discipline is one in which there is one answer and you

01:18:02 use certain kinds of reasoning to reach that answer then uh you will maybe have a harder time developing the habit of questioning uh where where this The Authority for uh for decreeing this answer comes from I I'm I'm handing you a you know an opening if I know you have have uh written about such things yeah that's exactly it I mean I think I mean I think schooling in general operates that way in large part

01:18:35 to sort of uh train us all to be okay to sit in offices all day and not question authority and contribute to economic growth I think we have more questions from people online Alex would you go go over to the microphone and hey so I'm just the uh avatar for these online viewers um we have John sidley thank you for this question we has two but I'll pick pick one um so he quotes

01:19:06 Henry George uh forer the virtue of a logical Pro oh sorry not that one there was a better wait hold on okay they quote Michelle shy hopefully I didn't bungle that name Michelle sholi on cons critiques the aesthetic feeling of beauty is a way of feeling life not my life alone but life as it is shared by Humanity so how are human maths or mathematics and Beauties Co evolving what a good question that's good that's yours very isn't

01:19:39 it well how how those mathematics and Beauty evolved is that it is that the question that's the basic question yeah well I mean the title of one of my books was quoted particularly the square root of minus5 now cardano 16th century discovered he had to to solve uh

01:20:09 cubic equations had to uh involve himself with square roots of negative numbers and he said um here's this computation he wrote here's this computation um and uh dismiss and the computation needed aunk of minus5 and he said dismissing mental tortures go through this computation even though you're going to make use of the square root of negative numbers and

01:20:42 you'll end up with an answer which you can then check okay so uh he viewed or he thought his readers would view imaginary numbers as mental tortures on the other hand uh you can go to anyone in the so to speak modern world including um Fineman in one of his

01:21:12 books where he sort of makes sort of absolute um pays homage to imaginary numbers to the square roots of minus positive numbers and um uh so there's a and the beauty of it now there has to be some Evolution going from cardano to uh the modern world and the evolution will be in terms of its uh uh

01:21:44 the perception of beauty in in in some sense in the same object is that I mean we just played the same thing during the conversation today when we said that uh when you described category Theory you all talked about how incredible it is and how that's the language although as Barry yourself discussed when it came about people said this is just nonsense so there is there's a yeah well the nonsense is a compreh is an intellectual nonsense the

01:22:14 cardano he hated this number and nowadays and he hated it because he thought it was ugly mental torture and yet yeah one of the definitions of uh of beauty is a sort of our appreciation of beauty is a free play of the imagination and I think in some sense that that comes in here quite literally Natalie yeah I want to say something about the current state of mathematic Aesthetics of

01:22:46 mathematics um not so much just to highlight I mean I think um what what you said said there's this interplay between sort of uh personal um feelings of of of beauty and mathematicians differ differ really broadly in that like some people really love to find unity in things and some people really love to find the exceptions the anomalies the counter um examples and and all of that and Mathematics that so there's different sort of uh personal

01:23:17 aesthetic preferences let's say and then there's disciplinary ones that um I think are more broadly shared and you see this in the Arts as well um that change over time when there's some shifts like in category theory that you were just describing or around um um irrational numbers um and I think that uh those all operate um to in ways that um interface with Society um so how uh there could be some um aesthetic value

01:23:51 or beauty in being able to help people outside of mathematics understand mathematical ideas how what about that wouldn't that be exciting or to show um some of the dangers of taking certain abstractions too far and what choices do we make in mathematics um to re in those dangers uh Whitehead was always very careful to say beware of your abstractions they're dangerous as soon as you forget what the contingency is you're in a lot of trouble so how is mathematics taking

01:24:23 that to heart in in that that could be a disciplinary thing or it could be an individual thing but it it matters I think to broader society as well in which case it becomes political because it's about who gets excluded and included in mathematics well uh thank you and then the next one by viewer whose YouTube channel is written in Arabic characters I I can't read that I'm sorry but thank you for viewing and for this question so what they wrote is what is the relationship between the permanence and

01:24:54 impermanence of mathematical concepts with the permanence and impermanence of mathematical methods and then they sort of reference as an example computer verified proofs for [Music] methods well well certainly I'm not going to uh comment on computer verified proofs right now but certainly there does seem to be uh one of the motivations for that is to fix the meaning of the concept

01:25:27 once and for all if it has been translated into a uh into a program and the program has been run and and checks then that means uh two things first that the the argument is valid according to the system that's been programmed and secondly you don't have to uh you don't have to think about it anymore because because because there it is it will be there that that's permanent if the that if this if the language in which it's written is is is permanent um of

01:25:59 course that you can see that as a hindrance to uh to Evolution because the uh uh if you were to adopt a different uh framework a different kind of language and the different languages uh available with which people work are not mutually uh uh don't don't communicate with each other don't have the same uh the same theorems in fact then if you but if you fix one if you fix the one that belongs to uh to uh Elon Musk for example

01:26:32 because he's he's uh he he's he's decid he's hired the person who is planning to solve mathematics so if that's if that's the uh if that's the the version of mathematics then you can imagine that that the the future uh there will be everything will be permanent uh like in in the way that death is permanent uh I mean that's that's that's one reaction but there could be others as well anything anyone

01:27:06 else yes please have question no no I mean I'll I'll keep an eye um I wonder I first I must confess that I I came to um to appreciating the beauty of mathematical thinking relatively late this Inner Room but I came through it through music and I wonder what you all have to say about this because I feel like

01:27:38 um mathematical thinking is very much like composing in music and to to get to the number five if you combine five violins or five trombones it still it will sound very different and both of the fives if you will have very different properties and I wonder if we in education just let students um recompose done compositions

01:28:08 or whether we give them the tools to compose their own and whether we have this aesthetic of um very much like this break between classical music and modern music if we're now at in a point in life where we can um Define a new aesthetic but through play rather than uh driving rules yeah um thank you for that I um I think uh there was in the early 90s a book

01:28:38 called The Art of problem posing by um no no um problem posing not problem solving that's the important part uh by um step Brown and um um Marian Walter and um it was specifically to get to this idea that uh all of math is basically about you solve my problem okay and so it's a bunch of exercises that somebody

01:29:09 else came up with and you it's just like if you had to play everybody else's music and actually most mathematicians would agree that one of the most creative parts of math is coming up with a problem posing a problem yeah um CU you get to set the terms and then you get to decide what you know a solution is so that book was giving teachers tools to how to encourage um their students to pose U their own problems and and when you do that you end up sometimes um encountering mathematical

01:29:40 ideas that definitely don't have are not in the curriculum and could be even Beyond uh you know their their level of understanding but not always there's ways you can sort of manage that in the classroom and there there are definitely a lot of people who um took that up and took it seriously as a part of humanizing part of mathematics and showing kids that like those kids who did that would not be surprised that there's new mathematics because you know they were making new math right and um but again it's something that's a bit

01:30:11 hard to do as a teacher if you're given a curriculum that you have to stick to on the one part and on the other part if you're not feeling very confident about what math is going to come out of this CU it's hard sometimes for teachers to say I don't know sorry um so it sometimes doesn't work very well I think with our educational system but I think there's a a lot of pockets of that that you can find um people doing you know music and Mathematics doing art and Mathematics uh um going out in the land

01:30:41 and doing mathematics there but they're they're small pockets and they tend to be at the elementary school level where the stakes are a little bit lower yeah um so I have a question that's directly related but just a just a few ideas um when you talked about the new math you reminded me so I'm Mark Mitten I'm a magician and I have a lot of friends in math through the Martin Garder Network and um so um I remember taking a long

01:31:13 walk in Princeton with the late Joe con and um he said you know what's terrible he said I knew all the guys that created the new math I knew them each personally and I know that each one fell in love with math through geometry right and so I thought of that then I thought of of your question just now I had the honor of um supporting Barry student mul burgara at Princeton in a math and Magic class and at one point he turned to the kids he goes you know this steam thing I don't really get

01:31:44 and because you know I guess I I fell in MA fell in love with math through music and poetry and so this gets into this thing of I wonder if part of the the focus on permanence is somehow related to this focus in culture on on novelty that comes in sometimes I call um science the negative space of Art and there's this been this big push in in science and and and art and then I I I'm a Craftsman I I

01:32:16 studied economics at college but then I get to study with old my old favorite magic Masters for many years in their 80s and 90s and it changed my life and um so in a in a a late night dinner with like Conway and Joe con and and um silven Capel I I gave this idea are we all getting confused is is the essence of mathematics much more like a craft and silven Capel said you know I think you're really on to something because I remember when I was a grad student there

01:32:48 was a grad student spinning his wheels and he couldn't get it together and Israel gelfant yelled at him he said don't you get it math is not headwork it's hand workk anyway but that's it's kind of a question I've been having this fantasy based on your reading your blog because you said you cited somebody who said that the the um the the one job that we're going to lose first because of AI

01:33:19 is the mathematician job that was a uh you cited it in one one of your that was not a a person who said that it was a study right and it was a study based on the characterizations of jobs by I think it's the Department of Labor they have a and they break down jobs into tasks and um so there's a list of nine or 11 tasks that make up the job of mathematician uh most of which uh most

01:33:52 of which uh I've never encountered or uh but that's the way that's the way it's understood so of course what what a mathematician is is not just you know if if if the NSA claims to be the the world's largest employer of mathematicians then maybe those people are doing that job right that that I we don't know the number because it's classified but the uh but it's the largest number whatever it is uh but but

01:34:23 then the people who wrote this study uh who are in AI for the most part admitted that they did not uh that they had hired some team of matching uh the tasks to things that uh what whatever AI can do whatever chat GPT in particular can do and uh said well this looks like this this job looks like what something that cat GPT can do but they admitted that

01:34:54 they were not experts in this field they didn't talk about mathematics in particular but they just nevertheless they they they uh there is the the problem with the study itself although they acknowledged that there could be these problems then there was a problem with the the the article in the press that synthesized this to say that the mathematicians are the job most at risk and somebody wrote to me uh somebody in New York wrote To Me look we're the most at risk so you know but that's uh but

01:35:26 that's not is that that I don't see how that follows from what Mark Mark's question though so I may well it's related to a thought experiment is if math wasn't the gatekeeper subject of our curriculum then it probably would be more like art and you could probably convince yourself of that by wondering what would art look like if it was the gatekeeper subject of our curriculum there'd be a lot of tests on whether you could draw straight lines or I don't know stuff like that perspective right yeah probably I

01:35:59 think it's not just the the in the curriculum itself I mean the example you just gave that all the people that actually uh work and started the new math for themselves fell in love through fell in love in mathematic through geometry I had this moment and I write about in book and one of the great mathematician of the kind of mid 20th century uh called stin Rod um and I looked at his paper when he was a student and it's all Di diagrams it's all like he's working on these curves he's like writing this P really passionate long letters about these curves um by the end of his career he

01:36:30 becomes known with kind of most abstract algebraic category Theory approach all the diagrams have completely disappeared from his writing you don't find it anymore and part of it is is the incentives of the field itself uh it becomes that the if that's the dominant within the kind of community if that's kind of the dominant approach you because there's an incentive structure uh you fall in line with the incentive structure of the community and they're all there always with mathematician even in the height of boraki that were doing

01:37:03 different approach right that were kept uh this kind of uh um more geometrical approach or this geometrical investigation but they were often often relegated in some way I I don't know how to quite explain I talked to I talked to another mathematician who who Des desing to me that uh his name is uh Thomas benof is I think he retired from Brown by now but he he was describing to me he wrote a lot of books about it uh that since he was a kid he was fascinated by the fourth dimension

01:37:35 he had this like he you that something that kept him up at night thinking about fourth dimension trying to think how can you think about the fourth dimension who would build this models like threedimensional models and draw and did all stuff and I remember that I had an interview with him and he told me that when he was coming into the field he felt that the way that he was trying to do math had no space because the dominant approach for the field was much more abstract uh and he kind of it was very hard for him to figure out how to um that's how we describe it how to find

01:38:06 you know a place for the sort of mathematics that he does uh so I think there's also kind of institutional um and structural uhthe as well to all of this but yeah there's a story about chali who was giving a lecture in a big uh undergraduate classroom who chali had to complain about the yeah so he's writing out his his uh proof of the theorem and don't know which one and he gets stuck at a certain

01:38:37 point and is like pacing and pacing and turns his back to everybody in the classroom makes a little diagram on the board erases it turns around and keeps on the doing the proof and it's obvious right well excl excluding himself from the yeah possibility of using diagrams that's true that's true hi I'm not a mathematician not a scientist but in school I really loved math and

01:39:08 um and I will I'm from former Soviet Union and Soviet Union M actually in high school and middle school was on college level here and um my question is uh probably I'm not sure if you would like it if you like this question about limits of M limits of this particular language for example you were saying about uh sometime long ago we thought

01:39:40 that sun goes around earth now we but we still say Sunset yes you see and in poetry if you read The Poetry you will never meet explanation no this is actually wrong it's Sunset is wrong you see so so to um come to some ideas we have we have just everyday language and it's really more permanent maybe than M

01:40:11 and like separate science Fields And even right now some people start say that maybe we uh come to too much of calculation because right now we have very difficult relationship with ecology with nature should we come with calculations or should we uh to to this problem to like climate change and

01:40:41 everything should we do it more like indigenous people you see who try to feel it and try to kind of be together not separately and calculate or for example if you try to optimize uh uh and calculate relationship with your children how it would go you see so so what could you say about limits and probably the for example calculations of

01:41:15 CO2 yes right now people start to say what we should reduce it and then we solve everything and then we come idea of car uh carbon um storage somewhere and some people say it's it's bad idea because it's we capture it and store it and it's create more problems so that's how that's the question about limits that's it and now and how it's go goes

01:41:45 to our relationship with our environment than I'm so glad I am not in charge of fixing the climate instead I do math I do not harm the world I create more of it and I communicate with it and it is limited in that sense like it's this very fundamental human thing I don't think it claims to be able to solve conflicts between people like a parent and a child you mentioned

01:42:16 um I think the question in that in essence was like what have we lost by thinking so mathematically all the time um but mathematics has definitely created um there are definitely mathematical tools that exacerbate um the current climate situation and um I'm thinking too about the book by um Kathy O'Neal called um the um destruction weapons of math destruction yes um so

01:42:49 it's not any mean mathematici Ian who has decided to engage these weapons of math destruction but the practice of mathematics of always um taking away the context and operating at the level of the abstraction where context often really matters in the real world when these algorithms for example are getting used to determine the lives of teachers and the livelihoods of of of people in general so I I do think it's related too to this aesthetic and political issue

01:43:19 that we were talking about is and you know the the role that math and not mathematicians but that math plays in um allowing us to make generalizations that wreak havoc on contextual issues you can't divorce a way the humanities from the mathematics just no you're trying to build a better Society you need both an example like the rise and fall of cryptocurrency and who is it Sam bankton freed it it represents like seems like a class of people who are going

01:43:51 to maybe create a try to create a Utopia by using math basically but I mean math is baked into how crypto works in the blockchain but they have forgotten about ethics and politics and power structures and Justice and everything so you I mean you talk about the limits of what math can do in society they are definitely there it is limited I you know your your comment makes me wonder whether it's ever

01:44:22 happened that children have sued their parents for uh bringing them up without calculating on on the grounds that economic rationality would dictate as as it's as it's defined by the people who Define that sort of thing uh would dictate that they uh base their upbringing on what will maximize the their uh their

01:44:53 uh their economic their property as when they become adults now I don't know I can imagine that that some lawyers would be willing to take on uh a case like that and uh what would be the counterarguments because uh the power is organized primarily by the people for whom economic calculation is uh the most important consideration

01:45:24 so what would you know what would uh so that but I don't think that's the ma the mathematicians are particularly responsible for that you know this is a this is a structure that the mathematicians have provided but I think uh and I I I would not I don't know that I've ever heard of any children bringing a suit against their parents but but it's not number and calculation that are the problem indigenous people use numbers too right so it's a it's what you said before I think think is the splitting of the mathematics from the ethical questions and the sustainability

01:45:55 questions so if if you're thinking in IND indigenous way you're thinking about well what are you're working in a collective way and discussing what are going to be the um consequences of using these mathematical Tools in terms of you know other things that we might care about so maybe it's not so much about mathematics but the way that we've siloed these different disciplines into you know places that can flourish and become really powerful but yet not be in

01:46:26 in uh communication with each other around overall goals well there I mean real there are real really economists who discount the future uh in the future effects of uh of uh uh climate change you know they they calculate that in fact it's uh more economically rational not to do anything about uh or not to do very much because if you consider the economic growth in the present that outweighs or you know building on it on

01:46:57 it then that outweighs whatever consequences it might have for so this is this is not purely hypothetical the uh and but I'm not again it's not the math it's not because they're using mathematics they're using mathematics in a certain way well can I sh one thing because I I mean I think you're right about it's I mean it's not because they're using mathematics but I do I think that um often with the use of all of these mathematical methods uh part of the problem is that because mathematics in

01:47:27 our culture is associated with objectivity and with true knowledge uh there is an there is a willingness to if something is presented in a certain kind of way with mathematics there's a willingness to assume that this is somehow representative of the truth of something that this is an objective study even though I absolutely agree with you that it's it's always limited I mean I teach right now I'm teaching an entire course course it's basically on that question which is trying to show how quantification is is a limit is have changed the way we think about different kind of social problems um and it's the

01:48:00 limits of it but I think there is something about because math because the place of math in our culture that then kind of gets translated to also its uses in some way because math is related in this culture you think that ethical Consciousness related to math is sort of under under state under mind under thought or something that's a good question I don't know if I it's not so this is a good question and I'm not quite sure if that's the the like that's

01:48:31 the link that's the the ca link rather there that I think that when we have become accustomed to the idea that if something is represented in a numerical in a very kind of Quantified way in a way that there's a calculation behind it we we we tend we we all tend we tend to agree that this is somehow a more accurate a more objective uh description of a social problem let's say value free it's a value free it's an objective uh and I think that as opposed to if I told

01:49:01 you about that problem and I will Spin and I will tell you about this person whatever the social problem and I will tell you stories about it there will be that will that's way of narrating a certain kind of problem the same s the same problem um has less legitimacy than uh translating the problem to a quantifiable and calculative Logics uh and I think that that's the problem that because because uh because we tend to associate certain kind of uh ideas with with those mod of modes of

01:49:32 visiting well I think um this um fabulous conversation was now will become part of a permanent record or at least as long as you at least as long as YouTube exists so I'm uh really so grateful to you all for this amazing uh talk and uh look forward to having you here back again sometime in the future thanks so much thank [Applause]

01:50:19 you