Theories of Everything with Curt Jaimungal - Dror Bar Natan on Knot Theory, Learning from Ed Witten, Eric Weinstein, and Quantum Field Theory

Episode Date: September 8, 2021

YouTube link: https://youtu.be/rJz_Badd43c Dror Bar-Natan is a Professor of mathematic who's interests include knot theory, QFT, and Khovanov homology Sponsors: https://brilliant.org/TOE for 20% off. http://algo.com for supply chain AI. Patreon: https://patreon.com/curtjaimungal Crypto: https://tinyurl.com/cryptoTOE PayPal: https://tinyurl.com/paypalTOE Twitter: https://twitter.com/TOEwithCurt Discord Invite: https://discord.com/invite/kBcnfNVwqs iTunes: https://podcasts.apple.com/ca/podcast/better-left-unsaid-with-curt-jaimungal/id1521758802 Pandora: https://pdora.co/33b9lfP Spotify: https://open.spotify.com/show/4gL14b92xAErofYQA7bU4e Subreddit r/TheoriesOfEverything: https://reddit.com/r/theoriesofeverything LINKS MENTIONED: PDF link of notes: https://tinyurl.com/4u4ru2yh TIMESTAMPS: 00:00:00 Introduction 00:03:29 Quantum Field Theory and Knot Theory 00:10:19 Another QFT / Knot Theory connection or analogy 00:20:46 Knot Theory and Gravity + Bar-Natan's PhD on Chern-Simons 00:24:08 Homology and Cohomology 00:32:54 Dror's average day / routines / laziness 00:35:23 What is "work" for a Professor? How much of it is thinking vs. reading etc.? 00:36:50 All the research is to support teaching 00:41:34 Motivated examples are what's missing in math teaching  00:47:35 Are you more creative when younger or older? 00:48:43 Being "systematic" with learning and research 00:50:37 Rant against theorems and proofs (emphasis) 00:54:51 What understanding a mathematical concept looks like 01:03:35 On laziness, strengths, and weaknesses in mathematics 01:07:47 Advice for the struggling mathematician 01:09:30 What did Dror learn from studying with Ed Witten 01:12:12 On Eric Weinstein 01:19:32 Knot Theory and Physics (again) 01:20:10 Why can't physics of 2 or 3 dimensions be extended to 4 easily? 01:34:21 Hard Problem of Consciousness 01:35:54 The problems Dror thinks about daily 01:37:49 What direction would he like mathematics research to head in? 01:40:26 Philosophy of math (ultra-finitism) 01:44:45 Pure mathematics in service to applied mathematics 01:47:14 How to learn a new field of mathematics 01:47:15 [the scientific mystical philosopher] Unreasonable effectiveness of mathematics 01:48:28 [Romain Gicquad] On IQ 01:51:06 [Tori Ko] ​What do you think about biology / psychology / philosophy 01:54:03 [Phill Thomas] What led him into the field of mathematics? Can anyone can learn advanced math? 02:03:52 [Tyler Goldstein] Do we need to make new research institutions not linked to academia? 02:06:04 Bias in academia 02:10:15 What's a mathematician? Who's a mathematician and who's not? 02:14:31 [Ashley Shipp] Does he think there is a connection between knot theory and protein folding? 02:19:49 [Jack Dysart] Is math a human construct? 02:22:50 [cx777o] Does he believe in a concept of god? 05:19:49 [Bill McGonigle] Ramanujan and his Goddess 02:23:47 Revealed story that Dror has about Curt and the Prisoner's Dilemma 02:30:27 [Harinivas P] Bible Code and applying advanced math in one's life 02:37:26 [Roy Dopson] Why does it take 360 pages to "prove" that 1+1=2? * * * Just wrapped (April 2021) a documentary called Better Left Unsaid http://betterleftunsaidfilm.com on the topic of "when does the left go too far?" Visit that site if you'd like to watch it.

Dror Barnatin is a professor of pure mathematics at the University of Toronto, previously specializing in knot theory, and more recently specializing in finite type invariance and Kovanov homology. We touch on quantum field theory's connection to knot theory, how to learn mathematics in general, we also touch on Ed Witten as Dror Barnatin's advisor for his PhD was Ed Witten, and Dror gives some anecdotes about Eric Weinstein, as while Eric was pursuing his PhD, Dror served as an informal advisor to Eric at Harvard. Professor Barnatin taught a course in linear algebra that I took while I was at the University of Toronto, although I was a terrible student and I regret that I skipped the majority of his classes because he

had an exceedingly intuitive approach. For example, there are vector space axioms, can I, I do, and they seem extremely disconnected from any reason or motivation. So what the professor would do would be before introducing a particular abstraction, he would say, imagine seven times a vector instead of imagine a times a vector, and then generalize it, which gave the abstraction some concreteness. For those new to this channel, my name is Kurt Jaimungal. I'm a filmmaker with a background in math and physics, dedicated to explicating what are called theories of everything from a mathematical physics perspective, but also the possible connections

consciousness has to the fundamental laws of nature, provided these laws exist at all and are knowable to us. This particular podcast isn't one to be listened to, or at least you can, but you're going to be frustrated as what's being referred to are one-dimensional objects embedded in a three-dimensional space in some configuration, and the verbal description leaves much to be desired. If you're listening to this on Spotify or iTunes, I recommend you click the link in the description and follow along. You can also download a PDF of the written notes from this podcast in the description as well. There are a couple sponsors of today's podcast. The first is Algo, which is an end-to-end supply chain optimization software company with software that helps business users optimize sales and operations, planning to avoid stockouts, reduce returns, and inventory write-downs

while reducing inventory investment. It's a supply chain AI that drives smart ROI, headed by a bright individual named Amjad Hussain, who's been a huge supporter of the podcast from his early days. The second sponsor is Brilliant. Brilliant illuminates the soul of mathematics, science, and engineering through bite-sized interactive learning experiences. Brilliant's courses explore the laws that shape our world,

elevating math and science from something to be feared to a delightful experience of guided discovery. You can even learn group theory, which is what's being referred to when you hear that the standard model is based on U1 cross SU2 cross SU3. Visit brilliant.org slash toe, that is T-O-E, to get 20% off the annual subscription. I recommend that you at least try the free trial and don't stop before four lessons and you'll be greatly surprised at the ease at which you can comprehend concepts that you previously had a difficult time understanding. If you'd like to hear more conversations like this then do consider going to patreon.com slash kurtjaimungle. There's also a crypto address and a paypal address if that's more up your alley. I now do this full-time and the patrons are what

allow for me to put out this content on such a frequent basis. Every dollar helps, certainly, and regardless, I'm grateful for your viewership. Again, that's patreon.com slash kurtjaimungal, C-U-R-T-J-A-I-M-U-N-G-A-L.

Thank you, and enjoy. I actually had you as a professor, I believe, in linear algebra. Mm-hmm. I hope you didn't suffer too much. My favorite aspect of the course was that you would use the word seven. You give an example of a number. So you'd say a certain field number times a vector.

Well, let's call it seven. So seven times a vector would be so-and-so. You know, it can get to extremes, right? You may end up doing things like the limit of e to the minus seven, seven goes to infinity, is equal to whatever it is, zero, right? Yes. What's your field of research? knot theory and from a very algebraic perspective and related algebra things. It used to be much more knot theory and quantum field theory, and it slowly moved to being much more algebraic, and the relationship with quantum field theory became

more and more remote. Okay, what's the relationship, let's forget about the remoteness, what's the close relationship between knot theory and quantum field theory? It's a bit hard to explain and somewhat surprising and that's why it took a long time to discover. Let me give you two answers that are somewhat unrelated. And one answer is that quantum field theory is about particles. that quantum field theory is about particles and knot theory is surprisingly also about the motion of particles. So, you know, let's draw a simple-minded knot. So, by the way, is this visible, my screen? Yeah. Okay, so here is a simple-minded knot.

It's actually the so-called trefoil knot. But, well, whatever it is. Now, what you can do is you can... so it's living in R3 and I could set up my coordinates so that this coordinate will be called T. Maybe you want to call it Z. But you know I like to call it T right now and the two coordinates here are X and Y and then I can slice this knot at equal time slices so basically I draw a plane which lives at some fixed height, so at an equal time plane, and slice the knot, and then the slice is four points. And as I move these points in time, so as I move the slice in time, the points move,

and they actually do a little lovely dance. And the dance is as follows. So if we call these points A, B, C, and D, then they start at a certain position, A, B, C, and D, then they start at a certain position, A, B, C, and D, right? And then as I move, well, you can see that C and B trade places. So they play a dance, they somehow dance around each other. So they play a dance. They somehow dance around each other

B goes this way and C goes that sorry B goes One way and C goes the other way and after a little while they trade places. So you have a C sorry a C B D and as you move the plane even farther they trade places again. So again you will have A, B, C, D. And if you move a little farther again they trade places. So you have A c b d and then something funny happens uh uh points uh well points well by this time this is point b is over here and point c is over here and points a is here and D is here, and what happens is that point C moves towards point D, and points A moves towards point B, and they coincide, and when they coincide they annihilate each other, because if you look at the plane which is above the level of the knot, you no longer see any points.

So what happens next is that these points move towards each other, these points move to each other, they meet and bang, they annihilate, and after that there is an empty set. And in fact I started at height, at a certain height had I started below, I start with the empty set, and instead of annihilation of pairs, you get a creation of pairs. So a pair of points get created here,

and a pair of points get created here, and then they perform this nice dance. So this was this particular knot but the same story is true for every knot. Every knot can be written as a certain dance of points in which sometimes pairs get created, sometimes pair meet and annihilate each other, and in between these times they all dance. Anyway, physics in general is about the motion of particles and it's not too surprising that there is a relationship. Physics also sees things like creation and annihilation of pairs of particles. And, I mean, so again, it's not so surprising.

But this is very, very, very loose. And, gee, the other explanation I can give you is even more loose. But in some sense, more closer to the truth. Well, okay, maybe you've heard that quantum mechanics is related to path integrals. So, you know, in classical mechanics, you know, if a path goes from point A to point B, sorry, if a particle goes from point A to point B, it goes along a specific path. In quantum mechanics, it has a probability of going from A to B, it has a probability of going from A to B, and you compute this probability

by doing a certain integral over the space of all paths. So the particle, or I mean, even my interpretation of quantum mechanics here is a bit loose because interpreting quantum mechanics is a very hard thing. But with this loose interpretation you sum over all probabilities of going this way or that way or that way or that way and by adding up all these probabilities or more precisely integrating over the space of all of those probabilities or more precisely

phase factors or amplitudes, you can compute the probability of going from A to B. That's quantum mechanics. Now, likewise quantum field theory, but in quantum field theory it's essentially the same except the integrals are more complicated. But the point is that already these integrals are not like integrals over Rn, right? Rn is the collection of n tuples of points, a1 up to an. So a point in Rn is determined by n real numbers. Whereas a function, the space we are integrating over now,

is the space of all functions. And a function is determined by infinitely many real values. You need to know the value of the function here, the value of the function here, the value of the function here, the value of the function here. So to determine a function you need to know infinitely many real numbers. So the integral that you end up doing is not 157 style, sorry now I'm using Toronto style language, it's not one variable calculus's not second-year calculus when you do n variable integrals. It's infinite dimensional, so infinitely many variables integration. This is what we need. So, quantum field theory ended up developing a bag of techniques for computing infinite

dimensional integrals. And what these techniques are is a whole different story, but quantum field theory ended up doing this. And then it turns out that infinite dimensional integrals are useful to topology. So the way that quantum field theory becomes useful in topology is indirect. It's not that the quantum field theory itself is useful. Even this is a lie because sometimes the quantum field theory itself is useful. But the main path by which quantum field theory becomes useful to topology is not direct but a bit indirect. Quantum field theory developed a bag of techniques and this bag of techniques applies elsewhere.

Anyway, why are infinite dimensional integrals useful to topology? The answer is again a bit complicated, but let me try to summarize it. So let me bring a knot. Here's a knot. Okay. Great. Well, actually, that's not a knot. Why not? What I'm showing you is a curve in R3.

what I'm showing you is a curve in R3. A knot is an equivalence class of such curves. So this is a curve. If I deform it a bit, it's still a curve, but it's considered the same knot. If I deform it some more, it's still a curve, but it's the same knot. If I deform it some more, it's still a curve, but it's the same knot. So a knot is not a single curve, but an equivalent class of curves. Now suppose you want quantities that are associated to knots. You want to be able to tell knots apart, so you need quantities that would be computable

from the knots and that you would use them to tell the knots apart, okay? Now, it's very easy to compute quantities that are associated with curves. So if you have a curve, you can measure how much it bends, so the radius of the bending. Okay, again, these things have technical names, the curvature, maybe the torsion. There are various names for various quantities you can associate with curves.

for various quantities you can associate with curves. So, you know, I can look at the maximal curvature of this knot, of this curve, and the maximal curvature will be the place where it turns the most, and that's probably around here, for this particular curve, around here. Okay? Now, but that quantity is a quantity which is associated with curves. It's a useless as a quantity for knots. It's useless as a quantity for knots because if we're talking about knots, you know, I could deform it and the maximal curvature could go somewhere else, could change, could

grow, could shrink, could differ. Right. So the way to get quantities that are associated with knots rather than with curves is to integrate over the space of all geometries. the space of all geometries. So instead of picking the quantity for a specific geometric manifestation of the knot, you integrate over the space of all geometries and then the result will be a quantity associated with the knot. Unfortunately, the examples are very very hard hard. So, you know, the famous example, the example that got written, well, one of the examples that got written very, very famous in topology or among mathematicians

is the so-called Chern-Simons theory or Chern-Simons quantity, Chern-Simons theory or Chern-Simons quantity, which is in itself some integral. So you integrate over R3, so the quantity itself is a finite dimensional integral, right? So you integrate over R3 something called the Chern-Simons form which is something called which is a wedge da plus two-thirds a wedge a wedge a whatever that is but this is you know related to stuff appearing in second year multivariable calculus in the most advanced versions. Okay, so wedge products and differential forms and exterior derivatives and stuff like that.

Then you take this quantity and you multiply it by i, multiply by k, exponentiate it, multiply by another quantity, the so-called holonomy along the knot of A, whatever that is. So I'm throwing names because I really have no choice because giving the precise definition is a matter for a graduate class. Yeah, yeah. Okay, and this is the quantity that you associate with a geometric knot. quantity that you associate with a geometric knot. And then to get an invariant of knots or to get a quantity associated with the knot itself, you integrate it over the space of all such a's. And I didn't really tell you what these a's are, but it's an infinite dimensional space. tell you what these A's are, but it's an infinite dimensional space. So I guess I haven't really answered your question by giving an example, but I just told you

the examples are hard. Okay? Anyway, well they're hard, but they're exactly of the nature that physics knows how to study or that quantum field theory developed techniques to study. And so one can study it using quantum field theory techniques and get all sorts of, well, quantities relevant to naught. The so-called Jones polynomial is an example,

and then there are many other examples. Are there any relationships between not theory and gravity, so general relativity, let's say? I'm not an expert. The answer is yes. But don't ask me to detail it. As I told you, I've actually moved away from that subject. What got you interested in knot theory originally? This subject.

So mainly you were interested in physics? Well, okay, I was a math undergrad. But then during my time as an undergrad and then a master's student in Tel Aviv University, I never finished the master's. I moved on to doing my PhD in Princeton. But during my time as a master's student in Tel Aviv, I got more and more interested in physics. And there is just wonderful mathematics inside physics. So I got interested in physics and the fundamental question of physics, right? What are the basic laws of the universe? I mean, you can't ask a better question. So I got more and more interested in physics and decided to do my PhD in mathematical physics. And I ended up working with Ed Witten, who is a very famous physicist or mathematical

physicist or mathematician, depending whom you ask. And originally, my interest was in physics. Or, you know, my aim was to do things that were relevant to physics. Somehow, that's not what happened. You know, at the end, you don't choose your career. Sort of the career, you don't choose your career. The career leads you. So I wrote a thesis which was somewhere in between math and physics. What was it called? What was it called?

Perturbative Aspects of the Chern-Simons Quantum Field Theory. And that's an approximate title because I don't exactly remember. But then it turned out that that was relevant to something called finite type invariance, and I thought I had new things to say about finite type invariance, and that's completely a non-theoretic subject. So I told myself I'll give myself a break from physics and write a paper or two on finite typing variants and then come back to what I really care about. And that paper or two became 10 years with several papers, many papers and many results.

And then a chance meeting led me to yet another subject. And, you know. What was the other subject it led you to? Well, no, the next subject after finite type invariance was Khuvanov homology. Anyway, you don't choose what you do. You know, you at most try to steer it. But it steers you much more than you steer it, like the subject. Do you have an intuitive explanation for people who, let's say they're at the second year level,

what is a homology? What's homology? It's a tough, a very tough question because it's many things or it's related to many things and I could answer it in many levels. You know, let me give you a summary answer. Sure. Okay. So often in mathematics, there are two ways to describe objects. You can describe them implicitly by telling what equations they satisfy, or explicitly by constructing all of them. Let me give you the simplest example.

So consider the unit circle. Sorry, I kind of care about the points of the unit circle. Sorry, I kind of care about the points of the unit circle. So here is the unit circle in the xy plane and here is a point xy. The implicit description of the circle, so the implicit is by writing an equation which the points have to satisfy and everybody knows what the equation is the equation is x squared minus sorry plus y squared minus 1 is equal to 0 okay but there is also an explicit description and again

I'm not sure which one is why the the naming is such, but it doesn't really matter. Namely, I can construct all the points of the circle. And the construction is you set X is equal to cosine of theta and Y is equal to sine of theta. Okay? Now, this is the general feature in mathematics, that many things can either be constructed, that's the explicit, or be defined by an equation. Now if you restrict your attention to linear algebra, if you do the same thing but in linear algebra, then the situation is that you have some space in the middle, maybe it's a vector space, and inside that space you want to describe

things. Actually, I'm not sure why I'm drawing it curved, because if it's linear algebra I should be drawing it straight. But anyway, you want to describe them. One way is by saying these things, well you construct a map into another space and you say these things are the things in the kernel of this map. So I don't know what let's call this map Phi and these things are the kernel of Phi. And then so this is the implicit description. This is you describe the things by an equation. And then there is also the explicit description, so you have yet another vector space, and you have a map, let's call it Psi, and you say the things here that are, well, the good

things, like the points of the circle, are the image of Psi. So you say these are the things that are the image of Psi. Uh-huh. Okay? And this corresponds to the explicit view. You construct the things, you present them as an image via some operation coming from some, you know, maybe some smaller dimensional space. Okay. So, okay.

Here's a funny thing. So, if your construction is reasonable, then everything you construct solves the equations. In other words, everything in the image of psi is contained in the kernel of phi. So once you have a situation like this, you know that you're going to have that the image of psi is contained in the kernel of phi. And in fact, if you're constructing the exact same things, then the image of phi will be equal to the kernel of phi, not just contained. Okay? And then you will be saying that the situation is exact. Things got... exactly the things that you've constructed are the things that you described by the equations. But that's not always the case. Sometimes you have equations defining something

and you want to know, did I construct all the solutions? So sometimes this inclusion is strict and then you want to measure by how much or how much did you fail. So how much did you not construct everything? And that's exactly the difference between the kernel of phi and the image of psi. And in linear algebra, the difference is quotient. So you take the quotient space. And this is the so-called homology. So somehow homology measures,

called homology. So somehow homology measures, so this measures how good are your constructions. Did you construct everything or did you leave some stuff out? Is the kernel, the implicit, the description, does it contain more things than you've constructed? So somehow homology measures how good are your constructions. And this is true in general. And then there are lots and lots and lots and lots of specific examples, and to some extent it's a bit of magic. I'm not sure I can give you a really good explanation why there are so many examples and why so much of mathematics end up having close relationships with homology. Maybe this is a partial explanation. So basically you always want to measure how good is what you've done.

Are you able to talk about what cohomology is and then homology's relationship to cohomology? Or is that a whole other can of worms? It's another can of worms, but in some sense it's not interesting. In some philosophical sense, it's not interesting. So when I described cohomology, it's essentially the same. So the difference between homology and cohomology, more than anything else, is a difference of conventions, goes from left to right or from right to left. It's from this perspective, from this very far away and very imprecise perspective, there's really no difference. Professor, what do your average days look like?

For example, do you wake up at the same time? Do you drink the same type of, do you drink coffee? And it has to be this particular type of coffee. Do you eat a certain breakfast? What does it look like? Paint the drawer picture. I'm a believer in the, you know, there was at some point a revolution, right? Before that, people used to work seven days a week, 12 hours a day, or maybe with a short break on Sunday morning, depending which culture you came from, I suppose. And then at some point there was this workers' right revolution. And somehow the working week shrunk to 40 hours a week or so, like five days of eight hours a day. And I'm essentially a believer in that. And furthermore, I'm a lazy bum.

in that. And furthermore, I'm a lazy bum, so during my eight hours of work a day, five days a week, I take coffee breaks, I, you know, browse the web, I cheat. But anyway, you know, I wake up at 630, I exercise, take a shower, have breakfast, check my email, do all sorts of unimportant things. Then around 9 usually I start working. I work until 5 or so. I'm sorry, I'm very dull. and then I try to have fun for the rest of the day whatever that may be might be during the pandemic it was it is a bit harder and then on the weekends I try to go biking hiking paddling mostly things I don't have to do with work. I'm lying a little bit because occasionally there is a reason to work on the weekend

and occasionally I do have a meeting with somebody. Like I have an Australian collaborator, so often our meetings are in the evenings. So, I mean, I've given you an idealized story. The truth is a little different. It, you. It varies. When you're working from nine to five, how much of that time is spent thinking versus reading someone else's paper versus answering departmental emails and requests?

What is work? Because work, obviously, for a professor, you have to teach. You also have to do research. There's perhaps 10. Work is is all of that and it depends on the season so basically when i teach two courses uh so you know there's basically fall spring and summer you know fall semester spring semester and summer okay uh in the fall and the spring I teach either two or one course per semester, in the summer I don't teach. So when I teach two courses per semester that's almost all my attention, almost everything I have goes into that and the rest goes into the regular meetings that I have to keep.

So I have several graduate students that I work with, several collaborators, and I'm trying to keep weekly meetings with them. So there is a certain amount of weekly meetings that I have to keep. When I teach one course, I have some breathing room, and when I teach nothing, namely during the summer, then a higher proportion goes into research. Even in the summer, the proportion is not very high. Firstly, is your ideal situation to simply be a research professor? I know some professors don't, they treat teaching as an aside. They don't particularly like to do it.

I'm curious about your case. Well, ideologically, I mean, I see myself as a teacher. I see myself as a teacher. Even the research, in some sense, is to support teaching. Even the research is so that I will remain in shape and even more so, and that the community will remain in shape and even more so and that the community will remain in shape so you know if if i do

research i am able to uh teach graduate students how to do research and then they are uh become uh stronger mathematicians and can teach mathematics better. So I even see research as mostly supporting teaching. So I definitely wouldn't say that I see everything as research, right? I mean research, again, teaching is very significant. Well that's the theory. In practice, you know, waking up every morning and preparing your class and this marathon of teaching where you have to prepare your class, you teach, you feel terrible about the things you missed during class, you don't even have time to digest it and you have to get ready for the next class already. So, you

know, it's very hard. I mean, people who haven't done that don't appreciate how hard it is, but it's emotionally very, very difficult. And after a semester of teaching, you have no idea how much I'm looking forward to the break. So it's not like teaching is, you know, when I teach, I want out. Do you find that to be true even if it's an undergraduate course? The reason I ask this is that I imagine as a professor,

the undergraduate material you could teach with your pinky. You don't have to think much about it because, first of all, you've mastered it. Second, you've taught it for years and years. I do not teach the same class for years. Like, I think the maximum I've ever repeated the class was like three years. So it's not. But but even then, you always have to prepare. You always blunder. I always blunder. Maybe perfect people don't.

But, you know, I always blunder here and there, and then I have to figure out ways out. I always find new things to do, new ways to do things. I always, even when I repeat, but most of the time I don't repeat. And the material is sometimes hard, you know, so and even when it's not hard, so you know, on day one of a class, you teach, of a linear algebra class, you teach the axioms of a vector space. I obviously know these axioms, right?

I obviously, when I see a vector space, I know it. When you give me a vector space, I know what to do with it. But that's not the same thing as getting up in front of a class and listing the axioms and making sure that you haven't forgotten one small axiom somewhere, or that you've phrased them perfectly so that even people who have never seen these things before will understand. So I mean even day one of a very

basic class requires preparation. Razor blades are like diving boards. The longer the board, the more the wobble, the more the wobble, the more nicks, cuts, scrapes. A bad shave isn't a blade problem, it's an extension problem. Henson is a family-owned aerospace parts manufacturer that's made parts for the International Space Station and the Mars rover. Now they're bringing that precision engineering

to your shaving experience. By using aerospace-grade CNC machines, Henson makes razors that extend less than the thickness of a human hair. The razor also has built-in channels that evacuates hair and cream, which make clogging virtually impossible. Henson Shaving wants to produce the best razors, not the best razor business, so that means no plastics, no subscriptions, no proprietary blades, means no plastics, no subscriptions, no proprietary blades, and no planned obsolescence. It's also extremely affordable. The Henson razor works with the standard dual edge blades that give you that old school shave with the benefits of this new school tech. It's time to say no to subscriptions

and yes to a razor that'll last you a lifetime. Visit hensonshaving.com slash everything. If you use that code, you'll get two years worth of blades for free. Just make sure to add them to the cart. Plus 100 free blades when you head to h-e-n-s-o-n-s-h-a-v-i-n-g.com slash everything and use the code everything. We're going to talk about mathematical techniques. In fact, I believe when we emailed, you said you don't like the emphasis on theorems and proofs. You like, well, we can talk about that later. As for teaching techniques, I'm curious, what have you gleaned?

What have you found to be productive? And perhaps even some behaviors you find to be unproductive that you see other people, other teachers making the mistake of doing i mean there is so much uh you know starting from come you know completely silly things like make sure that you're heard right the classroom is large and you speak without a microphone, people in the back are not going to hear you and then they're not going to follow. Make sure that you can be seen. So if you write with a tiny, you know, if you write tiny letters with a faded chalk,

people can't read what you write. The class is a failure no matter what you say. And then there are higher level things and basically everything should be motivated. So, you know, I mentioned the first class of a linear, the first class in a linear algebra course. Okay? So, you can give by, you know, operations and

properties and, you know, you write item number one to item number 15, where 15 is, I don't know, some axioms like alpha times beta times a vector, you know, whatever. So one of the axioms of a vector field, okay, of a vector, sorry, a vector space. And then the question arises, why and who cares? Right, right. Okay. So if you start your class this way, people aren't going to like it. So before this, there has to be a little section.

I don't know here where you explain. Well, for such and such reasons, we care about ventral spaces so there has to be motivation first ok and and that's true not only in the first class of the first linear algebra course. It's true in every class. Whenever you say something, there has to be a reason, and the people who are listening to you

should know that you've thought about it, should know that you're telling them these things for a reason. Okay? And that's often omitted. It's often too easy to just follow the definitions and the theorems and never say what they're good for. And not only you should say these things, you should say them up front. Likewise, when you write a proof, it's often easy to... well, to... so, I mean, often there is a picture behind a proof. The proof is telling some story,

and you want the picture of that story in the minds of the students. But it's often too easy to reduce it to a list of formulas or a list of statements. So a higher level thing is to make sure that everything is motivated. What else? What's another higher level aspect of teaching that you find is a common mistake

among the teachers that you see? I think maybe this is the biggest, not showing the motivation clearly enough. And by the way, that's part of the reason why research is a necessary part of teaching, because you have to know the motivation. You know, you can't say vector spaces are a great thing unless you've actually worked with them, unless you've actually seen them used.

So maybe vector spaces are easy enough so that you can, are good enough, are easy enough so that you can, you know, by third year you understand why you learned about them in first year. But if you want to be able to say something intelligent about homology, you have to have used it. The only way to use it is in research. So that's what I meant by you have to be in shape. Being in shape also means remember what can be done with these things. Do you find that you're more creative as you get older or that your creativity peaked at a certain age? I, you know, I wish I could answer. The truth is that I don't really remember. You know, I mean, it's very hard to remember myself at any other age.

So, I mean, certainly I have matured. So the little I can remember from 20 or 30 years ago is that I have matured. I'm more systematic, perhaps, than I was. Am I smarter? Am I more creative? Am I stupider? Am I less creative? I'm not sure. What do you mean

when you say you're more systematic now? Can you give an example of how you would look at a certain problem now or solve it versus 20 years prior? And also a subion, do you see being systematic as a good thing, or do you see there being innovation in being disorganized, or creativity embedded there? No, systematic is generally better. much more into the low-hanging fruits so and I was lucky that I found some but but basically I'd like wait for opportunities and then use them. Okay. And now it's much more I have a long-term goal. You know, I want

a theory of something. You know, your podcast is a theory of everything, right? I just want a theory of something. So I want a theory of something because I feel for various reasons that it ought to be there. And then I worked very hard on, well, very hard to

my standards. I already admitted being a bum. So, you know, I worked at least for a long time trying to find that theory. You know, even at a very, at a much lower level. So I think when I was first year student, you know, so, you know, math theorems and proofs were pressed into my brain was essentially an external process right so i went to classes and people pushed theorems into my brain and Some of them seemed like complete formalities, like a sequence of deductions that I could memorize and I could verify, but were essentially created by aliens. You know, I had no idea how they came to be. but were essentially created by aliens. You know, I had no idea how they came to be.

Right. And some theorems immediately became a picture. I immediately understood that the proof describes a certain geometric reality. Okay? Describe... I could see the intuition behind the proof. It's not always geometric, though. It's just you see the intuition. Not necessarily geometric.

But I could see somehow the motivation. I could imagine how the proof was discovered. I see. I would get the feeling that I understood the proof. And that was essentially a dichotomy. Some proofs or some concepts were of that type, made by aliens, and pushed into my brain by professors. and pushed into my brain by professors and other concepts, theorems, proofs, whatever,

were clearly intuitive and made sense and I understood them. And that was it. That was the world. There was this and there were those. And then sometime later in second year, I actually did my BA very, very, very quickly. So basically, I did it in essentially two years. So when I say second year, you should compare it maybe with year three and four of most students. So somewhere later, late in my studies, I realized that if I think harder about the theorems that,

or the concepts that appeared to be alien, then sooner or later I see the picture behind them. Sooner or later I realize that I am able to move them to the other class, the class of things that are motivated and understood. And then I became systematic about it. So every time I learn something new, I may learn it from a book that describes it as a formula. Okay, and then I'll struggle to see the intuition behind the formula. And so even at that level,

it's something that took learning. It wasn't, So even at that level, it's something that took learning. It wasn't, again, when I got to the university, you know, when I was in first year, I took classes and I got good grades in all of them. And nevertheless, there was a point where things became better, where I realized I can actually understand everything and not just repeat everything. And that came later. What does that process of finding the intuition behind, let's say, a formula that before you previously simply had to memorize as handed down from the gods or the aliens,

what does that process look like, the process of finding the intuition? Does it look like, I'm sure it's a variety of these, but does it majorly look like you copy it down with a pen and paper precisely and then you draw arrows and you say okay so what's an example of this what's a counter example of this or is it you go on a bike ride and you keep thinking about it what is that process look like perhaps it would be edifying if you took a specific example but if you can't then just give the broad strokes you know what what does it look like from the outside from From the outside, it looks like I'm

sitting on my chair and staring into infinity, right? Or sitting on my balcony and looking at like inside. It's very hard to answer without without going through specific examples, and the specific examples are... Far too complex? I don't know if they are far too complex, but they fit in a math class. And the other thing is you will never see these examples in my classes because when I teach something, I try to motivate it. Okay, so you will not see the alternative.

Okay, you will see the motivated way of explaining the implicit function theorem or the change of variables formula for multi-dimensional integrals, right? You'll see the motivated form. You will not see it. I will not be showing you the unmotivated formulas. For you, much of the time when it's difficult for you to understand a proof or a theorem or a concept, is it most of the time because it was unmotivated? Well, the intuition is not explained often. So in textbooks, often the intuition is just getting used to the concepts.

So sometimes intuition is not... It's just playing around until you're used to it. You know, you've just landed in a foreign city. You've just landed in Athens. Your hotel is two kilometers from the Acropolis. You got there at night. You wake up in the morning. You want to go to the Acropolis.

You punch Acropolis you punch Acropolis into Google Maps and Google Maps tells you you know go left go right go straight go left go right and here's the Acropolis okay you follow the instructions you didn't learn Athens you've learned the way to the Acropolis. To learn Athens in the afternoon after the Acropolis, you have to wander around and then you have to continue wandering around for the next two weeks. And then maybe you're starting to have a grasp of what Athens is

so there is similar there is something similar in in mathematics okay or in science or in fact in every field of learning okay i mean but in mathematics you know um, you don't learn... I mean, if you learn a proof of something, if you learn the proof of the most interesting result in a certain field, you haven't learned the field. Okay? You've just learned... And especially if you follow the directions

by somebody else. Okay? The learning happens if you wander around later. And in a way, you have to do it on your own. Right? This wandering around. So wandering around in mathematics means you experiment,

you change the conditions, you ask yourself what if you remove, what if you add conditions, what happens if you look at analogous situations. But these things... Well, so I... Like part of tourism you do with a tour guide, and a part you have to do on your own. And... So some part of being systematic, if you wish, is doing this on my own thing in a systematic way. Okay, this realization that I can't just jump into a topic by following somebody's guidance. I must sit down and think and ask myself many, many, many, many more questions and go on sidetracks until eventually I'm comfortable. Now, in this example, let me see if I'm understanding it correctly.

Let's say Athens is the city, and then you have the Acropolis, which is a landmark. So that's almost like a landmark theorem, and then the directions are the proof and the steps to get there. Would you say, and you're saying the learning comes from the wandering, would you say that when you first land in Athens, you should wander? Or should you learn the major landmarks and the directions to get there, perhaps forgetting the directions? You just get yourself there so that you see what the landmarks are like later you wander. The analogy would be you learn the major theorems and then later you try and get the intuition by playing around with them or do you play from the get-go? I mean there has to be a bit of both. Basically you know if you land in Athens and start wandering randomly you end up

you may end up wandering randomly in a completely uninteresting direction. Plus, there is a fear factor and you may fear the bad neighborhoods. Interesting. But there actually is an analogy in mathematics. No, I mean, you fear doing certain things and the guidance can help you overcome the fear. But maybe fear is not the right way. I mean, think that you have to explore Athens when it's dark. Okay, so you simply don't know which way to go. You just land, you know, I tell you, think about vector spaces. You have no idea what good questions to ask are.

What are the good questions to ask? And so you need guidance, or else you will be stuck in one neighborhood and never realize that there is more than that. But, you know, after the guidance took you from here to there, you have to go back and explore and explore more and explore more and until you until it's a solid and not just a single path. So you have to do your own Feynman path integral over the space of mathematical

ideas? There is a bit of that, yes. What's your greatest strength as a mathematician, or some of your strengths? Who says I have strengths? What's your greatest weakness? I'm a lazy bum. Are you being modest when you say that? Because I imagine someone who has an accelerated degree, two-year degree that should ordinarily take four years, it's not as if you spend the majority of your time playing video games and being unindustrious. Oh, you'll be surprised.

I mean, of course, I didn't play video games. That was back in the 80s. Video games came later. But I definitely killed a lot of time. So who are you comparing yourself to when you say that you're slothful? Gee, I compare myself to who I want to be. Anyway, what are my strengths? Strengths is...

I think I'm decently good at explaining things to others. Often that involves thinking about them. I mean, explaining is not just reproducing what you've heard somewhere else. I mean, it's first of all being creative about these things. But, you know, ask the students, ask the people who listen to me in lectures, when I give research lectures.

It's very hard to... You know, there's a famous story about Einstein that somebody complained to Einstein about his problems with math. And the reference was to high school math. Like, I never got high school math. And Einstein said, let me assure you that my math problems are way bigger. So sort of... Meaning what? What did he mean by that? You know, Einstein spent all his life struggling with mathematics, not just high school time. Okay. So in a similar way, gee, I'm sorry.

I mean, now it might sound like I made some analogy, which I do. I'm not making. I understand. Trust me. But still, you know, I like to implement. I like to compute things, implement them on the computer, which means that I spend a lot of time programming the concepts I work on, turning them into running programs. You could say it's my strength, but when I look at it, all I see is that I'm struggling and

struggling and struggling and failing, and here and there there is a minor success, which gets superseded a little bit later by somebody else. So yeah, maybe one of my strengths is the ability to implement things, to turn, to actually, you know, turn mathematical concepts into running programs. But it's also one of the things that keeps me depressed so much of my time. Some people who are watching this, they may be discouraged because they weren't particularly great at high school mathematics, or some people may have struggled during their undergrad. What advice do you have for those people? If they want to continue to learn mathematics or physics at a graduate, perhaps even larger

than graduate, PhD level, do they just continue plugging away and not feel discouraged because they feel like an imbecile on a daily basis what advice do you have for them there is chug along and there is realize when it's not working and i hate to tell you that i don't know. I mean, I can't give you a clear recipe to decide which one it is. Different people are different. Some people struggle and struggle and struggle until they succeed. Sometimes some people don't. Or they struggle and struggle and struggle and don't succeed. Sometimes some people don't. Or they struggle and struggle and struggle and don't succeed. You get feedback.

I mean, some people depend on daily feedback. Some people can live with a bit of feedback once a year. I'm not sure I have a clear answer. What did you learn from Ed Witten with respect to mathematical techniques or meta-mathematical, so how you should think about math or physics? Well, I learned specific mathematics from him, right? So, you know, I learned, like many other people,

I learned this formula from him, right? And how to read it and how to use it, interpret it, work with it, right? Do things with it. But I learned many other specific things with him, from him. I learned from him. He's actually, I mean, like, people think of him as a great researcher, but he's also a great expositor. So when you come to his lectures, when you come to his lectures, he is able to make even very complicated things appear simple. The problem is that often you will not see that, because he's turning the hyper complicated into merely complicated.

So you don't automatically realize what a great exposition work he had done. And I think I learned, well I don't know if I learned that from him, but I strive to learn that from him. And then in other ways, there are senses in which I didn't learn anything from him. Namely, he is so much more talented.'s just no way of no other way of saying it the guy is smart the guy is extremely productive if you try to imitate his brilliance you'll fail at I failed. If you try to imitate his productivity, you'll just be disappointed. So in some ways, I haven't learned. There are certain things I haven't learned from him and I just simply couldn't. Do you mind describing your relationship with

Eric Weinstein? When I was speaking to Eric, he said that you acted as a de facto supervisor to him. Because Raoul Bott wasn't particularly, although it was officially his supervisor, he didn't act as the supervisor for whatever reason. It was a strange relationship. I don't understand it. I'm curious, what do you think he means when he says that you acted as the supervisor? You gave him advice or? I don't think it's correct. It's more like a coach or, you know, I mean, um, so Eric, uh, Eric was very unbalanced, and he still is unbalanced, and I don't mean emotionally unbalanced. I mean, he has great talents in certain directions and fear of others. And there are certain things that one has to do, certain type of computations, certain type of writing. Basically, he has fear of turning his theories to concrete.

his theories too concrete. So, and that's part of the reason why he had difficulty with Raoul Bolt and with, basically with the normal system in Harvard. And basically with the normal system in Harvard. And my role was to help him pick something concrete out of his big theories

and push it enough so that it became a thesis. Yeah, I also helped him with some computations. You know, at the end of the day, there was some like, so his, you know, big theories lead to something very concrete and specific, which was a tiny part of the big theories, but we extracted that tiny part and made it more concrete. And at the end it became a question, you know, some big matrix, 500 by 500 or whatever it was, is it positive definite or not? And I helped him decide that. And that was decided by a computer? Like you ended up programming it?

That one was decided by computer. But that's the kind of thing he would have never done on his own, because he has this fear of, in a way, it's a fear of testing your theories, because they might fail. But in another way, it's just inability. He never learned how to do it. He never got good at turning the very abstract into concrete. So, yes, that was my role. Have you read his recent Geometric Unity papers or heard about Geometric Unity at all?

I've heard a tiny bit about it. I have not read it. I'm not in a position to comment. You talked about it's difficult to make a concrete or take something concrete from a theory, at least in his case. Do you mind giving an example of his strengths or an example of a brilliance of his that you saw? He intimately knows the small groups and their representations. The small groups and their representations. And how they interact with each other. So. The small groups, like the classical groups, or what do you mean when you say the small group?

OK, what I mean by the small is, so you know, there are several families of so-called symmetry groups. Okay, so there are S-U-N and S-O-N and S-P-N, and then there are a few sporadic ones. There is G2 and F4 and E6 and E7 and E8, and it doesn't go beyond 8. And all of these, so these are abstract things, these are groups. They have so-called representation theory. So each one of them can be written as a group of matrices, can be realized within the world of matrices, but in many ways. So Eric intimately knows these groups and the representations and how they sit inside each other for small values of n and, well, for all of those sporadic ones

which are all small. And there are symmetries and there are so-called outer symmetries. I mean, he knows all about these things. This is relevant to physics because, you know, electroweak theory is somehow about SU2 and the grand unified theory, so strong and electroweak is somehow about SU3 or... So, no, no, no, no, no, no, I have it wrong sorry electromagnetism is well electromagnetism is you one and then you know I I actually have not thought about these things for many many years so I could be saying things wrong, but his strength is that he knows these things really, really, really well.

Or one of his strengths is that he knows these things really, really, really well. Earlier when we were talking about particle physics, well, knot theory, you mentioned that for every knot, there's a corresponding physics story of particles dancing and so on. Is the opposite true?

So does the arrow go in the other direction as well? For every particle interaction, is there a corresponding not? For every particle interaction of a specific type in three-dimensional space, yes. But you have to be very specific. You have to specialize a great deal. Physics mostly happens in four-dimensional space. If this can be easily explained, why is it not simple to just extend the three dimensions to four dimensions and come up with answers, much like in vector, we're talking about vector spaces, it's trivial to go from R3 to R4. The same theorems apply.

Okay, so the answer is, first of all, certain things do extend. so you know if not theory is to some extent about dance

the dancing of points in two dimensional space so you know here is

a very rough kind of correspondence. Okay? So, knot theory or ordinary knots is roughly about one-dimensional things or curves or whatever in three-dimensional space. And as I mentioned before, it can be interpreted as points dancing in two-dimensional space. So points are zero-dimensional place space so points are zero dimensional so uh you can uh so you can interpret no theory as zero dimensional things right a single point is a zero dimensional thing

so it's zero dimensional things dancing in a two-dimensional space. Now, you can lift this one dimension up. So, you can say, let's, instead of zero-dimensional things dancing in R2, you can think of one-dimensional things dancing in R4 somehow correspond by the same correspondence, looking at sections, they somehow correspond to two-dimensional knots, so two-dimensional things are knotted in R4. Now you can ask, now what exactly are one-dimensional things dancing… Is that supposed to be R3? Sorry, not R4.

I wanted to add one. This is in R2. Adding one, you get R3. Sorry. So one-dimensional thing dancing in R3. Now you can ask yourself, what are one-dimensional things dancing in R3? And fortunately, I'm ready for that.

So here is a one-dimensional thing. And here is another one. And this is a loop it's a one-dimensional thing that's another one it's a loop it's a one-dimensional thing and they can dance around they can fly in r3 and they can fly in interesting ways that don't exist one dimension less namely they can fly through each other. Okay? So yes, there is a theory of one-dimensional things dancing in R3, and it corresponds to a theory of two-dimensional spaces knotted inside four-dimensional space. So to some extent the answer to your question is yes there is

higher there are higher dimensional analogs of knot theory. It turns out that some features are different. It turns out that in some ways and they're actually qualitatively so the theory of knots in one day so you know unlike vector spaces so vector spaces you kind of once you learn r3 r4 you've learned r to the 50 okay but but but here there are qualitative differences so so some things from the study here translate to things here and vice versa but not everything so myself I've spent most of my time actually on these two okay but then there is higher there is even higher stuff by the way, so, you know, linear algebra, when you take a first class in linear algebra and maybe also a second class in linear algebra, you get this impression that everything is

the same in all dimensions. Yep. But that's not true. Even in linear algebra? everything is the same in all dimensions. Yeah. But that's not true. Even in linear algebra? Even in linear algebra. So, well, not quite linear algebra, but in algebra. Okay?

So, you know, there is something called the real numbers. And then you can add I add I and you get the complex numbers and then you can say let's add J and K as well and you get the so-called quaternions but the quaternions are different than the complex numbers in that this is a commutative ring and this is a non-commutative ring. And then you can add, gee, I forgot what's the standard name for what you add. So I don't know, add in, in stands for nameless because I forgot the name, and you get octanions, and the octanions are no longer even associative.

So it's still sort of a ring, namely it has a multiplication, but it's not even associative, and then the process ends. There is just nothing beyond that. You can't continue. So even in linear algebra, so that's not quite well, I don't know, even in basic algebra, not all dimensions are the same. One is not the same as two, is not the same as four, is not the same as eight, and not the same as sixteen, is not the same as 4, is not the same as 8, and not the same as 16, where things

no longer exist. So it's just not true that things are independent of the dimension. It's only, they're only independent of the dimension in the first two years of undergrad. independent of the dimension in the first two years of undergrad. Now, in this example, we're going to different spaces by adding one or by adding whatever extra variables there are there. In the previous example, we're extending R4 to R5 or RN to N plus M. So we're in the same space, in a sense, except adding an extra dimension. So what's an example that holds for R3? Or actually, sorry, let's make this even simpler. What's an example that holds for R2, but doesn't hold for R3? Or you can say R3 to R4. Like,

what's an example of a quantity that drastically changes as soon as you go to R4 and becomes extremely complicated, but in R3 it's trivial? The motion of points is fundamentally different in two dimensions versus in three dimensions. So let me explain. So, right, suppose you have two points A and B in two dimensions and you want them to trade places by moving. Okay, there are two basic ways they can do it. They can trade place by B going above A, or they can trade places in sort of the opposite way by A going above B. So A will go to here and B will go, sorry, A will go, A will go take the upper route and B take the lower route. These two ways are somehow fundamentally distinct.

And to use technical language, you cannot homotope this way to that way. They're not homotopic. You cannot continuously deform this way of moving to that way of moving. But in R3 that's not true anymore. So in R3 points can trade places in only one way. So let me find two points for you. So I suppose, well, they're not points, but they're close to points. So here is red and blue.

So in R3, they can trade places going this way or they can trade places going that way but there are many other ways they can trade places. They can trade places by going that way or that way and in a sense all of these trading places ways are equivalent to each other because if you trade places, if the two points trade places they span a circle, right? So basically, let me use this, right? So, you know, they span a circle by, I wish I had a third hand, maybe I lose my... Sure, sure, sure, sure. Okay, they go like that, okay?

That spans a circle. And the other way of spanning a circle is by going the other way of spanning a circle is by going the other way around the same circle, but now I can flip that circle over and that homotox that deforms one way of trading places to the other way of trading places. So in R2 there are two distinct ways of trading places. And in fact, two are easy, there are actually a little bit more. But anyway, in R3, so if you're talking about three-dimensional space,

and you have two points in the three-dimensional space, there is only one way or one or all way or let no let me say different let me say differently all ways point is that a way of trading places is a circle, right? One point goes along one arc and the other point goes over the other arc. And you can switch from one way of trading places to the other by flipping the circle you can't do it if you're only in the plane since trading places is actually fundamental it makes but by the way it's also fundamental in physics so you know you you physics. So you might have heard of bosons and fermions and how they behave with respect to trading places, and then there are anions that behave yet differently with respect to trading places.

So basically, trading places or how things behave under trading places is a fundamental thing, and it's different it's it's different in two dimensions than in three right likewise there are things that are different between three dimensions and four and and so it's actually this kind of myth or not myth but this um the thing that you learn in first and second year that uh rn is the same rn no matter what n is is not actually true have you heard of the hard problem of consciousness i have not heard it in this language namely no no it has nothing to do with what's written on the board or maybe it does but not as far as i see this is sort of a clear philosophical problem right but uh with the added prefix the high and it's clearly hard but but but as a phrase

the hard problem of consciousness i don't know what you're talking about there's only one consciousness that i need to explain and it's my own. Right? From my perspective, you're not conscious. Right? You just react to things that I do. You know, if I tell you you're stupid, you get angry at me. If I give you a compliment, you get happy. But these are all... I don't see consciousness here. Okay? So there's only one consciousness that I have to explain, and only one person that I have to explain it to, and it's myself. But anyway... Having said that... Um...

Yeah, I'll keep the rest private. What keeps you up at night different things on different days whether she loves me or not whether I can complete the computer program I'm writing or not, or why is it not written, my classes for tomorrow, the theorem I don't know how to, I have no clue how to prove, the state of the world, COVID, I don't know, different things at different times.

Is there a through line? So let's say one that you've been thinking about for 15 years or more. You know, I'm politically aware and always been, have always been. So, you know, how to make the world a better place. But of course, that's a subjective thing. What is better to me

may or may not be the same as what is better to you. So may not be the same as what is better to you. So in a way when I'm saying how to make the world a better place, it's more like how to make the world more like how I would about it a lot and mostly my conclusion is that there's nothing I can do or there is much I can do. My powers are limited. What direction would you like math research to head? Math research to head. I think I mentioned that I like computation.

And I actually... So, I mean, it's not the only thing of value in mathematics, right? There are things in mathematics with great philosophical value, regardless of whether they are comput are often not. And often people just don't care about computations. just don't care about computations. They pretend that, well, they think of computation as something, as a technical thing to live for their graduate students. And I see it as a much more integral part of the work. So if I were to change, I would make computation a bigger part. Computation in the sense of implementation on the computer, not in the sense of writing by hand.

I would make it a much bigger part. And that happens at many levels. So when you teach a class in linear algebra, you teach about how to compute a determinant. Well, along with it, there should be a computer program to compute the determinant, and there should be a realization that there are maybe two formulas for the determinant, and one of them is way better than the other if you try to implement it. So I would emphasize computation much more. Yeah, maybe that would be the main change. Do you have a philosophy of math? So for example, there's Platonism and formalism? If anything, I would be an ultra-finitist.

But it's much more graded than that. So, you know, there is a hierarchy of of complexities of things or sizes of things in mathematics. So, you know, people in set theory and logic and, you know, talk about large cardinal axioms and then there are parts of mathematics that have to do with uncountable or bigger than uncountable. And then there are infinite parts. And then there are just infinite without being uncountable, right? Just countable, say. And then there are parts of mathematics that are finite but huge. So, you know, Ramsey theory, you have, you know, you prove that if a graph is large enough, then it has properties,

certain properties, but large enough means really, really, really, really, really, really, really, really huge, bigger than the number of 10 to the 12. I don't know what a computer can do in a fraction of a second to what a computer can do in an hour. Okay? And, you know, maybe you can go a little bit beyond to 10 to the 16. I don't know. 10 to the 16. I don't know, 10 to the 15. I don't know. Okay?

So there is this whole grading. And to me, the bigger things are, the less important they are. they are. And because they're farther away from our everyday experience. So they may have philosophical value and some of them do. So I don't want to completely say anything Anything about large cardinals is not interesting, or anything about infinity is not interesting. But the bar is higher. For something that genuinely uses infinity to be interesting, the bar is higher than if the thing is in the range of sizes that we can actually hold in our hands. So that's what I mean by an ultra-finitist. I mean, sort of the... So this is a form...

So an ultra-finitist would say the only numbers that exist are the numbers I could count to. And maybe by extension that my computer could count to. So maybe my computer can count up to this number, but 10 to the 100 does not exist in some sense, because my computer will never be able to count there. Okay, will never be able to get there. Okay, we'll never be able to get there. So I'm not an extreme, I would say sometimes there is value to this one too, and sometimes there is even value to infinity, and sometimes there is even value to the bigger form of infinity, but the bar has to be higher, the philosophical value bar has to be higher the bigger the size is. These things are always interesting.

These things are interesting only on special circumstances. Now, I'm not sure, do you classify yourself as a pure mathematician? Definitely. Okay, so for a pure mathematician who seemingly doesn't care about reality, it's supposed to be discontinued. Again, the valuable math is the math that I can, or the more valuable math, is the math that I can turn into a computer program. It's still pure math in the sense that I'm not looking for an immediate or a specific

application. But both from the application perspective and from the pedagogical perspective, I still prefer the math that can be implemented. Because the math that can be implemented, even if I'm not looking for an immediate application, is the math that has a higher chance of being applied. Because in the application you'll end up computing. And also, from the teaching perspective, from the pedagogical perspective, there is more value to the education you give if it comes with implementation, because you want your students to be able to implement the things that they learn. Yes, sorry, what were you asking?

Oh, I was saying you ultimately care about the practical aspect of math even though you're operating in pure mathematics? Ultimately, yes, though possibly very indirectly. So maybe the way the math I'm doing or maybe the way the math I'm doing, or maybe the way I care is in that the pure math I'm doing enables me to teach graduate students who will then become college instructors,

and their students will be able to actually implement the mathematics they're learning. So, I mean, it doesn't have to be immediate and direct. But yes, I do care about how at the end it gets applied or how at the end it becomes applicable. Great. Now we just have some audience questions I asked people beforehand. What are some questions you have for Professor Jorah Barnett? So the scientific mystical philosopher, that's the username he wants to know, he or she wants to know, why the unreasonable effectiveness of mathematics?

Sorry, why is mathematics unreasonably effective? Sorry. Why is mathematics unreasonably effective? Gee, I don't know. Ask God. She may or may not be able to give you an answer. be able to give you an answer. Yeah, it's weird. We live in a predictable universe. When we throw something up in the air, it always goes through the same trajectory. Nearly. I don't know.

I mean, that's a miracle. Roman Gekwad wants to know, have you ever had your IQ scored? And if not, what do you believe it is? No clue.

And also, you know, IQ scores are fun story that I've never verified it, but I heard it, okay? That when the people who introduced IQ tests first introduced them, they ran them on test subjects, and the women, the females, got higher scores than the men. That was back in the 50s, or I don't know exactly when, but that was a long time ago. They figured, hey, that's impossible. And then they tweaked the tests a bit. They changed the nature of the questions. They asked them differently until they finally were able to make men equal or maybe better, I don't know, than women.

And again, this is true to a certain system of education in a certain time. Right? Men and women did not receive the same education. It would have been easy to distinguish because they came, their backgrounds were different. So you could arrange such things, okay? that's about the value of IQ scores they were I mean

to some extent they measured something to some extent they were cooked to measure whatever people wanted them to measure. Not much to learn from these measurements. Tori Cole has a question. What do you think of biology slash psychology slash philosophy? I think they're foundational to understanding myself and therefore my experience of the universe. Do you think you would gain much of an understanding

of an exploration from those fields to mathematics? Again, biology. Biology, psychology, and philosophy. I mean, they're all interesting fields. I'm not sure what more to say, you know? Do you think there are insights there that if you were to read a biology book, a psychology book, perhaps a philosophy book, do you think there are insights to be gleaned from there that you can then

apply to mathematics? There are connections here and there. Much of mathematics is independent of that. So Much of mathematics is independent of that. So, obviously biology motivates parts of mathematics. So parts of mathematics you know, are about modeling biological systems. When we compute the Fibonacci numbers, we give examples of the reproduction rules of rabbits.

Okay? And then there is much more like that. And then likewise for psychology, And then, likewise for psychology. But, and likewise, and of course philosophy. So, you know, there are bits of mathematics of great philosophical value. So, you know, Goodell's theorem says that there will always be things we will not know. Or said differently, there is no algorithm to decide to find the proofs of theorems. These things have philosophical value.

For most mathematicians, it's not a day-to-day working relationship. My own mathematics benefits very little, if any, from contact with biology, philosophy, and psychology. Okay, Phil Thomas wants to know what led him, what led you, professor, to the field of mathematics, and is your field something you think virtually anyone can learn? What led me there? You know, it's a question about my life story, right? So what led me there is essentially... A sequence of coincidences and misunderstandings. I don't know. A number of things.

So to some extent I was good at that and that pushed me to do more of it. And that's a very common answer for very many things, right? People tend to do what they're good at. Misunderstandings? You said misunderstandings as well? Yes. To some extent, I misunderstood math. So, you know, there was a time that I really liked computers and electronics. And I, at some point, realized that there is a lot of kind of hard work in it. So writing a computer program is very hard. You have to write hundreds of lines of code and then debug them and they, whatever.

And similarly for electronics, right? You have to place those transistors on the board and solder them together and solder them together and you always get it wrong. And you know, there is a lot of hard work where mathematics is so much easier. All you have to do is to prove theorems. That was a complete misunderstanding. I mean, I thought it was like much purer.

At the time, I didn't realize how much hard work is involved. So in some sense, I got to mathematics by misunderstanding, by being lazy and not realizing that I'm getting trapped into a place where laziness is not always enough. And also, at the end, I ended up liking those parts that I first thought I was going away from, namely computation. So I ended up, I mean, computation to me was like a childhood thing and a later acquired taste.

There was a period in between when I thought I was pure. I would never compute anything. Right. What else? Did I answer or did I miss? There was a second part to the question. Do you believe that your field, either the field of mathematics or your subfield, the one that you're working in currently, do you feel that your field is something that virtually

anyone can learn? No. No. anyone can learn? No. No. So, some amount

of mathematics everybody can learn and everybody should learn and maybe it's more than what is taught in in our current educational system education system okay but but you know I will never catch up with Ed Witten ok the amount of stuff that he knows the depth of stuff that he knows,

the depth of stuff, the depth by which he knows things, I'll never get there. It's not, I'm not talented enough to get there. It might be that in my 30 years in the subject, I've learned things that most people also wouldn't be able to catch up with. So, you know, there is the Moshizuki proof of the ABC conjecture. Disputed.

Okay, Moshizuki claims it's a proof other people think it isn't I will never be in the position to understand that I mean possibly if I started early enough I'm not even. I happen to know Moshizuki. I remember him when he was young. We were friends when we went to graduate school together. So I remember him when he was young. He was smarter and faster than me. It's conceivable that after 20 years he's gone so far that I will never be able to catch up. It's equally possible that there are people who would never be able to catch up with me. This said,

some amount of mathematics is useful to everybody. of mathematics is useful to everybody. You know, we're now in the in the midst of a pandemic. Understanding what's the exponential function is something that is basic knowledge, is something that every human should have. Because it actually dictates, it tells us something about how to behave at this time. Likewise, making various probabilistic estimates. making various probabilistic estimates. You know, if the chance of such and such type of side effects is one in a million, what does it mean? What does one in a million means in practical terms for me? Should I fear it? Or is it so remote that I can ignore it?

So basic numeracy is an absolute necessary ingredient for everyone and can be reached by everyone. When you mentioned that you'll never be able to catch up to Witten and perhaps some people won't be able to catch up to you and so on. What is that a function of? Biology? You mentioned the word talent. So is that just innate? Is what's innate called intelligence? Is that associated with IQ? What are you referring to when you say you won't be able to catch up? It's a number of things. You can call it talent. You can call it IQ if you wish. But of course, there is no numerical measurement. And it's a much, much more complicated thing. So You know, so much of how far you can get in mathematics depends on how much you concentrate on mathematics.

So you know, maybe when you're... if at a certain part of your life you're able to completely isolate yourself and think on nothing but mathematics, you'll end up knowing more mathematics than if you are also a human alongside. So, so, so, so talent is also circumstance and, and luck and in some sense being narrow-minded somehow being narrow-minded helps in certain ways speaking of Eric Weinstein you mentioned him okay so so so he was very unbalanced. He had great talent for certain kind of things and great fear of others. And maybe he would have been a much more famous mathematician, physicist now if it wasn't for these coincidences. So, I don't know. I mean, you could call it innate, you could call it learned,

you can call it many things, but it's a very complicated combination of things. Tyler Goldstein has a question. Do you think we need to make new research institutions that aren't linked to academia? And I imagine what he's referring to is something that Weinstein has brought up and Wolfram as well. They're not huge fans of the peer review process, or at least they're not huge fans of only thinking that work is worth it if it's gone through the peer review process. And while that's associated with academia, I'm curious if you have any thoughts as to that. Well, there are things outside of academia. Well, there are things outside of academia. It's not... Arguably, most research is done outside of academia.

I mean, you know, there are a lot of research is done inside companies, inside... Well, a lot of research is done outside of the peer review process. But overall, I think the system is reasonable. Sorry, overall you think the system is... Well, I think the system is reasonable. So, you know, you can say bad things as much as you want about peer review, but it's better than the alternatives. It's better than having no review. You can say maybe Mathematica should be more hierarchical.

There should be big bosses who determine which theorems we should be proving, and then they should have their smaller bosses who would be responsible to specific theorems, and they would be the big bosses of other people who would be proving dilemmas. You could say something like that. It would work just as ridiculously as it sounds. current system is you know given given the limitations of humans is is reasonably good i believe there are some arguments to say that what you just outlined is somewhat like it happens given grant agencies they have some field that they fund and so then the research tends to go toward there so you can view them as the

large boss. And I believe Lee Smolin makes some arguments saying that that's why string theory is as large as it is, even though it shouldn't be because we haven't seen much success from it and so on. So do you see what you just outlined as the ridiculous scenario? Do you see that as having any bearing to how it actually is? Do you see that as having any bearing to how it actually is? So, yes, of course there is bias generated by the granting agencies, by the powerful people, be them the granting agencies or the editorial boards of journals. So, you know, if you have a result

that somebody on the editorial board of an excellent journal likes, that's close to the field of somebody in the editorial board of an excellent journal, then your result is much more likely to get published in this excellent journal. And then you're much more likely to get tenure. And then more work will be produced in that field. And you'll advise students who will produce then more work will be produced in that field and you'll advise students who

will produce even more work on that field. Yes, it all exists. But it's not clear that you can think of a better alternative. And also, there is still some amount of academic freedom. So the granting agency decides once every five years what my funding will be. will be. In those five years, some of my time I do what is necessary to get my next grant, but some of the time I also do things that I think are useful, that I think are useful. Likewise, you know, some of my career is determined by which journal I publish in, but some is determined by other factors, okay? By the results that aren't published, by the lectures that I give in conferences in China, you know, it's much more multidimensional than

than just having a boss and the boss is the granting agency even the fact that the editorial boards are different than the review boards of the granting agencies allows make some variety and again what is the alternative so you know,

one stupid alternative would be to say, to allow anyone to say, I am a mathematician. Here is my proof that pi is rational. Or that you can divide, you know, here is my proof of the Goldbach conjecture. Who's arguing with me? I'm saying it's a proof. My word is as strong as yours. It's much better not to lose control completely. You just used the word mathematician, and I'm curious, what do you see as a mathematician? Is it someone who's a professor of math? Is it someone who gets their master's or PhD?

Do you have to frequently publish research in order for you to call yourself a mathematician? Like what gives, obviously it's a nebulous term, but it's not extremely amorphous. It's defined. So what do you define it as? It's hard to say. You know, many definitions don't have sharp boundaries. In fact, only in mathematics there are sharp boundaries between things. Like either the series is convergent or it is not convergent, but only...

And there is a very, very specific criteria to decide it. Only in mathematics there are sharp boundaries. And now you're asking a metamathematical question. You're asking a question about mathematics, not a question in mathematics. So there aren't sharp boundaries. So basically, all of the above. If you teach mathematics, if you think a lot about mathematics if your way of

thought is abstraction uh proof um you know then that's mathematics but but but you can always you can always find the borderline things that will be hard to decide. Here's a reason why I'm asking, because not everyone can call themselves a doctor. I know that some people, they get their MD, and then they're no longer practicing researchers, they no longer are connected with the field, but then they'll give health advice, resting on the laurel that they have an MD, and they'll say, well, this comes from a doctor. And I'm sure the same occurs in mathematics, though, to a much lesser degree. Same with physics. Someone may say, well, I'm a physicist because I have a degree from a university in physics, but that doesn't necessarily make you a physicist. It doesn't mean that your physics work

should be taken more seriously because you have that degree so if you're looking if you're looking for a set of thumb rules to decide uh uh which experts are to be trusted and which should not be them experts in mathematics or experts in medicine, well, the answer is these questions are complicated. So, you know, in medicine, you probably have to look for the bigger name doctors the more you know the more established the doctors because the system actually does on average promote the better people even if ways okay and you should look for was it said by most doctors or by one doctor who hasn't have gotten their PhD. Yes, there are brilliant people who did brilliant PhDs in mathematics and ten years later lost their mind and are speaking nonsense. I mean, there is no look at this certificate and if somebody has the certificate then they're

trustworthy. Ashley Ship wants to know, is there a connection between knot theory and protein folding? And is it possible to use, well, if so, is it possible to use the artificial intelligence research that has been done on protein folding to inform knot theory? There is a connection between protein folding and between DNA and stuff like that in not theory, people... it's not very strong. Like in general, people like to say things about it, but the actual research, at least my own research, is very different from what you would end up doing in biology of DNAs or proteins by the way In some sense DNA the study or

Okay, so DNA molecules are really, really, really, really long strings. Chromosomes are really, really, really, really, really long strings. Namely, they are of length billions, or the ratio of their length to their width is billions. is billions, because they have billions of amino acids along them. So if you were to make it out of rope, it would be very, very, very thin rope, which is very, very, very thin rope, which is very, very long. But yet this rope is compressed into tiny, tiny, tiny regions. So they get knotted. And even your experience from a much smaller knot,

so here's an electric cable, if you have a little bit of an experience with letting loose of cables, not tying them nicely, as I tied this. Yeah. If you let it loose, it gets knotted, and it's nearly impossible to untie. it's nearly impossible to untie. Chromosomes are much worse. They're much, much, much longer in relative terms, and they get cramped into tiny, tiny, tiny spaces. So how come they don't get knotted?

And if they do get knotted, how come during various parts of the life of a cell, so when cells split, they split, they somehow manage to untie themselves and disconnect. So how is it happening? So there must be some huge brain inside every cell that can untangle knots which are much, much, much, much, much more complicated than the knots that you can untie. Or you know, the knots that you take hours to untie sometimes.

Okay? So there must be something there. And there is something there. And there are enzymes that actually, you know, so when DNA molecules need to annot, to untangle, there are enzymes that allow one molecule to pass through the other. So basically, you want to pass one string through another, so that they will come to position. There are enzymes that cut this one temporarily. It gets cut, it passes to the other side, and it gets reconnected. So in a sense, biology invented anti-knot theory enzymes and they are necessary because of knot theory.

Because if there were knots you wouldn't be able to divide cells. So biology invented some anti-knot theory enzymes that allow things to pass through each other, that nullify not theory. So in some sense, the most interesting not theory that happens in biology is somehow anti-not theory. That's a lovely story, but in practice, there isn't much. Jack Dissart wants to know, is math a human construct that we've devised to better understand the universe? Or is math a set of universal, unchanging principles?

I'm a human. How would I know about, you know, how non-humans would see math? Or how, you know, I don't know. But this said, I mean, I don't want. But this said, I mean, I don't want to answer the philosophical question, but I want to answer some practical things that comes out of it. Every bit of math. So, you know, pick up a math book here. Every bit of math in this book,

and not just this, I mean, every bit of math in this book, and not just this, I mean, every bit of math in every book was created by a human. That's a good thing to remember. It means that there you see a proof and you have no idea where could it come from, well you have no idea but somebody had this idea and you may be able to find it. So it's often extremely instructive to try to reverse engineer somehow, not just understand the proof, but understand how humans,

with all their failures, could have found them. So it's actually useful to know that math was entirely created by humans at the end. Do you think we don't spend enough time in books talking about the missteps? So, for example, here's how the person arrived at this proof. They originally thought of it like, so this didn't work. Or would that just bloat the book unnecessarily? It would probably bloat the book.

But motivation should be there. And when I say motivate, okay, so some story should be told. Like when you present a proof, there should always be some plausible discovery story. How could this proof have been found? It doesn't have to be a historically accurate story. But there has to be some story explaining that this proof could have been found by humans. It wasn't created by some supernatural entity. This question, CX770 asks, does he believe in the concept of God?

Nope. Okay, this one's a tangential question. Bill McGonigal asks, Ramanujan said that he learned from his goddess. Others suggest, well, Ramanujan said that he learned from his goddess by meditating and praying. Or it came to him in dreams, some of his formulas, esoteric formulas. What does Professor Jorah Barnetton make of this? Can't speak for Ramanujan. I tend to be a rationalist, atheist, no miracles. If I think hard about something, I either solve it or don't, but there are no miracles in the business. Another question, this one, this is my question. Are there some relatively elementary concepts that you've recently had a new insight on.

So for example, I don't know if you know my brother Sebastian. He's a professor at U of T as well in math finance. And he was telling me, Kurt, you know, only recently, maybe last year, I thought I realized, oh, the triangle inequality is actually about a triangle. He would only just use the formula. Also recently, okay,acobian determinant the jacobian is actually like it's like a linear map of of what is that point doing close instead of just okay think of it as an abstraction of slopes and stuff yeah so do you have an example sorry of what of things that I've only recently understood?

No, no, no. Of something you have understood, you've used before, but then maybe you've come to a different understanding of it and you say, oh, that's an interesting way of thinking about it. I didn't see that before. It happens all the time. Well, I wish it would have happened all the time. When it happens, it's a great day. Not all days are a great day,

but there is the occasional great day. You know, every time you learn something has higher meaning, or more meaning than you thought, it's great news. It happens all the time in the life, in the professional life of a scientist, though not every day, okay? Well, I'll give you an example which somehow, which I think also partially involves you. So a few years ago, I was asked to give a 10 minutes talk or something to a, I don't even remember what was the event, but you were there. Oh, you remember that. Mentorship event or something like that i don't remember that's what

that's right and i thought what would i talk about and i well okay the truth is that i have some list of uh ready-made topics that i can talk about you know if you ask me to talk in front of high school students i sort of have a number of topics that are ready. Okay. But nothing really fit. And also, sometimes I get bored of myself. So I have to occasionally add something. So I said, okay, why don't I talk about a prisoner's dilemma?

And so I mentioned the prisoner's dilemma in five minutes or so, and it's a very light math, right? It's hardly math. It's so light that it's hardly math. But then I realized that it has a profound... I guess the economists have always understood that. They always thought it was important. But I realized it only relatively recently.

So that was, what is it, three, four years ago? I only then understood how profoundly important it is. important it is. So, you know, the prisoner's dilemma is some numerical example where you have two sides, two people, A and B, and they can choose whether to cooperate or betray. Each one chooses independently whether to cooperate with the other or betray the other. And if you look at the numbers, each one, no matter what the other does, would prefer to betray. And as a result, both of them would betray. And the overall outcome is worse to both of them. Again, I'm not describing the whole thing.

I'm not telling the numbers. I'm just saying there is this situation where if each one acts selfishly, it's worse for both. And well, it has huge philosophical implications, you know. There is a whole political class that believes that people should be acting selfishly, because if each one acts selfishly, it's actually for the best of the… better for everyone. That's somehow the foundation of the belief in free markets so yes I am actually still a great believer in free markets I actually still greatly believe that free markets are a good thing but there is a little asterisk sorry that's not duality that's

an asterisk so that there is a little asterisk asterisk and the asterisk says provided somebody arranges so that free markets when each one acts selfishly you will not be led into prisoners dilemma dilemm dilemmas. You will be, you will be, you'll be, you'll stay out and, and your selfish acts will actually benefit the overall good. So, I mean, this asterisk here is the prisoner's dilemma. Sorry, I, well, okay. Yes, yes. Sorry, I... Well, okay. Yes, yes, yes, I get it. Anyway,

and you think it's a small asterisk, but the more you think about it, you realize it's actually a huge asterisk. It completely changes the picture. And it's much, much, much more common. You think of it as an esoteric example, but then you realize that it's much, much, much more common, you think of it as an esoteric example, but then you realize that it's everywhere. So it's actually a very, very big asterisk. And I think I realized it

relatively recently. I'm honored that you remember that. I didn't think that, I'm honored to even be a footnote in your memory. Oh, don't be silly. Okay. What... Have you applied... So this question comes from Harinavis... Harinivis. Sorry. Harinivis P asks,

has Dror applied the principles of mathematics to some other domain of his life? Like, let's give an example, a non-trivial example. Well, okay, I mean, I can give two answers, okay? One answer is, well, I don't know if it's the principles of mathematics, but, you know, people once, there was a claim going on and it's been going around and in fact it's still going around that the Bible contains predictions in code.

So if you read the Bible with letter skips, if you read every tenth letter, you will occasionally find a word. That's not surprising. But the claim was that these words were, that it actually happens more than statistics would predict and that these words have messages in them. And there was a group of people in Jerusalem that published a paper and in fact it passed peer review, so they published a paper in which they supposedly proved this fact they did careful statistical analysis again carefully in quotes and and and and and and and showed that the effect was real, and therefore the Bible must have been written by God, because no human could have put this information into the Bible.

It's information that did not even exist at the time that the Bible was written. And so God wrote the Bible, further, it's the Hebrew Bible. So there is a God, and that God loves Hebrew more than she loves Japanese. I understand, yeah. This would be the greatest discovery. If it was true, it would have been the greatest discovery in the history of science. And I'm not kidding, right? I mean, there's no longer an argument between atheism and theism.

Theism wins. My role was to debunk this. Is this real life or not? I don't know. Me and a group of people read critically these assertions that were good enough that they passed peer review in a respectable journal,

and we found gaps. And the gaps are, well, were good enough that the same peer-reviewed journal accepted our paper too. Okay, so, you know, in some sense, my role was negative, right?

What could have been the greatest discovery in the history of science, a proof that there is God and it's the Hebrew God, was debunked. Maybe it's a negative discovery, but well, I mean, maybe my effect, well, my part was negative. I've killed a great discovery. But that's also a valuable thing to do, I mean, to kill a... The other thing is, otherwise,

mathematics plays an extremely limited role in my life. Other than my professional life. If you cut out the eight hours a day that I work on mathematics, mathematics has no role in my life or hardly any role so I mean you do and in fact also almost no role in your life so basically you know if you you give change in restaurants, you, so, so, or in,

in shops, sorry, you add 15% tip, that's about the math you need in real life. Multiplication, you don't need, but, but if you're a carpenter,

you probably need it, right? You need to, tile, you know, if you're a carpenter, you probably need it, right? You need to tile. If you're a trade person, you need to tile a certain room. The room is five by six. How many tiles do you need? So multiplication, trade people need, but people like me don't need. Trigonometry, well, maybe architects need

trigonometry. Some parts of engineering, and we get more and more esoteric, need higher and higher mathematics. Without the very high mathematics, without general relativity, GPS satellites wouldn't work. So in some sense we use mathematics every day. But we're not the engineers who design GPS satellites. GPS satellites were designed by a group of, a very small group of people somewhere in some, you know, design bureau.

And they use high-level mathematics. So, but my real-life high-level mathematics. But my real-life high-level mathematics is almost nothing except once, sorry, the Bible codes and another time. And I've given talks with this title. So the title was The Hardest Mathematics I've Ever Really. Mm-hmm. Yeah. And if you Google this title, probably you'll be able to find a video of that talk, and you'll find that one time where it was actually in my professional life, but in a meta way,

so not my research, but something about how to present my research, I ended up using very high-level mathematics, hyperbolic geometry in fact. So one time it happened. Anyway, what was your next question? Why does it take 360 pages of abstruse symbols to prove that 1 plus 1 equals 2? Occam's razor says we can cut this. We can just say 1 plus 1 equals 2 because every single last one of us believes it does. So what do you make of that?

Why is it so complex to prove something simple like 1 plus 1 equals 2? Do we even need that? And this question comes from Roy Dobson. So, let me answer a different question. You know, so 1 plus 1 equals 2, so the reference is probably to the work of Russell basically if your axiom system is very very very basic and and very, very primitive in a way, very, very simple-minded, and you want your axiom system to be very, very, very basic, then it might be that even proving 1 plus 1 equals 2 would be difficult.

But it will be a part of the build-up. And later you will be proving harder things. But the truth is, it's silly. But let me give you another example. So, you know, there is the intermediate value theorem. The intermediate value theorem says that if you have a function, continuous function,

so if f is continuous and it's negative here and positive here, then at some point it crosses zero. We teach it in week seven, maybe, I don't know, of undergraduate calculus. Maybe it's just week five or maybe it's week nine, I don't remember, of undergraduate calculus, maybe it's just week five or maybe it's week nine, I don't remember, okay? You can ask, this is so idiotic. Why does it take five or seven weeks to prove the obvious that if you go continuously from here to here you'll be crossing zero? continuously from here to here, you'll be crossing zero. And the answer is that when you prove the intermediate value theorem,

you are not actually proving the intermediate value theorem. You are actually testing your definition of continuity. You're testing your definition of continuity. You're testing your abstract setup. You're testing your abstract setup, your language. You're testing the machinery of for every epsilon there exists a delta such that blah blah blah. If you could not prove the intermediate value theorem you'd be rejecting the abstract system, not the intermediate value theorem. You'd be rejecting this, not that. So the seven weeks were not about the intermediate value theorem, they were about

this. For every epsilon there exists a delta. And then you can ask, what's for every epsilon there exists a delta for? Why do we need this? Well, we don't need it to prove the intermediate value theorem. The intermediate value theorem was testing this. What do we need it for? And the answer is, well, there are later things like e to the x is equal to sum of x to the n divided by n factorial, which is, this, by the way, looks innocent and people think they understand it, but it's actually a miracle. And this miracle depends on, well, it doesn't depend, but I mean but there may be other ways to reach there but you will believe

this miracle after you've understood this system so the answer is in a way we didn't go from here to here to here but to here,

but more like this was known, it validated this, and then we could use it to do other things. The podcast is now finished. If you'd like to support conversations like this, then do consider going to patreon.com slash C-U-R-T-J-A-I-M-u-n-g-a-l. That is Kurt Jaimungal. It's support from the patrons and from the sponsors that allow me to do this full-time.

Every dollar helps tremendously. Thank you.