Terry Tao How To Think Like A Mathematician

TITLE: Terry Tao “How to think like a mathematician” presented by the UCLA Curtis Center CHANNEL: UCLA Curtis Center URL: https://youtu.be/kRcro90Aj0w

---TRANSCRIPT---

One thing that we’ll want to think about is this question of how do we get students to really think hard about these mathematical concepts. What we’re going to do is we’re going to have a speaker who will give us some food for thought. I guess it’s Pi Day, so pie for thought if you will. And we’ll have Heather Dallas, the director of the Curtis Center, to come up afterwards and lead us in this conversation. So I’m hoping that you’ll listen intently to the words that are going to be said, and that we’ll have a chance to discuss and have an interchange of ideas afterwards.

That being said, I do want to introduce our speaker for today, who’s a professor in the Department of Mathematics and the James and Carol Collins Chair in the College of Letters and Sciences. Um I want to give you 3.14 interesting facts about Professor Tao. Um interesting fact number one, he is the youngest participant to date in the International Math Olympiad, first competing at the age of 10. Fact number two, when he was 24, he was promoted to full professor at UCLA and still remains the youngest person ever appointed to that rank by UCLA. Fact number three, Professor Tao has published 16 books, but he has a new book coming out, Six Math Essentials, that’ll be his first popular math book. And interesting fact 0.14 is that today he’ll give us a talk entitled “What Does It Mean to Think Like a Mathematician?” So let’s give him a round of applause.

Thank you, Andre. That’s that’s a great introduction and I’m very happy to be here. Uh technically I think I was a colleague of Phil Curtis. I came here in ‘96. It was about the same time he was retiring. I’m not quite sure exact dates. So I never actually got to interact with him too much, I’m afraid. But he was a living legend. Well, he was a legend for our department.

So yeah, so I’m here to sort of bring the perspective of a mathematician to this event. And I think it is important we have the way of thinking like a mathematician, I really enjoy it. And it’s a very precious thing that wish more students would gain access to. But it’s just it’s often such a shame. And I think most people don’t have a handle of even what a mathematician does. You know, so maybe a lawyer or doctor or engineer, you have some idea of what they do. A mathematician, it is a little bit — the mental images people have in their heads are a bit inaccurate. And every mathematician has had this experience that it’s very hard it’s very awkward at parties. You know, like I ask, “What what do you do?” And so I’m a mathematician, and you always get this response, “Okay, I’m bad at math at school.”

Some people think that mathematicians are like wizards, that we gain access to these magic spells and with these weird symbols that we can somehow cast on people. Some people think that it’s really complicated technical skill and there’s only one correct answer and lots and lots of wrong answers and you make one mistake the whole thing collapses and we are somehow juggling all kinds of complicated equations. Or they think that we are like these Hollywood sort of — we see all these equations and I we don’t or maybe some people do, but I don’t certainly. We don’t see these things in front of our eyes when we do our math.

So we have all these archetypes which I’m not the norm really. I mean, most of us are pretty normal, I think. Which is unfortunate in many ways. I think on the one hand it is a little bit cool that sometimes we get this sort of reputation for being geniuses and so forth, but it does create issues with students, that maybe they read stories about mathematicians solving these great problems and being very clever, and then they have their own math problem to solve and they can’t solve it in one go and they quit math before it becomes fun.

Yeah, and so if you teach that mathematicians are happy geniuses, then and you don’t feel like a genius, then you don’t feel like a mathematician.

So I just wanted to sort of share what a little bit of what it’s like to feel like be a mathematician. This is something every mathematician knows about, but we don’t communicate it as often as we should.

So one thing — one of my sort of frameworks for thinking like a mathematician is that there’s actually multiple stages of thinking like a mathematician. And that’s part of why math is — mathematical thought is a bit hard to internalize because it is a somewhat complex thing to evolve. So I like to divide math into what I call the pre-rigorous, rigorous, and post-rigorous stages.

Pre-rigorous is roughly sort of K12, K14 type of education where you’re taught examples, intuition, formulas, computing, but you can make mistakes and you don’t really understand what’s going on. You just have sort of a vague handle on everything.

And then you go to college, and if you’re in one of the math majors, you’ll get taught these dreaded proof classes where you start learning how to think rigorously. And now there’s only one correct way to solve problems and lots and lots of incorrect ways and a lot of your old pre-rigorous intuition gets sort of pooh-poohed and they say, “No, no, that was for kids. Now this is the real math.”

But then what is not really appreciated because most people don’t reach the stage is that actually there’s a third stage, which I call the post-rigorous stage, where once you know how to do these things rigorously and precisely, you actually go back to revisit your intuition and think much more fluidly and informally, but knowing that now if you wave your hands and do something, you can convert it to a rigorous argument if you want and you can go back and forth. And this is basically graduate school. And that’s the fun part and it’s a shame that most people don’t see that.

So let me try to illustrate with sort of examples of what these phases look like. So let me talk for example, there’s a basic concept in math called proof by contradiction. And it’s considered sort of a not unintuitive and difficult concept for students to understand, but actually primary school kids teach this concept to themselves in fact at recess. For example, when I was a kid we had silly games like this. We would gather around and we would just name — who can name the biggest number? And so I would name 1 billion, 1 trillion, 1 quadrillion, and you just go back and forth. And this game would go on until someone realizes that no matter what number someone names, the next person can always name that number plus one.

So someone eventually figures this out. And at that point, they will realize there is no biggest number because no matter what number you can pick, there’s always you can always add one and get a bigger number. And they have used proof by contradiction. They proved that a biggest number cannot exist because if it did, it would be bigger than itself or be bigger than the number plus one, which is not possible. And this is something that they’ve discovered, but they can’t verbalize this. This is pre-rigorous. They don’t have the language.

So then you take some more advanced classes in math high school and then undergraduate, and then you start seeing proofs. And proofs come in various flavors. There will be direct forward proofs where you have some word problem and you have some hypotheses and then you just start applying various mathematical transformations and you get from A to B. You get from your hypothesis to your conclusion. So that’s a direct forward proof. There’s also direct backwards proofs where you have some hypothesis conclusion and now you start with what you want to prove and you reduce it, you transform it, you cancel terms, and you do all this algebra and you get back to what you started.

But then sometimes occasionally you’re taught a proof by contradiction and it’s so weird. You might be so — proof that say square root of two is an irrational number, that this number cannot be written as the ratio of two integers. And then you say, “Well, let’s suppose that it is a rational number.” And then you do something and you argue for a bit and then you see something say that what I the conclusion I got didn’t contradict something I said earlier. Therefore, my original conclusion was true that root two is irrational. And this is really hard for — some students get it, but definitely many students do not connect that with the kind of pre-rigorous experience they had with contradiction that they may have learned about themselves. But it’s the same concept, taught in a very different way.

So once you actually learn — you become professional mathematician — you use proof by contradiction all the time. And it’s not — we don’t view it as actually anything clever. Basically it’s just one move you can make. If you doubt that something is true we just — it’s very natural for us to assume it’s true. See what happens — if something bad happens, if we can get a contradiction out of it we know that it couldn’t be true so it has to be false. And it’s just a move in a game for us.

So actually G. H. Hardy the mathematician made this very nice quote. So reductio ad absurdum which is Latin for proof by contradiction is one of the mathematician’s finest weapons. It is a far finer gambit than a chess gambit. So a chess gambit a chess player may sacrifice a bishop or a rook to get some positional advantage but the mathematician can offer the entire game. Okay, and still win.

So that’s three different ways of thinking about the same concept.

Another one which I think people already mentioned — some of the previous panelists mentioned — is that math is a place where it is okay to fail which is actually the opposite of the way we teach it often especially in the rigorous phase where if you get a sign error wrong and so forth all these red marks come out and you lose these points. But actually compared to other disciplines math is actually — failure is very very cheap. If you’re an engineer and you’re designing a bridge and you make a mistake that’s an expensive mistake. If you’re a heart surgeon and you cut the wrong thing that’s also a very bad mistake. But if you have a solving a math problem and your proof doesn’t quite work out that’s a very cheap mistake. You just do it again.

Vladimir Arnold once said that math is even — you can think about it as the part of physics where experiments are cheap.

So because of this as a part of what you’re taught as a graduate student is just keep trying things and keep making mistakes and keep doing things even if you suspect they are likely to fail. Because the way in which they fail is often very valuable. And so this is a mindset that we internalize and it’s just we’re just used to it but like almost nobody else gets it. In almost any other discipline people are just afraid of making mistakes. But in math we have the freedom to fail and that’s actually very precious.

There’s a disconnect between the way we assess math solutions and the way we assess math process. So solutions should be correct but the process can have lots and lots of failure and that’s important.

So I’ll just give you a practical actual example. So the student came up to me a few months ago and was asking for help on a math problem. So he had a hint. So there’s a technical Taylor approximation. It’s a technique that would solve the problem. And he had learned about Taylor approximation but he had learned it three times. So he had three different books and they said there’s three different Taylor approximation formulas to use and he didn’t know which one to use. And so he was paralyzed. He says I’m stuck. I don’t know what to do. He experienced what someone has called analysis paralysis. There’s too many options didn’t know which one to pick. And I just told I said something very simple. It doesn’t matter if the first thing you pick doesn’t work. Just try one. Maybe it works maybe it doesn’t work. It may partially work. But then you can see what to do next. And this unblocked him. I didn’t give any further hints but basically just permission to fail was all he needed.

Niels Bohr has this great quote that what is an expert? An expert is a person who made all the mistakes that can be made in a very narrow field. If you haven’t made the certain mistakes — if you haven’t made them in the past you’ll make them in the future. So it’s actually important to make them now. So put them in the past so that in the future you do not make that same embarrassing thing again.

And it is a really really important part of our process. And it’s a shame we don’t report on that. So our culture is not perfect in some ways. When we publish our papers we don’t publish our papers often — sometimes a few of us do — about all the wrong turns and our feeling of getting lost which is the default state of being a mathematician actually. But we only publish our wins usually. And then you read the famous mathematicians or like for example Fields Medalist Maryam Mirzakhani — she’s more honest than most of us but even still her papers are full of wins. And then you read your own paper and you do your own work and your proofs aren’t working and so forth and you feel like an impostor. And so it would be better to normalize our disclose our failures a bit more often I think.

I’ll just give you one example. I worked several years ago on a problem in partial differential equations. There was a very famous mathematician Jean Bourgain who worked very hard on it and we got a partial result. We wanted the whole thing. And so we were kind of cocky and we thought oh we’ll do this in a few months. I worked with four other people and it worked. We tried something crazy and it worked. It was great. We were in fact we were trying to book a restaurant to celebrate with champagne. And we started writing up the proof and then one of my co-authors who was more careful than the rest of us actually noticed that we expanded this thing to 13 terms and we had controlled 12 of them correctly but we just forgotten about the 13th term. So oh yeah, I’ll deal with this. I checked that actually we could not control this 13th term. Actually it was the one term that was the worst and we had somehow dropped it. We thought it was a minor thing but actually no matter what we did we just could not get rid of this term.

And so we tried all kinds of things and we could not fix the proof. But by that point we’d spent like six months on this problem and we had to cancel this reservation. We were really invested. And so we just kept at it. It took us two years to solve this problem. We finally found a much different way to solve the problem. And actually this paper is one of the papers I’m most proud of. It won a prize. If we had not had this mistake of this early success we would have quit way before we had solved this problem. So actually sometimes making a mistake is even a positive thing. So there’s a paper in a little journal called Annals of Mathematics.

There’s a corollary to that which is that in order to make progress in math you have to ask lots and lots of really stupid questions. Often we condition our students to not speak up unless they’re really confident in their answer and that’s actually often the wrong approach. Students often ask silly questions when they’re solving math problems which if you know the answers then why would you ask this question? It’s dumb. But actually these questions are really important to answer.

Mathematician — some of the deepest progress in mathematics has come from mathematicians asking similarly stupid questions just at a slightly higher level. You can almost reduce every new breakthrough in mathematics to someone asking a stupid question.

Paul Halmos was really very famous for sort of teaching students how to think like a mathematician. Really emphasized that you should always not just accept what your teachers tell you as a static thing. You have to really fight it and really make it your own. And one way is just to ask your own dumb questions.

I’ll just give you one example from personal experience. My graduate advisor Elias Stein was an accomplished mathematician at Princeton. There was an inequality he had proven in all cases except one at the end point. And at some point I said I’m going to prove the end point case of Stein’s inequality. I tried both by myself and with some co-authors. We got some very weak partial results but it was one of my dreams to sort of complete this result that my advisor had worked on.

And at some point I was invited to a — I think it was 80th birthday conference — and I had to present something and I said well I was working on this and I had some partial results and so I presented what I had. And then a colleague of mine in the audience just asked a question, “Have you ever tried to disprove — Is there any reason to expect why this inequality is true? Is there a counterexample?” I had — this had never occurred to me that the thing I was trying to prove could actually be false. So I had no good answer to this question. I then spent the evening trying to see if it could be false, and actually within a week I had found a counterexample.

So sometimes it’s not necessarily you that have to ask a dumb question. Somebody has to ask the dumb question.

Just to summarize, these are three aspects of being a mathematician. You have to learn both intuition and rigor, but combine them eventually into some sort of post-rigorous mindset. You have to embrace the freedom to fail, and you have to ask dumb questions. Thank you very much.

[Q&A session follows with discussion of collaboration, AI in mathematics, mathematical pedagogy, the experience of doing math, and related topics.]