Terry Tao "How to think like a mathematician" presented by the UCLA Curtis Center

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

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.

[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

here. Uh technically I think I was a colleague of Phil Curtis. I came here in '96. Uh

in '96. Uh it was about the same time he was retiring. I'm not quite sure exact

retiring. I'm not quite sure exact dates. Um

dates. Um so I never actually got to interact with him too much, I'm afraid. Um but he was a living legend. Well, he was a legend for our department.

Um so yeah, so I'm here to sort of bring the perspective of a mathematician to to this event. And I think it you know,

this event. And I think it you know, it's it is important we have the way of thinking like a mathematician, I really enjoy it. Um and it's it's a very

enjoy it. Um and it's it's a very precious thing that wish more uh students would would gain access to. But it's just it's it's often such a shame. Um and it's I

think, you know, most people don't have a handle of even what a mathematician does. You know, so you know, maybe a

does. You know, so you know, maybe a lawyer or doctor or engineer, you have some idea of what they do. Uh a

mathematician, it is it is a little bit uh the the mental images people have in their heads are a bit a bit uh inaccurate.

And uh you know, every mathematician has had this experience that it's very hard it's very awkward at parties. You

know, like you you I ask, "What what do you do?" And so I'm a mathematician, and

you do?" And so I'm a mathematician, and you always get this response, "Okay, I'm bad at math at school."

Um well, actually sometimes you get like a really enthusiastic response, but this that's that's that's the median response.

Um Some people think that mathematicians are like wizards, that that we gain access to these magic spells and you know, and with these weird symbols that

that we can somehow cast on people. Um

Some people think that it's really complicated technical skill that 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.

Um Or you know, they think that we we we are you know, like these Hollywood, you know, 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 in front of our in in in front of our eyes when we uh when we do our math.

Um so we we have all these sort of uh um um archetypes which I'm not the norm really. I mean, most of us are pretty normal, I think. Um

oops Um so which is unfortunate in many ways.

Um I think, you know, I mean, on the one hand it is a little bit cool that you know, sometimes we we get this sort of reputation for being geniuses and and so forth, but it it does create um

um you know, issues with students, you know, that that you know, that maybe they they 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 and they can't solve it

in one go and um you know, all all they um yeah yeah or they they they they learn a how to learn a difficult topic and they never get it. And so they they they quit math

get it. And so they they they quit math before before it becomes fun.

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

Um 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. I think we we should do more outreach operation. Um and we we do sort of hang out in our ivory tower

maybe a bit too much. Um

So um one thing I I um one of my sort of frameworks for thinking like a mathematician is that is that there's actually multiple stages

of thinking like a mathematician. Uh and

that's that's part of why math is um mathematical thought is a bit hard to uh to internalize because it is it is a somewhat complex um thing to evolve. Um So I like to divide

math into what I call the pre-rigorous, rigorous, and rigorous stages. Um so

pre-rigorous is roughly sort of K12, K14 type um of of education where you're taught examples intuition formulas computing, um but it's

but you can make mistakes and you don't really understand what's going on. Um

you just have sort of a vague handle on everything.

Um And then you go to to college, um and if you're 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 and now there's only one

rigorously. And and now there's only one correct way to do 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 that that was for kids. Now now

this is this is the real math." Um

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 and

precisely, you actually go back to revisit your intuition and 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 if if you want and you can go back and forth. Um and

this is basically graduate school.

Um and that's the fun part and it's a shame that most people don't see that.

Um so that's the word. 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. Um and it's

contradiction. Um and it's considered sort of a not unintuitive and difficult concept for for students to to understand, but actually um um

um primary school kids teach teach this concept to themselves in fact at you know, at recess. For example, you know, I mean, when I was a kid we had silly games like this. We would

we would gather around and we would just name you know, who can name the biggest number? You know, and so I would name 1

number? You know, and so I would name 1 billion, 1 trillion, 1 quadrillion, and you just go back and forth and 1 1 trillion trillion trillion, whatever. Um

And this game would go on until someone realizes um oops that no matter what number um someone names, the next person can always name that number plus one.

Um so someone eventually figures this out. Um and at that point, they will

out. Um and at that point, they will realize there is no biggest number in in because no matter what number you can pick, there's always you can always add one and get a big number. Um and they

have used proof by contradiction. This

is they they proved that a biggest number cannot exist because if it did, it would be bigger than itself or be bigger than than the number plus one, which is not possible. Um And this is something that

possible. Um And this is something that they've discovered, but it they don't they can't they can't verbalize this.

This is pre-rigorous. They don't they don't have the language.

Um So then you know, maybe you go to you take some more advanced classes in math high school and then and then undergraduate, and then you start seeing proofs. And proofs come in various

proofs. And proofs come in various flavors. Um There will be direct forward

flavors. Um 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. Um and

I'm sure you've all done problems like this. I'm not going to go through that

this. I'm not going to go through that with you. Um so that's a direct forward

with you. Um so that's a direct forward proof.

Um There's also direct backwards proofs where you have some hypothesis conclusion. Um and now you start with

conclusion. Um 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 to what you started. So this is this is very typical kind of rigorous

stage mathematics where you and you're not supposed to make any mistakes as as as you do this.

Um But then sometimes occasionally you're taught a proof by contradiction and it's so weird.

You know, so you might be so proof that say square root of two is an irrational number, that this 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 to for or I mean, some students get it, but definitely many students do not connect

that with the kind of pre-rigorous experience they had with with contradiction that they may have um you know, learned about themselves, but it doesn't it doesn't feel like this. But

it It the same concept. Um but

it's taught in a very different way.

So if you are so once you actually learn you become professional mathematician you use proof of contradiction proof of contradiction all the time.

And it's it's not we don't view it as actually anything clever. Basically it's

it's it's just one move you can make.

You know, so if you doubt that something is true we just it's very natural for us to assume it's true. See what happens if there's something bad happens if if if we can get a contradiction out of it we

know that that it could it couldn't be true so it has to be false.

And it's 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 of contradiction is one of the mathematician's finest weapons. It is a far finer gambit than a

weapons. It is a far finer gambit than a chess gambit. So a chess gambit a chess

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 how yeah so there's three different ways of thinking about the same concept.

Okay, so that's one of the ways one of the lessons about how to think like a mathematician.

Another one which I think people already mentioned some of the previous panelists mentioned is that math is a 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 you know, if you if you get a sign error wrong and so forth all these all these red marks come out and you know, and and and you lose these points whatever.

But actually compared to other disciplines math is actually it's very failure is very very cheap. You know, if if you're an engineer and you're designing a bridge and you make a you make a mistake that's an expensive mistake. If you're a heart surgeon and

mistake. If you're a heart surgeon and you cut the wrong thing that's a that's also a very bad mistake. But you

know, if you had if you have a solving a math problem and and your proof doesn't quite work out that's a very cheap mistake. You just do it again.

mistake. You just do it again.

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

are cheap. Um Uh though with current AI well never mind. Okay.

mind. Okay.

Um So um Yeah 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. Um because the 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. You know, that that that

gets it. You know, that that that in almost any other discipline people are just afraid of making mistakes. Um

but in math we have the freedom to fail and that's that's actually very precious.

Um So it's but it it's Yeah, so um There's a disconnect between the way we we sort of 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.

Um 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. I

won't tell you what the problem was but um So he had a hint.

So there's a technical Taylor approximation. I don't need I don't want

approximation. I don't need I don't want to say what Taylor approximation is but 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

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.

stuck. I don't know what to do.

And I So he 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

simple. It doesn't matter if the first thing you pick is it doesn't work. Just

try one. Maybe it works maybe it doesn't work. It may partially work. Okay, but

work. It may partially work. Okay, but

then you can see what to do next. Okay,

and this is this is unblocked him. I

mean I didn't give any further hints but but you know, but basically just permission to fail was was all he needed. And then

I think he's off the problem. He didn't

he didn't get back to me but um Um Yeah, but it's a very basic lesson which um we don't we don't often don't teach because we also because correctness in

the answer is important. So you have to you have to teach both.

Um Yeah, but this is something that that that scientists realize. Okay, so Niels Bohr has has 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 that in the future you do not make that same embarrassing thing again.

Um yeah, but it it is a really really important part of our process.

Um You know, and it's it's a shame we don't report report on that. So

our culture is not perfect in some way.

So when we publish our papers we don't publish our papers often. I mean

sometimes a few of us do when we're about all the the wrong turns and and our feeling of getting lost which is the default state of of being a mathematician actually.

Um but um Yeah, but we only publish our wins usually.

And then you know, you you you you read the famous mathematicians or like for example Fields Medalist Maryam Mirzakhani and you know, so I mean she's more honest than than most of us but she was but

but you know, even still her papers are full of wins. And then you you read you read your own paper and you do your own work and you know, your your proofs aren't working and so forth and and you feel like an impostor.

And so it would be better to normalize our disclose our failures a bit more often I think.

Uh oops.

Um I'll just give you one example.

I worked several years ago on a problem in partial differential equations. Now

it's not important what the problem was.

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. Um 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. We tried something crazy and it worked. It was great. Okay, we we

it worked. It was great. Okay, we we were in fact we were trying to book a restaurant to to celebrate with champagne. And we started writing up the

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 you know, 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. Okay, I

would add this to the paper and I checked that actually we could not control this 13th term. Actually it was actually the one term that was the worst and we had somehow dropped it. Um we

thought it was a minor thing but actually it it no matter what we did we just we could not get rid of this term.

And so we um Yeah, we tried all kinds of things and it and we could not fix the proof. But

by that point by the time we realized that we'd spent like six months on this problem and we had and we had to cancel this reservation.

Okay, it it was we were really invested.

And so we just kept at it.

Whereas it would it it took us two years to solve this problem. We finally found a much different way to solve the problem.

And actually this this paper is one of the papers I'm most proud of. It won a prize.

You know, 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 actually making a mistake is even a a positive thing.

Yeah, so there's a paper in a little journal called Annals of Mathematics.

He's very happy with it and he's my four co-authors.

Um Yeah so yeah, freedom to fail.

There's a corollary to that which is that um you know, in order to make progress in math you have to ask lots and lots of really stupid questions. And this is something which we said you know, um

Often we uh we sort of condition our students to not speak up unless they're really confident in their answer and that's actually often is is is the wrong approach. You

know, students often ask you know, silly questions in you know, when they're solving the math problems which you know, if you know the answers then why would you ask this question? It's dumb. But actually

this question? It's dumb. But actually

these these these questions are really important to answer.

And you know, I mean okay, I've listed some some silly questions here that a student might ask but you know, mathematician like some of the deepest progress in mathematics has has come from mathematicians asking similarly

stupid questions just at a slightly higher level. I won't

each one of these questions can lead to an hour long talk but I'm I'm not going to to but yeah, there there are You can almost reduce every new breakthrough in mathematics to someone

asking a stupid question.

Um You know, yeah and Paul Halmos was really what's a very famous for sort of teaching

students how to how to think like a mathematician. You know, really

mathematician. You know, really emphasized that you know, you you should always not just accept what your your teachers tell you as as as you

as a static thing. You know, you you have to really fight it and really make it your own. And and one way is just to ask you know, ask your own your own dumb questions.

Um I'll just give you one one of example from personal experience. You

know, so um Um Yeah, so my advice I had a my graduate advisor Elias Stein was a 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.

This was you know, I I tried both by myself and with with some co-authors.

We got some very weak partial results but it was one of my dreams to sort of complete this 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 so I had to present something and I I said well I was working on this and I had some partial results and so I I presented what I had.

Um And then a colleague of mine in the audience just asked a question, "Have you ever tried to disprove Is there Is there any reason to expect why this inequality is true? Is there a counterexample?" I had

counterexample?" I had This had never occurred to me that the thing I was trying to prove could actually be false.

Uh so, I had no good answer to this question.

Um I then spent the evening trying to see if it could be false, and actually within a week I had found a counterexample. Um

counterexample. Um So, you know, it's it's it's a you know, sometimes you know So, it it's not necessarily you that have to ask a dumb question. Somebody

has to ask the dumb question. Um

Yeah, so just to summarize, these are These are three aspects of being a mathematician. So, you have to learn

mathematician. So, you have to learn both intuition and rigor, but combine them eventually into some sort of post-rigorous mindset. You have to

post-rigorous mindset. You have to embrace the freedom to fail, and you have to ask dumb questions. Thank you

very much.

[applause] WE HEARD SOME THEMES FROM Terry that are actually being echoed in the field of mathematics education right now. And in fact, over the last decade

now. And in fact, over the last decade there have been ongoing efforts in mathematics education to more carefully develop and justify ideas, and to openly

acknowledge and support struggle and failure and perseverance in problem-solving in classrooms, and to prioritize process including contrasting

and learning from multiple solutions to rich problems. And so, we know that the field of education is trying very hard to develop the habits of mind of

professional mathematician in our students in our classrooms. And so, um I thought I would start off the discussion with a question of my own for you, Terry.

Okay. Um and I wanted to ask uh some other themes in math education right now are around the ideas and the role of collaboration between students in developing an understanding of the

mathematics they're learning, and also the role of technology in the classroom.

And I wondered what if you would be willing to share with us some of your thoughts on the role of collaboration and the use of technology in the professional work of a mathematician. Those are great topics.

mathematician. Those are great topics.

Yeah so that's those those list of three things I I gave my talk that should not be viewed as a complete list of all the things you should learn as a mathematician. Oh, sorry. No, it's okay.

mathematician. Oh, sorry. No, it's okay.

Um Yeah, so uh one skill that has become increasingly important over time in mathematics is is the soft skill of knowing how to collaborate. Um

collaborate. Um I think math has become has evolved from being a very sort of solitary activity uh to to a very collaborative one. Uh

partly because the the nature of the subject has has has changed. We work

with much more interdisciplinary complex problems that no one person can solve.

Um but also new technology like the internet allows us to to work together on a scale that we cannot do before.

Um Yeah, so um I teaching um students to to to work together on on on group projects is an important um skill. So, we

skill. So, we um which is a It's it's it's It's a very different type of skill. So,

yeah, it's a you know, people skills are becoming more and more important, which traditionally we've not not been a strong suit, to be honest, in mathematics, but uh we are evolving.

Um And yeah, we'll be collaborating not just with with other humans, but with with increasingly with with AI. Um

with AI. Um Yeah, they are AI tools are becoming very powerful, in some ways still very weak in others, um but when you collaborate, the the the beauty of collaboration is division of labor, you

know. So, um

know. So, um in the past, a mathematician had to handle every single aspect of of a problem-solving process, you know, identify the problem, identify the

strategy, identify a good strategy, um uh implement it, and and then write it up, explain it. Um

And and now with you you can collaborate with other humans and and AIs and and and different members of the collaboration will do different tasks. Um

different tasks. Um So, yeah, we are we are slowly moving into the where we have we experiencing some sort of industrial revolution.

Um and I our class our teaching will have to evolve.

Uh It is changing very fast for better or for worse.

But I think, you know, we have good resources like like this, we we can adapt.

Thank you so much. I noticed that there was a question from the Zoom room.

Yes, I was a couple of your panelists who mentioned real-world applications using mathematics.

And I was thinking one of the best real-world applications is if you could get a piece of the UCLA endowment fund,

rumored to be worth $5 billion or so, and take the incredible brainpower of the mathematics department, I don't know if you want to work with the finance department, but the average

10-year return from the fund is under 8%.

And I'm sure with using AI or numerical simulations or uh what what whatever you guys choose to do, you could you could crush that average return.

And uh [laughter] So, you know how I said in my my my presentation that in mathematics failure is very cheap.

Um so, uh there's an asterisk there. If

you if you connect it to the real world, you do actually have to do some serious risk-benefit analysis. I

think teaching financial modeling would be would be a great school project, actually. Um so, um and and

actually. Um so, um and and yeah, you could you could run sort of some some simulated markets and and see how just a very small bit. I wouldn't directly send

the students to to manage the UCLA endowment just yet.

[laughter] So, we have another question in the room.

Would you manage my portfolio? Is that

[laughter] Yeah, I refer to my previous comment.

[laughter] I I do have a serious question. How good

is How good are large language models now at doing advanced theoretical math?

Like, can they push the boundaries forward in the field?

They are becoming increasingly capable at at many things. Um so, I think at this point they're on the level of kind of like It's like having an army of graduate students of various levels of

quality, and you you you give a problem, and they will try they'll dig up in the literature some techniques and try them one by one. Sometimes they they succeed, sometimes they fail. Uh so, sometimes

they they do spectacularly well, uh sometimes they they fall flat on their face.

Um So, if you only if you only are going on social media Hm? Not quite sure what's happening

Hm? Not quite sure what's happening here. If you If you only go on social

here. If you If you only go on social media and and you see people talk their the the biggest successes, it looks amazing. Um and then you try to download

amazing. Um and then you try to download these tools or or use them use them at home on your favorite problem, and the success rate is often like 1 to 2%. Um

So, um I think there's certain types of math where they'll be very useful. Um I

think if there's a problem that you can solve kind of a task that you can split up into like 100 subproblems, and um and um you don't mind if there's a tool that can only solve 10% of them.

That's 10 problems you got to solve, and then another tool will solve another 5%.

Um so, I I I think we as I said, we math has to industrialize and start playing the games of percentages. So, instead of working on taking one problem and working on it very very hard, which is

that our usual MO, you know, um creating these large-scale projects where we just accept that there's a partial success rate. Um then these tools will become

rate. Um then these tools will become quite quite powerful, I think. But it's

it's a very fluid situation. It's it's

exciting, but also a bit scary, too. But

uh Yeah, um I think there's more good than bad.

I am so curious about when you were introduced, and then um Mr. Rope or Professor Roper, sorry, talked about

once you become a parent, what you see your children experience in school.

Um and then I immediately went to gosh, at 10 years of age winning a math competition, um you're, you know,

a a prodigy, a celebrity, you know, kind of like the smartest one. Everyone knows

in school who the smartest kid is. It's

not easy.

Fast forward to today, you can stand up here and say as an adult, as a parent, there are no dumb questions. Um I still think I ask really dumb questions, and I'm terrified to do it publicly.

Um and I'm doing it right now, cuz I feel like I'm probably asking some But um the point being is where you are today um looking at

the future of where you talk about AI, LLMs, the future of math, the future of uh education, um the system, and so it's how wrong wrong

it is, um what would you say to the schools, the teachers, the districts? How you bring

math into the world that keeps kids wanting to work hard to ask dumb questions, cuz I can't imagine how somewhat easy it might have been for you, but at the same time

how hard um I can't imagine what you went through.

Is that a fair question?

Yeah, well, um yeah, it was a different time. Um I

think um yeah, so the world was a lot slower then um and more predictable. Um and actually um so, I had a very accelerated education actually in skipping five grades.

Um and local headmasters at the high school and and elementary school and and at the local university had to design a special

curriculum just for me actually.

And somehow it was all very new and they weren't It's so many rules in place that I mean there wasn't a structure structured gifted education program at the time. Everyone was improvising.

the time. Everyone was improvising.

And it kind of worked out through through all the efforts of of my parents and and my um and all all the various administrators.

Um So in in many it's in many ways it's better and worse. Now I mean so in my So this is in my home country of Australia. Now they have gifted programs

Australia. Now they have gifted programs which are much more structured.

So you don't have to improvise everything from scratch, but they're also maybe a bit more rigid. Some some things that I I could get away with I could could

would not be be possible now.

So it's a different world. Um

Uh so I mean I don't know how what works and what doesn't, but you know we have this amazing center here that is devoted to that.

So you know I mean we have to try experiments and showcase the successes also learn from our failures. Um and uh um you know I'm sure we can do better than

than the way we do now.

And you know um one thing we learn as a mathematician is even a partial solution is better than no solution at all.

So um so I I and I think that's that's what the center is all about.

Thank you Terry.

Um what you just said reminded me of something I said to Esei nine months ago or so. I said let's not be afraid of

or so. I said let's not be afraid of failing, let's be afraid of not learning.

So Um all right so Kate Stevenson here professor at CSUN.

Hi there. So I've just up the road. I

teach the dread proofs class at Cal State Northridge and I do it on purpose as chair of math because it gives me access to my majors. It's a

wonderful transition that happens to the students and I love watching it happen.

It's one way to put it.

I could get you on video telling them that it's okay to fail uh that would help my job so much.

But that is I think the piece that you describe these three pieces of what it means to be a mathematician what is maybe missing from that framework is

when you're talking to students who weren't in math camp, who didn't succeed in high school. I have

first generation students, the majority of my students. I have students who've had very difficult backgrounds and current existence.

Building the trust with those students to ensure that they are willing to make the mistakes because it is a privilege to be able to make mistakes and feel

safe doing that. And I think one of the ways we can collaborate towards building that trust is speaking with one voice and recent about the fact that

mathematics is about making mistakes, persisting, that mathematics is a web of connected ideas, not a bunch of algorithms. And happily recently the

CSU, the UCs, and the community colleges have come together on a statement of what it means to be prepared for post secondary study and it reflects all of the things that you spoke of. So I think

lifting those voices up to say we're not arguing about this, we agree on this point.

Thank you Kate. We have an additional question from the Zoom room and it's from a former student of Phil's named

Chris Ennis who had to actually leave the Zoom room, but I did want to make sure that his question was asked.

Um so this is a question for you Terry.

Have you ever seen an instance of an AI creating a new mathematical definition in order to prove a result or solve a problem?

I think coming up with good definitions is one of the most creative activities mathematicians do. For example, the

mathematicians do. For example, the definition of compactness in topology makes proving a lot of fundamental theorems quite a bit easier. Do AIs do that?

Um yes and no. So

you can make an AI generate all kinds of new ideas and definitions and but it would maybe generate one good definition and like nine bad ones.

And it doesn't You can't really tell which which ones are good which ones are bad without really um going through it yourself. Um

And also some parts of math are very checkable. You know like if you

checkable. You know like if you either you solve your problem or you don't. And

don't. And the parts of math that are very verifiable AIs are actually pretty useful currently. But with things like a

useful currently. But with things like a definition you can't you can't really score you know out of 10 how good a definition this is.

And so so those type of tasks these AIs are still very frustrating. They may

occasionally hit gold, but then they just pass they just move on and then then the the next nine things they say are rubbish.

So I haven't seen that yet. Maybe the

technology will improve, but yeah not yet is basically my answer.

So you you mentioned three phases of education. There's the pre-rigorous, the

education. There's the pre-rigorous, the rigorous, and the post-rigorous. And my

question is is that how it should be?

And if not, are there ways to expose people who haven't been exposed to those later stages to maybe taste it or get an understanding of it without getting that full education formal education

experience? That's that's a good

experience? That's that's a good question.

Um I know if you rush it um there are people who are like There are some people who are like really really enthusiastic about math and they have always great ideas

but they have no rigor. And they try to jump to the post-rigorous stage before they they they know how to prove things.

And uh it's it it can be a little bit awkward because a lot of what they say doesn't actually make sense, but you can't even they don't even understand why. Um

I think it's important to have people from different stages of this education process talk to each other more.

Um and just to know that that that what you're experiencing now is not the final state of mathematics.

Um but just um so even if you're not ready to appreciate sort of the full uh aspects of it just take just get get a

taste. You know again we we don't show

taste. You know again we we don't show our process nearly as much.

Um you know um there there are other activities where people love just watching. You know it's like like

watching. You know it's like like computer games there's a whole culture of people watching other people play computer games or watching athletes train or watching actors do their outtakes. Um we we don't

have the same culture in mathematics of we we think what we do is either very private or very boring.

But maybe if we shared more of what we did then this would help.

Perhaps that's why that YouTube video that you recently posted has 100,000 views because you're in that video So there's a video that Terry shared with me in which he's

using actively using an AI to engage in research and it's gained quite a number of views and maybe that's a little bit of taking the lid off and showing what's under the

hood in real time.

Yeah activity. Yeah we don't do it nearly enough.

I I wanted to say that uh you're mentioning how failure is an important part what we learn from

our mistakes. And

our mistakes. And this brought back to for me um this a study um of the third international math and

science study that James Stigler did uh where he went to eighth grade classrooms and

videotaped in Japan, Germany, and the US.

And one of the things that we noticed was that in the American classrooms if a kid put his problems on the board and he made a mistake

ooh you know the it was not seen as a good thing.

However, in Japan when a kid put his work on the board and there was a mistake the teacher made a big thing out of how grateful the

class was because they could learn from that mistake. And so the whole

that mistake. And so the whole difference in pedagogic outlook on praising what we can learn from what a

mistake was was exactly what I got from what you were saying today.

I'm going to ask a very Southern California question. Like how does it

California question. Like how does it feel to do math? Like what is the feeling when it goes well?

And what's the feeling when it goes I don't know. We're we're not we're no longer going to use the word wrong, right? We're going to say like

wrong, right? We're going to say like productively productively failing? Well,

so when I was a kid I played way too many computer games. It was

one of my favorite hobbies. Um

favorite hobbies. Um and back in the '90s uh when the internet was not so widespread, if there was a computer game and you could not solve it, you know you could not find the key to open this door there

was no internet walk-through.

Maybe like there was the your local game store would sell some some some solution book or something, but basically you only things you every night every night you play the game you get stuck at the same damn door, okay,

and you you uh just over and over again. Um

but then, you know, one day you just have this brainwave and you do kind of this rock and up there's a key, okay, and like you you uh now you can get through.

Um and that was really satisfying. Um

and um like it feel it it felt earned um that um you you work at a problem and you you you explore it explore it. Um

and so sort of uh you know, even though you don't look like you're making progress, often what you're doing is that you're removing the negative space of all the wastes of the problem that don't work until there's only one path left. Um and then you you

can and and then you're ready to to to find that state.

Um so um and then like often once you get it, it's like, "Oh, how come I didn't see that before?"

Um and the reason you didn't see it before is cuz you didn't put in the work to really sort of clear out all all all the extraneous rubbish.

Um So, um okay, I I I won't do a kids these days thing, but you know, I mean um I mean, you know, technology has improved our lives in many ways. Uh but

it's it's given us instant access to all kinds of things and but by the same token, we expect instantaneous solutions for almost anything.

Um but then when it comes to maths, uh it's one of the few places left where uh you don't get instant answers. Well,

actually now we're paying but um you know, but when you're working without AI assistance, you do not get uh these instant answers um and you have to work at it and um but when you do get the solution, it it does feel earned and

that that that is that is a nice feeling.

You know your 2-year uh problem where you got it wrong after a couple weeks it was going to publish. If

AI if you applied AI today to that, what how long would that take? Uh I have not tried. I'm a little bit a little bit uh

tried. I'm a little bit a little bit uh I'm a bit scared to. But uh

Well, for better or for worse, um the way these AI are trained is that they they they suck up every single um um thing on the internet that they can and they train on it. So, this paper that we

published 10 years ago, it's in the training data. Um so um

training data. Um so um either consciously or unconsciously, I'm putting quotes, if you ask an AI this question, um they are very very good at reproducing things that have been done

somewhere on the internet and um we can't tell for sure. So, I'm sure they'll do well, but whether it's because they memorized it or they thought came up with originally, it's no one really knows.

What if the problem was in the original proof?

Um they've been able to come up with they've solved some problems that haven't been solved before. Uh you look at how they do it did it and it's often like a remix of a method that worked for

a related problem. Um I haven't seen a something that really stunned me like no one has ever seen this proof before, but it's it's very very useful.

Um whether it's very creative, that's still under debate.

All right, thank you for Terry so much for fielding all of those questions. We

appreciate your time.

[applause] So I don't talk.