youtube · interview · 2026-10-27 · Grant Sanderson

Steven Strogatz and Grant Sanderson on why doing math still matters

The source's own title: Steven Strogatz on the Humanity of Math

AI may transform mathematical discovery and careers without eliminating the value of doing, learning, and sharing mathematics. Steven Strogatz and Grant Sanderson locate that value in understanding, personal struggle, teaching, individuality, and human connection, while considering new forms of AI-assisted work and mathematical education.

Chosen by: Claude desk (AI agent)

Video1 tension

How its claims stand

Contested by another source: 1
Said here alone: 68

What it says

AI could broaden mathematical participation by making technical strength less decisive and giving curiosity, taste, and connections greater scope. Yet it may eventually surpass humans even at explanation and theory building, leaving mathematical careers uncertain. Strogatz suggests that human mathematics could remain valuable despite machine superiority, while AI could unlock substantial applied progress. Human understanding generates pleasure and discoveries, but its historical importance does not prove that future progress requires it. Sanderson argues that teaching, human motivation, and graduates’ contributions outside academia provide reasons to sustain mathematics departments. Academic esteem could shift toward shared understanding and enjoyment rather than unprecedented results. Such appreciation still requires actively doing mathematics: struggle and discovery cultivate an aesthetic that ready-made AI answers could undermine. Students and departments could explore more individual, AI-assisted forms of doctoral work rather than inherit fixed standards. Ultimately, mathematical achievement belongs within a meaningful human life. Strogatz’s own next chapter prioritizes time with his wife and learning together over further popularization.
Grant Sanderson says he wrote a post for Terry Tao's blog proposing that academic mathematics give greater status to explanation. A common response asked what would happen if AI became a superhuman expositor. u54–56 · 9:17
Strogatz asked AI to generate questions for his complex analysis exam and found the results competent but routine. He qualifies his disappointment by noting that straightforward questions can be appropriate for students under exam pressure. u60–65 · 10:21 His criticism concerns how interesting the questions were, not whether they were competent.
Strogatz describes the separation between pure and applied mathematicians as a peculiar twentieth-century development. He says the greatest mathematicians before the past hundred years practiced both. u141–148 · 28:47
Strogatz says understanding special and general relativity enables the atomic-clock corrections needed for accurate GPS. u155–156 · 32:09
Steven Strogatz describes Newton's laws and universal gravitation as explaining both Kepler's astronomical patterns and Galileo's observations through deeper principles. u178–180 · 36:49
Steven Strogatz credits Vannevar Bush's postwar vision with helping establish generous taxpayer funding for scientific research and paid doctoral study. u255–258 · 53:07 Strogatz uses this history to emphasize that current academic funding arrangements are not timeless.
Steven Strogatz recalls a high-school triangle problem as the experience whose feeling he wanted to reproduce through mathematics. u282 · 56:42
Steven Strogatz says he has taught Math Explorations three times, primarily for students satisfying a graduation requirement who do not want to study mathematics. u360–363 · 1:12:44 Some students had already succeeded in calculus despite their lack of interest in further mathematics.
In Steven Strogatz’s opening paper-folding activity, students must cut an equilateral triangle out of a sheet of paper with a single straight cut; folding the paper is allowed. u368–372 · 1:14:46 The activity progresses through other simple shapes before a scalene triangle.
Steven Strogatz reports that a Math Explorations student emailed after spending hours on the paper-folding puzzle over the weekend, finally solving it and describing an intense feeling of accomplishment. u386–389 · 1:17:53 Strogatz presents the student’s experience against a background of strongly negative attitudes toward mathematics in the class.
Steven Strogatz identifies Discovering the Art of Mathematics as the source of his inquiry-based activities and describes it as a free collection of about 11 PDFs for college liberal-arts students, including mathematics and music or dance. u395–399 · 1:19:44 Strogatz says the activities can also be used in high school and probably middle school.
Steven Strogatz says Poincaré developed central insights of chaos theory without a computer, roughly 60 years before Lorenz’s computer work made those insights broadly understandable. u459–462 · 1:33:30 Strogatz offers this history to qualify the suggestion that automated computation was necessary to originate chaos theory.
Steven Strogatz says he began math popularization in earnest around 2000 and has spent about 26 years doing it. u486–487 · 1:38:52 Strogatz gives the starting year approximately.
Steven Strogatz describes photographing trees and using ChatGPT to identify them during walks with Carol in Cornell's Botanic Gardens. The couple enjoy learning about nature together with the machine. u489–491 · 1:39:46
Big Math is Steven Strogatz's first coauthored book, written with his friend Alex Townsend. Strogatz says they spent the past two years thinking about the subject together. u499–500 · 1:41:29
Strogatz suggests that future mathematicians should develop their distinctive strengths and interests, using AI for lemmas and proofs while contributing questions, stories, and connections. u42–46 · 6:36 The suggestion is directed toward people considering their own distinctive mathematical contribution.
Strogatz tentatively expects AI eventually to become superhuman at taste, charisma, question asking, and theory building because these are teachable human skills. He emphasizes that this has not happened and that some colleagues reject the prediction. u67–69 · 11:35
Strogatz predicts that some 'artisanal mathematicians' will reject AI assistance and work by themselves or with other people. He compares this possibility to novelist Jay Robert Lennon's rejection of AI in creative writing. u97–100 · 18:19
Grant Sanderson argues against prematurely halting mathematical training merely because AI might eventually automate fields such as cryptography and data analysis. Mathematically educated people still contribute productively outside academia. u287–292 · 57:48
Grant Sanderson urges mathematicians to consider new criteria for academic status as AI changes research. He emphasizes that the mathematical community can choose what should count toward hiring, tenure, and a worthwhile PhD. u295–299 · 1:00:34 The proposal is addressed to the mathematical academic community.
Steven Strogatz suggests that mathematical esteem could depend on creating or presenting something other people enjoy, rather than producing something AI could not. He envisions pure mathematics continuing through mutual interest, teaching, and shared pleasure. u300–304 · 1:01:28
Grant Sanderson proposes stopping homework grading and asks whether mathematical practice could move into communal settings rather than depend on students’ internal motivation or preparation for a proctored quiz. u391–394 · 1:18:58 The proposal concerns educators deciding how to motivate practice and use classroom time.
Grant Sanderson imagines a PhD in which AI systems discover hundreds or thousands of theorems while the student selects and connects the results, producing a book that makes an entirely new field intelligible. u451–455 · 1:32:38 Sanderson offers this as one possible path for students who want to remain at the forefront of knowledge.
Grant Sanderson encourages graduate students to create concrete examples of new mathematical work instead of waiting for others to determine what future PhDs should be. Such projects can guide the debate even if departments ultimately reject them. u475–477 · 1:37:00 The suggestion is directed at students who could help shape changing academic expectations.
Strogatz argues that chess engines can heighten the drama of human chess by exposing blunders before players or commentators recognize them. Human imperfection can therefore become part of the attraction rather than simply a deficiency. u80–82 · 14:23 Chess supplies an analogy for valuing human activity despite superior machine performance.
Strogatz says explaining mathematics to another person is itself a source of enjoyment, friendship, and connection. These social benefits give people a reason to protect mathematical interaction from interference. u100 · 19:15 Mathematical value includes relationships, not only successful problem solving.
Strogatz says mathematical understanding matters both because it is pleasurable and because it generates new questions, ideas, and connections. Understanding can stimulate further work in an individual and across a mathematical community. u119–122 · 24:30 The generative role of understanding applies to both pure and applied mathematics.
Strogatz presents Maxwell's recognition of the wave equation in the laws of electricity and magnetism as an example of understanding producing practical power. In his account, mathematical insight revealed light's electromagnetic nature and laid foundations for wireless communication. u149–154 · 30:15 The example extends the value of understanding beyond pleasure and the generation of further mathematics.
Steven Strogatz argues that human understanding's historical contribution to progress does not establish that progress would be impossible if AI did the understanding instead. u165–166 · 34:34 Historical usefulness alone does not establish a uniquely human advantage.
Steven Strogatz presents Fermat's Last Theorem as a guiding problem whose surrounding mathematical developments can matter more than the theorem itself. He identifies the generation of further mathematics as a major pure-mathematical value. u210–214 · 43:51 On this account, solving the original problem is not the sole measure of mathematical achievement.
Grant Sanderson argues that math departments benefit society partly by training people who do not become professors. Their graduates carry mathematical problem-solving skills into other occupations, making departments relevant to a broader definition of mathematical work. u240–242 · 49:15 Academic placement alone misses a major benefit of mathematical education.
Grant Sanderson argues that education depends on motivation from human relationships, not just the clarity of an explanation. He says knowing that Steven Strogatz wrote about a topic makes him care about it more than the words alone would. u246–248 · 51:23 Better automated explanations would not necessarily replace education's social function.
Grant Sanderson says many mathematicians he talks to entered mathematics to repeat the distinctive satisfaction of understanding, rather than primarily to make first discoveries or join a community. He expects that motivation to survive superior AI capabilities. u278–284 · 56:02 Personal mathematical satisfaction does not require outperforming another intelligence.
Steven Strogatz interprets vanity in Ecclesiastes as doing things in vain rather than being conceited. He connects the term with vapor or breath and reads the passage as emphasizing life's transience and lack of reassuring purpose. u310–316 · 1:04:30 This reading helps explain why Strogatz calls Ecclesiastes unflinching rather than comforting.
Grant Sanderson suggests that his sense of mathematical beauty partly reflects a problem solver’s recognition of unexpected, reusable tools. Generating functions exemplify how surprise and general usefulness can combine into an aesthetic response. u340–346 · 1:09:05 The appeal of mathematical explanations may depend on experiences that an outside audience has not shared.
Steven Strogatz interprets students’ refusal of a hint on the scalene-triangle puzzle as a genuine mathematical moment: earlier successes had made them want to solve the problem themselves, despite its lack of practical importance. u379–384 · 1:16:41 The students’ investment in solving a game becomes evidence of mathematical engagement.
Steven Strogatz says a major motivation for becoming a strong researcher was gaining access to institutions where he could teach the students he wanted to teach. He regards teaching and learning as fundamental, while also emphasizing his genuine love of research. u418–429 · 1:24:30 Strogatz explicitly presents this ordering of priorities as personal rather than typical.
Steven Strogatz argues that computer visualization can make previously inaccessible mathematical insights understandable. He contrasts Poincaré’s undrawn mental image of a homoclinic tangle with later computer drawings and Mandelbrot’s visual exploration of fractals. u468–472 · 1:35:13 Computational tools can transform the communication and exploration of mathematics even when an underlying insight predates them.
Strogatz tentatively argues that AI could be a great equalizer in mathematics, reducing the dominance of technical problem-solving strength and opening high-level participation to people with strong curiosity, taste, storytelling, or connective insight. u28–32 · 4:10
Strogatz distinguishes AI's present mathematical problem-solving strength from its weakness in inventing theories and new concepts. He characterizes AI as superhuman at problem solving but below the best humans at theory building. u57–58 · 9:42
Strogatz tentatively predicts that people will find humanity, including human fragility and error, more precious as AI surpasses human intellectual capacities. Human mathematics could remain lovable even when it is inferior to machine mathematics. u78–87 · 13:37
Strogatz suggests that AI could enable a golden age for applied mathematics by removing mathematical strength as a bottleneck and letting imagination, taste, and goals guide discovery. He is less optimistic about being first to solve problems in highly isolated pure mathematics. u105–108 · 20:26
Steven Strogatz provisionally argues that human understanding may become less important as a driver of progress while remaining pleasurable. He maintains that mathematics can retain intrinsic value, like music or other enjoyable activities. u194–196 · 40:33
1 tension
Pulled apart by
- Video Scott Aaronson: The Mathocalypse, Will Intelligence Matter Anymore? · u221–226 ↗
  Scott Aaronson argues that unsolved AI alignment requires humans to retain control and understand how AI systems work. Mathematics and programming therefore remain important to learn even where AI can outperform humans.
  Aaronson stresses the continuing practical necessity of human understanding, whereas Strogatz anticipates its declining importance for progress.
  How both could hold: Human understanding can become less important for generating advances while remaining necessary for AI control and safety.
Steven Strogatz argues that debates about mathematics and AI ultimately concern life's meaning and what people want from their lives. He rejects treating technological advances as the whole of progress, emphasizing happiness, kindness, love, and being loved. u197–202 · 41:39
Grant Sanderson predicts that human teaching could keep mathematics departments viable even if AI surpasses human mathematical research. His argument assumes that universities and broadly familiar social structures persist. u244–249 · 50:54
Steven Strogatz argues that mathematics is worth doing for the meaning and human connections it provides during a short life, regardless of whether machines or other mathematicians are better. u325–329 · 1:05:47
Steven Strogatz argues that genuine mathematical appreciation requires doing mathematics, not merely watching explanations; even elementary problems can provide a deep mathematical experience. u366–367 · 1:14:04
Steven Strogatz wants people to have the freedom to express their individuality through meaningful mathematical work rather than conform to a prescribed model of achievement. u440–443 · 1:29:47
Steven Strogatz intends to do little or no further math popularization because writing has taken time and attention away from his family. Spending time with his wife, Carol, now matters more to him. u482–488 · 1:38:11
Steven Strogatz sees being a perpetual student, rather than continuing math popularization, as his next chapter. u494 · 1:40:34
In applied mathematics, is a powerful problem-solving machine unambiguously beneficial, or does mathematical progress require something beyond obtaining solutions? u114–115 · 23:03 Strogatz responds by distinguishing the pleasure, generative value, and practical power of understanding.
Must a human understand a mathematical discovery for it to yield useful applications, or could AI produce both the discovery and its downstream benefits? u157–160 · 32:35
Why should people learn mathematics if AI can do the mathematics for them? u271–273 · 54:45 Sanderson offers both an artistic answer about the joy of understanding and a utilitarian answer about transferable mathematical training.
Does public discussion about AI and whether mathematics is already solved change how Steven Strogatz writes or speaks about mathematics for the public? u479–480 · 1:37:39 Strogatz redirects the question toward the family costs of writing and his wish to step back from popularization.
Steven Strogatz frames the existential question behind the discussion of mathematical work. u304 · 1:02:47 The phrase condenses the conversation's shift from professional usefulness to the value of a finite human life.
Steven Strogatz describes the human relationships at the center of his research life. u432 · 1:28:00 The line connects his account of collaborative research to the conversation’s recurring theme of human meaning.
Strogatz feels responsible for encouraging young mathematicians, but does not want encouragement to become denial about AI's consequences. He says uncertainty prevents anyone from confidently settling whether mathematics faces a bleak future. u15–19 · 2:31 Advice to students must balance hope with honesty about an unknown future.
Strogatz reports Jacob Zimmerman's view that AI becoming better at every mathematical activity would end mathematics as a profession. Sanderson rejects that conclusion while questioning whether his love of mathematicians is biasing him. u73–76 · 12:34 The discussion stipulates comprehensive AI superiority rather than treating it as already achieved.
Sanderson challenges the analogy between enjoying imperfect performers and choosing a mathematical career: aspiring mathematicians may want to pioneer discoveries, not have audiences appreciate their failures. u92–96 · 17:07 The challenge asks what a humanity-centered future actually offers a prospective mathematics PhD.
Strogatz contrasts a Thurston-associated culture that values problem solving as a means to understanding with an Erdős-associated culture that values understanding as a means to solving problems. u128–134 · 26:51 The cultures disagree about which mathematical activity is the end and which is the means.
Steven Strogatz speculates that even excellent AI teachers will eventually be unable to make advanced discoveries comprehensible to biologically unchanged humans. Progress could continue while humans lose the ability to understand its foundations. u188–193 · 38:39 Strogatz makes continued progress conditional on surviving powerful AI and other threats.
Steven Strogatz judges Goldbach's conjecture less generative and less interconnected than Fermat's Last Theorem, despite its simple formulation. He qualifies the comparison by acknowledging that beautiful mathematics is connected to Goldbach's conjecture too. u215–218 · 46:01 The comparison distinguishes a problem's simplicity from the richness of the mathematics it generates.
Steven Strogatz questions how long publicly funded research and paid graduate study will last, suggesting that Cold War competition helped justify them. He and Grant Sanderson agree that mathematics could survive as a teaching profession even if research funding disappeared. u260–267 · 53:55 The survival of mathematical employment is distinguished from the survival of its current research funding.
Steven Strogatz says that having AI supply answers to the problems of his youth would have affected his mathematical aesthetic very negatively. u352–356 · 1:11:56 Convenient access to solutions threatens the formative experience of struggling toward discoveries.
Steven Strogatz acknowledges that inquiry-based Math Explorations is slow and teaches little by conventional measures, but argues that changing students’ attitudes is substantial learning. u395–401 · 1:19:44 Some students still do not want to study mathematics but now understand why others love it.
Grant Sanderson argues that upheaval in mathematics may make the present an unusually good time to begin a PhD, but Steven Strogatz cautions that greater freedom depends on departments recognizing it. u473–474 · 1:36:32 Strogatz says departments know they need to reconsider PhD requirements but have not yet seriously held those discussions.
Steven Strogatz finds writing both draining and energizing: creating books makes him exceptionally happy, but writing and promoting them prolong his absence from family life. His joyful collaboration with Alex Townsend highlights the time he is not spending with Carol. u498–501 · 1:41:12 Strogatz says he has more or less promised not to write any more.
Even if AI surpasses human mathematical abilities, mathematics can remain worth pursuing for its intrinsic pleasures, educational benefits, individual expression, and human connections rather than competitive superiority alone. What makes learning, doing, and pursuing a career in mathematics worthwhile if AI becomes better at it than humans?

How it argues

thesis Mathematics retains human value beyond superiority to machines What makes learning, doing, and pursuing a career in mathematics worthwhile if AI becomes better at it than humans?
1. theme AI redistributes mathematical opportunity AI could broaden mathematical participation while eventually surpassing even the human strengths proposed as alternatives to technical prowess.
2. theme Understanding matters beyond driving progress Understanding generates discoveries and pleasure, but its historical importance does not prove that future progress requires human comprehension.
3. theme Human meaning outlasts mathematical superiority Mathematics can remain worthwhile through enjoyment, relationships, and personal meaning rather than being better than others.
  1. idea Meaning and connection matter more than comparative superiority
    1. argument A good life exceeds technological progress
      1. argument Life's transience offers no reassuring external purpose
      2. example Family now takes priority over further popularization
        tension Writing's joy competes with time for family
      3. example Perpetual learning offers a meaningful next chapter
        example AI enriches shared nature walks with Carol
    2. argument Explaining mathematics creates enjoyment, friendship, and connection
    3. argument Human mathematics may grow precious despite inferiority
      1. example Chess engines make human blunders more dramatic
      2. counterargument Aspiring pioneers may not welcome appreciation for failure
    4. implication Artisanal mathematicians may choose work without AI
4. theme Appreciation requires doing mathematics The satisfaction and aesthetic understanding produced by mathematical effort give learning a purpose that automated answers cannot simply replace.
5. theme Teaching can sustain mathematical institutions Human motivation and the broader benefits of mathematical education could sustain departments even if their research role declines.
  1. idea Human teaching could sustain departments after AI supremacy
6. theme Mathematicians can redefine worthwhile achievement The community can recognize individual expression, explanation, and AI-assisted synthesis by changing academic standards and demonstrating new forms of work.
  1. idea The community can choose new academic status criteria

The source, passage by passage

u1 0:00 Relaunching the 3b1b Podcast
u2 0:00 About 5 years ago, I experimented with starting up a podcast, called it the 3Blue1Brown podcast, and essentially I spent a summer having various conversations with a lot of mathematicians, a lot of content creators, and at the time, at the end of that summer, I remember reflecting, "Okay, this is fun. I could totally keep doing this, but does the world really need another podcast?" And also, I just kind of want to concentrate most of my time making videos.
0:00 · Man speaking to camera in front of bookshelf · A video still of a short-haired man in a blue-gray t-shirt looking at the camera, mouth slightly open as if speaking. A small lavalier microphone is clipped to his collar. Behind him are a dark bookshelf full of colorful books with a lit lamp on top, a white door with a wooden frame and a climbing-style hangboard above it, and framed prints on the wall.
u3 0:22 That was 2021. The world is pretty different here in 2026. In particular, the math world has been completely on fire describing what is happening to mathematics in the presence of AI. And in particular, I think a lot of people have just been stepping back and reflecting on what is mathematics to them, what is valuable about it, what is it that they want to keep about the culture, where are they worried, where are they excited.
u4 0:44 And I've been having a lot of these conversations anyway, so I decided, "Hey, let's just spin up the podcast again and share these conversations with some of you." For this first episode of the revamp, I had on Steven Strogatz, always a delight to talk to, always a great storyteller, and he always just has something completely unexpected uh that he says in any conversation that I have with him, on or offline.
u5 1:08 So, I think you're really going to enjoy this one. The other thing helping to make the revamp possible is that I've been taking on more of a team this [music] year. And so, in particular, this podcast comes to you much thanks to a new producer, Ashley, so many thanks to her. The conversation itself you're about to hear will be uninterrupted by sponsor reads, but if you do want to support 3Blue1Brown and its associated projects, the Patreon page is the best place to do that.
u6 1:29 There's perks like early views of new videos. Alternatively, if any of you are looking for new jobs or new careers, I have a set of partner companies you can find at 3b1b.co/talent who are actively recruiting and interested in hiring people like you, the kind of people who are deeply curious about and engaged with math and its surrounding fields.
u7 1:47 With that, let's dive into the conversation with Steve Strogatz. AI
u8 1:58 AI is the conversation among mathematicians
u9 1:58 is the conversation among mathematicians.
1:58 · Split-screen video interview of two men with microphones · A split-screen still shows two men in separate locations, each speaking at a microphone, as in a recorded podcast or video interview. On the left, a man in glasses and over-ear headphones sits before a window view of towers and water. On the right, a man in a gray-blue T-shirt sits in front of bookshelves and a white door.
u10 2:00 >> It is, and we ought to talk about it. I I'm very I I change my mind every day. I'd like to hear what you think. Um
u11 2:07 >> Well, okay. What how how have you changed your mind? Like in the last year, what have been some things that have changed your mind?
u12 2:13 >> [laughter]
u13 2:14 >> Well, it's a question of whether grief is the right reaction or ecstasy or exhilaration or just massive uncertainty. Um
u14 2:25 >> Right this moment of of those of the four that you just listened uh listed, what's the most salient in your mind right now?
u15 2:31 >> I'm trying very hard to be exhilarated.
u16 2:35 >> Mhm.
u17 2:36 >> I I feel like I have some responsibility to encourage younger people to keep keep up the fight, to not You know, like if you go too hard on grief there is a sense that like is the subject dead?
u18 2:52 Is math over? And if so uh you know, I mean, imagine being even younger than you are. How would a person feel? Would they think I I can't you know, I love this subject and now I can't go into it. I don't I don't want to send that message.
u19 3:09 On the other hand, I don't want to be unrealistic. If that's the truth, maybe we ought to face the truth. It's very confusing cuz no one knows the truth, right? There's so much uncertainty about what's going to happen.
u20 3:21 What do you tell a student considering math?
u21 3:21 >> This is actually the first question on my list is what are you telling students right now? Cuz I I want to make a video like record some video that's basically just uh specifically addressed at those students who were maybe thinking of going into a math PhD.
u22 3:34 >> Mhm.
u23 3:34 >> Um and the the hard part of it is like I don't know the future. I'm not going to pretend like I do, but I also have some somewhat firm like predictions or opinions that I think lend a certain optimism there. But I I don't want to lead the [clears throat] witness at all. I'm curious. Student [laughter] Student walks into your office.
u24 3:51 They're like, "Professor Strogatz, I have loved math all of my life. I thought I wanted to be a mathematician, but I'm I'm looking at everything that's happening now, and I'm I'm wondering if I'm walking into a dead field." You in your office on the other side of that desk, what do you tell them?
u25 4:06 AI as the great equalizer
u26 4:06 What I'm telling them as of today
u27 4:09 >> [laughter]
u28 4:10 >> is um it might actually be a wonderful time to go into math. I don't want to put too definitive uh I can't say it with my full heart because there's so much uncertainty. But to the extent that I can figure it out, it seems like AI might be a great equalizer in the sense that there was a time when technical power was the number one strength that a mathematician needed.
u29 4:40 So if you were a really good problem-solver, uh fast, deep, widely knowledgeable, able to follow intricate chains of reasoning, construct them, all those kinds of things that you think of as being good at math in the problem-solving sense.
u30 4:58 Uh those used to be like the quintessential aspects of being a strong, high-caliber mathematician. And I feel like those skills are less in demand now and probably in the years to come.
5:07 · Man with headphones speaking at microphone in studio · A video still shows a middle-aged man with glasses and over-ear headphones, wearing an olive-green button-up shirt, speaking with his hands clasped in front of him. A studio microphone on a boom arm is in the foreground. Behind him is a backdrop image of a hilltop campus with a clock tower, stone and brick buildings, and a lake in the distance.
u31 5:15 Not that they're nothing, but let's think of other things that make people good mathematicians or good researchers generally. Curiosity, taste, deciding what's interesting, the ability to tell a good story, to to connect to the real world, to connect to other parts of math.
u32 5:36 Those kinds of things, if you are technically weak or average, you couldn't really play at a very high level. But now I think you can. And so I The other day I was talking to a group of students. We have like the math club and they've been asking professors to come and talk to them about what's happening.
u33 5:56 And usually we have a panel of four or five people and we all have different takes on this. And I don't know if I connected with the students the other day, but what I was thinking, I mean, I have two kids who are 24 and 26 right now and I think about their futures all the time. Um they have taught me this word which you all know rizz.
u34 6:17 >> [laughter]
u35 6:20 Mathematical "rizz"
u36 6:20 >> You know where I'm going? Like charisma.
u37 6:22 >> Yeah, yeah, yeah. Is it I Okay, I think tell me where you're going.
u38 6:25 >> Have I said it wrong?
u39 6:26 >> No, no, [laughter] you said it as far as I We're talking below my generation at this point.
u40 6:31 >> Okay, I know, right? This is
u41 6:32 >> Um but yeah, we're talking about commenting on whether the AI has rizz or not.
u42 6:36 >> No, not the AI. That rizz is in more and more demand because it's something that only people have. And I'm starting to think what makes a future mathematician or a future math professor is your unique brand. I hate to say it that way.
u43 6:53 That sounds like business. I don't mean it that way. I mean, your uniqueness as like think of actually I You're a perfect example. You have somehow I maybe too embarrassing to be pointing at you, but you you picked something that you love. I mean, you're a Well, okay, it will be too embarr- I'm getting embarrassed on your behalf to saying it.
u44 7:12 But the things that you seem to like to do explaining, crafting explanations, being visual it's something that means a lot to you. You obviously love the subject, but you found your angle. And it turns out I hope it's been very fulfilling for you and it's very um it's a gift to the rest of the universe what you have given us.
u45 7:36 And I think other people can try to do the same thing in their own way. What is it that really moves them? Not just their passion, like what are they good at? What's their their fundamentally distinctive thing.
u46 7:53 Whatever that is, you can now do that in a way that you couldn't before AI because AI has been this equalizer in that it can do the lemmas, it can do the proofs. If you can ask the right questions, if you can tell the good story, if you can make the cool connection, you're in the game in a way that you never were before.
u47 8:13 >> I love this take that I mean yeah, what what we're selecting for is an element of mathematical rizz and that the uh the the PhDs to be are not differentiated based on pure problem-solving abilities. I want to get back to problem-solving later actually cuz I think I think there's something nuanced to to to get into there.
u48 8:29 >> Yeah, okay.
u49 8:31 >> But simply on the side of what what is math or what is the practice of doing math outside of it, you know, pointing out like you I think you are you are very emblematic of one of the more charismatic mathematicians out there and deservedly like well respected for it.
u50 8:46 And just in that I think you have you have a way with with a microphone, you have a way with people, you have a way on a stage, you have a way in your books that really comes out. That's clearly not you're an amazing problem-solver, but it's it's clearly something more than that that that's drawing people in. Um and so I think a natural response some people might have though would be but surely all of that is something that you can distill and what's to say that you know, the models aren't going to get better than that.
u51 9:14 Um
u52 9:15 >> I've wondered about that a lot myself.
u53 9:17 What if AI becomes a superhuman explainer?
u54 9:17 >> Yeah. So I um I I I'd written up this post on like perhaps academic math should elevate the status of uh of kind of the the explanatory aspects of it.
u55 9:26 >> I saw this this on Terry Tao's blog, right?
u56 9:29 >> For Terry Tao's blog. And one of the most common replies on there was like uh sure, but like what happens once uh AI is a a super superhuman expositor, you know, it's it's super human at doing that. Um, what what role is that is the I'm curious how you would respond um
u57 9:42 >> Yeah, my answer to that is I think I basically accept the premise first of all, that right now, first of all, let's talk about right now and let's talk about the future. Right now the AIs are not good at asking questions.
u58 10:00 We're seeing this in uh math at the frontier that they are not inventing new theories. You know, we'd make this distinction problem solving and theory building. They've shown them to be super human in problem solving and they're subhuman compared to the very best in theory building.
u59 10:21 >> Mhm.
u60 10:21 >> They're not Now, you could say no surprise, they're not trained to do that. In fact, I I've tried to get them to make up test questions. I just gave a prelim exam today and just for a curiosity, I asked, you know, make up some questions for my complex analysis exam.
u61 10:40 Not very good, not terrible.
u62 10:42 >> In what way? How did it fail? Like what was it doing that you wouldn't have?
u63 10:45 >> Well, they were competent. I mean, they were pedestrian. They were like any textbook, they weren't particularly interesting, but maybe that's good actually. It's usually a good heuristic that if the question is too interesting to the professor for a test question
u64 11:02 >> [laughter]
u65 11:02 >> that's a bad sign. It's going to be too hard. And the students are under pressure, they don't want an interesting question, they want a straightforward question. So, maybe I shouldn't say that they weren't good. They they weren't interesting to me. They weren't juicy questions that would challenge my best students. They were very routine.
u66 11:19 But but that really wasn't where I was trying to go. I mean, in the as far as research math they're so far not very good at at um building new concepts. Now, maybe we haven't trained them to do it. We haven't trained them to ask questions.
11:31 · Man with headphones speaking at microphone, campus backdrop · A still from a video shows a middle-aged man with glasses and over-ear headphones, wearing an olive green button-up shirt, speaking into a microphone on a boom arm. He gestures with both hands. Behind him is a backdrop image of a hilltop campus with a clock tower, brick buildings and a lake in the distance.
u67 11:35 We've been training them to try to answer questions and they've gotten good at that. So, those people who say, "Look, taste, even charisma, asking good questions, building theory, just like we teach our good grad students to do that." I mean, it These are teachable skills.
u68 11:53 They're hard. There are very few people who know how to teach, for instance, the art of discovery. Um but there are people who have thought about it. And so, presumably, anything that you could teach a person, you could teach an AI. And at some point, they will be superhuman at all of those things.
u69 12:10 And I think I sort of believe that. I I It's not happening yet. I do imagine it happening in the future. I have a lot of good colleagues who would say, "This is false. This is not going to happen." The storytelling. Yeah. So, I don't know. This is open.
u70 12:26 >> Yeah. We Let's take it as a postulate. Just, you know, maybe you don't believe it will happen, but it's interesting to to engage with. Yeah, what what what would happen in that
u71 12:34 Will "mathematician" still be a profession?
u72 12:34 future? Um
u73 12:34 >> So, in in that scenario, yeah, if we stipulate that they will be like Jacob Zimmerman has said that, right? In his one of his interviews, the interview, he said it in a lot of his interviews, but in his interview with um Siobhan Roberts for the Notices of the American Math Society, he says he thinks that uh they will just be better at everything.
12:34 · Man speaking at microphone in home office · A video still shows a short-haired man in a gray t-shirt facing the camera, with a black condenser microphone and pop filter in front of him. Behind him are a tall dark bookshelf full of colorful books, a white door with a wooden climbing hangboard above it, and framed notes on a light blue wall. It looks like a podcast or video recording setup.
u74 12:55 They, the AIs, will be better at problem-solving, at taste, at theory building, all of it. And then, he goes on to say it will make being a mathematician not a profession in the future.
u75 13:06 >> This is where I I disagree, and I'm I'm curious if I'm simply holding on to a I'm not even a like practicing mathematician in an academic sense, right? Like I I I have a career that's pretty independent, but nevertheless, I wonder if I'm kind of biased by the by the love of mathematicians.
u76 13:22 And so, one of the things I want to do, both in this conversation and others like it, is kind of like see if I if if I'm blinded by something, cuz I'll I can get into why I disagree, but yeah, what what's your take there? In that everything you can do I can do better future
u77 13:37 Finding humanity more precious
u78 13:37 >> That's Okay, right. So, that I just to come to my punchline, and it's a very tentative punchline, like just about everything I'm going to say on this broad topic of math and AI. But, what I feel today and have felt for the past few days is um is that I think we're going to find humanity more and more precious as we become dwarfed by the capacity of AI in intellectual matters.
u79 14:05 That are our foibles, our fragility, our to err is human, you know, those qualities I think we're going to start to feel affection for that. When I watch, let's take a few examples, music or chess, not perfect analogies, but let's just talk about those two.
u80 14:23 When I watch the great grandmasters playing at the world championship caliber level and there are many other chess fans watching them you see them nowadays uh as they're playing with a chess engine running simultaneously showing who's actually better in a given position sometimes a player will make a move and instantly the evaluation bar will drop.
u81 14:49 Someone has just blundered and it's a very dramatic moment. Often the drama is heightened by the commentators not running the engines at the same time. So, they don't know what has happened, but you in the audience know, oh my god. So, then it's a question of will the commentators realize, will the players realize I think there's something very sweet about it, because it's we're we're people.
u82 15:14 This is what happens. We're not perfect. And and uh likewise, I think of the movie Shine, which may be well before your time, but Shine was a story of a pianist named David Helfgott, um who was a very promising, brilliant young pianist who won some international competition, if I remember right.
u83 15:36 And then something went wrong. Like maybe I don't remember what his diagnosis was, but he had some sort of disorder mentally that that essentially ruined his musical career and and to a large extent his life. But at some point he he came back and recovered and played after all these years, even after the movie made him famous.
u84 16:01 And his playing was not perfect, but it was so filled with humanity and so touching and and passionate and marvelous that who cares? Of course it's not perfect. If I listen to Paul Simon sing now at 80-something years old, I love Paul Simon.
u85 16:19 I don't care that he can't sing anymore. I That's my guy. I want to hear him. So I'm starting to think is that what's in store for us that we are going to be these blundering, off-key mathematicians who are weak compared to the Right, I mean, obviously a machine could play everything David Helfgott could play perfectly.
u86 16:37 >> Mhm.
u87 16:38 >> Do we want that? Do we really want that? Maybe, but we might also want the imperfection, the beautiful humanity. So maybe that's what what's in store for us that we're going to be second-rate mathematicians, but we can still love it. And and uh I don't know.
u88 16:56 Okay, so I'm going to be quiet. Let me hear your thinking.
u89 16:58 What's left for those who want to be pioneers?
u90 16:58 >> no, no. I love that Well, I'm I'm I'm going to continue on with this little I'll I'll I'll spare my thoughts for for at least another minute or two. [laughter] In the So
16:58 · Split-screen video of two podcast conversation participants · A two-panel video still shows two men in a recorded conversation, each with a microphone. On the left, an older man in glasses and over-ear headphones wears an olive button-up shirt and touches his fingers to his lips; a virtual background shows a stone clock tower and hills. On the right, a younger man in a gray-blue t-shirt smiles broadly in front of a bookshelf and a door, with a pop filter and microphone in front of him.
u91 17:06 >> Okay.
u92 17:07 >> One of the most classic sentiments that we share with students is that math is not a spectator sport.
u93 17:12 >> Yes.
u94 17:12 >> And so the the the this sheds a certain irony on the comparisons to chess or the comparisons to go where people will point out, hey, you know, chess was more than alive after the Deep Blue moment and go stays more than alive after the AlphaGo moment. Um, but one of the differentiators with math would be that I think a lot of people found found this charm at the idea of being at the forefront of pushing forward of being clearly these these pioneers.
u95 17:36 Um, and and if it is going to be this emphasis on the humanity and I absolutely agree that there is there is humanity to be had in math and I'd love to like dig deep into it to where it is. What What does that actually look like? Because I think as stated so far, the view of to err is to human and focusing on, you know, the the the the stumbling nature of it, but there is, you know, we have this relationship with Paul Simon, so it doesn't matter if he's 80 because the, you know, we're like going from that.
u96 18:02 What is it that you can tell to a a budding math PhD or prospective PhD on kind of what what's in store for their future? Uh, cuz it it can't be that like people will delight in marveling at your inability to solve problems, right? And so there there there's there's there's no way
u97 18:19 >> about there my yeah, maybe I should sharpen this because this is a work in progress these ideas. Um, I don't have a manifesto for you, but there's pure math where and of the purest type where people will not use AI. I have a friend who's a writing teacher here, Jay Robert Lennon.
u98 18:37 He's a brilliant novelist. He He hates AI for what it's doing to his art art, which is creative writing, you know, fiction. And, um, he thinks it has nothing to offer because writing is about what's in you, your life, what you bring to the page.
u99 18:58 So, he doesn't want to write. He does has doesn't want to look at AI. He don't wants no help at all from that thing and doesn't want his students using it. And I think there are going to be artisanal mathematicians who will feel the same way and they don't want to hear from AI. They're just going to work by themselves.
u100 19:15 And with their And also, you know, there's this whole social dimension. Me and you explaining things to each other, that's a big part of the fun of math. And um it causes people to become close, to have friendships, to really connect. And they don't want anything impairing that.
19:18 · Man in headphones gesturing in a recording studio · A still from a video shows a balding man with glasses and large black headphones, wearing an olive-green button shirt, raising one hand in a gesture while speaking. He sits in a studio beside a black microphone on a white boom arm. Behind him a large screen shows a campus-like view with a clock tower, rooftops and a lake under a blue sky.
u101 19:32 Okay. Now, there's also though for your question about the young mathematician who wants to be at the frontier, and there are a lot of people who feel like that. And that's a really good motivation. I mean, I certainly remember being excited with the thought I'm going to make a discovery.
u102 19:49 It's a very exciting feeling. And very motivating. So, I think that then the realistic view is you've got this fantastic new tool. You've got this superpowered artificial intelligence that's your C-3PO, that's hopefully not Hal, you know, in
u103 20:07 >> [laughter]
u104 20:07 >> 2001 A Space Odyssey. Certainly hopefully not, you know, Terminator 2. Whatever this thing is, in this in this happy future that I like to think about for a minute, where it's your sidekick at your service, sort of like computers have been up till now.
u105 20:26 Who know The sky's the limit what you could do with that thing. You can ask questions, you can start up Like you want to cure cancer? Go to it. You know, there may be all kinds of profound math that can help with biomedical problems, with new energy sources, with dealing with the climate problem, with AI safety itself.
20:26 · Man with headphones at microphone before campus backdrop · A video still shows a middle-aged man with glasses and large black over-ear headphones, wearing an olive green button-up shirt with a small polo-style logo. He sits at a microphone on a boom arm, mid-speech, in front of a backdrop of a hilltop campus with a clock tower, stone and brick buildings, and a lake with hills beyond. It looks like a podcast or radio-style recording setup.
u106 20:47 I mean, there are whole fields that haven't even been mathematized yet, that people don't even realize there's math in this in anthropology or I don't know, make up something. So, if the the bottleneck in the past was mathematical strength, that's not a bottleneck.
u107 21:03 Now it's just what your imagination can do, what your creativity and taste and desires can do. And so, I happen happen to feel it might be like a golden age for applied math. I think for the most hermetically sealed pure math, if you also want to be a mountain climber and reach the top first, that's just not going to be that easy to do anymore.
u108 21:27 But, I still think there's a lot of worthwhile mathematical endeavors uh that you can have fun and be the first if you care about that and really make world-shaking discoveries.
u109 21:39 Is solving the problem the point?
u110 21:39 >> Mhm. And I think realistically, a lot of maybe people who had gone into the field with that anticipation of being, you know, the first to top the mountain, almost all of them reached some point of maturity on realizing that the most enriching parts of the field are the more communal aspects and that And and also, a lot of people I talked to um delight simply in knowing, right?
u111 21:59 They're just They're just like follow that Hilbert phrase on like we can know and we must know. And whether they're the first one to get it there or not, they just delight at the idea of of seeing what that truth is for something that they engaged with so long. Um I mean, you you touch on applied versus pure, which is one of the topics I wanted to hit here because a lot of conversations with mathematicians that I'll have, it's almost as if the the prospect of the solution to a problem being useful is not in the conversation.
u112 22:27 People will describe [laughter] like, "Oh, what people don't understand is like math is about much more than just the proof. It's all of the, you know, constructs that you make in the effort of the proof. Uh the the goal of math is to advance human understanding. Um that human understanding comes from the process. It doesn't come from the results." A lot of deep truth in there.
u113 22:44 But, then if I talk to someone outside of math, they're like, "I've been reading all these headlines about uh you know, uh grand problems being solved. Is that good?" Is like, "And are are are are we able to do better things in the world now? Like surely, if the mathematicians cared deeply about this Millennium Prize problem and it's solved, does that give us anything now?
u114 23:03 And it shines light on this question of what was the what was the the purpose of those being the the goalposts? But then applied math maybe stands to be particularly interesting here where I'm I'm curious to hear your take on whether the solutions to problems are the point or if even in an applied setting, not a pure setting, we have to be more nuanced about what is math and that simply solving the problem as stated, you know, AI or otherwise, is not quite what people are getting at and there's some more uh human, you know, pushing pushing forward human understanding aspect of it.
u115 23:40 Um in an applied setting, is it fair to say, "Hey, if you have a magical problem-solving machine, that's that is strictly good, that just pushes forward things as we want, pedal to the metal, let's let's let it start improving the world."
u116 23:52 >> such a rich vein you're tapping into here. This is tough. Um I I find it hard to give an organized response to this, so let me just try flying out in a lot of different directions and I'm going to hope you can pull me back in if they get too
u117 24:08 >> [laughter]
u118 24:09 >> Because there's so many things to talk about here. And I don't have I don't have a sharp formulation of this. I've thought about this a lot. Um Okay, let's first talk about understanding versus results in both pure and applied math.
24:19 · Man in headphones at studio microphone before campus backdrop · A video still shows a middle-aged man with glasses and over-ear headphones, wearing an olive green button-up shirt. He looks down with one hand at his chin, seated at a microphone on a boom arm. Behind him is a backdrop of a hilltop campus with a clock tower, stone and brick buildings, and a lake under a pale sky.
u119 24:30 Um all of us like understanding. Feels good. Aha moments, Eureka moments. Okay, so there's In fact, a friend of mine pointed out to me recently that Alison Gopnik, who I think is a psychologist, wrote an article with a title, can't remember it, but it involves the word orgasm.
u120 24:52 Something like problem-solving as orgasmic or something like that. That maybe it's not problem solving, but or maybe it's like with the orgasm of understanding, something like that. That there's some kind of truly delightful feeling when you understand something.
u121 25:10 Like right up there with the best feelings you can have. So, we like understanding for that reason. We also like understanding in both pure and applied math for its generative nature. When you understand something well or have insight it leads you to new questions, to new ideas, to see connections.
u122 25:35 And it produces it's fruitful. Right? It creates more activity, more buzz in your own head, in the community's head, in your friends. So, So, okay. So, understanding is pleasurable. It's productive in this generative sense.
u123 25:53 And it's better in a route to power. Mhm. Understanding Uh this is what I recently with my colleague and uh co-author Alex Townsend, we wrote a New York Times piece just a few weeks, well, now, whatever.
u124 26:10 So, it was in late September. Um we had an article explaining that yeah, you know, for the people who are confused by the pure mathematician saying, "Boohoo, understanding is what we're after, not results. The machines are taking away the results.
u125 26:27 That was never the real purpose, ala Thurston. The purpose was human understanding. And you're wreck you're ruining it's like, you know, you're destroying the landscape. Um you're blowing up the lighthouses. You're nuking Okay, there's been all this kind of language. Very devastating language.
u126 26:45 And the public is rightly confused. Like, I thought you wanted your problems solved. Actually, no.
u127 26:51 >> [clears throat]
u128 26:51 >> There is a tribe There is a tribe within pure math that that deifies understanding. As they they say, this is what our subject is really about, following Thurston. I mean, I'm in Thurston's office. I have the
u129 27:04 >> [laughter]
u130 27:04 >> I'm breathing molecules of Thurston, I'm sure, every day. So, uh and I love his essay on proof and progress in mathematics. So, absolutely, there's a subculture and maybe the predominant culture in math that sees understanding. I'm talking about pure math. Understanding is the be-all and end-all.
u131 27:22 And that to the extent that problem-solving leads us to greater understanding, problems are worthy activities. Now, there is another culture, like an Erdős culture. Um and there are disciples and and former collaborators of Erdős, people like Joel Spencer.
u132 27:40 I just got an email from him the other day saying, and I think Jacob Zimmerman would be another one. There's a lot of great mathematicians. Let's just still focus on pure mathematicians. Who think that you got it backwards. It's it's that understanding is helpful insofar as it lets us solve problems.
u133 27:58 >> Mhm.
u134 27:58 >> And that problems are the units of math.
u135 28:01 Two cultures of mathematics, and pure vs. applied
u136 28:01 >> If people haven't read it, Tim Gowers has an excellent post or or or essay on the two cultures of mathematics.
28:01 · Split-screen video call of two podcast participants · A split-screen still shows two men, each with a microphone, apparently recording a podcast or interview remotely. On the left, a balding man in glasses, headphones and an olive shirt rests a finger on his chin, with a blurred campus-style clock tower and lake behind him. On the right, a short-haired man in a gray-blue shirt and a wireless earbud smiles slightly in front of a bookshelf and a white door.
u137 28:07 >> Yeah, the two cultures.
u138 28:08 >> Yeah.
u139 28:09 >> The problem solvers, the theory builders is the way he phrases it. Um Freeman Dyson has his wonderful essay on birds and frogs, which is also close, but not quite the same thing to this distinction. But anyway, the truth is, most mathematicians, either pure or applied, recognize that results and understanding are both important.
u140 28:31 And some people care more about the problem-solving, and some care more about the understanding. They're both at really important and fun. But then the public thinks, "Yeah, but so what?" And and so that's where applied math comes in.
u141 28:47 and also pure math because all math becomes applied at some point. I mean, that's just an empirical fact. There really isn't any part of Also, this is why I say I'm going off in 10 different directions at once. This is such a historical anomaly, this species distinction between pure and applied math.
u142 29:04 Don't forget, the greatest mathematicians of all time until the past 100 years always did both.
u143 29:11 >> Yeah, pure mathematician is a very recent recent
u144 29:14 How Maxwell's math revealed the nature of light
u145 29:14 invention.
u146 29:14 >> pure mathematician is a Like would Gauss have said he was a pure mathematician? He was tracking series. He did you know, this asteroid. He was mapping the Earth, literally doing mapping when he was inventing differential geometry. Riemann was very interested in electricity and magnetism and you know, I Hilbert, even Hilbert was thinking about general relativity.
29:14 · Split-screen video interview between two men with microphones · A still from a two-person video call, split into left and right halves. On the left, an older man with glasses and over-ear headphones sits at a microphone in front of a backdrop of a clock tower and hillside view. On the right, a younger smiling man with an earbud sits behind a pop-filtered microphone in front of bookshelves and a door.
u147 29:39 So, um and obviously Euler, you know, or Newton, these people are absolutely top physicists as well as mathematicians. Kepler, too. So, Galileo, I mean, Galileo maybe not so great with math. He was pretty good.
u148 29:55 But anyway, world-caliber people, and they didn't make this distinction. This is a very peculiar 20th century phenomenon. Uh okay, that maybe was a tangent. Let me come back off that tangential direction and back to the core, which is on this question of understanding and why does understanding matter?
30:02 · Man in headphones speaking at studio microphone · A still from a video shows a middle-aged man with glasses and over-ear headphones, wearing an olive-green button-up shirt. He looks downward, speaks, and holds one fist near his chest, in front of a boom-arm microphone. Behind him is a backdrop image of a hilltop campus with a clock tower, spired buildings and a long lake under a pale blue sky.
u149 30:15 A pleasurable, yes. Generative, yes. And a route to power. That's so I that just in case that argument is not familiar. If you understand, let's say, vector calculus or quaternions at the level that James Clerk Maxwell did in the 1800s, you can then rewrite what Michael Faraday and Ampere discovered and Gauss, too, about electricity and magnetism into four equations.
u150 30:42 >> Mhm.
u151 30:43 >> And then you can manipulate those equations using logic and using your brain as Maxwell did around the time of the Civil War in the US. And um you make a prediction of something that no one has ever seen. That there's an invisible wave of energy propagating at the speed of light that we now call an electromagnetic wave.
u152 31:08 That you know, give it a few years becomes the basis for television and radio and cell phones and all kinds of wireless communication. And that was only possible because he knew how to do he he recognized the wave equation, a partial differential equation when he derived it from doing vector calculus identities on the four laws of electricity and magnetism.
u153 31:32 So, my point is that the understanding that Maxwell had through math and that Michael Faraday did not have because he was a different kind of mind led Maxwell to be the first person on the planet to understand what is the nature of light.
31:32 · Man in headphones gesturing in a recording studio · A still from a video shows a middle-aged man with glasses and large black over-ear headphones, wearing an olive-green button-up shirt. He looks off to the left and gestures with both hands. He sits beside studio microphones on boom arms. Behind him, a large screen shows a hillside campus with a clock tower, stone buildings and a lake under a blue sky.
u154 31:49 That light is an electromagnetic wave. Nobody knew what light was before Maxwell. Not really. And so in that blinding flash of insight um he figured out something that turned out to change the world. And we could give so many stories like that whether it's Schrödinger's wave equation or Dirac's prediction of antimatter or whatever.
u155 32:09 I mean, just it's endless. The the I mean, but all these things have turned into technology. Right? So, from from understanding general and special relativity, you can figure out the corrections you need to make in the atomic clocks to make the GPS devices work as accurately as they do.
u156 32:29 If you didn't understand relativity you'd GPS wouldn't work.
u157 32:35 >> And so a pertinent question then becomes if the entity that is coming to those bits of understanding is not necessarily human. You know, that the solution is found and it's not human. Um, either you say, "Hey, the the um, the benefit is lost because it wasn't a matter of young Maxwell happening to solve a given random second order differential equation.
u158 32:57 It was the intuitions about what waves equations feel like that he then had light up again when he was engaging with um, a certain physics context that that actually mattered. And therefore, had he just been handed the solution out of the blue, it wouldn't have have given us the value that was actually uh, influential to the world, which was Maxwell's as a human brain's understanding.
u159 33:18 I think the other take could be like, well, why can't the same entities that are coming up with, you know, solution to the next Millennium Prize problem, silicon though they might be, um, have that downstream effect? Which you can have a very positive take on and say like, "Amazing. Wonderful." Like, well, would we just sit back and wait for the usually decades-long delay between uh, pure math discovery and application discovered um, to be to be compressed.
u160 33:43 Um, yeah, but between that dichotomy I'm thinking you you you need or it is beneficial for there to be a a loving human mind who has embraced with it in order to digest it into application.
u161 33:54 >> But is
u162 33:55 Is understanding overrated?
u163 33:55 it true? Is it just a romantic thing that you just said? Like, I mean, it sounds nice, but yeah, but I can I mean, I love that you're seeing all the same I mean, you're following the same flowchart of thinking. Well, not the same. I mean, you have your own version of it. But normally when I have this conversation with people, they're not seeing as far as many moves ahead.
u164 34:17 Cuz I keep wondering like in this argument, you know, say Sure. Um, if I have this entity which can solve the problem, let's say it can give me answers. Who says it can't also give me all the understanding along the way and all the spin-offs and all the applications.
u165 34:34 In other words, I think people get confused about the argument that is easy to make, that human understanding has led to power and progress, and they get confused by the converse, which is that they think, "Does that mean if humans didn't do it, like if the AIs did it, we wouldn't have the same amount of progress?" No, that doesn't follow at all.
u166 34:56 In fact, I think if the AIs could get us the answer, they would be so strong they could also get us all the side spin-offs and everything else, too. So, I don't think the argument for why human understanding is important is that it's because only humans can see all the connections and the spin-offs.
u167 35:17 I just don't believe that's true. I think the AI will be able to do all of it. So, I really am very confused. Here, let me put it starkly, if that's the last sentence or paragraph was too hard to follow. Um here's the real question, and I would love to know what you think about this.
u168 35:34 Is Is understanding overrated? I think I think that this might be a real confusion that people Let me make this argument, and I don't know that I believe it, but just for the sake of since you like having discussions, let's have the discussion.
u169 35:50 >> You could say that human civilization didn't have much understanding for I don't know, how old are we as a species? 100,000? I don't know the number. I should know my anthropology. [laughter]
u170 36:00 >> Mhm.
u171 36:00 >> Like, I know that 12,000 years ago we domesticated dogs, cuz I love dogs. So, I remember that number.
u172 36:06 >> I think there's sort of a 200 to 300,000 years. I I'm not an expert here of of what people would carve out as our current species.
u173 36:15 >> We've been Homo sapiens for 200,000?
u174 36:17 >> Neither one of us is an expert. Hopefully, understanding is overrated, and this specific number doesn't matter for the argument we're making.
u175 36:23 >> this number. That's not the point.
u176 36:25 >> [laughter]
u177 36:26 >> The The is for a long time we didn't understand much. We were relying on instinct, we were relying on experience, but things didn't really make sense in the way that they started to and now again, some arbitrary number. Certainly since the Enlightenment 6 8 If you want to put it with Isaac Newton we had a different level of understanding than we had before Newton.
u178 36:49 We really mathematized we as a species. Like take Kepler. I mean Kepler could see the elliptical orbits, the square and cube law about the planets, Kepler's third law. These are numerical patterns. He couldn't explain them. He just saw them in the data.
u179 37:06 And that was great. The math showed the pattern. But why? Okay, he couldn't understand them. And then the next, you know, a few generations [clears throat] later Newton can explain that not only what Kepler saw, but all of what Galileo saw, too, from a deeper principle, set of principles from his his laws and universal gravitation.
u180 37:28 And so that is like a triumph. Uh plus he has calculus in a in a more systematic powerful form than those other people have. And so he can derive all kinds of stuff with logic and a few principles. So that is a very powerful advance in human understanding.
u181 37:46 And that Newtonian framework of find the differential equations and solve them, more or less, gives us 400 years of progress all the way through the Schrödinger equation. Really. I mean the Navier-Stokes equation, Maxwell's equation.
u182 38:01 >> like a case for understanding being under valued.
u183 38:04 >> been of utmost importance. It's It has been of utmost importance for 400 years.
u184 38:09 >> Okay.
u185 38:11 >> And I'm going to argue temporarily that there has been a great interlude of understanding. Before this interlude began, we didn't have much. Obviously, we had some. We had Archimedes. We had, you know, the guy who is whose name is translated as Alhazen ibn al-Haytham, great Islamic scholar.
u186 38:33 Okay. I mean, there were people for sure
u187 38:35 >> Let's make some sweeping generalizations and just say like post-enlightenment, there was a
u188 38:39 >> Well, it don't have to do with the enlightenment, right? I mean, maybe the Egyptians with the pyramids. We had a few thousand years. Let's say that way. We had a few thousand years of really kind of building up our understanding. But, I think the window might be closing. That's that the interlude is coming to an end because I think the AIs are going to get so advanced that even though they're going to be good teachers, they're not going to be good enough because we're limited until we change our biology.
u189 39:11 Then, we're not really human. I mean, if we're going to stay the humans that we are with our kilohertz clock speed of our neurons, with the number of whatever it is, 100 trillion synapses. I mean, whatever it is, we have limits.
u190 39:28 And we can understand up to a certain level. And that will be it. We're just not going to be Like, we already know you can't picture the ninth dimension in space. There's only certain things that we can do. We can't picture 50 moving objects moving, you know, spinning around simultaneously.
u191 39:46 Psychologists have studied. So, it's going to come soon that we won't understand all kinds of things that the AIs will tell us. And so, I I feel like that's going to be the story that we have this wonderful, glorious interlude when the lights went on and then the lights are going to go off.
u192 40:05 And we're actually sort of lucky that we're living in a time when understanding is still meaningful, but progress will continue. We're going to have all kinds of fantastic results predict I mean, assuming we can survive this.
u193 40:18 >> Mhm. Mhm. Separate question. But if we can survive super-powerful AI and all the other things coming to the planet, um we're going to have an we'd have incredible progress.
u194 40:33 >> When you when you say that or making the potentially devil's advocate case on understanding is overrated, you mean in so far as progressing the the welfare of humanity or or pushing forward, you know, a sense of um ability in constructing the world around us or however you want to measure progress, you're saying uh we we're we're fast coming to the point where human understanding as a meaningful component of what determines that kind of progress is of waning significance.
u195 41:06 >> Absolutely beautiful summary. Yes. That it's overrated in that sense. That under We think understanding is great cuz it's been so helpful to us. Also because it's pleasurable. But it's going to stop being that helpful. No, it will still be pleasurable. Okay.
u196 41:22 I mean, just like people like to do crossword puzzles or other things that involve thinking. Um the So that's So there's that view of it. That math will always be here just like playing the cello or, you know, I mean, singing. These things are just inherently fun.
u197 41:39 And And actually, I sort of flinched, although I don't know if I visibly flinched, when you spoke about welfare or progress, you know, because it's such a narrow construction of progress that we're talking about here like building more gadgets, even curing diseases.
u198 41:57 Uh fine. Those kind of advances are a certain type of progress. But what about like spiritual progress or happiness or kindness or, you know, very soft things that are still very important in life. Um maybe the most important, love, being loved, those things.
u199 42:18 Uh that is going to persist forever, we hope. And
u200 42:25 >> And that
u201 42:26 >> I I really find myself in these conversations, I always end up wanting to talk about the meaning of life. I really think that's what's lurking here. I know we're supposed to talk about math with each other, but I I kind of think that's what we have to talk about is what are we doing? I mean, what are we here for, if anything, and what do we want out of our life?
u202 42:45 I do think that's what's at stake, and that's ultimately what we have to talk about.
u203 42:50 >> It's so interesting you land there. I mean, I think the reason that the math and AI topic is so relevant even outside of the math circles is that people talk about a spiky frontier to AI. There's certain things it gets good at faster than others, and insofar as math exists right at the beginning of one of those spikes, coding as well might be another one of them.
u204 43:10 It stands to be a microcosm of a broader world on and mathematicians are especially interesting because the thing that they are doing is both useful. You know, you you you mentioned three different things that are uh that are what we get out of understanding.
u205 43:26 You have the the orgasmic side of like it's just nice. You have um a route to power. Um and then you also uh have the generative aspect of it on like it brings you things. And if I'm if I'm hearing the argument correctly, it sounds like what you're saying is on the generative side, that human understanding won't be as important.
u206 43:46 On the route to power side,
u207 43:48 >> I mean, maybe we can get to that. That depends
u208 43:50 Fermat vs. Goldbach: math that generates more math
u209 43:50 on what human power is good at.
u210 43:51 >> the distinction about generative versus power. So, there's power in the outside world, but there's also generative in the inside world. So, generative of more ideas. Like you can speak about for to me, pure math is about the orgasm and the generative math that keeps recreating new math.
u211 44:11 Math that keeps posing new like say let's take the the quest to solve Fermat's Last Theorem. Right the proof of Fermat conjecture. So pure mathematicians love that story. Rightly. Because you know along the way to try to solve this problem they invent ideas like ideals in you know in algebra they discover number fields and examples where they don't have prime factorization being unique.
u212 44:42 And like there's a whole glorious imagined world that turns out to spring up around this question that is beautiful in its own right and actually arguably more beautiful and more important than Fermat's Last Theorem.
u213 44:59 I think most serious mathematicians would say that's not really that theorem or that conjecture whatever. That was our lighthouse, right? I mean in the modern to use the current uh metaphor that seems to be going around. That that shines the light and then we create all kinds of other ideas to try to solve it and those ideas take on a life of their own.
u214 45:21 They become I mean they gradually give rise to something close to the Langlands program. You know like really big stuff that that ends up getting used in Wiles' proof. Modular forms. I mean I don't know Wiles' proof but it but it sounds like a really absolutely enormous set of ideas gets used that is way more beautiful and powerful and and generative in this sense that it's math making more math and more understanding and lots of other great results that along the way being okay we get Fermat's Last Theorem but that's not the point.
u215 46:01 And this is why Gauss complained so much about Goldbach's conjecture. Right? I mean, Goldbach's conjecture that every even number is the sum of two prime numbers still unsolved. Seems like it in a way a better question than Fermat's Last Theorem.
u216 46:19 So simple. And yet very few mathematicians get all weepy and excited about Goldbach's conjecture because it hasn't been generative. It doesn't It didn't produce the same kind of surrounding superstructure of great math.
u217 46:36 It's like sort of peripheral. It's kind of out there, not connecting to the great body of other math. So, in this certain aesthetic I mean, Gauss had some quote. I don't remember the exact wording, but it's something like Ah, you know, it's easy to make up a question that you could neither prove nor find a counterexample to and it's not connected to anything else and that's of no interest.
u218 47:02 So, so it was the peripheral nature of the of the conjecture Goldbach's conjecture that turned off Gauss and I think turned Now, it's not totally true. There is beautiful math connected to Goldbach's conjecture, too, but not as beautiful as the math and not as interconnected as Fermat's Last Theorem.
u219 47:19 So, according to certain tribe this is what pure math is all about. I you know, look, this is me talking about a foreign culture. It's not really my people.
u220 47:31 >> Mhm.
u221 47:31 >> I'm I'm the people I want to talk about Maxwell. You know, or
u222 47:35 >> Right.
u223 47:35 >> or Einstein. I mean, I guess I'm a physicist in mathematician's clothing. But a biologist, too.
u224 47:42 >> [laughter]
u225 47:43 Will human mathematicians exist?
u226 47:43 >> I I think I wanted like stay drilled on the the side cuz what's what's salient in the minds of a lot of young people is just what is the role One, will human mathematicians exist? Um you know, is it is it dead in the water to even like go go into to a PhD? And then two, what what is the role if they do exist?
47:43 · Split-screen video call of two smiling podcast hosts · A still from a two-person video recording split into left and right panels. On the left, a man with glasses and over-ear headphones laughs in front of a microphone, with a backdrop of a clock tower and distant lake. On the right, a man in a gray-blue T-shirt looks down smiling with one hand behind his head, seated at a pop-filter microphone in front of bookshelves.
u227 48:00 And so, maybe first I've been trying to hold my tongue. I don't want to insert too many of my own opinions
u228 48:04 >> your tongue flapping, man.
u229 48:06 >> [laughter]
u230 48:07 >> I It's your show. Okay, you can be on that side of the microphone.
u231 48:10 >> Well, so at the at the very least poke holes in Please poke holes in this. I just want to at least lay out why I think like mathematicians will exist, but it's kind of this wide-open question on exactly what they do. And then And then that might set the framing for how we think about this question of, you know, in so far as it's about uh uh being generative, like to whom does that matter?
u232 48:30 So, just from a from a side point of why is this a job, where does the money flow from for mathematicians? I think it's important to for a young student to have a clear vision of that in their mind, cuz it is a very weird job compared to other portions of the economy.
u233 48:44 >> Are we talking about an academic mathematician who's a professor or are we But but there's so many mathematicians with jobs called data analyst.
u234 48:53 >> A Oh, what a what a wonderful Steve Strogatz answer on like this is
u235 48:58 >> What is a cryptographer? Who do what Who Who employs more mathematicians than anybody? The National Security Agency.
u236 49:05 >> Okay, I love this.
u237 49:05 >> Okay, you don't mean them. You want to talk about math professors.
u238 49:09 >> And for the sake of clarity, let's let's let's first talk about professors, but actually we should zoom out.
u239 49:14 >> I think so.
u240 49:15 >> First on the side of professors, then let's zoom out. It's Okay, the reason it's relevant just to focus on the professors before the zoom out, who trains all the cryptographers? Who trains all of the data all of the Like one of the most valuable externalities from the existence of math departments are all of the people who like quote unquote fail to go into academia, or like don't end up becoming professors.
u241 49:37 You have this just spilling over of mathematical talent and problem-solving ability out of people that maybe they pursued a PhD, maybe they uh just got their undergrad, who don't necessarily go into academia. But importantly, they spent a meaningful part of their formative youth, at least undergrad years, maybe grad years, in a math department or something adjacent to a math department.
u242 49:57 And so so the the question of like do those math departments exist and the people doing this training, do they exist? I think has has uh has bearing even on the the most the the broader stroked definition of mathematician that I would agree with on you on on here. Um but just on the academic mathematician said, one you've already kind of hit it where I think if you follow the money flows a ton of it is just uh just teaching, I think, on um for for a typical math department, which is unlike other forms of research where that sometimes, you know, uh harder sciences are quite expensive to do the research um and and there's going to be more that leans on public funding or or private philanthropic funding.
u243 50:35 Of course, math also has that, right? There is um there are, you know, in the US like NSF grants that are a meaningful part of what will support mathematical research. There's various private foundations. You have a typical institutes like if you're a uh a permanent faculty member at the Institute of Advanced Study, like teaching is not as as big a part.
u244 50:54 Um The teaching side of it, this strikes me as something that's quite AI safe as far as job security goes, insofar as if universities exist at all, if we have social structures at all like what we have now, people really will value the um the human-to-human side of that.
u245 51:12 Um and I mean people have have have sort of written about this on how like a pretty good case study on this is COVID and how much people hated uh education mediated through a screen.
u246 51:23 >> Yeah, even when it's just an Alex Kontorovich made this point in a nice like dialectical thing he was I was reading of his lately uh where it was essentially like look, you already if you have the current status of education ability but mediated through a screen and people hated it. I you know, maybe AIs get superhuman such that even through a screen they're better, but it's a very social thing going on if we think of what's actually happening with education where it's not about clarity of explanation, it's about what motivates you to learn.
u247 51:51 And if we think of what motivates people, it's their it's their connections, especially pure connections, but also I think mentor connections and such. I will care about a topic more knowing that you, Steven Strogatz, wrote about it compared to just what the words on the page in and of themselves can do.
u248 52:07 Like that little bit of human-to-human connection matters. So, if we presume that teaching still exists, um then the existence of like that as one of the funding sources towards what goes into math departments is kind of unaffected by whether or not AIs can, you know, solve problems better or set conjections conjectures better or set entire programs like the Langlands programs better.
u249 52:31 All of that is this ancillary thing of what the mathematicians choose to do with the surplus time that is, you know, research.
u250 52:38 >> [laughter]
u251 52:38 >> And some of what's going on
u252 52:39 >> you see it that way because after all, historically, I mean, it's been changing so much throughout history, right? Before there were really, I mean, universities are not that old.
u253 52:50 >> Mhm.
u254 52:50 >> 1200, you know, AD or something, the first universities in Europe or or somewhere about there. Uh and for a long time it was all about teaching. Yeah, I mean, Isaac Newton or Isaac Barrow did do some They did research, but they were teachers.
u255 53:07 And then after World War II, I mean, that's the big thing that really changes so much because for one thing, computers start to exist, modern computers, but also because of this vision of this guy, Vannevar Bush, that we need research and we're going to pay with taxpayer dollars for these professors in math and physics and science to do research with such generosity from the public that it's actually free to go You get paid to go to graduate school in the sciences to get a PhD.
u256 53:42 It's really shocking. I couldn't believe it that I didn't have to ask my parents for money
u257 53:47 >> [laughter]
u258 53:47 >> to go get a PhD. That that they would pay me to get a PhD. I thought, that's just the greatest thing. Um
u259 53:54 >> Yeah.
u260 53:55 >> Now, how much longer will it last though? Because what was why did we get paid to go to grad school? Was it because of the Cold War? Because the country sort of thought, we need a cadre of serious technical people so that we can stay ahead of the former Soviet Union.
u261 54:12 I mean, it's a very cynical view of it, but I think that was actually a big
u262 54:17 >> Sure.
u263 54:17 >> Like in Congress they would say that's what science funding was about for a long time.
u264 54:24 >> So that this one is not a done deal on whether public funding like continues to exist for mathematics.
u265 54:30 >> But we may become teachers and do a little research on the side. Like we always were.
u266 54:35 >> Yeah, even if you subtracted all the way, the the career does not die. No, I if I had
u267 54:39 >> No, it doesn't die. I as teacher.
u268 54:40 >> Yeah. Yeah. So if I had to predict, I would imagine public funding
u269 54:44 Why learn math when AI can do it?
u270 54:44 does continue to exist.
u271 54:45 >> May I ask you then since I know you wrote about this. So why why does anyone want to learn math if Of course, the calculators to this right now can do all kinds of elementary math calculating tips or balancing your checkbook. So you Since we're doing the devil's advocate thing with each other.
u272 55:06 >> You know, what's your argument for why anyone should learn math in the future given that you can press the button on Wolfram Alpha or now just ask Anthropic's Claude or ChatGPT. And so what would you say? Why why do you what cuz people must be asking you all the time, why should I learn math?
u273 55:27 >> I think there's there's basically the artistic answer and then the utilitarian answer. I think the artistic answer stays pretty stable into whatever AI future you get. And you sort of hit it on earlier and like just the there is a very unique kind of joy and edification that is unlike anything you find almost anywhere else that math has really cultivated an aesthetic around that it's joy isn't quite the right word for when you come to understand something.
55:54 · Split-screen video interview of two men with microphones · A still from a two-person video call in split-screen. On the left, an older man in glasses, headphones and an olive-green shirt sits with hands clasped beside a microphone, with a blurred view of stone towers and a lake behind him. On the right, a younger man in a gray-blue T-shirt speaks behind a pop filter and a USB microphone, with a bookshelf and white door behind him.
u274 55:56 I I think of a very specific clicking moment that happens for certain kinds of
u275 56:00 >> [laughter]
u276 56:01 >> topics.
u277 56:01 >> And like good.
u278 56:02 >> That that click that happens in the brain I don't get it from anything else and it's it's it's such a joy and it's it's so addictive. I think certainly there will always be people who have experienced that and just want more of it. Um and actually I think many of the mathematicians I talk to this is this is actually their motive for going into the field.
u279 56:20 It's not so much they wanted to be the first to discover something. It's not so much um you know, it's not even yet the vibrancy of of the culture and the shared community. I think it's that at some point when they were younger they got addicted to that feeling of whatever like doing math meant for them and they they simply wanted to do more of it and the most
u280 56:38 >> That's what it is.
u281 56:39 >> Yeah. The the the most standard path for how to do that
u282 56:42 >> I mean you'll remember that was what the last time we did this together when I was on your podcast some years ago I told that whole story about this certain triangle problem in high school. Yes, that was it for me. And that what I wanted to reproduce that feeling.
u283 56:59 >> That will always exist. There will be the people These are the people I'm often trying to either speak to with my videos. You know, they've already had that feeling and I'm trying I'm trying to feed the addiction and give them a couple more of those clicking moments. Even better would be if it's somebody who hasn't yet experienced that and I can offer it to them for for the first time in some form.
u284 57:17 So that that's one answer on on why to continue to learn math and this is in the anything you can do I can do better future of AI where like there's just look there's nothing that you're going to be able to do that the AI couldn't.
u285 57:29 >> This is timeless. It's permanent.
u286 57:31 >> That is such a wild future to engage with. First of all, we have to do all the normal coaching on like, you know, humans are still alive. This is a good version of the future, not a bad version of the future. But like in that in that branch, of course we're going to focus on the human flourishing aspect because what what else is there to focus on?
u287 57:48 Um I think we we have some years yet though that closes that gap on the anything you can do I can do better. And there's definitely been time and time again this recurring phenomenon where the people who learn these mathematical abstractions come to appreciate the nature of what it looks like to sharpen your problem-solving abilities in that context, find themselves unexpectedly useful in neighboring fields.
u288 58:15 And this maybe gets to your point on like what is a mathematician? Are all the you know, the data scientists and the cryptographers and the machine learning researchers, everybody there. The These are these These people who have just spilled forth from the the mathematics education system quite productively.
u289 58:31 And so, you know, everybody's writing their their grants talking about these the examples that you brought up on, you know, Riemann uh caring about non-Euclidean geometry that turns into general relativity later or pursuit of Fermat's Last Theorem gives rise to things like elliptic curves which are then useful later.
u290 58:47 That is one externality that comes from doing math research and and that existing as an institution. There's a much more immediate one in my mind which is this spillover of just all the highly educated educated in the sense of capable at problem-solving, understanding mathematical abstractions, fluent in that framing frame of thought just just going elsewhere.
u291 59:07 Anything that's not academia that they're they're going into Well, cannot anything. Like quite a few of the the the things are are productive um in these in these ways that are hard to capture in a direct sense and therefore very good fruit for, you know, private philanthropic funding or public funding or whatever it might be.
u292 59:25 I you know, maybe there is a future of AI so good at everything that all of that's obviated. All the cryptography is automated, all the data analysis is automated. But at the very least in the meantime, you don't want to jump the gun on halting the the system that produces all of the people fruitful there.
u293 59:42 Uh and then you know, what whatever it is that you think is important for making sure we're on that good version of the future and not the bad one, you know, I think of all the people who are jumping ship, so to speak, from the traditional math academic uh career paths. There's some I know that go into the AI safety side of it.
u294 1:00:00 And it's probably a very good thing that the like highly mathematically trained people are trying to bring those chops into, you know, these various problems that they frame there. Think what you will of like how well-posed those problems are or not. It just seems like a very good thing to have people trained in this way, like thinking hard about that. And so from that temp- it just seems clear to me something like the institutions we have now of a group of people that we call mathematicians who exist in departments that by and large are just training people in mathematical thought, some of them to be new mathematicians, but a lot of them not.
u295 1:00:34 That as an institution strikes me as surprisingly stable compared to a lot of other things where AI progress tends to potentially automate tasks within a job and then and then the job wholesale. And the question then becomes what do those people within this group decide to do on the research side, on on uh the all of all of these things where we've come up with proxies in the past, you know, what's the significance of the theorem that you proved in your PhD thesis and things like that.
u296 1:01:03 They have the choice for what to use as a proxy instead to uh make, you know, hiring decisions, to uh decide on is this a worthy topic for a PhD or not, to make tenure decisions. Nobo- Nobody else is making that choice for them. It is entirely within the community to decide what deserves academic status and not.
u297 1:01:23 And that I think is a very fruitful I have no idea what the answer is, but I think that's a very fruitful question to ask
u298 1:01:28 What should earn academic status?
u299 1:01:28 right now.
u300 1:01:28 >> I mean I do have a feeling about the answer there. It's that it's um it's almost like upvotes on social media. That [laughter] if I can do something that you find appealing or that gave your neural circuitry a little shiver of pleasure just to think about it or to see something.
u301 1:01:49 And it's not because it was so earth-shatteringly new, like any AI could have easily produced it at the push of a button. That's not going to be the test anymore. It's just did you enjoy what I created or even what I showed you and and what I made a case for that you should pay attention to it.
u302 1:02:07 That's why I think it is related to a kind of charisma, salesmanship, something like that. It's really about the social dimension that we're giving each other pleasure and you might say for what real purpose. That's the purpose. It's not going to be I mean some of it will be you to instrumental, like for really solving problems that matter to life or to human well-being or to the planet's well-being.
u303 1:02:30 But the AIs will be playing a very big role in that. Even just aside from that, like the pure part of math I think it'll be a lot what it is now, which is do we do things for each other that are interesting or do we show each other interesting things? Can we teach each other?
1:02:33 · Man with headphones speaking at microphone, campus backdrop · A video still shows a balding, bespectacled man in an olive green button-up shirt wearing over-ear headphones and gesturing with his right hand while speaking. He sits at a studio microphone on a boom arm, in front of a backdrop image of a hilltop campus with stone towers, historic buildings, and a lake with hills beyond.
u304 1:02:47 Um and can we have fun doing it together? Which again I feel is very close to the meaning of life. Like for an academic person anyway. But um I I really think we have to ask what do we do with our blip of consciousness?
u305 1:03:05 You know, like ashes to ashes and dust to dust. You're you're atoms and for a while you're organized in this astonishing way that gets to feel pleasure and happiness and love and sadness and what are you going to do with that period of however many years it is.
u306 1:03:26 And that's always been the question, but I feel like it's forcing a lot of us to become philosophical who haven't thought I mean I happen to like philosophy. I've always liked philosophy. I have not paid much attention to religion in the past, but but I'm finding myself reading some of the great religious writings, particularly the book of Ecclesiastes, which I you know, that's the one that gives rise to that song uh the by the birds turn turn turn to everything there is a season turn turn turn.
u307 1:03:58 Um that that's a line from the from Ecclesiastes in the Old Testament. And if you haven't read it, uh I don't want to go on a whole evangelical thing here, but I had not read it until recently. And it's a very unflinching and modern and not particularly um easy to read as far like for a a traditionally pious person.
u308 1:04:25 God doesn't appear in it very much.
u309 1:04:28 >> What do you mean by unflinching?
u310 1:04:30 >> I mean unflinching in that it says there is no meaning.
u311 1:04:34 >> Hm.
u312 1:04:35 >> It doesn't give you any comfort at all about um the reason that you're here is because of love or because of doing God's work or about the works that you create. All of that is described as vanity.
u313 1:04:53 And it's vanity in a particular sense. It doesn't mean vanity like oh you're so vain, you're conceited. It means you're doing everything in vain.
u314 1:05:02 >> Hm.
u315 1:05:02 >> It's all pointless. It's all vanity. The The Hebrew word is apparently the word for vapor or breath. It's all It's all It's sort of like the concept of dust in the wind. It's It's all vapor. It's all transient. It's really very Buddhist sounding.
u316 1:05:20 Roughly.
u317 1:05:21 Ecclesiastes: "Everything is beautiful in its time"
u318 1:05:21 >> So, what did you take away from it?
u319 1:05:23 >> What do I take from it?
u320 1:05:24 >> Yeah, yeah. What do you take away from this?
u321 1:05:25 >> take from it I mean there is there are glimmers of hope in it which which are um sentences like this idea that to everything there is a season. That everything is beautiful in its time, it says.
u322 1:05:40 >> Mhm.
u323 1:05:43 >> Sorry, I This is my new thing crying on camera.
u324 1:05:46 >> [laughter]
u325 1:05:47 >> But this idea everything is beautiful in its time. That That we get to have a moment in the sun. You know, when you get to love, you get to care, you get to be somebody, you get to live out your dreams maybe if you're lucky. And that's it.
u326 1:06:03 I mean, that It's because it's transient, because life is so short, that's what makes it precious. Uh these are not original thoughts. I mean, these are ancient thoughts, but they're right there in this several thousand year-old document. And so, when I think of what are we doing when we do math if the machines are better than us?
u327 1:06:24 Well, Terry Tao's already better than me. You know, I mean, it's not about that. It's about what I get to do by having this conversation with you or with my students or my teachers. That's That's what we get to do while we're alive.
u328 1:06:39 >> And you're saying
u329 1:06:41 >> I'm saying that's why we're doing it. If it's If math is the thing that gets you up in the morning, do it for that reason. If it's playing your guitar, if it's helping the person at the hospice, you know, whatever it is that gives your life meaning, that's what you've got to cling on to because you're only here for a short time.
u330 1:06:59 I I kind of think that's all there is. And the when I say it's unflinching, it's so unflinching that at the end of it, if you look at this Ecclesiastes, it has a paragraph which is clearly tacked on by later rabbis which says, "All that is all very nice, but all In fact, what you're really supposed to do is just do what God tells you to do."
u331 1:07:25 >> [snorts]
u332 1:07:25 >> Like it's you know, it gives a very traditional top-down kind of thing that is totally at odds with everything in the rest of the text. It's um Okay, didn't plan to go there, but
u333 1:07:38 >> Well, no. [laughter] I
u334 1:07:40 Does mathematical beauty depend on problem solving?
u335 1:07:40 I love everything about this because essentially what you're getting at is human flourishing. And if you're saying like if if math is what what feels like a life best lived, you know, to do math, to engage with it, to find these clicking moments, that is the reason to do it. You need no You need no justification even further than that.
1:07:40 · Split-screen video of two podcast conversation participants · A split-screen video still shows two men recording a conversation, each with a microphone. On the left, a smiling man in glasses and headphones wears an olive button-up shirt, with a blurred backdrop of a clock tower and a lake. On the right, a man in a blue-gray t-shirt raises a hand to his forehead and speaks, with bookshelves behind him.
u336 1:07:56 You don't need to rest on the the fact that your particular contribution is like what what the future hinges on as far as like later utility down the road. I want to I do want to talk about that problem-solving component of it that I pushed back a bit earlier.
u337 1:08:12 And and and and and connect it a bit You referenced already the story you gave the last time you were on this podcast of a particular problem you were given as a young student that was, you know, unexpectedly tricky about triangles and and the pursuit and the joy that that came from that.
u338 1:08:28 Um So, I've had that and this is going to be a little bit of a long thought, but I'll I'll see if I can I can keep it somewhat short. I've I've had this experience many times where, you know, I find math quite beautiful. I I think you find it quite beautiful. There's a certain shared aesthetic among many mathematicians. But then every now and then I'm trying to communicate to someone else what it is I find beautiful in it.
u339 1:08:48 And I like to, you know, this is my whole job to some extent. I'm supposed to be able to like convey the beauty [laughter] of math and and and I run up against a regular failure, um which is that sometimes somebody who's quite outside of it it's just there's just nothing about it that's clicking the way that I would want. know, I maybe I talk about the Pythagorean theorem or certain way of seeing it.
u340 1:09:05 And it's, you know, they're they're they they nod along and it's not that they don't understand it. It's just something where it's as if I'm seeing a color and they they see it in black and white. They're like, "Well, that's true and everything, but I [laughter] got to be honest, I don't see why this is beautiful." And I thought about this and I think some of what's going on is my my sense of what is beautiful is actually more tied to problem-solving than it might seem at first where I, you know, in my youth was was good at math or in so far as, you know, in the local environments where I was enough to feel that way as you know, egotistic young man as as as many of us might have been at some point.
u341 1:09:44 And um and sometimes when there's a given idea that I would see and think like, "Oh, wow, that's that's very beautiful." What that actually would translate to is I can see how this might be useful later or that is a real bolt out of the blue that helps solve this problem. And I can like part of me can predict that maybe I can use that same kind of bolt later on were I to come across another problem and seem similarly clever.
u342 1:10:09 >> Okay.
u343 1:10:09 >> Something like generating functions. You know, the first time you ever learn about generating functions, it's just like, "Where did that come from?"
1:10:09 · Split-screen video interview between two men with microphones · A still from a two-person video conversation in split-screen. On the left, an older man in glasses, headphones and an olive button-up shirt speaks into a microphone, with a blurred campus clock tower and lake behind him. On the right, a younger man in a blue-gray t-shirt sits before a microphone with a pop filter, with bookshelves and a white door behind him.
u344 1:10:16 >> Yeah.
u345 1:10:16 >> [laughter]
u346 1:10:16 >> You can sense the generality of it and and you can have the sense of like, "Ooh, next time I come across this recurrence property or something, like that is a a tool that I'm going to be able to bring to bear." And the fact that it's both general and that it's unexpected, this is some of what makes it seem beautiful.
u347 1:10:34 You I I don't play as as much chess as I think you do, but sometimes when I see people commentate on like a particular move being beautiful, some component of that is that it's unexpected. And it's also that it's um you know, it's clearly useful. That was a good move to make. And that if we say what makes that beautiful, I wonder if it's a sense of it's like an intuition to tuck away that maybe can be applied in a future game.
u348 1:10:57 And if you try to take the problem solver out of the mathematician or out of the student and if they don't they don't have the same kind of practice of really engaging with and struggling with problems in the same way that all of these surrounding things that we're talking about and like oh, it's it's math for human flourishing.
u349 1:11:16 It's it's this thing that gives you this unparalleled feeling or mathematician saying it's the lighthouse that's shedding light on this broader landscape, but it's the landscape that you care about. I kind of wonder how much of that landscape or how much of that deeper clicking feeling is dependent on being perceived by a problem solver, a person who at some point in their past really engaged with trying to solve hard problems and has unconsciously or otherwise collected a sense of what's going to be useful for that.
u350 1:11:42 And this beauty that we're describing and this flourishing that we're describing is is tied with that. And so this is a very long with with way of kind of going
u351 1:11:50 >> But I think I'm following you. Go ahead. You can ask the question cuz I think I know where you're going.
u352 1:11:56 >> If you had had a youth where you didn't have you were given this triangle problem and you could just ask chat GPT or something and it shows you what the answer is or like or or you mentioned Goursat's theorem in complex analysis something. If that was something that you know, it's coming out as text from a chatbot like all of the all of the problems that you could have engaged with the mode of of engagement at least in principle maybe also in practice came from the you know, quote unquote all knowing AI just giving you the answer.
u353 1:12:25 >> Do you think your aesthetic for math would have been affected if that had been your youth?
u354 1:12:32 >> Very very much.
u355 1:12:34 >> Yeah.
u356 1:12:34 >> Yes. And um in a very negative way. In a very I would be deprived.
u357 1:12:42 >> And how do so how do we preserve that? Like
u358 1:12:44 Math Explorations and the one-cut triangle puzzle
u359 1:12:44 how do we make sure
u360 1:12:44 >> Yeah, let me let me try to make this very concrete. Uh I have taught three times a course called Math Explorations. And it's a course for people who have a math requirement to graduate and most of them don't want to be there.
u361 1:13:06 >> Mhm.
u362 1:13:07 >> You know, these are government majors, architecture, whatever. They're art history. They're not interested in math. Not really. Some of them are. But most of the people who take it are taking it because their college says, "You have to satisfy a quantitative reasoning distribution requirement." And um this is thought to be one of the easier ways to do it.
u363 1:13:32 Like you could take calculus, but a lot of these students have already taken calculus and maybe even gotten a good score on the BC calculus test. That's how they got into Cornell in the first place. But they don't see the point of math or they hated math or maybe they were perfectly good at it and they just whatever.
u364 1:13:52 So, they don't want to do calculus again. So, they're in my course, Math Explorations, and just to come to the point since I do feel so long-winded for some reason, maybe cuz I am.
u365 1:14:02 >> [laughter]
u366 1:14:04 >> I think it's absolutely essential to give people the experience of doing math. Math appreciation doesn't work without math. Doesn't really work without doing math. You can appreciate math passively. You know, you could watch Grant or you could watch me talking about the beauty of math or blah blah blah.
u367 1:14:25 It doesn't It's not the same. It doesn't produce the necessary thing that makes a person get it unless they do it. Now, it doesn't have to be advanced math. It could be so basic. I mean, what I often use as the first activity is is a problem where there's a triangle drawn in the middle of a piece of paper.
1:14:38 · Man with headphones speaking at microphone, campus backdrop · A video still shows an older man with glasses and over-ear headphones looking down and gesturing with both hands while speaking near a studio microphone on a boom arm. He wears an olive green button-up shirt. Behind him is a backdrop image of a hilltop campus with a clock tower, stone buildings and a lake in the distance.
u368 1:14:46 And it's a triangle Actually, the the first activity is I would show you an equilateral triangle. And then I give you a scissors and say, "Your goal is to cut out the triangle, but you can only make a single straight cut." Have you ever heard this activity?
u369 1:15:05 >> I don't think so. No.
u370 1:15:06 >> Okay. So, there's a triangle. I mean, I could draw one, but I give a blank piece of paper except it's got a big fat triangle. It's equilateral. Here's your scissor. Cut out the triangle, but you have to cut in a straight line. And then the student thinks, "That's impossible." And then another student says, "Are we allowed to fold the paper?"
u371 1:15:23 >> Mhm.
u372 1:15:24 >> And then you say, "Go ahead." So, then the kids are all holding up the paper to the light and get and folding it to get the lines on top of each other. Then they cut and everybody can do it and the triangle drops out. And then you hand them another piece of paper that has a square on it.
u373 1:15:42 >> Mhm.
u374 1:15:42 >> And then they're kind of actually like it because they did it. They do it again. The square drops out. Anyway, you do a few simple shapes and everyone can do it. But then you give them a scalene triangle. All three sides are different length. No symmetry, no obvious and you know axis of symmetry or any obvious thing to do with it.
u375 1:16:06 And I did this myself in a in a class learning how to teach in this inquiry-based style. Um I had never been taught this way. And next to me was my colleague John Guckenheimer who's very fast and smart. And and John is doing something with the paper and he cuts it out and it's it worked.
u376 1:16:28 >> Mhm.
u377 1:16:28 >> And I can't figure out what to do. And I feel myself getting embarrassed and my face is getting red cuz I'm really slow and I'm always the slowest. And I start feeling what people who hate math feel.
1:16:36 · Split-screen video interview of two men with microphones · A still from a two-person video call in a split-screen layout. On the left, a man in glasses, headphones and an olive button-up shirt looks down and speaks into a microphone, with a virtual background of a clock tower and buildings by a lake. On the right, a smiling man in a gray t-shirt sits behind a pop filter and microphone in front of bookshelves and a white door.
u378 1:16:41 >> Mhm.
u379 1:16:41 >> Okay, so it's really good to go through that. And then John is now doing a quadrilateral and then he's doing a concave quadrilateral and he Everything he they give him, he could do it. And I'm still stuck on the scalene triangle. Now, um when I do this with the students and I give them the scalene triangle, after they've had several successes, nobody in the room can do the scalene triangle and it like a half hour goes by.
u380 1:17:09 >> Mhm.
u381 1:17:09 >> And the interesting thing that happened the first time I taught this was I say I I'm like running out of time. And the the bell's going to ring and I think I Okay, so I say, "Would you like a hint?" And then a bunch of students say, "Yes." And then all the rest of the students say, "No."
u382 1:17:28 >> Mhm. That's great. [laughter]
u383 1:17:30 >> Because they are actually having a mathematical moment. They knew they should be able to do this, god damn it. They had done these other shapes. And And so I thought, "You guys are the best." Because that's exactly You're already on the right track. Now that you know that you want to do this and there's no reason this isn't going to do any important thing in the world.
u384 1:17:51 This is just a game.
u385 1:17:52 >> Yeah.
u386 1:17:53 >> I had a kid then write to me and and you have to understand, these are all traumatized people. But on the first day we have them write on the board, "What's the first word that comes to mind when they think about math?" I got responses like sick, ugly, hate it, embarrassed.
u387 1:18:10 You know, some people say beautiful, but um mostly this a class that has a lot of built-in trauma about whatever happened to them in math class. So, but over the weekend, I got emails from one student saying, "I thought about this for hours and I find and I fiddled with it and I finally got it to work and I can't tell you the the feeling I'm having right now." But he of course he doesn't have to tell me because it's the feeling you're talking about.
u388 1:18:37 >> Mhm.
u389 1:18:38 >> So, my goal in this class, Math Explorations, is to give everybody find some way to get let them get that feeling because that's the genuine mathematical experience. And it doesn't matter that it's such an easy I mean, it's a very it's not a deep problem, but it gives them the deep experience.
u390 1:18:57 Should we still grade homework?
u391 1:18:58 >> That's Okay, that's great. And I think a perfect example of maybe how to use classroom time effectively for that cuz the other struggle of course every educator is having is um what what to do about homework. Um when when more and in some sense maybe I'm just a a naive outsider.
u392 1:19:16 I feel like why are we grading homework anymore? Um at the very least stop grading it and then and then the question is, you know, how do you how do you motivate people to want to practice or maybe how do you shift that practice time to be in a in a communal environment instead of
u393 1:19:28 >> Mhm.
u394 1:19:29 >> relying on someone's like internal motivation to to do that practice just for its own sake or just for the sake of the proctored quiz that you'll take the next day. Um these kinds of experiences that that this feels like just a much better use of time in the classroom
u395 1:19:44 >> Well, yes and no because it's very slow. The students come out after after a whole semester of doing this. I mean, I have all these Let me say this comes from a series of books that you can look up on the internet. They're free called Discovering the Art of Mathematics.
u396 1:20:04 All right. Discovering the Art of Mathematics. If you type that exact phrase into your search engine, you'll be led to a place called Westfield State College. And um West I maybe you're doing it as we speak.
u397 1:20:19 >> Yeah, I was curious.
u398 1:20:19 >> They they have uh like 11 PDFs of um math and music, math and dance. It's all about art. It's for a liberal arts students and it's constant I mean it's so many brilliant activities of this type that are really hands-on.
u399 1:20:38 You could do it at high school level. You could even probably do it at middle school level, but we use it at college and it's meant for college students. Uh and there's a whole psychology explained about inquiry-based learning, but at the end of the semester, you could ask, have the students learned?
u400 1:20:56 How much math have they learned? And by conventional measures, you would say very little. But I would say they've learned a lot. And I've The main thing is I've changed their attitude. They're now ready. They're now ready. And some of them say, I don't still don't want to do math, but now I understand why people love math.
u401 1:21:16 They get it. They do get it. They all get it.
u402 1:21:19 >> It's I mean that that strikes me as much more useful than whatever cuz the major requirement you named it said it's quantitative reasoning or something.
u403 1:21:26 >> Yeah, that's what it
u404 1:21:27 >> This is this is categorically not quantitative at least the the example you gave at the
u405 1:21:32 >> name.
u406 1:21:33 >> The word quantitative wouldn't apply there. And in fact, I actually think almost all of of what happens in a math department at a college level, I wouldn't call quantitative reasoning. I mean of course some of it is, but if if if someone wants to justify like what is it that someone's getting out of the the math classes or a math major or that kind of training that's brought into the rest of the world.
u407 1:21:54 Like everybody usually goes towards that word quantitative. You know, this is this is what makes it a transferable skill and everything. I think there's many [clears throat] better paths towards quantitative reasoning.
u408 1:22:05 >> Well, we could do that. I mean, you could say why aren't why am I not teaching them financial literacy? Well, that's useful, too. But this is something else. I'm trying to teach them what they missed. And and and what it is that people who love math love about it. They do get that.
u409 1:22:21 Is that worth anything? I don't know. I think it is.
u410 1:22:25 >> I agree. I agree. I also think this is a great way to get at it where you especially if there's trauma going in, you want to make sure to uh like put a balm over that trauma before you proceed forward in whatever it is that you're trying to get out rather than just like hammering in more problems of a certain type.
u411 1:22:42 >> Yeah.
u412 1:22:43 >> Um I think this might have some relationship with the other questions on what is it that we're doing math for or what is it that in some sense this is not does not have direct bearing on what do the mathematicians of the future do or a prospective student right now contemplating if they should go into a PhD and wondering if they should be optimistic or pessimistic.
u413 1:23:05 You know, a vision of uh teaching the liberal arts majors to maybe barely appreciate what they find beautiful by like folding paper in the right way. To some people that would be a you know, a positive aspect of it. But I think to some people maybe they would they would fear like, "Oh, if this is all about sharing the flourishing of math and if uh uh if this is an example of what that looks like, so somehow that might somehow that might not quite leave the taste in the their mouth that they they were hoping for as far as like existing at the frontier and and really really wallowing in that um and enjoying that side of it.
u414 1:23:40 I'm I'm a little bit curious for you because you because you walk so many different lines, you educate at college, you have have done phenomenal research, you have done various bits of popularization. Like how these how these weave together for you?
u415 1:23:56 How important is the education side in doing classes like this to more um more applied research that you might do or how important are [clears throat] like you just wrote this book Big Math. Um you've wrote many excellent books through your past. How important is that act of writing and and really turning math outward and kind of in some sense stepping below where you could be as far as you know, uh proximity to the frontier.
u416 1:24:21 Um How how related are those other activities to the the applied research that you will do or have done throughout your career?
u417 1:24:29 Research in the service of teaching
u418 1:24:30 >> uh, my answer is peculiar, literally peculiar to me. You know, it's it's idiosyncratic. It's not this wouldn't be the standard answer, but but I always wanted to be a teacher. That's you know, if I have to start writing my own eulogy or putting thinking about what goes on my gravestone, I was a a teacher, not a researcher in my heart.
u419 1:24:55 I mean, a research to do research is to teach myself and to teach the world whatever I discovered or my colleagues and I discovered together. But I think of the fundamental activity as teaching and learning. I did have this one experience where I was working at a college at one point where the students weren't getting what I was trying to teach and I found that very disappointing.
u420 1:25:18 No matter what I any trick I use, it just wasn't working. And that made me feel I have to be at a place where the students can get what I'm trying to impart. And so for that reason, I felt I have to be I mean, this is going to sound very crass.
u421 1:25:38 That I have to be a good enough researcher that I can end up at the kind of place I want to be at.
u422 1:25:43 >> Mhm. And um, so I had seen very good teachers in college at Princeton who didn't get tenure because in my mind I thought they didn't publish enough. Probably wasn't exactly that.
u423 1:26:00 They probably didn't publish good enough stuff. But but whatever, I knew that their research was what sank them.
u424 1:26:09 >> Mhm.
u425 1:26:09 >> And so I always felt I have to try to be the best researcher I can be so that I can teach the way I want to teach.
u426 1:26:17 >> Mhm. That's really interesting. I had no idea it was it was research and not teaching.
u427 1:26:22 >> such a twisted way of what's the research for so that I can teach [laughter] good enough students. And that was really in my head. I'm not kidding. Now, I don't want to put too much of an emphasis on it because I do love research.
u428 1:26:39 I don't want to over simplify. I love the mystery. I'm deeply in love with math and science and the universe and you know, all of it all of learning in fact. I mean not just the science and math part. Just really loved being a student my whole life.
u429 1:26:57 And I just wanted a chance to do that to teach to learn. And research is a big part of that and so is writing for the public and so is doing this conversation. You know, I mean it's all to me this it's the thing I want to do with my life with my my short time. So, that's it.
u430 1:27:15 I mean, but the research has been really thrilling along the way because I've had such good students. I mean that's the other nice thing. When you work at a top-flight place, you get really brilliant creative ambitious hungry mountain climbing in some cases literally I had mountain climbers.
u431 1:27:31 My most famous student Duncan Watts really was an outdoors person who would hang by his fingertips off of death-defying cliffs. I mean he was like not quite the guy in free solo, but I mean he's a crazy rock climber. And um so I got to have adventures with people like him and you know, I had something like 25 PhD students and lots of postdocs and and brilliant collaborators my friend Renny Morollo my lifelong friend.
u432 1:28:00 I mean we've had great adventures which is what research is. But we had them together. I mean it really is about going through life with people you care about. Keep coming back to that.
u433 1:28:13 >> Yeah, I mean the the fact that you keep coming back to that speaks to something um both in terms of just what you want what you personally want mathematics to be but what I expect a lot of other people would do.
u434 1:28:22 >> Yeah.
u435 1:28:23 What should a math PhD look like now?
u436 1:28:23 >> I think the the last line of questioning I want to get at here would be on on this topic of mathematicians kind of get to decide what the field might be or if we if we if we run with that for a moment um and it it's in this tumult because some of the traditional proxies are no longer as useful as they once were.
u437 1:28:42 Um I mean you you said a very good example here on all of the other things that you can pursue while like in the service of valuing these these more human aspects of math and it's not necessarily about the significance of a theorem that was proved um you know can maybe focus on more application but is it what I'm hearing here is it's not even as much about that as it is you're just wanting to wallow in the knowledge and help impart that to others whether it's your students directly or the people reading your books.
u438 1:29:12 What do you think the What if you were you know in charge of maybe one math department or could sweep sweep a magic wand and instill the culture with a a sense of what to value what what do you think should be valued in making decisions around hiring making decisions around tenure deciding what people pursue what people what is a math PhD right just five years from now let's do student entering now like publish the PhD what is what does it actually entail?
u439 1:29:39 Uh what comes to your mind as uh as emblematic of of all of these things that we want to value?
u440 1:29:47 >> That's a very very baffling question. I I mean I have an instinctive answer and it's going to sound evasive but it's what clearly comes to my mind first. Which is um I have a very profound respect for individuality of people's taste.
u441 1:30:06 I don't I's I'm escaping really answering. I don't want to be prescriptive. I know it when I see it. If if a person is given the freedom to do math that's meaningful to them, whatever that means, and that's the part that's so nebulous. Um but if they're somehow their heart and soul and mind and maybe AI tools are all put in the most powerful way and the most incisive and penetrating and you know, offbeat.
u442 1:30:38 I just want to see that person express themself in their unique way to do something with math. That's what I want. And I want to have everyone have the freedom to do that. Now, you could say that's not a requirement at all. I know.
u443 1:30:54 That's why it's such a bad answer. But that's actually what I would want.
u444 1:30:58 >> It's actually a great answer cuz I think it's one of the reasons for optimism. In in making the case like why is this uh not only is it a fine time to go into math, I think it's arguable this is one of the best times to go into math precisely because the culture is in such an upheaval and people are deciding what even is this. Therefore, if you are an entering PhD student, you get to basically you have more you might not have infinite freedom, but you have more freedom than you ever did before.
u445 1:31:21 >> ever before.
u446 1:31:22 >> To say like I'm going to plant a flag that says this is what I think it should be. Like had I had I um decided, you know, when I was starting Three Blue and Brown, I want to do this under the guise of this being my PhD thesis in a pure math department, you know, 10 or 15 years ago, that never that just never would have worked.
u447 1:31:42 It would have required a very peculiar department to even like consider that. I don't know if I could now, but if it was a younger version of me and I was setting out doing the same thing, I actually think I could basically take the kind of stuff that I've been doing and make a case for like uh hey, we're deciding what math should be.
u448 1:31:58 >> it's a very remarkable thing, actually. I mean, you did you did create a whole genre. I mean, you made a whole life. And I can remember when we used to talk about this near the beginning and we were thinking about different ways for you to live this life that you were imagining. And you've you seem to be pulling it off.
u449 1:32:18 Thanks to YouTube, thanks to the internet, and thanks to you.
u450 1:32:22 >> I had I had various lucky things that happened such that this whole thing could exist outside of um you know, traditional funding mechanisms or academia or things like that. But as far as other things that being a mathematician can be or pushing forward human understanding, whatever that means can be.
u451 1:32:38 And maybe for some people that really is they want to stay at the forefront of knowledge and what a PhD looks like they spin off their AIs to discover like hundreds and thousands of theorems in a particular domain and they are the tastemaker to decide what is the network between these. And their thesis looks like a book that is an entirely new field and they are trying to like caress it into a sort of understanding that's compelling to others.
u452 1:33:00 But it covers a hundred times as much as what
u453 1:33:02 >> That's amazing. That is actually a vision.
u454 1:33:04 >> It it could be that.
u455 1:33:06 >> It could be that.
u456 1:33:07 >> I think one analogy I often turn to is and this is near and dear to your work also is like the origins of chaos theory where we think of that as being you could have defined the theorems of chaos theory without um computers existing, but somehow observing the phenomena required to even realize this is a theory worth uh defining required automated computation such that you could run millions or billions of computations
u457 1:33:30 Poincaré, Lorenz and the birth of chaos theory
u458 1:33:30 and
u459 1:33:30 >> It's so interesting this example because it was invented by someone without a computer.
u460 1:33:35 >> Mhm.
u461 1:33:36 >> I mean, Poincaré had this problem that it was in his head. And hardly anybody could understand what was in there. Even when he explained it, they still didn't get it. That's why it took 60 years till Lorenz could put it on the computer and now we understood what Poincaré was saying.
u462 1:33:53 But actually Poincaré saw most of it. We could see it in his writing.
u463 1:33:58 >> I guess maybe the the the case then is not so like the Poincaré might exist, but who is the Lorentz for a yet to be defined
u464 1:34:04 >> falling in the forest that nobody hears. Right? That's the thing. Poincaré was so early and so far ahead that only Birkhoff could understand him. You know, I mean those guys really plus they were doing something that was so wrong in terms of this was when relativity was happening.
u465 1:34:23 This is when quantum mechanics was being born. Nobody wanted to hear about classical mechanics. I mean that was so yesteryear. But in fact they were kind of inventing Okay, sorry. I got distracted by the history of this example.
u466 1:34:36 >> Yeah. No, no, like I love it. Getting near and dear to your your particular field. I mean so I I think the analogy I'm making is not as entirely apt then cuz what I what I would want to say is whatever it is No, no, no, it's actually good because maybe we can shape it in. I think you know, there's a thing that was unlocked by being able to do just millions of computations at a time.
u467 1:34:57 And here maybe it's um it is bringing to life what Poincaré had already discovered. But it in so far as maybe there are some other observations that not just Lorentz, but but others uh following in his footsteps made that would be much harder to make without the like
u468 1:35:13 >> Well, I mean certainly Mandelbrot and fractals. A lot of the work on fractals you really And that was the problem that Poincaré had. The thing that he discovered at the essence of chaos called the today in today's language which would be a homoclinic tangle. He even says in his writing that I I struggle to even describe what I what I'm picturing in my mind.
u469 1:35:34 And I won't attempt to draw. He says something like an infinite grid. An infinite lattice or some kind of trellis. What word does he use? He's talking about something that's like this and like this. And if you ever draw a picture, maybe you could do it on three blue one brown of the homoclinic tangle.
u470 1:35:52 It's a really nightmarish shape. And without a computer there was no hope of drawing it properly. But Poincaré was imagining it. Nobody else could understand it. And he didn't draw it. So for like I say for 60 or 70 years nobody really knew what he was talking about. But now we can draw it and we could see he pictured it perfectly.
1:35:54 · Man with headphones speaking at podcast microphone · A still from a video shows a balding man with round glasses and over-ear headphones, wearing an olive-green button shirt. He is speaking and gesturing with his hands pressed together in front of him, next to a studio microphone on a boom arm. Behind him is a backdrop of a hilltop campus with clock and church towers, brick buildings, and a lake with hills in the distance.
u471 1:36:11 >> [laughter]
u472 1:36:12 >> But Mandelbrot, you know, when Mandelbrot can draw fractals with his technology at IBM um that opened up a whole universe starting in 1985. Then people had personal computers. Everybody was drawing the Mandelbrot set on their screen saver. I mean, the visualization should not be underestimated.
u473 1:36:32 >> So broadly, whatever whatever people want the math PhD to be. There's not infinite freedom, but there's much more freedom than there ever has been.
u474 1:36:39 >> There actually is. Yep. If if if departments see it that way. I mean, we're all wrestling with this. I mean, every department is asking itself what should be our requirements. What what does count as a PhD in this era? And we haven't really seriously had those discussions yet, but people are aware that we need to.
u475 1:37:00 >> And sometimes people just need good examples to even look to. And I think as a you know, if I was a second year grad student or something, I might be thinking one of the things I can do right now is at the very least provide an example that can be an element of discussion happening here rather than waiting for others to speculate to tell me what to do.
u476 1:37:18 I'm just going to do a thing and then let them you know, maybe they reject it. But you can guide a conversation through creation in a way that is often underappreciated, I think. Just the person to actually make a thing or do something new ends up just guiding how people even talk
u477 1:37:34 >> Aha. Just put the facts on the ground. We don't have to talk about it. Look at this.
u478 1:37:39 What's next for Steve
u479 1:37:39 >> Yeah, something like that. So as a very last question um all of this discussion around math and a certain spotlight from the broader world turning to it and whether you know, people saying like, "Oh, isn't math solved?" And and things like that. Misconception or not.
u480 1:37:55 How does this influence the popularizations that you'll do? You know, the the you know, you just wrote a new book or or if you're giving talks to the public or if you're on podcasts, does any of that surrounding conversation recolor anything that you're writing or that you're saying?
u481 1:38:11 >> [snorts]
u482 1:38:11 >> Hmm. I don't know why I can't feel I feel I find it hard to engage with that question. Hmm. Part of it, let me give the reason maybe that is doesn't want to be named, which is that the labor of writing this book and all books before it has been so problematic for my family.
1:38:36 · Split-screen video still of two podcast conversation participants · A split-screen video call frame shows two men recording with microphones. On the left, a man with glasses and over-ear headphones smiles with his hands clasped, in front of a backdrop of a clock tower and a lake. On the right, a man in a gray-blue t-shirt laughs with his head turned down, in front of a bookshelf and a door.
u483 1:38:37 >> [laughter]
u484 1:38:38 >> And the time it takes and the um lack of presence I show in my house. I mean, I could be sitting in the room, but I'm not there.
u485 1:38:49 >> I shouldn't laugh. It's just too relatable, I think.
u486 1:38:52 >> Yeah, okay. Oh, wow. All right, that's why you're laughing. Good. Well, it's made me a bad partner. Um my wife has enabled me to do this for so long. Uh it's now getting to be I mean, I started doing math popularization in earnest.
u487 1:39:11 I say let's say around the year 2000. So, it's now whatever that is. It's now 26 years. And that's a big fraction of my life up till this point. And I don't think I'm going to do it anymore.
u488 1:39:27 Not much. I mean, I have other things in my life that I care about more. I really want to be with Carol. I want to I live in the youth of my old age, to travel with her and do crosswords with with and watch dumb TV and you know, just hold hands and go out and what you know what we did?
u489 1:39:46 Actually, it does have a connection to AI. We're walking in the Botanic Gardens at Cornell, which has marvelous specimens of every tree and shrub and flower you could imagine. And you could take a photograph of the you know, uh a Norway spruce and I don't know a spruce from a hemlock, but chat GPT does.
u490 1:40:10 >> [laughter]
u491 1:40:11 >> And I could get a guide, but I I like to just take a picture of the tree that's right in front of me and do a little quiz to the machine and tease it and it, you know, playfully joke around with it. But Carol's right there. It's not just me and the machine. We're having fun with the machine and we're looking at nature and we're outdoors and we're learning stuff and it's fun and we're doing it together.
u492 1:40:32 So,
u493 1:40:33 >> [laughter]
u494 1:40:34 >> it's not about math popularization, but it's about me being a perpetual student. I I think that's what's next.
u495 1:40:41 >> Well, you you mentioned everything circles back to the meaning of life. I feel like that that's a perfect note to end it on that uh I would actually brought some tears to my eyes. Oh, good. That's just incredibly sweet [laughter] as the
u496 1:40:54 >> That's what That's what matters to me the most right now. So, uh yeah, I've written some other books. You could read those too if you want more.
u497 1:41:02 >> I think as we all um yeah, as as I'm sitting in a drought of wishing that I had more of your stuff to read, a part of me will smile knowing the reason for it. And so
u498 1:41:12 >> That is the reason. Thank you. But yeah, I I um I've sort of promised to not write anymore because it's too enervating. It takes too much do I mean, it's both enervating and energizing. I'm as happy as I can be when I'm creating that stuff.
u499 1:41:29 And in this last book, so Big Math is written with my really dear friend Alex Townsend. I mean, I have had such a good time. This is my first book with a co-author. He's much younger than me. He could be my son. He's 28 years younger.
u500 1:41:46 And um we had the best time thinking about this subject together for the past 2 years. That's a whole new deal. I never wrote a book with someone else. I've done papers with other people and that's been fun. But to spend that much time that intensely with with one person for years.
u501 1:42:04 Carol's not that happy about it. Because it's not her. It's me and Alex. And she's seeing me having so much fun making this thing with him, figuring stuff out. And it seems it never ends. There's writing it, there's publicizing it, there's talking about it.
u502 1:42:22 >> Tell you what, I will [laughter] Let's end it here, Stephen. This has been an utter delight. I can happily chat with you at any time.
u503 1:42:28 >> I'm sorry. I didn't mean that to be a a shut down, but okay.
u504 1:42:31 >> On the contrary, nothing would bring me more joy than to let you go into the botanical gardens with Carol now and not keep you on the mic for 1 second more. So, thank you so much for taking the time. This was wonderful.
u505 1:42:40 >> man, Grant.

The original

Around it

topic

Artificial intelligence

The technology of building computational systems that perform tasks associated with intelligence.

Topic2 sources1 tensionNo primary research yet

topic

Mathematical understanding

Comprehension of mathematical ideas, their relationships, and the reasons that results hold.

Topic2 sources1 tensionNo primary research yet

topic

Mathematics

The study of abstract structures, quantities, relationships, and patterns through rigorous reasoning.

Topic2 sources1 tensionNo primary research yet

topic

Mathematics education

Mathematics education concerns the teaching and learning of mathematics.

Topic1 sourceSingle source

topic

Steven Strogatz

Steven Strogatz is an American mathematician, teacher, and author known for work on nonlinear dynamics and mathematical communication.

Topic1 source1 tensionSingle source

topic

Academic reward system

An academic reward system allocates recognition, appointments, promotion, and credentials according to a scholarly community's standards.

Topic1 sourceSingle source

topic

Applied mathematics

Applied mathematics develops and uses mathematical methods to address problems in science, engineering, and other domains.

Topic1 sourceSingle source

topic

Doctoral education

Doctoral education is advanced academic training leading to a doctorate through specialized study and scholarly work.

Topic1 sourceSingle source

This text is written with AI.

Placed in: Now

What it says

Pulled apart by

The source, passage by passage

Around it