Nibbling on the Infinite Cheese of Knowledge

The American mathematician Barry Charles Mazur is among the pantheon of greats who has transitioned between seemingly disparate areas of mathematics with deceptive ease, while at the same time leaving an indelible mark in each one of these. As the citation of the Chern medal which Mazur received in 2022 reads, “His numerous fundamental contributions place him squarely within the ranks of the greatest mathematicians of the 20th century’’.

Attracted to mathematics by virtue of it being the “philosophy of electronics’’ as Mazur recalls, he was admitted to a PhD program in mathematics at Princeton University despite not having any prerequisite degree requirements for the same. From his initial contributions to the area of geometric topology, Mazur moved onto algebraic geometry and then onto arithmetic number theory where his unique perspective yielded profound results that were the cornerstones in Andrew Wiles’ proof of Fermat’s last theorem.

As a mathematics professor who has also taught philosophy courses at Harvard, Mazur’s view of the subject is broad and unfettered and stresses on the importance of imagination even in a field as rigorous as mathematics. In his own words “When it comes to mathematical inspiration, the imagination could be considered the ghost in the rational machine of proof-making.’’¹

One has to only read Mazur’s book Imagining Numbers: (particularly the square root of minus fifteen) to recognize that he is equally gifted at exposition, with an ability to communicate complex mathematical ideas to students at all levels. He also remains a pivotal figure in the mathematics community having guided a long list of students who have gone on to become distinguished mathematicians themselves.

At 87, Mazur remains as active as ever. In this long insightful conversation with Vijay Patankar at his home that reflects his philosophical outlook on life and learning, he talks about his Jewish upbringing in war time New York and the impressions it left on him, his atypical approach to teaching, the role of computers in the future of mathematics, his encounters with luminaries such as Jean-Pierre Serre and Robert Oppenheimer, as also the constant rapport with his wife Gretchen whom he considers integral to his success.

On behalf of the team Bhāvanā, I want to thank you for agreeing to talk to us. While researching for this interview I came across this absorbing documentary² that Oliver Ralfe did with you in which you talk about your childhood. I would like to begin with the same. Could you tell us a bit about your early childhood?

An early photo of Barry Mazur. Barry Mazur

BM: Well, I spent most of my childhood in the Bronx, New York. I was largely brought up by my maternal grandmother who initiated me into the Orthodox Jewish tradition under the tutelage of my great uncles (who were ultra-Orthodox). I went to yeshiva³ when I was a kid. I started school when I was about four years old. I loved it. That time (which included the full extent of the Second World War) was intense. War orphans from Europe would, every so often, appear in our classroom as new students. I am not sure how much money the yeshiva had; it was a ramshackle place. Many of the teachers, all rabbis, were, of course, deeply affected by the Holocaust. For an educational experience, I was perfectly happy. Looking back, I am amazed. I have no idea how they could have done such a good job with essentially everything else going on at the same time.

What was taught at that school?

BM: I would come in at six in the morning; there would be prayers before breakfast, and then we’d have breakfast; and then we had religious studies, the Bible, and in later grades, a good deal of the Talmud until lunch, which was at about 12 pm or 12:30 pm, followed by a recess of about an hour, followed by secular studies: English, science, etc., from 2:30 pm to 4:30 pm, and that was it – just two hours of non-religious studies. The whole thing somehow seemed to work, even though, as I said, a lot of the teachers were emotionally very affected by the catastrophe in Europe.

They managed to remain stable, perhaps.

BM: Oh, they were not terribly stable. They were expressive. For example, there was a teacher, his name was Rabbi Moseson, who the kids always called Mosey, or Black Mosey, because whenever he got annoyed, he would shout “Schwartz Yahr”, which means “black year”– I don’t know what it exactly meant to Mosey; possibly: by our misbehavior we had cast a black year on him, or perhaps: we were so disobeying or inattentive, he would cast a black year on us. And at times he would just shout “Anti-semit”. Anti-semit meaning, perhaps, that our misbehavior was staining our religion—or, perhaps, meaning nothing, a kind of Tourette’s⁴ response. So, that was Mosey. At the same time, of all the teachers I’ve ever had, he was one of the most impactful. I have mentioned this incident at other places because it is so important to me: Once, he was so exasperated that he just plopped into his chair and said, “Look, what we are faced with is an infinite cheese of knowledge. And we’re all, all of us, we’re all little mice; you are, I am, and we together are nibbling on it. We will never devour it. But it is the effort of nibbling on the cheese that makes what we are doing meaningful and worthwhile.’’

There is something that I had read and I don’t know if you have seen this. This is in an interview of Gian-Carlo Rota with Science magazine.⁵ Science asks him, “Why do some scientists remain active while others burn out?’’ And Rota says, “I have a one-word answer. It is the word `Kultur’. A broad cultural background, the learning at an early age in the family of the values of things intellectual, is the main factor that keeps people from burning out. That’s the reason many first-rate intellectuals come from Jewish or Chinese backgrounds, where they were exposed to intellectual values at an early age. The Talmud and the Tao are good trainers of scientific minds.’’

BM: That’s very good. That is brilliant. I didn’t know that Rota said that.

How does exposure to a religious environment—whether conservative or not—shape one’s values, resilience, and worldview, sometimes even unintentionally, through observation and presence alone? I am asking this in the context of the impact of this teacher at the yeshiva that you mentioned, who was obviously an intense person.

BM: Yeah, Mosey’s collegiality at the end. His basic, or his fundamental, collegiality; thought of himself as on the same level as we are, because everyone is on that level; and it’s not so much result-oriented, because we’re only going to nibble. We’re not going to consume that cheese, but nibbling is enough. The act of nibbling congenially is more important than consuming everything. This has very much a Talmud feel: one rabbi will say X, where this X is a tiny thing; then the next rabbi will say Y, which doesn’t really refer to the X much, but you can imagine it being connected to it somehow. So, there are little things which join, but they don’t merge. There’s a kind of way in which two or many thoughts join to produce a tenor of thought without necessarily having a kind of Hegelian synthesis.⁶

We are all little mice faced with an infinite cheese of knowledge. We are just nibbling on it, but will never devour it

Like a sheaf of thoughts.

BM: A sheaf of thoughts (laughs). There you are. Without bothering to find the cross-section (laughter).

How do you comprehend our universe? That is, how does one go about understanding it? Could we for example say that this universe, or this entire thing is God?⁷ Is that one good way of looking at it?

BM: That’s the Spinoza aspect.⁸ On that topic, there’s a book by David Mumford, called Numbers and the World: Essays on Math and Beyond. It’s not a well-known book. I got a copy, was very impressed by it, and especially his chapter on Spinoza, and I emailed him saying, “Hey, this is a good book.’’ And he said, “Well, nobody has reviewed it. Why don’t you review it?’’ I’m writing a review. It’s a very, very rough draft, and I haven’t finished all of my review which has a little section giving a sense of what Mumford thinks what Spinoza thinks God is like.

It appears to me that you have greatly benefited from such thoughts, and in the process, you have also gained spiritual strength. Do you think the intense religious and scholastic environment at your home and yeshiva helped you? Do you think these can or should be part of the upbringing of future generations?

BM: It certainly helped me. Now that you ask me, first, it never occurred to me that we didn’t impose or put any religious training on our son. On the other hand, he became a great scholar of Gnosticism and early Christianity. He wrote about Neoplatonism in particular, which has a religious intensity. Our son’s great works, I mean these absolutely beautiful and important books he’s written, are about that. So whether one should or shouldn’t impose religious training on one’s children, I just don’t know. In his case it didn’t seem to matter. I remember at one point, my son said that he was inspired by what he was reading in Plotinus.⁹

Getting back to your early days, what were some of the other things that you enjoyed learning about at school. In the documentary by Ralfe, you talk about the philosophy of electronics.

BM: Well, if it was anything, it would be electronics. I mean, there was something basic that I didn’t understand about electronics. There’s something called standing waves.¹⁰ Imagine that I have a pair of wires from this couch to that chair; and here, on the couch, there is a power source with alternating current that is led by that pair of wires to a light bulb by the chair. The power source produces light. But try to understand how the energy of the power source moves to the lightbulb, noting that the current is alternating, the electrons are moving back and forth, back and forth. If the distance is just right, you get what are called standing waves. There may be distances between the couch and the chair where the voltage between pairs of points that are at that distance will vanish. But you could put a wire between those two at those pairs of points, and the light bulb would still light. An older friend of mine told me this. “That”, I thought, I’ve got to understand. How is it that somehow energy is moving, and you can void it at one point in its journey and it still moves unperturbed past that voided point? I thought of this as more and more of a paradox. Also, electromagnetic waves—the `receptacle’ of the energy that is transferring from radio transmitter to radio receiver—I believed, was just a gimmick that Maxwell dreamt up. (I wrote about this when I was at MIT.) A gimmick to conserve the law of conservation of energy, because the energy would have just disappeared if there was nothing, no intermediary between transmitter and receiver. So those are things that got me going. That’s why I said in the other interview, I felt myself as a philosopher of electronics, I wanted to get to the heart of that.

electromagnetic fields are magic

What was the most fascinating part about it?

BM: The paradox. You want to understand something, and the minute you get a sense of understanding, you find that there’s some fundamental paradox in it. You are there, you’re ready, you’re trying to understand, but there is that paradox. It is a wonder, even magic if you want to call it that. I mean, electromagnetic fields are magic.

Then you moved to college – you went to MIT, to pursue your wish to understand physics, or electromagnetism to be more precise. Soon thereafter, you realized that the philosophy of electronics was mostly mathematics and you turned to maths. Do you recall any teachers from those times, friendships, or other experiences of lasting impact?

BM: Yeah, I had a few very close friends in high school. They were largely science-interested people. One of them was George Zborowski. The two of us had the idea that we wanted to “understand America.”

I really did want to understand it. There’s a classic novel called USA by John Dos Passos. The main character in the novel wanted—as we did—to understand America. He would go off and take jobs everywhere. He would take a job for a month or two and become totally absorbed in the surroundings and the nature of the job and the nature of the people who do that sort of job. The book was peppered with `newsreels’, inviting the reader to “understand America’’ via the daily news as well as via personal interactions.

The character in the novel would put books under his pillow (e.g., Henry George’s Progress and Poverty). And we thought, well, that’s what we want to do too, to take jobs everywhere. We couldn’t do this in school time, but we could do it in the summers. I went and took a job on a farm, on a dairy farm. I was a door-to-door salesman. I worked on a pier as a stevedore,¹¹ a longshoreman, i.e., someone who carries loaded dollies from trucks on the pier to boats, to ships. We, both of us, also thought of ourselves as `writers’. I don’t think either of us wrote a word, but we were still `writers’ trying to do this. I did this throughout my high school.

Ten years later I met up with George again. He had continued to take all sorts of jobs. In Cambridge he worked in a biological lab at MIT. After a few years he continued his journey—going off to the South Pole on an expedition to discover the origin of life.

I lost contact with him again. Every year on my birthday I go on the internet, looking for some trace of him. The only thing that appears on the internet is his picture in the high school yearbook. (I’ll try again soon.) He shaped my way of thinking about a lot of things.

This is so touching! George is a kind of truth-seeker in a sense. Very free-spirited, and yet very strongly tied to values.

BM: Oh, absolutely. Yeah, that is true, that’s well expressed.

Is it possible to be a good philosopher-mathematician and yet have time for family and other things? How do you manage that?

Barry Mazur in Brookline Woods. Barry Mazur

BM: Yeah, that is a good question. I think one way for me to do that is by not doing any project unless Gretchen says I should do it.

If anybody asks me anything, I send the request to her and she decides whether I would have the time to do that away from my family. And it’s her decision and if she writes “No”, it’s a “No”; I mean I don’t have a second thought about it. And the “Yes” is rare. That’s how I organize my program and then, you know, that leaves time for activities, but of course, one always wants more activities.

I believe Gretchen has a science background too – a PhD in cellular and developmental biology from Harvard, and is now a fiction writer. Have you two collaborated formally on any of your projects?

BM: No. But as such, everything I write, I pass through her and she will give me comments, and she does the same. But we have never written a joint collaborative thing.

In your article “The Consolation of Math in Plague Time’’,¹² you write: “Happily, in good times, there is such exhilaration in doing mathematics—and in bad times: consolation.’’ And you talk about the intention to learn a bit about the mechanisms of forecasting data science and handling data. You also talk about something called gems of constancy in that article. What are these gems of constancy?

Barry Mazur, Gretchen and Bernard Saint-Donat in Vezelay, France in 1970. Barry Mazur

BM: These are fine details of life that sparkle, and they will always sparkle. Now, what could they consist of? Well, they might consist of playing the piano. It’s on that level. Here’s an example. When I play the piano, though it depends on what I’m playing, I very often find myself imagining playing with my father who was a fine violinist. He was a professional at one point. And we played one thing together, nothing much, and not much of it, but that resonates. The one tiny thing that we played together, that’s a gem of constancy. So, that is an example.

So via these and your work, you were able to continue to do maths, or keep thinking without so much of a fixation. And there too, I’m sure, Gretchen’s presence and her filtering, if you like, helped.

BM: Oh, filtering. Yeah, she does it. Also, every once in a while, if I haven’t played the piano in a while, she says, “what about some piano?” It doesn’t matter what I am doing. I could be Zooming with someone and I’ll say, “Okay, I’ve got to go. I need to do some piano.’’ Because I guess she feels that these are gems of constancy, if you want to put it this way.

The other thing is we write… It depends on the time. In the summers, we very often, after dinner, write poems. The poems could be just poem sketches. It’s not as if one spends all night doing it. But we put our minds to that for a few minutes, maybe for half an hour. And then, when she has a birthday, I send her a hundred of them.

That’s wonderful. That is so nice and so interesting! Do you have any advice for young people who wish to pursue a career in mathematics but aren’t yet able to make up their minds, or convince their parents?

BM: For young people who are not being pressured by their parents, here’s a thought. Pay attention to the questions that arise in you. Not questions asked of you, but arise in you, that you’re curious about, but have a kind of mathematical perspective. That’s a great sign. And don’t ask: “Do I have talent or not?” It hardly makes any sense. Rather ask: “Do I have a question? Does it occur to me naturally from time to time to ask a question out of my pure curiosity? Am I developing my own mathematical sensibility?” If so, that’s a pretty good sign.

don’t ask: “Do I have talent or not?” Rather ask: “Do I have a question, out of my pure curiosity?” That’s a pretty good sign

Many students struggle with maths and are therefore put off by it. What is your advice to students who are studying, or planning to study the subject?

BM: Take something that you’re working on. It doesn’t have to be deep. It doesn’t have to be anything that would impress anyone. It can be as simple as possible. And do two things. First, enjoy that you’re doing it and then appreciate that, once you’ve done it. Appreciate yourself, in the sense that you now have something in your understanding that will help later, will radiate. And certainly, don’t worry about the people who seem to have gone much further than you. Just appreciate what you’ve done fully, noticing how it’s going to lead you to do some next thing.

Does your learning happen more by reading and by talking to colleagues, or as the need arises, possibly while solving a problem itself? Could you please reflect on the processes of learning?

BM: Learning things that seem important, or surprising, whether or not they’re connected to what I already know, or what I’m working on, is always a delight for me, as I’m sure such an activity is a delight for most people. In times when I’m in search of an answer to a question or a conglomerate of questions, my first impulse is just to learn from examples. Often from just focusing on a single example. Enlightening examples are triggers for further exploration—and the simpler the example the better. Also I try to isolate the `first case that I don’t understand’ and keep it in mind. I often recommend this to students. But I love to read anything that’s relevant. And talking to people about what I’m working on—or about mathematics in general—is always an education.

don’t worry about the people who seem to have gone much further than you. Just appreciate what you’ve done fully.

The perception about mathematicians is that, generally, they are direct and that they find it difficult to make untrue statements. First question: Do you have any comments on that? Second question: Do you find it difficult to deal with people who are obtuse?

BM: Isn’t it good to find some difficulty when you make an untrue statement? And if by “obtuse” you mean slow to learn things, well, since my job as a teacher is to teach, I don’t worry at all about finding it difficult when I encounter such a student… it’s just, perfectly happily, part of the job.

Is it lonely being a mathematician, given that only a handful of people truly understand your work?

BM: I don’t think so. But that presents itself as a serious question. What does it mean to be lonely? Does it mean: I want to have a broad broadcast of everything I think? Or does it mean that I want to have a full conversation with however many people are interested about everything I think? Now, to have a conversation, you need a certain number of people in the room. But if you have too many, it’s not going to be a conversation. If the word lonely comes in, it has to do with the rest of your life and not with what you’re doing in mathematics. I mean, I can easily imagine someone saying, “Oh my God, here I am, doing mathematics, nobody understands me, or nobody reads me or listens to me.’’ That would be terrible, no matter what you’re doing. You don’t have to be doing mathematics for that to be a sorrow. But even though there’s a limited range of people who can absorb the mathematics you’re doing, it’s enough for a conversation: a sufficient critical mass.

Sufficient critical mass. And often it is a dialogue with oneself too, I suppose.

BM: Yeah, often it’s a dialogue with oneself, exactly.

Many consider maths to be the language of the universe, through which we understand everything. Well, most things. What are your thoughts on this?

On a picnic with Jean-Marc Fontaine and Kenneth Ribet. Barry Mazur

BM: Well, I would remove the word everything. I mean, there are modes of understanding; mathematics is one of those many modes. Some modes of understanding require feeling and understanding at the same time. Mathematics is a bit of that too. Understanding needn’t stand alone. It’s a part of connecting with aspects of the world. And these connections can be understanding, can be feeling, can be merely enjoying the beauty of things. Well, my feeling is that Mathematics is not the only mode of understanding.

Any views on the works of Ramanujan, Chowla, and Harish-Chandra?

BM: Oh well, Ramanujan is simply central to everyone’s work in number theory. It would be an interesting exercise to try to find a number theorist whose work has no connection, even if admittedly tiny, with Ramanujan and his thoughts and his attitude towards mathematics.

Chowla, I am certainly influenced by his work, his work with Selberg, yes. Harish-Chandra, equally so. With Harish-Chandra the connection is an interesting one, that dates back to when Gretchen was 18, and the two of us were in Paris for the year, so that is 62 years ago.

My younger brother, Joseph, had been working part-time as an architect, The two of us would constantly discuss mathematics. He became more and more interested in the subjects we were discussing—and at one point Joe said that he wanted to become a mathematician.

So I said, “Hey, we are going to Paris in a few months, why don’t you come to Paris, sit in on mathematics classes, so you will learn French and mathematics at the same time?’’ Now that is something I would never tell any of my students nowadays. Anyway, Joe came along, he learned French, he learned mathematics and he went to a course of Harish-Chandra, who was lecturing there, and I think they became friends; or, if not friends, at least sort of more than just a student-teacher acquaintance. So, if there is a connection between Harish-Chandra and me, it would be through my brother.

So is Joseph a mathematician too?

BM: Oh yeah, he is a mathematician. He got his PhD from MIT on a topic in algebraic geometry called Henselization. Then he taught at Marlboro College for his whole career. He was a great teacher, he was the inspiration of generations of mathematicians. And now he writes wonderful books on mathematics. And other things. Recently he has changed his direction by writing a column called “Understanding War” in one of the journals called The World Financial Review.¹³

And again, perhaps he was also influenced by the same teachers, Mosey and others, at your yeshiva?

BM: He went to the yeshiva, but somehow he did not take to it much. And he did not go all the way through the primary grades of the yeshiva.

Any other memories or special memories from visits to other places, especially outside the US? For example, you have for long been a frequent visitor to IHES.¹⁴

BM: Right. We would visit every summer. And we would stay there one semester out of four, over decades. We went there until our son was about ten. We figured we shouldn’t be taking him out of his school in Cambridge that much.

And there was Grothendieck, who was wonderful. I wrote a very short piece about my personal feelings for Grothendieck. It’s called “Thinking about Grothendieck”.¹⁵

Mathematics is not the only mode of understanding.

Did you form any lasting friendships in Paris and at the IHES, during your visits there?

BM: Well, my close friend from topology days is Valentin Poénaru. I call him Po. He is a great mathematician. I knew him very early in my career, it could have been 1959 or 1960 when he was still in Romania. We were working on very similar things. I went to a conference in Prague, I think, where I had a conversation with a friend of his, who told me that Po wanted to leave Romania where he was persona non grata. Nevertheless, when Po was invited to the International Congress in Sweden he was allowed to attend. And he came. Gretchen and I also came to the Congress with the intention of meeting Po, and bringing him back to Paris with us. When Romanian mathematicians were allowed to attend International Congresses at that time, they came supervised by guards who were in charge of them, to keep them from wandering off, and to make sure that they returned to Romania. He had come with suitcases of books and other things. He managed to avoid his guards and went underground for a while. We took his suitcases to the airport when we left the conference. There, I saw the mathematician Jean Dieudonné checking in at a counter different from where we were. I said to Gretchen, “Hey, that’s Dieudonné!’’ A few minutes later, Dieudonné saw us, came to us, and pointed to a woman and said, “Hey, that’s Ingrid Bergman!”

Oh, that’s such an interesting and amusing anecdote!

BM: Happily, we made it to Paris. Po lived with us for a while and was connected to the IHES as we were. Then he came to the United States and taught at Northeastern for a while, and later went back to Paris.

I suppose you had already finished your thesis at Princeton in topology and were employed at Harvard, and visiting IHES. Serre, who had also started his career in topology and moved to number theory, was in Paris. Any memories about Serre that you would like to share?

BM: Serre is an amazing figure. He is wonderful. Powerful. I mean, I think in some way, it is not so much that he personally inspired me—he may have done that—but just his being; his lectures and his writings inspire all mathematicians, I think. Not just me.

Here is one thing that I was told he did, and that was a kind of revelation to me. He had written a draft of a mathematical paper and put it in his drawer. Weeks later, without looking at that draft, he made a new draft. He did this a few times, and it was only when his newest draft was, word-for-word, identical to the previous one that he felt it ready to be published. This shows such intensity and such striving for a level of excellence of exposition. It is remarkable.

In an interview with Anthony Bonato,¹⁶ you speak about an anecdote with Oppenheimer when you were a postdoc at the Institute for Advanced Study, Princeton. He drove you to J.W. Alexander’s house.

BM: Yes, Oppenheimer did. It was very moving that he would do this. I mean, I was young; if I was invited to J.W. Alexander’s house I’d happily find a way of getting there. But Oppenheimer was Openheimer¹⁷ and here he’s taking time off from his absolutely demanding schedule to drive me!

How did it come about?

BM: He just said he thought it’d be good if I had a conversation with J.W. Alexander.

It was his initiation. Remarkable!

BM: Yeah, he was remarkable. And he was very happy just sitting there, as Alexander and I talked.

When did you start seeing yourself as more of a philosopher, or thinker, than a mathematician?

BM: I don’t know whether I think of myself as a philosopher, but I do think that mathematics has an aura which is way beyond what we label as mathematics. So in that sense, I am working not in a specific field but in mathematics that includes the aura, and I think all mathematicians have a bit of that kind of philosophy in them. Whenever someone makes, let us say, a striking definition, there is a philosophical turn opening up the subject. For example, the concept of Dedekind cuts was quite a novel way of thinking of real numbers. Richard Dedekind in his 1858 essay “Continuity and Irrational Numbers” was himself very appreciative of the fact that what he has put among the rational numbers, what he called “Punktindividuen” (“point individuals”), are far more numerous, more plentiful than the original rational numbers. Isn’t that interesting that he thought of it as a broadening of a perspective of a certain sort?

Very much the way Saunders Mac Lane and [Samuel] Eilenberg thought of their own attitude towards mathematics in general as a broadening. We normally say abstraction. There’s a difference between abstraction and a broadening that changes the perspective of things. So wouldn’t you call that a bit of philosophy right there?

Mathematics has an aura which is way beyond what we label as mathematics.

Yes, certainly. What role do naiveté, fearlessness, determination, and skill play in doing mathematics? I will explain why I am using the word naiveté: In the Biographical memoirs on Harish-Chandra by Langlands,¹⁸ he talks about knowledge hindering creativity or imagination. So perhaps the element of naiveté, of fearlessness, is playing a role?

BM: That is a broad order. I can say something here. When I was a graduate student at Princeton, there were a few of us, a tiny group, who, instead of doing reasonable mathematical work, were focused on point set topology which even then was slightly out of fashion; but we were very much into the classical Polish point-set topology fervor, and our main joy was to produce counterexamples. Curiously, every one of our counterexamples somehow revolved around the unit interval. One way or another, when we finally found yet another counterexample, or found that we couldn’t deal with some counterexample, we would delight ourselves by saying, “we don’t even know the unit interval”(laughs). That was the refrain: “we don’t even know the elementary unit interval.”

It reminds me of what we did in yeshiva: every time we finished reading certain specific portions of the Bible, we would shout, “Chazak, chazak v’nizchazek!”, which means “Be strong, be strong and may we be strengthened”, and we’d immediately begin reading the first sentence of that portion of the Bible again. So the shout “we don’t even know the unit interval” has the same feeling. You think you’re done but you’re not done and there is more that is yet to come.

The cyclicity and the nibbling at the infinite cheese.

BM: Yeah, yeah, that’s right.

How about the role of luck in maths or life?

BM: It just occurred to me at this moment that there is a difference in mood in saying luck or saying gratitude. In other words, something happens that maybe shouldn’t have happened or perhaps wouldn’t have been predicted to happen but happens happily because it’s a good thing. Then you could say two things. Boy, am I lucky? Or boy, am I grateful? So they’re different. Of course, one is a salute to the randomness of things. And the other has a slight tinge of the theological, of the transcendental. It’s interesting. Never occurred to me. To go all the way, one could be a move towards an appreciation of God. And the other could be a move towards the comprehension and appreciation of randomness.

You have many collaborators. I counted 65. Mathematics is generally considered a solitary activity. While that is so, and indeed, you have authored quite a few, very long solitary articles, it is also amply clear that you enjoy collaboration. Would you like to make any comments about working with others?

BM: I think it is so important to have conversations with others. No matter how broad you think your viewpoint is, it is not all the way. The minute you have someone else in the room, it produces something that is broader, more interesting maybe, and it is an education. It is an education for each of you, if you have a real collaboration. One of the things I did, and I have been doing, in my mathematics classes, is I have the students do presentations, but the requirement is that they can’t do it alone. They have to do it with one other person—making a team—and the two students have to consult with one another. It’s a conversation and in the presentation, one student can do half of it and the other one the other half, but how the thing actually gets presented is up to them; the collaboration has to be part of the project. That has two advantages. The first is the real one, which is they are learning what someone else thinks and how that may shape how they think. And the other is, of course, if you are going to have a presentation every hour, you need as many hours as there are students. That may be too many. But that is a minor detail. The point is, it gets them going. It gives them a real community.

No matter how broad your viewpoint is, the minute you have someone else in the room, it produces something that is broader, and it is an education

On the mathematics genealogy webpage¹⁹ – you have 59 students and 347 descendants! That is a pretty long list of stellar students and grand-students! Would you like to say a few words about any of your students, for example, Ofer Gabber?

BM: Well, about Gabber, he is so different. He was my youngest graduate student who began at age 16. He lived with us for about the first six months at least, perhaps a bit more because we brought him to the IHES. He is still there.

Do you give or suggest projects and questions to your students or do you expect them to find on their own? Do you change your strategy as per the student’s personality and skill set?

BM: I want them to find them themselves.

Depending on the student, yeah. I had [Vladimir] Voevodsky, who came from Russia, as a student. I did not tell him anything at all. Often, I would say to students that the main thing they should be focusing on is to develop their sense of questioning. It’s the art of questioning; I don’t think there’s any textbook that has that. There are some hints, like in George Polya’s Mathematics and Plausible Reasoning.

If you read that book, you get a sense, you get a good sense, of what it might mean to develop your instincts and your initiatives for questioning. I wrote a paper somewhere called “Questioning Answers”,²⁰ where I said you should be doing this, no matter what level of mathematics you’re at. I don’t think it’s a broad educational program anywhere in mathematics, but it should be mentioned and emphasized.

Writing diary with granddaughter Naia, Westport 2022. Barry Mazur

Here’s a neat example of what can be done in early math education—this happened in a (zoom) 3rd grade math lesson. During the pandemic, my granddaughter Naia did most of her classes here in this room by Zoom. I would be working beside her doing my mathematics, often impressed by how marvelous her classes were. It’s just a public school, not a private school. One of the things they did in their covid-zoom-days program was to reserve a few minutes before class, where on the main screen were animals doing things, but on the side of the screen was a chat-column the students could control. If it was, let’s say, the

n

-th day of the COVID business of virtual schooling, they were asked to produce a question with that number

n

as an answer. Now, if it were

101

, no kid would write

100 + 1 =101

. There would always be something like,

90

minus

10

divided by

8

times

10

plus

1

101

, and so on. The task was to express

101

or arrive at

101

by using different numbers and various operational jugglery between them. It was fun to see the students doing it, and as they did it, it would appear on the screen, one after the other, one after the other. It was just an amazing assortment of arithmetic questions with the answer

101

. That was such a stroke of genius on the part of the teacher of this class.

That’s wonderful. Have you had the opportunity to collaborate with non-mathematicians too?

BM: Non-mathematicians? Oh, loads. I don’t know whether I mentioned this earlier but I’ve been giving courses in the philosophy department. Initially, I gave a course in the Law School with a professor of law, Noah Feldman—he’s terrific. The title of the course: “The Nature of Evidence”.

The theme of that course was that evidence is an important word for many different subjects, and it means such different things, in those different subjects.

What does it mean for people in a given field to come to agreement that the evidence gathered is sufficiently conclusive to declare that they have achieved a specific result? To get a sense of its larger profile we invited guest speakers from various fields. Each of these people would give a reading list for our students, and we would have a session of the course where we would talk about that reading. Then the next session of the course would be a lecture by that guest speaker along with a seminar discussion. The following session would be a post-mortem about the reading, the discussion, and the lecture. So, it would be a triple for each subject. We did it over the semester for quite a number of subjects. The next to last session was mathematics—that was the session that I ran. The last one was law, run by Noah Feldman. It worked very well.

The main thing students should be focusing on is their sense of questioning

That was the first such course I gave—centering on the broad meaning of a single concept. There were many similar courses afterward: a number of history of mathematics courses, maybe a decade of them which I co-taught with various people.

And then came the philosophy department courses taught by the three of us: Amartya Sen, whom you must know, Eric Maskin, who is a mathematician-economist, and me. We would focus our course on some single word. For a student to be accepted in our course, the interested student would have to send us three things: their CV, a reason why they want to take the course, and what they would bring to the course from their background, their interest, and their experience. We would shape the course based on the background and interests of the students we admitted. So if there were more mathematicians there, the course would be more mathematical.

Barry Mazur at San Simeon perhaps 2001 Barry Mazur

The `word’ for the first one of these courses was Models. For example, mathematical models, economic models, medical models, biology-related models, and so on. Then the second was Utility. Utility and all the graphs of utility and the St. Petersburg paradox. Loads of paradoxes, in fact, about utility—and the variants of the notion utility in other contexts as well. Then the third course was Axiomatic reasoning. The fourth was, Subjectivity/Objectivity. The fifth was Truth. The sixth, Rationality, the seventh, Equality, and the one that occurred this fall semester was Intuition.

The mission of the students is to engage with the general discussion, contribute specific insights that come from their field of study, their area of experience, and—of course—write their final paper. They were asked to write an initial draft that would, in some way, incorporate or involve the seminar discussion, the conversation that we’ve had. So it would have to be a good deal broader than if they were writing a technical essay regarding the meaning and use of `the word’ with in their own subject.

If you go to my webpage,²¹ I have my handouts for all of those courses.

So the final thesis or the essay that they write, would include all the different facets?

BM: Well, largely, moving in the direction that they know. But may—in fact, should—incorporate other aspects as well.

What does hard work mean to you? And, what is your usual work schedule at Harvard?

BM: I suppose that any one of us has, at one time or another, found ourselves staring again and again at a question and yet nothing comes to us—or, worse, where we find ourselves incapable of even framing a reasonable question that might help in clarifying the obscurity of some issue we are interested in. Doesn’t our `ineffective staring and our intense concentration without any result’ count as `hard work?’ Very often: necessary work, though. Sometimes even a prelude to the emergence of understanding. There’s also a more menial genre of `hard work’ when we’ve entangled ourselves in a too complicated definition or clumsy prior formulation and we need to do the `hard work’ of disentangling.

As for my daily work schedule at Harvard, of course there are courses, committees, office hours, that interlace the schedule of anyone in academia. The intermittent times—besides the time devoted to exercise, and meals with people—well, those are my work times, often a collection of half-hour, or single-hour moments during the day.

Dinesh Thakur mentioned to me that a University Professor at Harvard can teach any course anywhere within Harvard!

BM: Yeah. I can teach anywhere. So, I do, but I never do it alone. I always do it with another faculty member. I taught a course where we also had a single-word course called “Explanation” with Mark Schiefsky, who is in the History of Science Department. That was a wonderful course. And, even earlier, I taught a course in the German and Comparative Literature department with Michel Chaouli, who’s now a very close friend of ours. The title of that course was simply: Reading the Critique of Judgment of Immanuel Kant.

Reading of Kant’s critique of judgement? Wow. Lucky students, I would say. About where the future of maths is headed, the role of AI: This is an obvious question to ask. Recently there was news about Google AI solving IMO (International Mathematics Olympiad) problems, and getting a silver medal. I wanted to know what you think? Also, there is this formalization, and verification and clarification of proof that Peter Scholze used related to his liquid tensor experiment. It seems it provided him with a certain pathway and a clarification of the proof that he had. Or, it might have provided some steps. In any event, the Lean²² language system kind of clarified certain aspects about the proof and definitely helped with some details. Do you have any comments about that?

BM: It’s absolutely wonderful, the Lean framework. Yes, verification and clarification—and unification. One of the many things it does is it simply gets us to reorganize some of our thoughts or gives us another perspective. I don’t think I’m in any position to make any statements or prophecies about the future of it, beyond just saying that it will be neatly incorporated in the broader context of mathematics. I mean, just as LMFDB [The

L

-functions and modular forms database]²³ is.

AI and Lean are further computational developments that will allow us to live with mathematics more easily

Well, with LMFDB, it doesn’t matter how much maths you know, well almost. You can just walk through the forest of data and learn something that you would never have learned theoretically. Not because you’re learning something statistical or you’re learning something computational. You’re getting sort of a real sense in your central nervous system of the data that is immediately available all around you. You can type into LMFDB: how many fields of degree seven have the property that they’re monogenic?²⁴ That’s something that you in earlier times could not even get a sense of without loads of effort, and time. But if you’re interested in learning a little bit about the forest of mathematical objects that you’re going to work with theoretically, and if you just want to walk through it, well, you have it all right there with a click of a button in the LMFDB site and you get a vivid picture of those mathematical objects. I think that type of picture is of a different nature than a purely statistical sense. You get a better feel of the subject than any theorem that gives you a statistical statement. It has a different feel. It’s neither better nor worse, but a different feel, just by wandering through the terrain of particular numerical examples so easily with their salient properties displayed; getting a sharp sense of them—even though you are merely browsing through. Now, that’s so new. And LMFDB will give you a load of detailed invariants vital for the understanding of any modular form that interests you. The fact that we have AI and we have Lean, I don’t think of it as a revolution in the very nature of mathematics. It is rather a further computational development that is going to allow us to live with mathematics more easily.

One of the future possibilities is that mathematicians will provide ideas, steps, pathways, and computers, while Lean and similar frameworks and proof systems will provide proofs by filling in technical details.

BM: In some cases, yes. But, if not, you can also use Lean, or a similar proof system as one of your many tools. Here’s a curious feature of the Lean language mentioned in a throwaway line in one of Kevin Buzzard’s articles. It’s that the definite article The and the indefinite article A seem to have switched roles in at least some Lean framework. Definitions of a concept that begin with the article The usually are abstract axiomatizations that give a broad view of the concept, while definitions that begin with the article A or An are specific constructions.

Receiving the National Medal of Science from President Barack Obama. Barry Mazur

You co-authored a book with William Stein about the Riemann Hypothesis titled Prime Numbers and the Riemann Hypothesis. Would you like to talk about it? Do you think it will have an algebraic proof or an analytic proof? Or would it be difficult to predict or expect any specific methodology or possible approach to it?

BM: I’m perfectly happy with everything that William and I write there, but there is nothing in our book that goes towards a proof. I suspect the proof is going to require a much deeper comprehension of what lies below the integers, “what’s below”

\textrm{Spec},\mathbb{Z}

. I don’t like the notation

\mathbb{F}_1

, for the field with one element,²⁵ even though I expect that there is some structural concept that’s “what’s below.” And we probably need to understand “what’s below” to open things in a broad way.

On the broadening of perspectives, what do you think would be the role of recent developments in topology in the future of number theory? Especially the notions such as the infinity category, and the higher homotopy theory…. I ask you because I think it was you who brought in the notion of “arithmetic duality’’ in your IHES notes.

BM: Oh, I think it’s absolutely essential that we really understand the close connections, and we’re beginning to. The amazing thing, of course, is if there’s a connection, and I certainly believe there’s a connection between the idea of, say, a circle—or a knotted closed curve—in

S^3

and a prime number. Think of all of the complex structures and invariants we get out of prime numbers from Iwasawa theory, and think of all the complex things we get from knots. In the quantum correspondence of knots, and even just the number of cross links, there are so many structural concepts in knot theory that work so beautifully in explaining knots, and there are so many structural concepts in number theory that work so beautifully in explaining Iwasawa theory and prime numbers. But if I take one of these concepts, say in knot theory, and ask what’s the correspondent concept in number theory? we often discover wonderful things in the process. The most classical such correspondence is the Alexander polynomial in knot theory related to the Iwasawa polynomial in Iwasawa theory. The number theoretic correspondence of some invariants in knot theory remain a mystery to us—and similarly the other way. There could very well be a framework that encompasses both number theory and knot theory synthesizing the two theories explaining the mysterious invariants in the structure of either side as somewhat reconceived invariants in the structure of both sides at the same time.

How do you approach maths? Do you, for example, see examples as seeds, and nurture them, you kind of broaden around the examples, which grow into trees?

BM: It can be true that there are examples that just open doors. An example that does this is this theorem: you take two Alexander horned sphere solid balls, and you put them together at their boundary. Now each one, what do they look like? Both of them are rather horrible. But you put them together and they become (topologically equivalent to) the usual

3

-sphere. You can put it this way. There is a very interesting involution of the

3

-sphere with fixed point set an Alexander horned sphere. (By an `involution of the

3

-sphere’ one means a continuous mapping of the

3

-sphere to itself which when applied the second time gives the identity map.). Now the usual involution of the

3

-sphere sends the `North Pole’ of that

3

-sphere to the `South Pole’, with the equator being the

2

-sphere, the ordinary

2

-sphere. That is certainly an example of an involution of the

3

-sphere. But, as I said, there is another, with a totally wild equator: the Alexander horned sphere. Now that is an example that inspires thought; something going on that you would not have expected. I do not know whether Bing or Alexander or anyone else expected this at the very beginning, but they did discover it. That there are really interesting involutions. To say nothing of the discovery of the non-standard differential structure on the sphere, that there are many differential structures on the topological

7

-sphere, that too is amazing. These are examples. Now what do you do with those examples? You try to explain them. You introduce a whole theory that does that.

One has to have an eye for such examples in some sense. The intuition perhaps, that there is much more to it….

BM: As I mentioned, I have recently co-taught a philosophy course called “Intuition’’. I began with a statement of Einstein about the intuitive mind, “The intuitive mind is a sacred gift, and the rational mind is a faithful servant.”

Perhaps related to this, you talk about the role of conjectures as well.

BM: Yes, I wrote an article called Conjecture.²⁶

Communicating an idea to someone with a different background forces you to rethink that idea and gives you a new perspective

How did the Fontaine–Mazur conjecture come about? Was this during one of your visits to IHES, or was it done over a long period of time?

BM: It must have been while we were at IHES. It had always seemed to me that the crystalline structure was very, very strong. It is a strong structure, in some slant sense very `geometric.’ It is just hard to imagine that a random Galois representation in the complement of the fibre at

p

can just be extended to have the crystalline property at

p

. So it seemed to me that the conjecture was natural.

What do you feel is your most easy and highly influential work?

BM: Oh, definitely my thesis, On embeddings of spheres.²⁷ That’s pretty influential. The idea behind the whole proof is describable in English.

It’s basically a `take’ on the following easiest way of looking at the [crux of the] proof: The infinite sum $1+(-1)+1+(-1)+1+(-1)+\cdots$ can be interpreted to be equal to $0$ , because $(1+(-1))+(1+(-1))+(1+(-1))+\cdots = 0+0+0+\cdots$ . At the same time it can also be seen to be equal to $1$ , because $1+((-1)+1)+((-1)+1)+((-1)+1)+\cdots = 1+0+0+0+\cdots$ . These interpretations are, of course, not valid for real numbers (since this particular series does not converge), but in an appropriate context in geometric topology where it is valid,²⁸ it was used in my thesis to give a partial proof of the Schoenflies Conjecture. There’s a certain amount of technicality to deal with, in the summing of certain objects in topology, and to have it make sense at infinity, but other than that, the crux of the proof is just what I said.

Was point set topology, the kind that you spoke about earlier, helpful here?

BM: I don’t know, it must have been so. This is close to Kuratowski in some sense, but I was after the Poincaré conjecture, not the Schoenflies conjecture. I called the Schoenflies conjecture Lemma 1, because it’s the first step to what I hoped would be a proof of the Poincaré conjecture, but I never got further than Lemma 1.

And then gradually you moved from topology to number theory. How did that happen?

Barry Mazur, Philadelphia 2018 Barry Mazur

BM: I think I described it a little bit earlier (someplace else). So, this – the process of moving from topology to number theory – almost has a hint of some sort of dynamical system going on! I became interested in dynamical systems. And I proved this theorem²⁹ with Michael Artin. Let me not give it in an English sentence, but I proved the theorem, which was a big theorem in dynamical systems, with the use of the theory of Nash manifolds. Nash manifolds end up being nothing more than connected components of the real locus of algebraic varieties. But anyway, that’s real algebraic geometry and so naturally I became interested in algebraic geometry. I worked on homotopy theory and algebraic geometry with Michael Artin. We wrote a book [Etale Homotopy] on it together. It brought in essentially Etale homotopy theory into algebraic topology in some way and consequently I became interested in algebraic geometry per se and studied algebraic varieties, and, then it occurred to me – as scheme theory was being developed – that if you’re doing algebraic varieties, you’re doing schemes. Then I moved to being interested in arithmetic things. Also the primes-as-knots analogy was something that allowed me to move from what I knew, which was knots, to primes, to Iwasawa-theoretic issues in primes, and to broader issues in number theory.

The scheme theoretic techniques and perspective show up very strongly in the Eisenstein ideal paper Modular curves and Eisenstein ideal.³⁰ Based on my understanding, it is one of the first papers that uses the entirety of scheme theory to address a very number-theoretic problem of bounding torsion on elliptic curves over the rationals. Do you have any comments as to how that came about?

BM: So many ideas, at that time, were new, were vibrating with potentiality. Yes, Grothendieck’s scheme theory of course and the foundational work of Michael Artin, Pierre Deligne, Nick Katz, David Mumford, Ken Ribet and Jean-Pierre Serre. The fine structure of finite group schemes was being developed by Jean-Marc Fontaine, and Michel Raynaud. And directly in connection with the project of my paper was the inspiring conjecture of Andrew Ogg (in celebration of which, Jennifer Balakrishnan and I have just written a survey article.)³¹

Is there any other question in mathematics (including number theory) that you have been thinking about, and to which you keep returning to again and again?

BM: Karl Rubin and I have been investigating over a period of years the notion of “Diophantine stability”, a curious possible phenomenon regarding abelian varieties defined over a given number field

K

. Namely, an abelian variety may have the feature—relative to a specific field extension

L/K

—that it doesn’t acquire any `new rational points’ as one passes from the base field

K

L

. Such Diophantine stability happens more frequently than one might expect, and it often has striking consequences for Hilbert’s Tenth problem, as Karl Rubin, Alexandra Shlapentokh and I are currently pursuing.

How about questions that you keep coming back to but that are not yet yielding?

BM: A lot of things. Well…. There is something I am working on with Michael Harris and Tony Feng, and I hope to understand it more. There is this new attitude towards everything in mathematics, at least in cohomology issues, where derived cohomology brings into the picture infinity categories. We’re writing a paper about that, and in connection with that paper I gave a lecture, a year ago or thereabouts, called derived class field theory. That is a theme that I think is going to be developed in a much broader fashion than what we are doing at the moment but may work its way into the primes and knots issues as well. I don’t know. We are thinking about something very modest actually at the moment, just classical class field theory, not even in any way more sophisticated way than classical, but with a derived aspect to it.

Fascinating! Could you please comment on the role of patience and steadiness of thinking in the process of creativity?

BM: Well, it’s about forgiving yourself for being slow to grasp something you are trying to grasp, how about that? That is: patience, patience is a self-forbidding instinct from blaming yourself for not grasping. Just being gentle to yourself, and saying that this too is interesting – that I don’t understand, it is not I can’t understand, it is just I don’t happen to understand at the moment. One needs to be forgiving for not knowing.

I think this is not trivial. This cannot be taught. Perhaps this stems from your training at the yeshiva where you learned it – without being taught. One walks through rain and much later realizes that one is soaked!

BM: Yes, could be! It’s also reminiscent of: “we don’t even know the unit interval!”

One needs to be forgiving for not knowing

Do you think the process of writing expository articles or even (technical) mathematics helps you understand the very concepts you wish to expound?

BM: Don’t you think so? The exercise of communicating an idea to someone who may have a different way of understanding the subject, or who might have a background different from yours forces you to rethink that idea. This will give you a new perspective. Writing, of course, clarifies—even though Socrates, who radiated clarity, may never have written down a single word.

How many projects do you work on at any given time?

BM: At the moment, I am working on three. I have just finished four. But that is not usual. Usually, it is fewer.

At the end of Ralfe’s documentary, you talk about questions that should not be definitely answered. The questions that carry you, carry us, that are about the meaning of life. What are those questions?

BM: It is not so much that. Rather, I would say, whenever we think about some broad concept, we should keep whatever reflections we have, open for revision. That is to say, we should not have a firm decisive conviction about, in fact, any broad issue. We should leave a certain amount of flexibility and allow for the possibility that we will know something better later than we do now.

Now, for a very practical question about the role of exercise, or regular physical activity, towards maintaining good health and building mental stamina.

BM: Oh… essential. I mean, that would be more important than proving a theorem, I think. I do a kind of stretching and an assortment of other exercises every morning for about 18 to 20 minutes. And then we have a serious physical training session once a week. We have Tai Chi, as we did this morning, by Zoom. Twice a week. Our aim is to walk for an hour a day. Our aim is that, not always executed. And we have another, less intensive, body conditioning, once a week. And that is pretty much it.

Is there any specific award or honour that made you happier than others?

BM: Must have been the first, the Veblen prize.³² That is for my thesis.

Here is a question related to the existing proofs of Fermat’s Last Theorem that bugs me – we have two or three (not totally unrelated) but different proofs of the Shimura–Taniyama–Weil conjecture about the modularity of Elliptic curves defined over the rationals. So, even though there are these proofs, it is still not clear as to why such a thing should be true! It is very mysterious. Do you think that this kind of question as to why this is true is wrong to ask? Or is it something that people have understood now and that it is now totally obvious?

BM: No, No…. It is not totally obvious as to why it is so!

We should leave a certain amount of flexibility, and allow for the possibility that we will know something better later than we do now.

Why is it expected that, something on the one hand such as the $L$ -function (attached to a given elliptic curve $E$ over the rationals), a complex valued function of a complex variable—a kind of a generating function—which is defined using point counting data at all the prime numbers, is a modular form, or simply put, has to satisfy certain very precise symmetry properties? I find this very bizarre and mysterious – what do you think?

BM: I agree, mysterious certainly! For any motive you could ask: is there an automorphic aspect to that motive? That would be the hope that automorphic representations `dominate’ (in some sense) much of arithmetic algebraic geometry, right? Just as modular curves actually dominate elliptic curves over

\mathbb{Q}

. It’s saying that there’s a richer structure that overviews motives—concepts that we think of as, themselves, basic structures. And what is underlying all that? Maybe it’s giving us more hope that something underlies

\textrm{Spec},\mathbb{Z}

(Laughs heartily!). But, of course, I don’t know.

I want to mention to you a simple calculation based on a probabilistic formalism and how that proves the formula $\sum_{n=1}^{\infty} (1-p) ^n = \frac{1}{p}$ . Let us consider the experiment of tossing a coin until one gets a Heads (assume that the probability of getting a Heads for a single toss of the coin is $0 < p <1$ . Since, upon successful tosses one will eventually get a Heads, the probability of getting a Heads is equal to $1$ . On the other hand, the probability of getting a Heads after exactly $n$ tosses is $(1 -p)^{n-1} p$ . Hence, we get another much more transparent proof of $\sum_{n=1}^{\infty} (1-p) ^n = \frac{1}{p}$ .

BM: Oh, right, that is good.

By analogy, something similar may also be true regarding Taylor–Wiles’ Theorem (Shimura–Taniyama–Weil conjecture) – perhaps we do not have a fully satisfactory proof of it as yet?

BM: I see, that is a good point. So, you are saying that even though one can go through a proof, involving lots of steps to prove a statement, some probabilistic argument might show that the statement is obvious. And that is an explanation, that is an emphasis on the obviousness of a statement, (even though it was already proved easily). Now, is there a similar emphasis on the obviousness of the Taniyama–Shimura conjecture, which we have not seen yet? That is your question?

Yes! And at the same time, I wonder about the power of formalism that leads to a certain “obvious” understanding of truths that are non-obvious and mysterious, and yet the entire structure is based on “simple” axioms. I mean, it is a mystery to me that it can be like that.

BM: Yeah. Could it be that what you are asking is a question, a natural question that anyone would imagine or naively ask, but you are asking it in a different way? You are saying, “I prove a theorem and I think I am done, but am I done? I may not be done because there may be a way of showing that the theorem is really a tautology.” There are two things, there is the formalism and the why. And you are asking what is the relationship between the two of them? Yes, it is mysterious!

In the video titled “Barry Mazur on Algebraization”, you talk about “finding $x$ ” and the associated deep idea of Kronecker. Later on in the same video, you mention the analogy “Algebraization” in literature; that is, language retreating from content, and by purposely keeping the content covered, as a secret, leading to revelation of a new attribute or a new invariant, that was until then invisible – and that is different from any previously associated “attribute”. As if, this act of hiding it as $x$ in a velvet bag keeps away any previous biases, thus saving us from getting blinded by the known bright attributes, thereby forcing us to use our “inner light” and the $x$ to interact directly to possibly reveal something new about $x$ . Would you like to say something about this?

BM: Here’s a simple example of the power of “hiding an

x

in a velvet bag” that illustrates the power of algebra. Take the square root of $2$ . How might you have come across this concept? How might you deal with it? Well, you might have found it as the ratio of the length of a diagonal to the side of a square figure. Or you might be dealing with it as a real number, so it has a decimal expansion

1.414159

…. The algebraic move—to face this concept—is shockingly simpler: just give it a name. Call it, say,

x

. If you are starting from scratch, the only thing you know about this

x

is that its square is $2$ . But that’s OK. Having named it, you’re on the road to analyzing the rich properties of the `thing’ that you named. This is, after all, akin to the starting `move’ in High School algebra: “Let

X

be…”. It is so ingrained in our nervous system that we don’t notice its depth. You make that `move’—e.g., by giving a name, say

x

, to the square root of two—and you immediately have at hand the ring

{\mathbb Z}[x]

of polynomials in

x

with integer coefficients—a ring with deep and revealing structure—offering a framework, and a powerful setting for finer exploration, it being the initial object in the category of rings that contain an element whose square is

2

. A similar move encapturing structure is the more modern Model Theory that, when dealing with a rich constellation of substantial concepts, sets aside meaning or any hint of semantics and focuses exclusively on the individualized syntax appropriate to that “constellation of substantial concepts.” That is, one works with a “model.” The surprise is that this is often a powerful tool to get into the essence of the substance of interest.

As we come to the end of this fascinating conversation, I wonder if you have plans of visiting India. It could provide an opportunity for Gretchen to enjoy listening to live Indian classical music?

BM: We have no plans, but what you describe sounds wonderful.

Thank you very much Barry for your patience, and a lively and insightful conversation – dwelling on the kind of creative processes that have shaped your academic bearings and how they led to a rich, prolific, and accomplished career. It was absolutely delightful to listen to you, and to enjoy your and Gretchen’s very warm hospitality.

Acknowledgements We are grateful to Shilpa Gondhali for her ever willing help with timely transcribing of the audio recordings. We thank Sneha Patankar for immense help with editing the raw transcript. $\blacksquare$

Footnotes

https://www.nature.com/articles/483405a. ↩
Available on YouTube at https://youtu.be/bEf26NFur3w. ↩
A yeshiva is a Jewish school or college where students study religious texts. ↩
Tourette syndrome (TS) is a neurological disorder that may cause sudden unwanted and uncontrolled rapid and repeated movements or vocal sounds called tics. ↩
Available at http://giancarlorota.org/essays/rotasharp.html. ↩
A Hegelian synthesis is a philosophical concept that refers to the resolution of a contradiction between a thesis and an antithesis to create a unified whole. ↩
The opening line Īśāvāsyamidaṃ sarvaṃ yatkiñca jagatyāṃ jagat of the Vedic text Īśāvāsyopaniṣat, which [in translation] says “each and every entity in this universe is enveloped by the Lord”, perhaps conveys almost the same sentiment. ↩
Baruch Spinoza was a Dutch philosopher of Jewish origin. Considered one of the most influential philosophers of the seventeenth century, Spinoza has deeply influenced modern Western thought, including that of leading scientists such as Albert Einstein. ↩
A Greek philosopher, regarded by modern scholarship as the founder of Neoplatonism. His teacher was the self-taught philosopher Ammonius Saccas, who belonged to the Platonic tradition. ↩
It is a wave whose peak amplitude does not move in space, and hence “standing’’. It is usually produced as a combination of two waves moving in opposite directions, each having the same amplitude and frequency. ↩
A stevedore is a person employed at a dock to load and unload ships. ↩
page 40 of the book, Math in the Time of Corona https://link.springer.com/chapter/10.1007/16618_2020_8. ↩
https://worldfinancialreview.com/category/columns/understanding-war/. ↩
https://www.ihes.fr/en/. ↩
Available at http://nrs.harvard.edu/urn-3:HUL.InstRepos:34798399. ↩
Page 123, Chapter 10, Limitless minds: interviews with mathematicians by Anthony Bonato, AMS, 2018. ↩
Oppenheimer was the Director of the IAS at that time. ↩
Published: 01 November 1985, in Biographical Memoirs of Fellows of the Royal Society, Volume 31. https://doi.org/10.1098/rsbm.1985.0008. ↩
https://genealogy.math.ndsu.nodak.edu/id.php?id=11730. ↩
https://bpb-us-e1.wpmucdn.com/sites.harvard.edu/dist/a/189/files/2023/01/Questioning-answers.pdf. ↩
https://sites.harvard.edu/barry-mazur/. ↩
https://lean-lang.org/. ↩
https://www.lmfdb.org/. ↩
https://en.wikipedia.org/wiki/Monogenic_field. ↩
https://en.wikipedia.org/wiki/Field_with_one_element. ↩
https://www.jstor.org/stable/20117628. ↩
https://api.semanticscholar.org/CorpusID:37896421. ↩
This technique has come to be known as the Eilenberg–Mazur swindle (see https://en.wikipedia.org/wiki/Eilenberg\%E2\%80\%93Mazur\_swindle). ↩
On Periodic Points, M. Artin, B. Mazur, Annals of Mathematics, Vol. 81, No. 1, Jan., 1965. ↩
Publications Mathématiques de l’IHÉS, Volume 47 (1977), pp. 33–186. ↩
Ogg’s Torsion conjecture: Fifty years later—https://arxiv.org/abs/2307.04752v2. ↩
Oswald Veblen Prize in Geometry – awarded in 1966 along with M. Brown and S. Smale. ↩

Vijay M. Patankar , is a professor at FLAME University. His main research interests are in number theory, arithmetic geometry, and algebraic complexity theory. He likes trekking, long-distance bicycle touring, watching magic videos with his daughter and listening to Hindustani classical music. He is a corresponding editor of Bhāvanā.