Prof. Stuart Geman was born in 1949. He was educated at Dartmouth Medical College (MS, Neurophysiology, 1973), and the Massachusetts Institute of Technology (PhD, Applied Mathematics, 1977). Since 1977, he has served for Brown University as a faculty, and he was elected to the US National Academy of Sciences in 2011. Prof. Geman made great contributions to computer vision, statistics, probability theory, machine learning, and the neurosciences.

TPL: First, thank you for coming. We have worked you hard this time.

SG: This is true, but I have enjoyed every interaction.

TPL: Can we say that you are applied mathematician?

SG: I would maybe modify it for a little for my purposes. I think of myself as a scientist. Clearly my union card is in mathematics, especially applied mathematics.

TPL: So your primarily goal or purpose is to understand scientific phenomenon and so on.

SG: Yes, yes. That has always been my ambition.

TPL: But what makes you choose mathematics as a tool?

SG: Well, I felt that I needed a tool. I needed the best tool to understand the things I was wrestling with at the time. When I went to graduate school in math I was actually in medical school, but focusing on the brain and studying physiology, and I had many thoughts and ideas. Most of which proved to be nonsense. Nevertheless, even to establish they were nonsense I needed the help of a powerful tool, mathematics.
Furthermore, as I looked at the literature in physiology, and in what to me is a related area, computer vision, I had the uncomfortable feeling that I was making judgments in part based on the mountain I had to climb to understand some of the approaches. So, for example, there would be something very mathematical using tools that I was unfamiliar with, and my instinct was to dismiss it, to find a reason why it wasn’t important. I didn’t like that so for that reason as well I felt I had to learn enough mathematics to be able to drill down right to bedrock to understand exactly what assumptions were being made and what the implications were.

TPL: Of course there are other tools in scientific investigation, but you choose mathematics. Could we say that it is because you were good at mathematics?

SG: Maybe, I never know which comes first. You like something because you are good at it or are you good at it because you like it. I did take to it, in many ways I fell in love with it.

TPL: When you are in elementary, high school, you were good at math and of course you were good in other things as well.

SG: Yes, I was good at math and sciences.

TPL: But supposed that you pursued your career in medicine. That would be okay with you now?

SG: No, I don’t think so. I don’t even think that was my plan when I went to medical school. I had been a physics major, and had gotten quite deeply into physics, which I have always loved, and still do, but I wasn’t quite sure what I wanted to do. I was torn. I had learned a little biology and was taken by that and given that I wasn’t certain, I decided that going to medical school was a case of closing the fewest number of doors, leaving the most number of options open. With that in mind I went to medical school.
I wouldn’t say to bide my time, because it was a wonderful experience. That was an education I wanted, at least through the basic sciences, but it wasn’t with the intent necessarily of being a neurosurgeon or a family physician or anything like that. It was with intent of poking around and learning some more science. It was a very different era, certainly in America in the early 70s, late 60s, you didn’t feel as anxious about your future. It was more of a land of opportunity than any place is today including America. So I took advantage of that.

TPL: May I say that you just simply have a powerful mind. You say you poking around. I do that also, but I wouldn’t say that poking around is going to medical school because that’s a serious undertaking, studying medicine is a lot of work.

SG: Medical school is a serious undertaking, I certainly agree with that. I am not sure I would take it as such. I think so much of it is an attitude of curiosity, and the luxury to feel that there was time to explore these things. I stumbled. I went to medical school and it was very different from anything I had ever done. I assumed I could learn medicine the way you learn physics or math. You sit down, and you sit still and you make sure you understand every single line. I mean it is almost a prescription. It may take you a long time, but it’s really in a way simple, just make sure you understand every step. But that doesn’t work in medicine, you start reading Gray’s Anatomy1, which is a few thousand pages of small print with a stunning number of facts and figures, and you can’t power through it. It’s hopeless. It took me a little while to learn that.
Instead it is a more passive kind of learning. You go to what would today be Starbucks, or some places where there are activities. You are not too bored or lonely. You just read, don’t underline, don’t make lists. At least for me just read, read over breakfast, read over lunch, read at bedtime, thumb through anatomy books, physiology books. The whole business is better absorbed for me in a by way of immersion. Don’t try to power through it; you would end with infinite lists. You end with the book almost totally highlighted. That doesn’t work.

TPL: Could I speculate from what you have just said, having gone through the process, you have become, in some sense more objective, willing to accept the fact as it is. In mathematics we can be more subjective.

SG: Actually, no, that’s an interesting observation but I would say the opposite is true. There’s another kind of calibration that has to be made as you move from one field to another. I think perhaps medicine is an extreme in the following sense. You cannot tell from the certainty from which a statement is formulated, how certain the underlying science is. As far as I can tell, they are uncorrelated. There is something in the culture, of at least the education of medicine that makes everything definite. Everything they teach you is the last word; we understand it. There is no way to know what’s a little bit confused, what’s uncertain, what’s likely or unlikely to be overturned by the next experiment. When that book is written, that medical textbook, it is just written as though it were the way things are, instead of the way things are as we under-stand them now.
So, it took me quite a while to develop the proper amount of skepticism. Now, usually when you read a theorem in math, usually it really is a theorem and it is not likely to change tomorrow. I think of the number of statements about neurophysiology and neurochemistry that I put into my brain that had to be taken out again, because they were just wrong! One after the other of the dogmas of neurophysiology has fallen and that’s true in physiology, in biochemistry, certainly in genomics. When I went to medical school, it wasn’t very complicated. These genes, probably a lot of them, were translated into proteins, and proteins folded and gave a structure that gave a function. And there you had it. Only four nucleotides and 20 or so amino acids; it was pretty simple. But in fact it is nothing like that. The more we look, the more we see. So that’s another kind of calibration but also part of what makes science exciting; especially brain sciences these days.

TPL: What you just said sounds very encouraging. There are research directions that are possible which not only have not be pointed out but also not even suspected of, in biology, you seem to indicate that.

SG: I believe that very strongly. I think it is, in some ways truer today than it has ever been. Of course we have accumulated knowledge and most of it is real and will last, but at the same time, it seems that the questions become more profound and fundamental. All the time. The basic properties of genetics, inherent and epigenetic, are still fundamental puzzles. As far as I’m concerned, though many of my colleagues who are neurophysiologists would disagree, we are missing the most basic purposes of neuro-architecture. I think what we teach, and the way we look at the brain, misses the mark. I think there is something going on that is really stunningly beautiful and that can be best described again with mathematics.

TPL: I am so impressed with the clarity and consistency of your thought.

SG: Thank you. I don’t know if I deserve that.

TPL: Perhaps we could go to the subject that you are actively participating. Exactly, what would you say you do in research?

SG: I would go back to an earlier part of the conversation to say that I look at scientific questions and puzzles and try to think hard about them and am often led to mathematical formulations. That for me provides at least a way to think about them. Now and then it leads to a mathematics question that I get stuck on independent of where it started, but my main interest at the moment, as you might guess, is the brain and what the neuro-circuitry is doing. Related questions are in computer visions, which I feel hasn’t quite gotten off the ground despite all the remarkable applications and many successes. I feel it has not made much of a move in the direction of matching biological vision. So there is a gap there that fascinates me, despite all the computing power and all the examples that we have in places like Google. Our computers are a very poor second to the real thing, the biological vision system.
So, those two related fields fascinate me. And because of this researcher in your department, Hwang Chii-Ruey, I got seduced into looking at financial models in the past couple of years. And I must admit, Chii-Ruey showed me some data that is so remarkable in its stability and in its indication of some kind of scaling property, that I couldn’t put it out of my mind. And this has sort of brought me into questions of what are the fundamental models and are they right. Are they really consistent with the data when we look at the data? Not just marginal data but as true temporal processes. So that’s an interest as well at this point. And I can’t resist keeping my eye on, and talking to people who are in genomics, because it’s evidently mysterious and exciting and the tools are so powerful. It is as we would have said in the sixties, a mind expanding experience to look at genomics right now.

TPL: Vision is important because it is a vital form of communication? I am such an outsider.

SG: I think it is important to me for a couple of different reasons. For one thing, it has evident commercial applications. So it has become a part of our lives in everything from industrial automation to search engines to all kinds of security. So it is important for that reason. It perhaps is important as well because it shines a light on the inadequacy, at least to my way of thinking, of the models we have for brains and how brains work. After all, brains are very good vision systems. In fact it is often said that about 50, 60, 70% of your brain is primarily concerned with vision. Now that has to be taken with a little caution, that statement, because blind people are using their brain but I don’t think those are inconsistent. It’s just that much of vision concerns the representation of what we are looking at and the manipulation of those representations. But at the same time, you have a relatively very large sensory cortex called the primary visual cortex, and the basic pathways of vision are a major part of the biological system. So, it may be important as a window into the principles of neuro-representation and neuro-computation. I think it is.

TPL: This how brain communicates to us?

SG: It’s a great deal how they communicate with us.

TPL: It’s like you want to understand black hole or supernova, we look at the radio waves.

SG: Yes.

TPL: And we want to understand the brain, we try to understand how it operates on vision and so on.

SG: Yes, I believe that because of all the technical interest in computer vision. We are better able than we were thirty or forty years ago to articulate what is so hard about the vision problem. That then gives us a more informed view about what we don’t know about the brain. You have to know what questions to ask. We are so expert at vision, at language, at communication. We’re so expert at developing motor skills and so on that it is hard for us to look at ourselves and even recognize what is hard about what we’re doing. But when you try for fifty years to build an impressive computer vision system and fail utterly (to my mind), maybe you at least achieve a sense of why the problem is hard.
So, sometime in the 60s when we got really ambitious about computer vision, the epicenter being the MIT AI lab, and we were in for a lot of surprises. There was a memo that came out of the AI labs sometime in the mid-60s that was entitled ’The Summer Vision Project’. This is a good thing to go back to because it gives us perspective. People at the time, the researchers at the time felt they had basically understood the fundamental principles of intelligence and what they needed was a demonstration of the power of their theory for intelligence. For a demonstration, they proposed a summer vision project. They would marshal the army of graduate students who were available during the summer. And very carefully over the course of a well scheduled three-month research activity, build a vision system. Then it’s laid out first, what would be the first third, the second third and third-third what the milestones were. And needless to say, it never got through the first third, because nobody had a clue about why the problem was so hard. What the essential ingredients were. I think maybe now we do. I think as I said, they shine a much more interesting and curious light on the neural system, the biological system.

TPL: So that’s the difficulty? That’s a bad question I think.

SG: No, no. It’s a wonderful question. I would answer that at many levels. Let me start at the highest level and maybe that would be the right place to start for our purposes. I would mention two words, representation and computation. The representation is rich enough to accommodate hierarchy, the composition of reusable parts. We learn as children basic components; we compose them, say into letters, letters into words, and words into sentences and so on. We organize our knowledge compositionally, and similarly there are reusable parts everywhere in vision, L-junctions and boundaries, T-junctions and strokes and surfaces and so on. Cylindrical objects that form the legs of a chair or a person’s legs, the legs of a desk. There are motifs or themes that are reused constantly. But the great challenge is capturing those representations in a workable way, in a way that’s mathematically coherent, that captures the likelihood of various compositions and that can measure the difference between the coincidence of arrangements, and a true composition, that can compare one hypothesis to another. That’s difficult and I think critical.
What’s easy to forget is how ambiguous images are locally. One thing I did for my son to try to impress this on him, was that on a Mac, I took a picture of a living room from the web and I scaled it way up, blew it up, so he was just looking at a little part of it, and I allowed him to navigate around. I said ‘tell me what you are looking at’; it was hopeless. He couldn’t even get off the ground, he couldn’t recognize any part of the scene. Exactly. It’s quite striking how ambiguous local things are. That means that to recognize some things we need to recognize its context. It’s more or less meaningless out of context and that is where this hierarchy of compositions comes in.
So representation at all its contextual levels, I would list as one component. And then the second component is easier for us to understand as mathematicians. If we look at the associated inverse problem, you give me an image and I want to infer the representation, the hierarchy, the labels of the scene and the decompositions, building made up of windows and bricks. The windows made up of parts. The bricks made up of different things and different layers and so on. In order to actually compute or parse the scene, i.e., infer the hierarchy. If we write that down formally it’s NP-hard (non-deterministic polynomial-time hard).
It’s basically the covering problem on steroids (the covering problem being a famous prototype NP-complete problem). And this one is somehow worse, though I don’t mean that literally. In any case, it’s a nasty, nasty covering problem. Another way to say it is that the computing has to be very contextual, non-local. So we have representation and computation, and in particular within representation we have the problem of context. We have to represent it and at the same time we have to be able to compute. We face with what my brother likes to call a context-computation dilemma. On the one hand, human vision is clearly contextual. On the other hand, context is clearly a great challenge to computation. How does the brain do it?

TPL: I begin to understand why you say that mathematics is a powerful tool. This reminds me, there was an accident that happened to my close relative and he lost one eye. At the beginning he could not even close the cover of his fountain pen. But after a while he could, and I guess it was about the learning process and he has to compile all his labor. So I see, it’s a complicated issue we are talking about.

SG: Quite spectacular. I would say that even that is a clue, the fact that he could achieve it. That’s right, because he lost his stereo, he lost his ability to tell distances in the usual way, but there are many, many other clues about distances. I would speculate that what made it achievable was that the basic representation of the world was already in place, and topological. He already had an almost literal representation, his mind eye, that could map out the pen, could marry the pen to its cap. So it was relatively achievable. I won’t say easy because I’m sure he struggled, but it was achievable exactly because the fundamental underlying representation was unchanged.

CRH: So is that the reason you moved from Dartmouth to MIT. You chose MIT for the AI people there?

SG: Exactly, MIT looked to me like a candy store. I mean with so much science, it was really a wonderful place to be during my math education. Of course wonderful applied mathematicians, a number of spectacular pure mathematicians. Beyond that there was the so called HST program which is still running. Health science technology program which is a part of the medical school, associated with the Harvard Medical School. I loved the way they taught medicine and so I took a course in cardiac physiology there. I heard Snyder2, then future Nobel laureate in neuroscience, talk about morphine receptors in the brain, just the beginning of a hint that there were endorphins - endogenous morphine. I heard Dirac3 talked about cosmological constants. I met with Chomsky4, spent quite a bit of time talking to him. I met Marvin Minsky5 of artificial intelligence fame. That was the perfect place for me at the time. Jerome Lettvin6, the neuroscientist, has an absolutely remarkable mind. I met and talked to him and was very privileged to be able to interact with him. So I think I chose very well.

CRH: When you went there did you have in your mind that you had some goal, I mean did you already have a problem for your thesis?

SG: I did. Definitely, I did have a goal. It was a little unusual in retrospect.

CRH: Herman7 said that he just signed it. I mean you had your own problems.

SG: We had an unusual and funny association that we are both fond of remembering. So yes, I had been working in neurophysiology and had actually finished a masters in neurophysiology while in medical school. And I went to MIT specifically to learn enough mathematics to get to the bottom of the kinds of models that I was exploring. These were models for neuro-circuitry, and that made me a little bit unusual among my classmates in graduate school. It seemed very natural to me as I gained more tools to explore this, and I did. It occurred to me to talk to professors. I remember talking to Dick Dudley8, a wonderful, powerful, probabilist over something that had come up in my research, and I benefited so much from Sigurdur Helgason9‘s fantastic lectures in analysis. But it didn’t occur to me to look for an advisor in the conventional sense. It really was not on my radar screen.
So I worked through these problems and got to the bottom to my satisfaction and it looked to me like it was roughly a thesis. But then there was the issue of actually getting your PhD, and that involves having an advisor. So I knocked on the door of the most natural advisor, since this was about random differential equations. It has to be somebody in probability or statistics, and Herman Chernoff seemed like the perfect person. Not really perfect, because he wasn’t particularly interested in the area. But I used some of the tools connected to his name and he had the knowledge base that seemed to be appropriate. So I knocked on his door and said “Would you be my advisor?” He said “Who are you?” I said “I’m Stuart Geman, I’ve written a thesis.” And I put 200 pages on his desk and he looked at it, then looked at me and told me to come back tomorrow. He had a glance at it. He didn’t understand a word of it. So he said I should lecture to him. So we set up lectures, regular two hour lectures. Michael Woodroofe10 was visiting from Michigan that year, and so Woodroofe and Chernoff were my audience. I lectured and Hermann was famous for falling asleep during lectures, and then waking up and asking wonderful questions. I watched him do that many times. But it was a little awkward when he would fall asleep and it was just the three of us. So I would have to wake him up. “Professor Chernoff, Professor Chernoff” He would wake up and listen and finally after many such sessions he signed my thesis and wrote me a letter that must have been very supportive. I had offers during that tough time.

TPL: That was what year?

SG: That was 1977. I’m infinitely in his debt. Now of course I have a totally different perspective. I can’t imagine being so generous. You know we are all wrapped up in ourselves one way or another. To delve into this topic that he had no real understanding of at the scientific level. Neurophysiology wasn’t his area. And he had no interest particularly in the kind of mathematics I was using. To delve into it, to give me that much of his time, it was a gift.

CRH: Hermann, he is very generous and very modest.

SG: He is modest, generous, and all round fantastic. I have thanked him many times but I can never thank him enough.

TPL: That’s a beautiful story.

CRH: I’d like to ask you about Chomsky, because in linguistics, Chomsky has his theory about how kids learn languages in modules and things like that. Does his idea have some effect on your thinking? Because you are talking about how to use language, context free, context sensitive to describe scenery of brain.

SG: I seem to have had a very peculiar interaction with Chomsky, which is to say that I knocked on his door, walked into his office, and we sat down and talked, as I remember it, for many hours. All the stories I have heard since then about how difficult he is. He is completely remarkable and original. I have heard many times that he is the most quoted scientist of the 20th century which I am quite sure is true. There are many things about that one afternoon of interacting that were peculiar.
The reason that I knocked on his door was that I came across his early 60s papers that laid out the Chomsky hierarchy, this hierarchy of grammars, the next one containing the earlier ones and ever more general to universal grammars. I was truly struck by that, it was just so beautiful, it looked like divine conception to me. I had never seen anything like it. And it had such implications in terms of computation theory and representation theory, so I was very struck by it and at the same time I had been forming the opinion that the way we think about one area, say medicine, is not so different from the way we think about another, say mathematics. They may appear to be very, very different but it was inconceivable to me that this machine, the brain, could have totally separate solutions to all these different challenges. With that in mind I went to talk to Chomsky, and we seemed to agree that these, whatever representations we were building and whatever modes of computation we used, they must cross cognitive boundaries. Playing chess cannot be completely different from doing mathematics or writing an essay or diagnosing a disease. I still feel very strongly that way; I think that’s a lesson the brain is trying to teach us, by its homogeneity. We make much of how different areas of the brain are different but to a much greater degree, they are virtually identical. More on that perhaps later, but the brain is very homogenous and I think that is a big clue. And Chomsky seemed to agree. He seemed to resonate on that. I cannot to this day resolve that conversation I had with him with his insistence that there is a special “language module”. So, evidently there are subtleties to the way he looks at this that I was not ready then to appreciate.

CRH: Don’t you think that those modules are reusable parts in some sense.

SG: I do, I think hierarchy and reusability, those are the principles of complexity in nature from atomic physics to cosmology, from atoms and molecules to complex polymers and proteins, from single nucleotides to genes and their combinations and their regulations, it’s always reusability and hierarchy. I do believe that, I don’t see how it could be otherwise.

CRH: I remember in the 90s you were talking about compositionality, and build things bottom up instead of top down because even when you were talking about molecules, atoms and going downstream. But your compositionality is to use parts to build things from bottom up right? Yes, I think that is the difference.

SG: You mean compared to the search for elementary particles?

CRH: Yes, because you are doing things bottom up. Try to build up things bottom up.

SG: I guess I would say that every time we think we have a sufficient collection of elementary particles, we immediately build bottom up. We build the system from there up. We keep learning that there are more elementary particles, although maybe that is finally running its course, maybe we’re there now. But it has been a history of finding every more elementary particles. I think that the principles of compositionality are very similar, to make a stretch and talk about cognition in the same breadth as physics. For example in vision these days, when we play with compositional models we have to start with relatively coarse “elementary particles.” They are already fairly complex little sub-scenes and the reason for that is computation. We don’t have ten or a hundred billion (nonlinear) computing units like the brain. We don’t have a system that has the stunning connectivity of the neural system, in which every processor, if you will, is connected to ten thousand, to as many as fifty thousand, others. And each connection is modifiable and nonlinear. We don’t have that.
We have a great deal of computing power but it’s not organized in the same way and that puts certain restrictions on what we do. So we start at a level and we work up. Yes, we build a compositional hierarchy. That what we do in our thinking about the problem, and occasionally when we are being practical and trying to build a system. But I don’t think for a minute that we are jumping in at the right level. If I look at the retina or the first station after the retina, which is the lateral geniculate nucleus, or let’s say the first station after that which is in the cortex, the primary visual cortex, what have the physiologists found? They found elementary particles that are much more elementary than the image patches that we use for elementary particles in our experiments. They find wavelets basically, (we used to call them Gabor filters but are much the same thing), neurons that represent wavelets and combinations of wavelets. That’s the right level to start at, something much more local, much more elementary. We’re not ready for that.

TPL: The current mathematical education, we have calculus, linear algebra, complex analysis and so on. We produce undergraduate with mathematics major. Now, with your highly unusual career path, what do you think about possible redesign of undergraduate mathematical education?

SG: I always feel cautious. I think we all need to be because it’s so easy to believe too much in a kind of self-serving idea about education. We have to really step out of ourselves. I feel I have benefited so much from science, from physics and biology and it so shaped my thinking. But at the same time I was a pretty serious student of mathematics as an undergraduate, enough to see the beauty of it and enough to get in a habit of going for a walk and thinking deeply about something. I feel like I have benefitted from all these things. There are so many ways to contribute and be excellent. I wouldn’t want to say that our pure math majors learn too much math in the absence of science. I wouldn’t want to say that because I think we all benefit from the consequence in many cases.
When I am advising an undergraduate math major I will nudge them in the direction of a physics course or to a biochemistry course, something physical. Something mechanical, even a computer circuits course so they get a sense of the way things work. I guess I think there is a good chance that these things can inform even the purest of mathematics. To my mind David Mumford11 in some ways is the ideal pure mathematician. Of course he’s not so much a pure mathematician anymore. He always had this interest in neurophysiology and in application. But I think even when he does/did his pure mathematics, it was very much influenced by a sense of the way things work, a picture of the world. He was always a student of science and I think that has served him very well. Of course there are plenty of spectacularly powerful, important mathematicians who did just fine without any physics, or even being suspicious of physics as many mathematicians are.
I don’t have a strong opinion. Here is what I would say to an undergrad. (I guess I do.) Have your eyes open. At least recognize you’re making choices. You’re making some decisions about how you are going to think, about what is going to influence your taste, and your solutions. That’s the best we can do. I have an opinion going earlier about high schools. I have developed a perhaps too strong opinion about science education or math education. I think for the vast majority of students, even many of whom are going to become great mathematicians, we teach too much math too early. I would replace second, third semester calculus, I might even go so far as to replace first semester calculus, with puzzles. Something fun that only requires thinking and not so much knowledge in getting things straight to think about. I think there are many wonderful physical puzzles, thought puzzles, games that you can play when you are in middle school already. They are infinitely hard. There is plenty of “elementary” stuff that is really challenging, and I think that’s time better spent 11David Bryant Mumford (1937- ), American mathematician, awarded the Fields Medal in 1974, the Wolf Prize in 2008, known for the distinguished work in algebraic geometry. for most young people less than, say, 18 or 19 years old.
The reason is that I feel like at least in my experience at Brown that the students have arrived with more and more impressive credentials as freshman, with amazing amounts of mathematics, often at a college level, great scores on the SATs and the AP exam. But there has been a steady decline in their understanding of calculus. It’s a remarkable phenomenon, but I’m quite sure it’s real. We can quantify if we tried hard enough. They don’t quite get it. Maybe they’re not ready to get it. That wouldn’t be so bad if it wasn’t so hard to undo something once you’ve learnt it one way. When you’ve leant things one way, it’s a big job to undo it. So misconceptions or misrepresentations, I guess I’m obsessed with that word representation. I guess people build structures to understand calculus or whatever, the notion of area or creating a volume by rotating a curve, the fundamental theorem and what’s really going on there in the picture in their mind for how it works. If they don’t get that right then it is going to be hard. A year or two later, it seems that people can pick it up very easily. They are much more ready.
I don’t mean calculus is ever easy. I agree with your statement the other night that it’s always healthy to teach it and remember how hard it is, especially the second and third semester. But I think people, when they are ready, they can do a much better job at it. It’s far more efficient because they’re ready to get it right. So that’s the only strong opinion I have about education. I think there are many sides to all these other debates. For example the debate about whether we should be teaching at the high school or college level more utilitarian subjects. Whether we should add utilitarian mathematics and have a bigger emphasis on it. As, for example, for the great majority of people it is more important to understand compound interest, basic statistics, because that’s so dominant in our lives now. So they can interpret what the doctor tells them. Even for the doctors, so he or she can interpret what they read. They need to understand basic statistics.

CRH: So is that what the reason you put a lot of puzzles in your office?

SG: Well that’s what I meant about it being dangerous and self-serving. But yes, I do love puzzles. I don’t know if these things are ever innate. That gets to puzzle of understanding the difference between nature and nurture. Maybe it’s even a false dichotomy. But from a very early time I played with puzzles. I just saw some old movies that my uncle had taken and I don’t remember if I was even two years old and I’m playing with that tower of Babel, trying to work it out. There was some kind of focus on puzzles quite early.

CRH: Tai-Ping you know, in his office he has more than twenty puzzles. Lo-Bin12 told me that you cannot solve one of them and Lo-Bin said that he solved all of them.

SG: He solved them all!

CRH: You were very mad about this.

SG: He’s the first to solve all of the puzzles in my office.

TPL: Education is about the brain, as you point out early. we know very little about how our brain functions so I guess it’s true that, there is an education reform by some of the smartest guys, but that is very dangerous.

SG: I think so. Like I said, I think we are at a stage where the best we can do for our students is to lay out the issues so they make their choices with as much perspective as we can give them.

TPL: Students should realize that they need to make decisions on their own future.

SG: It’s more likely to work better for them.

TPL: They have to take this in their hands. This reminds me about what Mark Twain said “I have never let my schooling to interfere with my education”.

CRH: For example, even people agree that maybe we shall have a statistics course for undergraduate. But the way of teaching, the reason I raise this is because you were famous. You attracted more than 100 students in the statistical computation class. I mean to teach the basic course. It depends on how you teach it.

SG: It does. I have to say though; famous may be a bit strong. I did attract a crowd when I taught these undergraduate classes some years ago. Though recently I’ve been humbled because I tried to teach the most elementary statistics course, not something I’ve ever taught before. It’s a level below the one I taught for example when you were at Brown, and yes we had a great enrolment, over 200 for that class. This is much harder.

CRH: Elementary one is much harder.

SG: For me it was. I’m torn because part of me wants to really get it right; and of course part of me wants to run away from it. It’s a big job, it’s a hard job.

CRH: The reason I am mentioning this is because for teaching these basic courses one has to take a serious look, even calculus or statistics. I know that Prof. Geman teaches very well even elementary courses. When I was there, you said your student now, what is his name?

SG: Matt Harrison13, oh his teaching is remarkable. He’s carrying the flag now. He has taken over many of those classes that were introduced earlier, like computational probability. It’s like the university can’t get enough of his teaching. He’s fantastic.

CRH: So even the way to teach elementary courses is very important. Not just to give the course but one has to prepare for it.

SG: I agree. I think we had a conversation about this. It’s always easy to underestimate how much work it is to teach well. I never was totally comfortable with this notion of some of my colleagues are great teachers; some of my colleagues are terrible teachers; because as far as I can tell, the terrible teachers put proportionally less time into it. Now, it’s again a case of which comes first, but it seems to me there are same basic principles of organizations. Nothing wrong with an outline, nothing wrong with repeating it several times during lectures so students, even if they get lost, at least they can be reacquainted with what’s going on. These are very simple principles that work. I mean even down to it really helps if they can read your handwriting.
I had a terrible time with that at MIT (Massachusetts Institute of Technology), I taught a six-week sections of a differential equations class. I took it very seriously and worked very hard and was quite convinced it was going great. I had such a good feeling about it. And then I got the evaluations and maybe two thirds, maybe three quarters, maybe more said they couldn’t read anything I wrote on the board. I was stunned, why didn’t they tell me? So anyway, I guess there are simple things that are addressable but they all take time. The careful organization of the material really takes time. Thinking through what comes first, always trying to remember what they know and don’t know. That’s hard because as you get better at these things, as you become an expert, more familiar, you forget where the hard stuff is. You forget what they don’t know. That’s work.

TPL: There is nothing elementary about just anything.

SG: That’s right, not when it comes to human beings and interactions.

TPL: Human communication is something never elementary. Write a complete sentence on the blackboard horizontally, that’s the first step.

SG: That’s a good start and often ignored.

TPL: I remember I had a very important interview because when I about to graduate in 1973 I did not have a job until April. At that time you present your talk on blackboard. So a friend of mine who is not in mathematics helps me to prepare the interview, which is life and death interview. So the most important advice was that when you erase the blackboard, you should go from top down and top down so that it’s very clean. It’s a very useful tip.

SG: That’s a useful tip, as are many others about managing the blackboard. I still teach exclusively with a blackboard, and it is especially effective when a classroom has several of them. I don’t think anything beats it for teaching math, what to leave up, what to take down. Even then if you don’t think about it before you are likely to mess up. You know, I’ll circle things in my handwritten notes that this is going to stay up for the whole lecture or for half the lecture, put it up in a corner somewhere. But preparing all of that takes time.

TPL: We usually ask the first question; something like when did you get interested in mathematics. This is a rather late stage to ask you that question. Is this still a good question for you?

SG: When did I get interested? Well, like I said before, it’s so hard to know whether you’re good at something because you enjoy it or whether you enjoy it because you’re good at it. And I don’t know how to tease those apart or even if it’s meaningful to tease those apart. Starting in high school, I wasn’t off to a very auspicious start. While many of my close friends were taking the fast track, the more advanced math I was deemed to be not ready, so I had to work a little more intensely. I had a private tutor, actually one of the teachers; they told me I had to work with him. I wanted to get into the regular math track. So that forced me to focus on it and realize that I enjoyed it and was good at it.
Again I really don’t know how to pick those apart. So I got interested and by the last year in high school when we hit the calculus, I was quite taken by it. I had the benefit of having my older brother who was just starting graduate math, but was living at home. Whereas my calculus teacher genuinely did not understand a delta from an epsilon and could not keep the order straight in a definition, my brother understood it. So I was able to not just learn it but help the class. The teacher was wonderful in that regard; he let me take a crack at explaining the definition of continuity, the delta epsilon business. So that was great. You learn something else there of course. You learn that there is no better way to learn than to teach. You are not getting up in front of those people unless you understand it. Furthermore, just the act of thinking about explaining it, tells you what you don’t understand. So by then I was pretty well hooked, although as I said my real love was physics.

CRH: Was that the reason or because your brother Don14 was in mathematics that’s the reason you did not go to mathematics first.

SG: That’s a very difficult question. So, did I run away from Don and later chase him and then did he run away from me and stay in pure math longer than he would have liked (according to him) and then chase me in application? I don’t know. I really don’t. I think my interest came very early on. I was one of those kids who tinkered in the basement, took apart radios, took apart everything. Pin ball machines were a very big deal in my suburbia and I became, at a very young age, “Mr. Fix-It”. I would go and fix your pinball machine and mostly the one we had I would take apart, look at the relays and build little computers and so on. Electronics and electricity were fascinating to me. So I think I had a natural interest in the physical sciences. I think my brother had probably none, exactly none. He was probably more naturally suited to mathematics. Although he too switched fields. He was in graduate school in writing, in creative writing specifically. His interests have certainly been very broad.

TPL: When we talk about teaching, there are certain things that really just have to focus on and improve. My English is terrible, that I know. But there is also sometimes the attitude which comes from difficult upbringing experience. Namely you want to survive and you do only just enough to survive, not viewing or learning a language as culture even as a joy. Instead I’m going to learn business English or this English from a book. Just learn just that.

SG: That’s related to what we talked about before. There’s no overstating how much the world has changed and how much it has shrunk and become more competitive, even if it’s just an attitude. It’s different now. People feel busy already when they are in their first day of college. They are concerned. They are planning and worrying. My colleagues will often say all they care about is the grades and getting a job. It’s easy for them to say. They grew up at a different time when you could afford to think more broadly. See I agree with your point completely that they don’t have the time. They don’t see the utility of learning a language. It wasn’t quite so much part of our vocabulary, our way of thinking. We were lucky.

TPL: No, no, but I do appreciate your attitude that we are lucky, self-serving. And this is very a setback. It is very dangerous to make a very conclusive statement. You don’t do that. That’s very nice. Now that we have worked you so hard during this trip, perhaps we can come back for part two of the interview when you come the next time.

SG: I would like that.

TPL: Next time we will work you less hard.

CRH: Because we only ask you very few questions.

SG: Probably it’s my fault for giving such long answers.

TPL: That has been a very happy experience in the past hour. Thank you very much.

SG: Thank you, I look forward to my next interview.

• Tai-Ping Liu and Chii-Ruey Hwang are faculty members at the Institute of Mathematics, Academia Sinica.