I first met Tadashi Tokieda at the end of June 2011 during a mathematics summer school for young students. Apart from the august presence of Conway and his disdain for his own game of life, T2 was one of the most memorable presences at the event, tantalizing us with his intriguing toys—which he defines as “objects of daily life we can find or make almost instantly, yet, if played with imaginatively, exhibit a behavior so surprising that it will leave good scientists puzzling over it for quite a while.” We would spend entire afternoons in the hope of piercing through their thin yet hard shell of apparent simplicity. As a young Oxford mathematician still in the wake of post-adolescent arrogance and with no proneness for subtlety, I might have dismissed T2’s toys as pure curiosities, deemed of the shelves of a curiosity shop. But I still vividly remember the mesmerizing quality of his talks, his clarity of thought, and his use of language—poignant and full of purpose and determination. These are indeed trademarks of T2’s talks of science popularization, which he gives with delightful fluency in your language of choice, whether that is English, French, Japanese, or whatever else lies on his polyglot spectrum.
In this age of over-specialization, T2, who is currently Director of Undergraduate Studies and mathematics tutor at Trinity Hall, University of Cambridge, and Professor of Mathematics at the University of Stanford, is the embodiment of an almost extinct homo universalis. Born in Japan, he was a painter child prodigy with acclaimed exhibitions. After persuading his reluctant family and the even-more-so French consulate, he moved to France at the critical age of fourteen, where he fell in love with the language and dedicated himself to philology. Later on, while working though a Russian collection of problems, he learned basic mathematics, which led him into doctoral studies in topology at Princeton University.
Following the talks he gave at OIST in March 2016, a conversation ensued over lunchtime. Before we plunged into our discussion, T2 noted that “scientists and especially mathematically-minded people are terrible, because they never give straight answers” and gave us two prefaces: “First is that I really don’t like to philosophize and pontificate. Not because it is morally wrong, rather that such discourses tend to be really boring because people tend to say completely trifle things, and you don’t learn anything new. The second preface is that I think in science you do whatever you like. It’s up to your taste, and of course your taste is different from my taste, and therefore your taste is bad and my taste is good.” Thus T2 added, “It’s important in science to distinguish two things: what you really like, scientifically and otherwise, versus what you have been led, bullied by various influences, into believing that you should like.” In light of this, emphasizing a lack of poignancy and universality of his discourses, he answered his first question.
The discussion has been excerpted below and edited for length and clarity.
On legitimate scientific questions to answer.
What I find most entertaining and the best use of my time is when I work on phenomena that are surprising. What I mean by this is the following: often in mathematics there are periods of your learning and in fact areas of mathematics where there is a result that you write down, which actually you find fairly intuitively obvious, but you have to prove it rigorously—you have to justify it. But at the end of the proof sometimes, not always, but sometimes, all you have established is the reassurance that it is true. And the fact of the matter is, unless you keep working in that field, after several years you forget what the proof was. All you remember is that you proved it once. […] But you see, in some sense I have not become any smarter. There is some sort of philosophical satisfaction that some mathematical result, which people weren’t sure about, is now a sure result, but personally I have not become any smarter. That, of course, is a part of science that is very important, but many of the really fashionable, cutting edge subjects tend to have lots of incremental research at this time. The program is already set. People think that this is really an exciting direction, this is the most important conjecture, these are the most relevant results that you should establish, and so on. But there is actually no real surprise; whereas, a chain of metal beads in free fall suddenly standing up—that is actually surprising. It might be modest; yes, it might not be so important in the grand cosmic scheme of things, but it is actually surprising, something that you didn’t expect would happen, you never thought would happen, and there is suddenly a small, but genuinely new part of the universe before you, and I feel I have become smarter. I have become a richer person. So surprise is very important. And I like to pursue things that, even if modest, genuinely surprise me.
On incremental work in science.
It’s a delicate balance, especially since it couples into your current career stage. And of course the further ahead you are, the more freedom you have, but at the same time here’s maybe a more practical way. I can’t deny, of course as I said in my final speech, that it is extremely important to do incremental work, technical work that guarantees that things are working properly and so on. But I think in any such activity in life, you can still try to find some surprise, and it will make life a little more interesting for you, and probably will help your research. And also when you write a paper it will make for more interesting writing. Somehow the most practical policy that I have is “am I going to spend ten hours of my life on this, or maybe ten days, or ten months of my time on this? Am I going to become any smarter after this than before?”
On language and science.
When you learn science from textbooks and school curricula and take examinations, you have the impression that there is a very set syllabus. And of course because the book is finite, there’s only a finite amount of material, it’s ordered in a certain fixed way, and so forth. Also when you learn languages, if you’re learning from a textbook, or in a classroom setting, the experience is very finite. But then you go out and live the life of that language and you discover that there is an infinite variety of the use of language: you meet different people, they speak differently, you have different situations. I think that is the greatest similarity. Human brains are wonderful adaptation machines: you give the brain only a finite amount of information, and then somehow it is able to use it in an infinite variety universe. It is not clear why that should be the case; you’d think its input should be more or less equivalent to its output. Maybe there’s a factor. Maybe there’s a dependence. But somehow finite input gets out as an infinite output, and it’s really strange.
That is something common to both languages and sciences, and it leads me to the following suspicion. Especially in mathematics, but also in physics and other sciences, you have this traditional ideology that somehow the description of science should be complete—it should consist of definitions, a logical system, and the rules by which you play within that system. Somehow everything is explicit. You want to exclude human subjectivity and imagination from science, or in other words you want a computer program—but, I think that’s false. We come to science with a lot of personal bias, most of which we are not aware of: I respond differently to the same formalistic system from the way you respond. Somehow that creates a huge diversity in the application and use of science. Even learning from the same book, the same teacher, and taking the same examination, two people can emerge with completely different views. Mind you, they both worked very hard and both understood everything. So I would say that despite the ideological official line that science must be explicit and everything must be written down, actually, most of science is very implicit and within human subjective experience. Do you find it regrettable? I don’t know, but because I like to have fun with science, I’m comfortable with that kind of situation.
Though taking the metaphor a bit too far, we couldn’t help but notice the connection with Gödel’s Incompleteness Theorems, as you cannot have a formal system being both complete and consistent, and moreover you’d be false to think we, as humans, are fully consistent.
[Gödel’s Theorems] say there are true statements that you cannot prove. Incompleteness results, while absolutely true and relevant, are still the negative results, and [much] goes beyond them. In some sense, this very formalistic and finite system called science, [and] very formalistic and finite system called language, have within themselves much more intelligence than was designed; this extra intelligence somehow comes from the resonance between that and individual humans.
The reason I’m telling you all this is that I think with science, this is really a heresy. People simply do not accept this, and in fact mathematics is an extreme example because you have to write down everything according to a formalistic system. But I think the truth of the matter is really different from what this ideology says. With languages people have a much easier time accepting all this, and in fact I have never heard any claim that language has to be explicit.
I also want to mention another similarity between languages and science, but again in the case of language, it’s much easier to understand. […] When you learn a language you really have to live the life of the language, but I think it is the same with science actually. You keep hearing about it, and scientific biographies are very bad with this, as they tend to mislogically [sic] glorify some geniuses. All of that helps—if you’re a fast thinker, if you have good memory, and if you somehow have an aptitude for this and that—but there is a path that no genius can hurry, which is living the life of a science. I am sure you have had this experience—I have had this many times—I don’t understand a piece of science, I find it very uncomfortable, so I get fed up and I give up. Two or three years later, I’ve done nothing on this, but when I come back to it, suddenly it all becomes obvious and it’s part of myself! But it’s somehow the same process as learning a language; we only begin to understand its details when cultivating human relationships. Then you’ve domesticated it and made the language part of yourself. So there are lots and lots of similarities between languages and science, but usually with languages people understand the situation much better. Science is really ring-fenced behind a fortress of myths and received ideas, while the process is actually much more flexible.
On popularizing science. T2 refers to one of the gifts that he kindly offered us: the unexpected shapes that result from cutting various combinations of paper loops and Möbius strips. Before you go on we recommend you watch the Numberphile videos in which he shows us these surprising results.
This [Möbius strips experiment], this is very good science in the sense that even the best scientists usually don’t see it coming. And yet it’s something that you can do. Then you can start asking, for example, what happens if you glue two Möbius strips of the same chirality. Now that’s an interesting situation. I’m usually dissatisfied with lots and lots of popularizations, even by the great examples of such popularizations. They usually have the attitude that “Oh, we are guardians of this great temple of science, and there is some really difficult and complicated stuff, I’m going to simplify for you, because you are too stupid to understand complicated science, and I tell you what to be excited about, and you shall be excited about this.” I’d rather give them something that they themselves find exciting.
Everyone—and it can be a schoolchild, or a seasoned scientist, or a road worker who walked in, and I’ve done this in an African village with completely random farmers—asks the same question. For once, and it doesn’t happen very often, you have as much access to the truth of the universe as I have; you can try this and your discovery will be as good as my discovery, Professor of Mathematics at Stanford University. It’s true that in principle science is all over the place, but it is hard to find a nice instance of surprising science from daily life. It exists, and when it does that is wonderful material to share with people because now they take possession, and it’s their thing, and they find it surprising. And not because they’ve been told that it’s something to be surprised and excited about. But the real challenge is finding such material; it requires as much imagination and as much scientific savviness as doing incremental research at the forefront of science.
In general I really don’t like the metaphor, the language of the “cutting edge” or “frontier” of science. That’s a picture of science where you have completely cleared part of the jungle and there’s an edge, beyond which you don’t know what is going on, and the only imperative is cutting more trees. But I don’t think it’s like that at all. Of course there are things that can progress only because you have bigger machines and more expensive and faster computers, but science is not just happening on the edge. Just under your feet sometimes there is unexplored wilderness. I like this image of science where not because you are cutting more and more trees you expand the clearing, but actually you are feeling and discovering the riches just under your feet. I think for popularizing science this realization is very important. Otherwise you become a merchant of standard things.
The way we manage science doesn’t favor change of attitude, because, you see, when you submit a paper to a journal or to a funding agency it can be rejected in one of two ways. One common way, which is very reasonable: the referees say that your work has been done before. Another way in which it can be rejected, which is also common, is “no, we cannot accept this, because it has never been done before.” You laugh, but it’s actually a very common ground for refusal. It’s a very delicate balance, but that’s how most of the funding works, and we are very reliant on funding.
You’d ask, “Tadashi, what do you mean by this? Where do we go from here?” There are lots of projects that need enormous amounts of money and human power, but I think it is important to fund the kind of science that needs a little bit of support for lots of people. Ask them to just go out there, do whatever strikes their fancy. In many parts of the economy and organization of, shall I say, capitalistic society, this approach is not unknown. But somehow science is completely stuck in the, shall I say, communistic, planned-economy approach, and it’s not conducive to [changes of attitude]. I think a mixed strategy is much better. But I’m a voice in the desert—I don’t think it’s going to happen any time soon.
On striking the balance between creative and traditional approaches.
Extremely important and difficult question. Of course there are stages and areas of science where you just have to slog your way through. But even then it’s very important and helpful, and it’s nice for your personal life, to look for surprises. In mathematics you often find situations where some obvious or intuitive result must be proved rigorously. Now, that’s of course a necessary part of science, but it’s also nice to have surprises. When you meet a surprise and then understand what’s happening behind the surprise, you become a little smarter.
However, let me draw another instance of the Romantic analog: people say the life of Lord Byron was very romantic— he went to Greece and fought in the war, and he was in love with so many women—and that is why he could write such wonderful poetry. I don’t actually like Byron’s poetry so much. But ordinary people, who for example work in a factory and then come home and live in a little apartment, live a very grey and miserable life; well, is it all miserable? Sometimes there are wonderful things in those lives as well. It’s a mystery. I think in science also, if you don’t forget, you can still keep looking for surprises and somehow water the plants, as it were, and then make something that is really nice every once in a while. Not every day. It’s not ecstasy after ecstasy after ecstasy, but nonetheless you don’t have to have the adventurous, romantic life of Lord Byron to have romance in life. In fact, you can find surprises and pleasure in everyday tasks—a pleasure as worthy as the greatest adventures of Lord Byron. By all means, you have to finish your next assignment, but again, keep looking for surprises. It’s all a mystery—and I wish you all the best.
Drawings by Tadashi Tokieda
Adi David is a PhD student in the class of 2015. A trained mathematician, he spent his undergraduate at Oxford, and before coming to OIST he lived in Romania, France, Germany, and Japan. He is an avid traveller and likes to engage with the arts, contemporary literature, and social issues—tending to take on more projects than feasible.