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The field is a fractal

In a blog comment, Winston Lin points to this quote from Bill Thurston:

There is a real joy in doing mathematics, in learning ways of thinking that explain and organize and simplify. One can feel this joy discovering new mathematics, rediscovering old mathematics, learning a way of thinking from a person or text, or finding a new way to explain or to view an old mathematical structure.

This inner motivation might lead us to think that we do mathematics solely for its own sake. That’s not true: the social setting is extremely important. We are inspired by other people, we seek appreciation by other people, and we like to help other people solve their mathematical problems. What we enjoy is changes in response to other people. Social interaction occurs through face-to-face meetings. It also occurs through written and electronic correspondence, preprints, and journal articles. One effect of this highly social system of mathematics is the tendency of mathematicians to follow fads. For the purpose of producing new mathematical theorems this is probably not very efficient: we’d seem to be better off having mathematicians cover the intellectual field much more evenly. But most mathematicians don’t like to be lonely, and they have trouble staying excited about a subject, even if they are personally making progress, unless they have colleagues who share their excitement.

Fun quote but I disagree with the implications of the last bit. The trouble with the quote is the implication that there is a natural measure on “the intellectual field” so that it can be covered “evenly.” But I think the field is more of a fractal with different depths at different places, depending on how closely you look.

If we wanted to model this formally, we might say that researchers decide, based on what other people are doing, which parts of their fields are worth deeper study. It’s not just about being social, the point is that there’s no uniform distribution. To put it another way, following “fads,” in some sense, is a necessity, not a choice. This is not to say that whatever is currently being done is best; perhaps there should be more (or less) time-averaging, of the sort that we currently attain, for example, by appointing people to long-term job contracts (hence all the dead research that my colleagues at Berkeley were continuing to push, back in the 90s). I just want to emphasize that some measure needs to be constructed, somehow.


  1. Winston Lin says:

    Thanks for your comments, Andrew! I agree, and I’ve wondered if Thurston was fully serious in that next-to-last sentence.

    Minor correction: You’ve accidentally inserted an “is” in “What we enjoy changes in response to other people.”

    Finally, I’ll use your post as an excuse to quote the rest of the section (“What Motivates People To Do Mathematics?”), and encourage people to read the whole essay.

    “In addition to our inner motivation and our informal social motivation for doing mathematics, we are driven by considerations of economics and status. Mathematicians, like other academics, do a lot of judging and being judged. Starting with grades, and continuing through letters of recommendation, hiring decisions, promotion decisions, referees reports, invitations to speak, prizes, . . . we are involved in many ratings, in a fiercely competitive system.

    “[blank line]

    “Jaffe and Quinn analyze the motivation to do mathematics in terms of a common currency that many mathematicians believe in: credit for theorems.

    “I think that our strong communal emphasis on theorem-credits has a negative effect on mathematical progress. If what we are accomplishing is advancing human understanding of mathematics, then we would be much better off recognizing and valuing a far broader range of activity. The people who see the way to proving theorems are doing it in the context of a mathematical community; they are not doing it on their own. They depend on understanding of mathematics that they glean from other mathematicians. Once a theorem has been proven, the mathematical community depends on the social network to distribute the ideas to people who might use them further—the print medium is far too obscure and cumbersome.

    “Even if one takes the narrow view that what we are producing is theorems, the team is important. Soccer can serve as a metaphor. There might only be one or two goals during a soccer game, made by one or two persons. That does not mean that the efforts of all the others are wasted. We do not judge players on a soccer team only by whether they personally make a goal; we judge the team by its function as a team.

    “In mathematics, it often happens that a group of mathematicians advances with a certain collection of ideas. There are theorems in the path of these advances that will almost inevitably be proven by one person or another. Sometimes the group of mathematicians can even anticipate what these theorems are likely to be. It is much harder to predict who will actually prove the theorem, although there are usually a few ‘point people’ who are more likely to score. However, they are in a position to prove those theorems because of the collective efforts of the team. The team has a further function, in absorbing and making use of the theorems once they are proven. Even if one person could prove all the theorems in the path single-handedly, they are wasted if nobody else learns them.

    “There is an interesting phenomenon concerning the ‘point’ people. It regularly happens that someone who was in the middle of a pack proves a theorem that receives wide recognition as being significant. Their status in the community—their pecking order—rises immediately and dramatically. When this happens, they usually become much more productive as a center of ideas and a source of theorems. Why? First, there is a large increase in self-esteem, and an accompanying increase in productivity. Second, when their status increases, people are more in the center of the network of ideas—others take them more seriously. Finally and perhaps most importantly, a mathematical breakthrough usually represents a new way of thinking, and effective ways of thinking can usually be applied in more than one situation.

    “This phenomenon convinces me that the entire mathematical community would become much more productive if we open our eyes to the real values in what we are doing. Jaffe and Quinn propose a system of recognized roles divided into ‘speculation’ and ‘proving’. Such a division only perpetuates the myth that our progress is measured in units of standard theorems deduced. This is a bit like the fallacy of the person who makes a printout of the first 10,000 primes. What we are producing is human understanding. We have many different ways to understand and many different processes that contribute to our understanding. We will be more satisfied, more productive and happier if we recognize and focus on this.”

    • jrc says:

      “Soccer can serve as a metaphor.” – you had me at “soccer”.

      But to add something – I find this interesting to think about in the context of Quine-ian Holism. Consider the metaphysical/epistemological statement:

      “The totality of our so-called knowledge or beliefs, from the most casual matters of geography and history to the profoundest laws of atomic physics or even of pure mathematics and logic, is a man-made fabric which impinges on experience only along the edges. Or, to change the figure, total science is like a field of force whose boundary conditions are experience….But the total field is so undetermined by its boundary conditions, experience, that there is much latitude of choice as to what statements to re-evaluate in the light of any single contrary experience.”

      Sounds like a fractal, no? Bounded by experience with reality at the exterior and the denseness of reality on the interior. Anyway…

      Thurston’s work (such as you excerpt it) strikes me as a kind of sociology of holism, a discussion of when and where mathematicians choose to extend the boundaries or deepen the inter-connections within their sphere. An economist would conceive of this in terms of trade-offs (difficulty of work, usefulness of contribution, professional incentives, etc.), but the idea is the same – an accounting of how the realities of the academic world (sociological, economic, psychological… human) partially determine when and where the web of knowledge that is science gets strengthened, revised, reconstituted, re-conceived.

      That said, I think in general I favor the idea (that I read in Thurston’s quote) that academic work be guided more by its potential contributions to human understanding than simply by doing whatever is easiest or right in front of our eyes. But that is really hard. It is too bad Entosophy isn’t around to tell me how big a failure I am because it isn’t always easy for me to see where new, productive, interesting work can be done. I usually enjoyed those rants.

    • Martha says:

      Speaking from my experience as a mathematician: I don’t think any one mathematician can speak for all. I take what Thurston says as speaking about himself more than about all mathematicians. I think about mathematicians I know, and see that different things motivate/turn them on in different degrees. Sure, often we’re influenced by “what’s in the air”– but by differing degrees in different people. (And yes, this is a statement about myself — that I tend to see differences.)

  2. Rahul says:

    I don’t think Bill Thurston quite literally meant “evenly” in the sense of a perfectly uniform distribution.

    I think it’s asking for something less spikey than the current status quo.

    • Andrew says:


      Maybe. But one could also make the argument that science is not spikey enough, that too many people work on uninteresting, old-fashioned problems just because they’re following the lead of old textbooks. Maybe a bit more trendiness would be even better. I don’t know, but it’s not obvious to me which direction would be better. I guess it depends on the field of study; I’d guess that some fields are too trendy for the reasons Thurston gives, while others are not trendy enough.

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