[Thurston's 17 pages distilled to about 5 pages of highlights]
[Mention the somewhat "dissident" Cornell mathematician DWH who first brought to my attention, and emphasized the importance of..to a whole class) Thurston's paper and how so many strong reactions following in letters to AMS Bulletin that they closed down the entire column, or something like that]
** *** **
Responses to the Jaffe-Quinn article have been invited from a number of mathematicians, and I expect it to receive plenty of specific analysis and criticism from others. Therefore, I will concentrate in this essay on the po sitive rather than on the contranegative. I will describe my view of the process of mathematics, referring only occasionally to Jaffe and Quinn by way of comparison. In attempting to peel back layers of assumptions, it is important to try to begin with the right questions:
1. What is it that mathematicians accomplish?
It would not be good to start, for example, with the question How do mathematicians prove theorems? This question introduces an interesting topic, but to start with it would be to project two hidden assumptions: (1) that there is uniform, objective and firmly established t heory and practice of mathematical proof, and (2) that progress made by mathematicians consists of provin g theorems. It is worthwhile to examine these hypotheses, rather than to accept them as obvious and proceed from there....
...Rather, as a more explicit (and leading) form of the question , I prefer How do mathematicians advance human understanding of mathe - matics? This question brings to the fore something that is fundament al and pervasive: that what we are doing is finding ways for people to understand and think about mathematics.
...If what we are doing is constructing better ways of thinking, then psychological and social dimensions are essential to a good model for mathe matical progress. These dimensions are absent from the popular model. In caricature, the popular model holds that: [...]
..We might call this the definition-theorem-proof (DTP) model of mathematics. A clear difficulty with the DTP model is that it doesn’t explain the source of the questions. Jaffe and Quinn discuss speculation (which th ey inappropriately la- bel “theoretical mathematics”) as an important additional ingredient. Speculation consists of making conjectures, raising questions, and mak ing intelligent guesses and heuristic arguments about what is probably true. Jaffe and Quinn’s DSTP model still fails to address some basic issues. We are not trying to meet some abstract production quota of definitions , theorems and proofs. The measure of our success is whether what we do enables people to understand and think more clearly and effectively about mathematics. Therefore, we need to ask ourselves:
2. How do people understand mathematics?
People have very different ways of understanding particular pieces of mathemat- ics. To illustrate this, it is best to take an example that pra cticing mathematicians understand in multiple ways, but that we see our students str uggling with. The derivative of a function fits well. The derivative can be thou ght of as:
[Thurston lists 7 ways of defining or thinking about it...infinitesimal, symbolic, logical, geometric, as a rate, as a [linear] approximation at a point, and 'microscopic' (zooming-in more and more)]
This is a list of different ways of thinking about or conceiving of the derivative, rather than a list of different logical definitions . Unless great efforts are made to maintain the tone and flavor of the original human insights, t he differences start to evaporate as soon as the mental concepts are translated in to precise, formal and explicit definitions. I can remember absorbing each of these concepts as something new and inter- esting, and spending a good deal of mental time and effort dige sting and practicing with each, reconciling it with the others. I also remember co ming back to revisit these different concepts later with added meaning and unders tanding.
The list continues; there is no reason for it ever to stop. A sa mple entry further down the list may help illustrate this. We may think we know al l there is to say about a certain subject, but new insights are around the corn er. Furthermore, one person’s clear mental image is another person’s intimidati on...[he gives a hypothetically numbered "Definition #37" from a far more advanced specialized field, defining "derivative"]
Human thinking an d understanding do not work on a single track, like
a computer with a single cen tral processing unit. Our brains and
minds seem to be organized into a variety of sep arate, powerful
facilities. These facilities work together loosely, “talk ing” to each
other at high levels rather than at low levels of organization. Here
are some major divisions that are important for mathema tical
(1) Human language. We have powerful special-purpose facil ities for
speaking and understanding human language, which also tie in to readi
ng and writ- ing. Our linguistic facility is an important tool for
thinki ng, not just for communication.....
(2) Vision, spatial sense, kinesthetic (motion) sense. Peo ple
have very powerful facilities for taking in information visually or
kinesthet ically, and thinking with their spatial sense
(3) Logic and deduction. ...
(4) Intuition, association, metaphor. People have amazing facilities
for sensing something without knowing where it comes from (intuition);
for sensing that some phenomenon or situation or object is like
somethin g else (associ- ation); and for building and testing
connections and compar isons, holding two things in mind at the same
time (metaphor). These facilit ies are quite important for
(5) Stimulus-response. This is often emphasized in schools ; for
instance, if you see 3927 × 253, you write one number above the other
and draw a line underneath, etc. This is also important for research
mathematics: seeing a diagram of a knot, I might write down a
presentation f or the fundamental group of its complement by a
procedure that is si milar in feel to the multiplication algorithm.
(6) Process and time. We have a facility for thinking about pr ocesses
or se- quences of actions that can often be used to good effect in
math ematical reasoning. One way to think of a function is as an
action, a pro cess......Mathematically, time is no different from one
mo re spatial dimension, but since humans interact with it in a quite
differ ent way, it is psychologically very different.
(1) Human language. We have powerful special-purpose facil ities for speaking and understanding human language, which also tie in to readi ng and writ- ing. Our linguistic facility is an important tool for thinki ng, not just for communication.....
(2) Vision, spatial sense, kinesthetic (motion) sense. Peo ple have very powerful facilities for taking in information visually or kinesthet ically, and thinking with their spatial sense
(3) Logic and deduction. ...
(4) Intuition, association, metaphor. People have amazing facilities for sensing something without knowing where it comes from (intuition); for sensing that some phenomenon or situation or object is like somethin g else (associ- ation); and for building and testing connections and compar isons, holding two things in mind at the same time (metaphor). These facilit ies are quite important for mathematics....
(5) Stimulus-response. This is often emphasized in schools ; for instance, if you see 3927 × 253, you write one number above the other and draw a line underneath, etc. This is also important for research mathematics: seeing a diagram of a knot, I might write down a presentation f or the fundamental group of its complement by a procedure that is si milar in feel to the multiplication algorithm.
(6) Process and time. We have a facility for thinking about pr ocesses or se- quences of actions that can often be used to good effect in math ematical reasoning. One way to think of a function is as an action, a pro cess......Mathematically, time is no different from one mo re spatial dimension, but since humans interact with it in a quite differ ent way, it is psychologically very different.
3. How is mathematical understanding communicated?
The transfer of understanding from one person to another is n ot automatic. It is hard and tricky. Therefore, to analyze human understandi ng of mathematics, it is important to consider who understands what , and when . Mathematicians have developed habits of communication tha t are often dysfunc- tional. Organizers of colloquium talks everywhere exhort s peakers to explain things in elementary terms. Nonetheless, most of the audience at an average colloquium talk gets little of value from it. Perhaps they are lost withi n the first 5 minutes, yet sit silently through the remaining 55 minutes. Or perhaps th ey quickly lose interest because the speaker plunges into technical details without presenting any reason to investigate them. At the end of the talk, the few mathematici ans who are close to the field of the speaker ask a question or two to avoid embarrassment.
... Books compensate by giving samples of how to solve every type of homework problem. Profe ssors compensate by giving homework and tests that are much easier than the mat erial “covered” in the course, and then grading the homework and tests on a scale that requires little understanding. We assume that the problem is with the studen ts rather than with communication: that the students either just don’t have wha t it takes, or else just don’t care. Outsiders are amazed at this phenomenon, but within the math ematical com- munity, we dismiss it with shrugs..
[Insert mention here, of my own experience in one seminar in matheamtics [admittedly an extrme examle but still somewhat representative] at Cornell, [oops, last 15? 30? minutes error] versus an applied math talk at other end of campus, at Cornell, in 1990s -Harel]
Much of the difficulty has to do with the language and culture of mathematics, which is divided into subfields. Basic concepts used every da y within one subfield are often foreign to another subfield. Mathematicians give up on trying to understand the basic concepts even from neighboring subfields, unless t hey were clued in as graduate students. In contrast, communication works very well within the subfie lds of mathemat- ics....
Mathematical knowledge can be transmitted amazingly fast within a subfield. When a significant theorem is proved, it often (but not always ) happens that the solution can be communicated in a matter of minutes from one p erson to another within the subfield. The same proof would be communicated and generally under- stood in an hour talk to members of the subfield. It would be the subject of a 15- or 20-page paper, which could be read and understood in a few h ours or perhaps days by members of the subfield. Why is there such a big expansion from the informal discussio n to the talk to the paper? One-on-one, people use wide channels of commun ication that go far beyond formal mathematical language. They use gestures , they draw pictures and diagrams, they make sound effects and use body language. C ommunication is more likely to be two-way, so that people can concentrate o n what needs the most attention. With these channels of communication, they are in a much better position to convey what’s going on, not just in their logical and linguistic facilities, but in their other mental facilities as well.
.....But people knowledgeable in multiple fields can also have a negative effect, by intimidating others, and by helping to validate and maintain the whole system of generally poor communication. For example, one effect often takes place during coll oquium talks, where one or two widely knowledgeable people sitting in the front r ow may serve as the speaker’s mental guide to the audience
4. What is a proof?
When I started as a graduate student at Berkeley, I had troubl e imagining how I could “prove” a new and interesting mathematical theorem. I didn’t really understand what a "proof" was.
By going to seminars, reading papers, and talking to other gr aduate students, I gradually began to catch on. Within any field, there are cert ain theorems and certain techniques that are generally known and generally a ccepted. When you write a paper, you refer to these without proof. You look at ot her papers in the field, and you see what facts they quote without proof, and wha t they cite in their bibliography. You learn from other people some idea of the pr oofs. Then you’re free to quote the same theorem and cite the same citations. Yo u don’t necessarily have to read the full papers or books that are in your bibliogr aphy. Many of the things that are generally known are things for which there ma y be no known written source. As long as people in the field are comfortable that the idea works, it doesn’t need to have a formal written source. At first I was highly suspicious of this process. I would doubt whether a certain idea was really established. But I found that I could ask peop le, and they could produce explanations and proofs, or else refer me to other pe ople or to written sources that would give explanations and proofs. There were published theorems that were generally known to be false, or where the proofs wer e generally known to be incomplete. Mathematical knowledge and understandin g were embedded in the minds and in the social fabric of the community of people t hinking about a particular topic. This knowledge was supported by written d ocuments, but the written documents were not really primary.
To avoid misinterpretation, I’d like to emphasize two things I am not saying. First, I am not advocating any weakening of our community standard of proof ; I am trying to describe how the process really works. Careful p roofs that will stand up to scrutiny are very important. I think the process of proo f on the whole works pretty well in the mathematical community. The kind of chang e I would advocate is that mathematicians take more care with their proofs, mak ing them really clear and as simple as possible so that if any weakness is present it will be easy to detect. Second, I am not criticizing the mathematical study of formal proofs, nor am I criticizing people who put energy into making mathemati cal arguments more explicit and more formal. These are both useful activities t hat shed new insights on mathematics
5. What motivates people to do mathematics?
There is a real joy in doing mathematics, in learning ways of t hinking that explain and organize and simplify. One can feel this joy discovering new mathematics, rediscovering old mathematics, learning a way of thinking f rom a person or text, or finding a new way to explain or to view an old mathematical st ructure.
This inner motivation might lead us to think that we do mathem atics solely for its own sake. That’s not true: the social setting is extre mely important. We are inspired by other people, we seek appreciation by other p eople, and we like to help other people solve their mathematical problems. What w e enjoy changes in response to other people. Social interaction occurs throug h face-to-face meetings. It also occurs through written and electronic corresponden ce, preprints, and jour- nal articles. One effect of this highly social system of mathe matics is the tendency of mathematicians to follow fads. For the purpose of produci ng new mathematical theorems this is probably not very efficient: we’d seem to be be tter off having math- ematicians cover the intellectual field much more evenly. Bu t most mathematicians don’t like to be lonely, and they have trouble staying excite d about a subject, even if they are personally making progress, unless they have col leagues who share their excitement.
In addition to our inner motivation and our informal social m otivation for doing mathematics, we are driven by considerations of economics a nd status. Mathemati- cians, like other academics, do a lot of judging and being jud ged. Starting with grades, and continuing through letters of recommendation, hiring decisions, pro- motion decisions, referees reports, invitations to speak, prizes, . . . we are involved in many ratings, in a fiercely competitive system
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-credit s 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, t he mathematical community depends on the social network to distribute the id eas to people who might use them further—the print medium is far too obscure an d 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 migh t only be one or two goals during a soccer game, made by one or two persons. That do es not mean that the efforts of all the others are wasted. We do not judge player s 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 mathematici ans 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. So metimes the group of mathematicians can even anticipate what these theorems a re likely to be. It is much harder to predict who will actually prove the theorem, a lthough there are usually a few “point people” who are more likely to score. How ever, 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 o f the theorems once they are proven. Even if one person could prove all the th eorems in the path single-handedly, they are wasted if nobody else learns them . There is an interesting phenomenon concerning the “point” p eople. It regularly happens that someone who was in the middle of a pack proves a theorem that receives wide recognition as being significant. Their statu s in the community— their pecking order—rises immediately and dramatically. W hen 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 a ccompanying increase in productivity. Second, when their status increases, peop le are more in the center of the network of ideas—others take them more seriously...
This phenomenon convinces me that the entire mathematical c ommunity would become much more productive if we open our eyes to the real val ues in what we are doing. Jaffe and Quinn propose a system of recognized roles di vided into “specula- tion” and “proving”. Such a division only perpetuates the my th that our progress is measured in units of standard theorems deduced. This is a b it like the fallacy of the person who makes a printout of the first 10,000 primes. Wha t we are producing is human understanding. We have many different ways to unders tand and many different processes that contribute to our understanding. W e will be more satisfied, more productive and happier if we recognize and focus on this
6. Some personal experiences
....[after expressing some uneasiness with sharing so personally] ...An interesting phenomenon occurred. Within a couple of year s, a dramatic evac- uation of the field started to take place. I heard from a number of mathematicians that they were giving or receiving advice not to go into folia tions—they were say- ing that Thurston was cleaning it out. People told me (not as a complaint, but as a compliment) that I was killing the field. Graduate studen ts stopped studying foliations, and fairly soon, I turned to other interests as w ell
..Today, I think there are few mathematicians who understand a nything approach- ing the state of the art of foliations as it lived at that time, although there are some parts of the theory of foliations, including developments s ince that time, that are still thriving.
I believe that two ecological effects were much more importan t in putting a damper on the subject than any exhaustion of intellectual re sources that occurred. First, the results I proved (as well as some important result s of other people) were documented in a conventional, formidable mathematici an’s style. They de- pended heavily on readers who shared certain background and certain insights. The theory of foliations was a young, opportunistic subfield , and the background was not standardized. I did not hesitate to draw on any of the m athematics I had learned from others. The papers I wrote did not (and could not) spend much time explaining the background culture. They documented to p-level reasoning and conclusions that I often had achieved after much reflection a nd effort. I also threw out prize cryptic tidbits of insight, such as “the Godbillon -Vey invariant measures the helical wobble of a foliation”, that remained mysteriou s to most mathematicans who read them. This created a high entry barrier: I think many graduate students and mathematicians were discouraged that it was hard to lear n and understand the proofs of key theorems. Second is the issue of what is in it for other people in the subfi eld. When I started working on foliations, I had the conception that wha t people wanted was to know the answers. I thought that what they sought was a coll ection of powerful proven theorems that might be applied to answer further math ematical questions. But that’s only one part of the story. More than the knowledge , people want personal understanding . And in our credit-driven system, they also want and need theorem-credits
..In reaction to my experience with foliations and in response to social pressures, I concentrated most of my attention on developing and present ing the infrastructure in what I wrote and in what I talked to people about. I explaine d the details to the few people who were “up” for it. I wrote some papers giving the substantive parts of the proof of the geometrization theorem for Haken manifol ds—for these papers, I got almost no feedback. Similarly, few people actually wor ked through the harder and deeper sections of my notes until much later. The result has been that now quite a number of mathematicians have what was dramatically lacking in the beginning: a working understan ding of the concepts and the infrastructure that are natural for this subject. Th ere has been and there continues to be a great deal of thriving mathematical activi ty. By concentrating on building the infrastructure and explaining and publishi ng definitions and ways of thinking but being slow in stating or in publishing proofs of all the “theorems” I knew how to prove, I left room for many other people to pick up credit. There has been room for people to discover and publish other proofs of the geometriza-ion theorem. These proofs helped develop mathematical con cepts which are quite interesting in themselves, and lead to further mathematics . What mathematicians most wanted and needed from me was to lea rn my ways of thinking, and not in fact to learn my proof of the geometriz ation conjecture for Haken manifolds. It is unlikely that the proof of the [more] general geometrization conjecture will consist of pushing the same proof further
Not all proofs have an identical role in the logical scaffoldi ng we are building for mathematics. This particular proof probably has only te mporary logical value, although it has a high motivational value in helping support a certain vision for the structure of 3-manifolds. The full geometrization conject ure is still a conjecture. It has been proven for many cases, and is supported by a great d eal of computer evidence as well, but it has not been proven in generality. I a m convinced that the general proof will be discovered; I hope before too many more years. At that point, proofs of special cases are likely to become obsolete.
..In this episode (which still continues) I think I have manage d to avoid the two worst possible outcomes: either for me not to let on that I dis covered what I discovered and proved what I proved, keeping it to myself (pe rhaps with the hope of proving the Poincar ́e conjecture), or for me to present an unassailable and hard- to-learn theory with no practitioners to keep it alive and to make it grow
..On the other hand, I have been busy and productive, in many different activities. Our system does not create extra time for people like me to spe nd on writing and research; instead, it inundates us with many requests and op portunities for extra work, and my gut reaction has been to say ‘yes’ to many of these requests and opportunities. I have put a lot of effort into non-credit-pro ducing activities that I value just as I value proving theorems: mathematical politi cs, revision of my notes into a book with a high standard of communication, explorati on of computing in mathematics, mathematical education, development of new f orms for communica- tion of mathematics through the Geometry Center (such as our first experiment, the “Not Knot” video), directing MSRI, etc. I think that what I have done has not maximized my “credits”. I have been in a position not to feel a strong need to compete for more credit s. Indeed, I began to feel strong challenges from other things besides proving new theorems. I do think that my actions have done well in stimulating mathematics.