2012 saw the death of Bill Thurston, leading geometer and Fields Medalist. Learning of his death led me to re-read his famous 1994 AMS paper on the social nature of mathematical proof. In my opinion, Thurston demolished the views of those who thought mathematics is anything other than socially-constructed. This post is just to present a couple of long quotes from the paper.
The first statement should lead anyone advocating that one-way broadcast communications medium known as MOOC to tread warily in mathematical subjects:
Mathematicians have developed habits of communication that are often dysfunctional. Organizers of colloquium talks everywhere exhort speakers 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 within the first 5 minutes, yet sit silently through the remaining 55 minutes. Or perhaps they 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 mathematicians who are close to the field of the speaker ask a question or two to avoid embarrassment.
This pattern is similar to what often holds in classrooms, where we go through the motions of saying for the record what we think the students “ought” to learn, while the students are trying to grapple with the more fundamental issues of learning [page-break] our language and guessing at our mental models. Books compensate by giving samples of how to solve every type of homework problem. Professors compensate by giving homework and tests that are much easier than the material “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 students rather than with communication: that the students either just don’t have what it takes, or else just don’t care.
Outsiders are amazed at this phenomenon, but within the mathematical community, we dismiss it with shrugs.
Much of the difficulty has to do with the language and culture of mathematics, which is divided into subfields. Basic concepts used every day within one subfield are often foreign to another subfield. Mathematicians give up on trying to understand the basic concepts even from neighboring subfields, unless they were clued in as graduate students.
In contrast, communication works very well within the subfields of mathematics. Within a subfield, people develop a body of common knowledge and known techniques. By informal contact, people learn to understand and copy each other’s ways of thinking, so that ideas can be explained clearly and easily.
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 person to another within the subfield. The same proof would be communicated and generally understood 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 hours or perhaps days by members of the subfield.
Why is there such a big expansion from the informal discussion to the talk to the paper? One-on-one, people use wide channels of communication that go far beyond formal mathematical language. They use gestures, they draw pictures and diagrams, they make sound effects and use body language. Communication is more likely to be two-way, so that people can concentrate on 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.
In talks, people are more inhibited and more formal. Mathematical audiences are often not very good at asking the questions that are on most people’s minds, and speakers often have an unrealistic preset outline that inhibits them from addressing questions even when they are asked.
In papers, people are still more formal. Writers translate their ideas into symbols and logic, and readers try to translate back.
Why is there such a discrepancy between communication within a subfield and communication outside of subfields, not to mention communication outside mathematics? Mathematics in some sense has a common language: a language of symbols, technical definitions, computations, and logic. This language efficiently conveys some, but not all, modes of mathematical thinking. Mathematicians learn to translate certain things almost unconsciously from one mental mode to the other, so that some statements quickly become clear. Different mathematicians study papers in [page-break] different ways, but when I read a mathematical paper in a field in which I’m conversant, I concentrate on the thoughts that are between the lines. I might look over several paragraphs or strings of equations and think to myself “Oh yeah, they’re putting in enough rigamarole to carry such-and-such idea.” When the idea is clear, the formal setup is usually unnecessary and redundant—I often feel that I could write it out myself more easily than figuring out what the authors actually wrote. It’s like a new toaster that comes with a 16-page manual. If you already understand toasters and if the toaster looks like previous toasters you’ve encountered, you might just plug it in and see if it works, rather than first reading all the details in the manual.
People familiar with ways of doing things in a subfield recognize various patterns of statements or formulas as idioms or circumlocution for certain concepts or mental images. But to people not already familiar with what’s going on the same patterns are not very illuminating; they are often even misleading. The language is not alive except to those who use it.” [pages 165-167]
And in a later section Thurston discusses the nature of proofs in mathematics:
When I started as a graduate student at Berkeley, I had trouble 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 graduate students, I gradually began to catch on. Within any field, there are certain theorems and certain techniques that are generally known and generally accepted. When you write a paper, you refer to these without proof. You look at other papers in the field, and you see what facts they quote without proof, and what they cite in their bibliography. You learn from other people some idea of the proofs. Then you’re free to quote the same theorem and cite the same citations. You don’t necessarily have to read the full papers or books that are in your bibliography. Many of the things that are generally known are things for which there may 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 people, and they could produce explanations and proofs, or else refer me to other people 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 were generally known to be incomplete. Mathematical knowledge and understanding were embedded in the minds and in the social fabric of the community of people thinking about a [page-break] particular topic. This knowledge was supported by written documents, but the written documents were not really primary.
I think this pattern varies quite a bit from field to field. I was interested in geometric areas of mathematics, where it is often pretty hard to have a document that reflects well the way people actually think. In more algebraic or symbolic fields, this is not necessarily so, and I have the impression that in some areas documents are much closer to carrying the life of the field. But in any field, there is a strong social standard of validity and truth. Andrew Wiles’s proof of Fermat’s Last Theorem is a good illustration of this, in a field which is very algebraic. The experts quickly came to believe that his proof was basically correct on the basis of high-level ideas, long before details could be checked. This proof will receive a great deal of scrutiny and checking compared to most mathematical proofs; but no matter how the process of verification plays out, it helps illustrate how mathematics evolves by rather organic psychological and social processes.
When people are doing mathematics, the flow of ideas and the social standard of validity is much more reliable than formal documents. People are usually not very good in checking formal correctness of proofs, but they are quite good at detecting potential weaknesses or flaws in proofs.
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 proofs that will stand up to scrutiny are very important. I think the process of proof on the whole works pretty well in the mathematical community. The kind of change I would advocate is that mathematicians take more care with their proofs, making 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 mathematical arguments more explicit and more formal. These are both useful activities that shed new insights on mathematics.” [pages 168-169] (Italics in original)
Let me emphasize several of Thurston’s key statements in the above text:
“Mathematical knowledge and understanding were embedded in the minds and in the social fabric of the community of people thinking about a [page-break] particular topic. This knowledge was supported by written documents, but the written documents were not really primary. [pages 168-169]
. . . .
I was interested in geometric areas of mathematics, where it is often pretty hard to have a document that reflects well the way people actually think.” [page 169]
In our text-dominant culture, the idea that written documents are not primary in a domain would be shocking to many educated people. Yet some institutions of higher learning in mathematical subjects, such as those in the countries of the former USSR, still examine advanced students by oral, not written, exams. (See here for a discussion of the move to written exams in mathematics at Cambridge University.)
William F. Thurston : On proof and progress in mathematics. Bulletin of the American Mathematical Society (New Series), 30 (2): 161-177. Available here.