Communication of mathematics

Next topic: Communication of mathematics: Hilbert, in his famous speech at the International Congress in 1900, in order to make a point about mathematical communication, cited a French mathematician who said: "A mathematical theory is not to be considered complete until you have made it so clear that you can explain it to the first man whom you meet on the street". In order to pass on to new generations of mathematicians the collective knowledge of the previous generation, how important is it that the results have simple and elegant proofs?

ATIYAH: The passing of mathematics on to subsequent generations is essential for the future, and this is only possible if every generation of mathematicians understands what they are doing and distils it out in such a form that it is easily understood by the next generation. Many complicated things get simple when you have the right point of view. The first proof of something may be very complicated, but when you understand it well, you readdress it, and eventually you can present it in a way that makes it look much more understandable - and that's the way you pass it on to the next generation! Without that, we could never make progress - we would have all this messy stuff. Mathematics does depend on a sufficiently good grasp, on understanding of the fundamentals so that we can pass it on in as simple a way as possible to our successors. That has been done remarkably successfully for centuries. Otherwise, how could we possibly be where we are? In the 19^th century, people said: "There is so much mathematics, how could anyone make any progress?" Well, we have - we do it by various devices, we generalize, we put all things together, we unify by new ideas, we simplify lots of the constructions - we are very successful in mathematics and have been so for several hundred years. There is no evidence that this has stopped: in every new generation, there are mathematicians who make enormous progress. How do they learn it all? It must be because we have been successful communicating it.

SINGER: I find it disconcerting speaking to some of my young colleagues, because they have absorbed, reorganized, and simplified a great deal of known material into a new language, much of which I don't understand. Often I'll finally say, "Oh; is that all you meant?" Their new conceptual framework allows them to encompass succinctly considerably more than I can express with mine. Though impressed with the progress, I must confess impatience because it takes me so long to understand what is really being said.

Has the time passed when deep and important theorems in mathematics can be given short proofs? In the past, there are many such examples, e.g., Abel's one-page proof of the addition theorem of algebraic differentials or Goursat's proof of Cauchy's integral theorem.

ATIYAH: I do not think that at all! Of course, that depends on what foundations you are allowed to start from. If we have to start from the axioms of mathematics, then every proof will be very long. The common framework at any given time is constantly advancing; we are already at a high platform. If we are allowed to start within that framework, then at every stage there are short proofs.

One example from my own life is this famous problem about vector fields on spheres solved by Frank Adams where the proof took many hundreds of pages. One day I discovered how to write a proof on a postcard. I sent it over to Frank Adams and we wrote a little paper which then would fit on a bigger postcard. But of course that used some K-theory; not that complicated in itself. You are always building on a higher platform; you have always got more tools at your disposal that are part of the lingua franca which you can use. In the old days you had a smaller base: If you make a simple proof nowadays, then you are allowed to assume that people know what group theory is, you are allowed to talk about Hilbert space. Hilbert space took a long time to develop, so we have got a much bigger vocabulary, and with that we can write more poetry.

SINGER: Often enough one can distil the ideas in a complicated proof and make that part of a new language. The new proof becomes simpler and more illuminating. For clarity and logic, parts of the original proof have been set aside and discussed separately.

ATIYAH: Take your example of Abel's Paris memoir: His contemporaries did not find it at all easy. It laid the foundation of the theory. Only later on, in the light of that theory, we can all say: "Ah, what a beautifully simple proof!" At the time, all the ideas had to be developed, and they were hidden, and most people could not read that paper. It was very, very far from appearing easy for his contemporaries.

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