Assorted
THE DICHOTOMY BETWEEN STRUCTURE AND RANDOMNESS, ARITHMETIC PROGRESSIONS, AND THE PRIMES, TERENCE TAO, math/0512114
Abstract. A famous theorem of Szemer´edi asserts that all subsets of the integers with positive upper density will contain arbitrarily long arithmetic progressions. There are many different proofs of this deep theorem, but they are all based on a fundamental dichotomy between structure and randomness, which in turn leads (roughly speaking) to a decomposition of any object into a structured (low-complexity) component and a random (discorrelated) component. Important examples of these types of decompositions include the Furstenberg structure theorem and the Szemer´edi regularity lemma. One recent application of this dichotomy is the result of Green and Tao establishing that the prime numbers contain arbitrarily long arithmetic progressions (despite having density zero in the integers). The power of this dichotomy is evidenced by the fact that the Green-Tao theorem requires surprisingly little technology from analytic number theory, relying instead almost exclusively on manifestations of this dichotomy such as Szemer´edi’s theorem. In this paper we survey various manifestations of this dichotomy in combinatorics, harmonic analysis, ergodic theory, and number theory. As we hope to emphasize here, the underlying themes in these arguments are remarkably similar even though the contexts are radically different.I began to treat the oppositional pair lawlike/happenstantial here, but there's clearly much more to be said.
Another quotation from MacIntyre which is very relevant to the themes of How Mathematicians May Fail to be Fully Rational. This one is from The Essential Contestability of Some Social Concepts, Ethics 84(1) 1-9, 1973 (available on JSTOR)
Now this was written at the end of that fascinating period in the philosophy of science when the protagonists fought tooth and nail to establish their representation of science through historical case studies. Since that time there has been a steady trend of separation between philosophy of science and history of science. I don't think this is a happy state of affairs.Consider...the continuing argument between Kuhn, Lakatos, Polanyi, and Feyerbend, an argument in which what is at stake includes both our ability to draw a line between authentic sciences and degenerative or imitative sciences, such as astrology or phrenology, and our ability to explain why "German physics" and Lysenko biology are not to be included in science. A crucial feature of these arguments is the way in which dispute over the norms which govern scientific practice interlocks with debate over how the history of science is to be written. What identity and continuity are recognized will of course depend on what side is taken in these latter debates but since these debates are so intimately related to the arguments about the norms governing practice, it turns out that the dispute over norms and the dispute over continuity and identity cannot be separated. (p. 7)
History and philosophy of mathematics have never come that close together, despite Lakatos's efforts. At the Mykonos conference at which I first read my paper, subtle tensions emerged which need to be treated. Leo Corry (historian) made a distinction between historians' history, on the one hand, and fictional mathematical writings (e.g., Uncle Petros and the Goldbach Conjecture) and mathematicians' histories (by, e.g., Bourbaki), on the other. Poetic licence is dangerous when a history is presented as factual. But there's a dissatisfaction that runs the other way. Barry Mazur (mathematician) suggested that we don't yet have a good history of Euclidean mathematics, despite the reams of pages written by historians on this period. So here's the question: Is there a way of writing a truthful history of mathematics which would fully satisfy the mathematician? (Both Corry and Mazur's papers are available here.)
From a very good blog in the field in which I an currently working, John Langford gives his vision of what the Web could do to facilitate research in machine learning. His remarks seem to me just as applicable to mathematics.
5 Comments:
For now, just a boring comment: you should make it easy for people to read the math papers you're talking about, by attaching a link whenever you mention their arXiv numbers. There's a huge difference between what people will read if they can get there with a single mouse click, and what they'll read if they happen to know how to use an arXiv reference number to find a paper on the arXiv...
and most philosophers probably don't know how to do that.
Let's see if I can do it here:
THE DICHOTOMY BETWEEN STRUCTURE AND RANDOMNESS, ARITHMETIC PROGRESSIONS, AND THE PRIMES, TERENCE TAO, math/0512114.
Good. It works.
Btw, if you can delete the stupid thing saying "This post has been removed by the author", that would be nice.
Your question:
So here's the question: Is there a way of writing a truthful history of mathematics which would fully satisfy the mathematician?
My response:
Hartshorne's Geometry Euclid and Beyond may give a clue to the type of history a mathematician might favour. Basically a mathematician might favour a view which, e.g., sees the discovery of the ancients still, by in large, as being results for today. (I would suspect that Hartshorne would accept this view, with the understanding that Euclid needs to be updated in a number of ways which were definitively mapped out by Hilbert in his Foundations.) Euclid's proof of the Pythagorean theorem still seems right to many mathematicians, despite the insistence of historians that Greek mathematics can only be understood in the context of ancient society. Histories which maximize historical relativism at the expense of the claim that lasting truths are discovered ultimately undermine even the most evidently correct mathematics (e.g., many results in finite mathematics). (Historical relativization of mathematics has contributed to, in some countries, the almost complete theoretical victory of an extreme constructivism in mathematics education in which it is claimed that the work of students must be considered right, even in the face of established mathematical authority.) So an answer to your question is this: a "natural" mathematical way to write a history of mathematics would be to accept the results of the past as results for today as well (if they remain convincing).
Perhaps this isn't so important in these days of fast connections, but remember that there are mirror sites. The American ArXiv site should tell you your nearest mirror.
Dennis, what you say points to one important battle-line. I would not want this to be the only one, however, because I also think it the concern of philosophy to consider how a result is thought about in a given historical era, whether it's seen as essential to a field or peripheral, etc. Any history must, if only implicitly, depict the relative importance with which different portions of mathematics are imbued. The question then arises as to whether changes in this attribution of importance can be said to be rational.
David, although I may have pointed to a “battle-line”, that was not my aim. I just wanted to observe a dynamic which can be set in place by historical relativism. What stance to take toward robust historical relativism (in mathematics education and elsewhere in mathematics) is another question.
I would think that a history which accepts the results of the past as results for today would need to sort out what is essential in any given era. Can such a stance accommodate rational changes in attribution of importance of different portions of mathematics? Perhaps this question can be addressed by addressing another question: whether such a stance can be historically objective. To address this question one might consider investigating the conditions for the possibility that such a history can be objective. These conditions might be rational or, at least, impose rational constrains.
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