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October 1 - October 14, 2024
No: mathematicians want to know it exactly, if they can. Not just because they are weird obsessives, but because they know from experience that getting that exact value often opens unexpected doors and throws light on the underlying math. The mathematical term of art for this exact representation of a number is “closed form.” A
How (asked Zeno) is motion possible? How can we say that an arrow moves, if, at any given instant, it must be somewhere? If all time is composed of instants, and motion is not possible in any given instant, then how is motion possible at all?
Twenty-six decades of effort by some of the best minds on the planet have failed to prove or disprove this simple assertion (which has inspired at least one novel, Apostolos Doxiadis's Uncle Petros and Goldbach's Conjecture29). There are a thousand conjectures like this in arithmetic30; some proved, most still open.
They know, too, from the history of their subject, that even failed attempts can generate powerful new results and techniques. And there is, of course, the Mallory factor. When the New York Times asked George Mallory why he wanted to climb Mount Everest, Mallory replied: “Because it's there.”
The moral of the story is that an infinite series might define only part of a function; or, to put it in proper mathematical terms, an infinite series may define a function over only part of its domain.
France was, then as today, a highly centralized nation.
Andrew Hodges wrote a beautiful book about him (Alan Turing: The Enigma, 1983), and then Hugh Whitemore made a fascinating play based on the book (Breaking the Code, 1986).
“A great deal more is known than has been proved.”
In 1890, in fact, Henri Poincaré published a definitive paper on the three-body problem, making it clear not only that the problem has no closed-form solutions, but that it has another, even more disturbing quality: Its solutions are sometimes chaotic. That is, if you vary the initial conditions of the problem—the numbers equivalent to M and a in my two-body example—very slightly, the resulting orbits change drastically, beyond all recognition. Poincaré himself commented that one set of conditions produced “orbits so tangled that I cannot even begin to draw them.”
