Philosophy of mathematics should be tested against five kinds of mathematical practice: research, application, teaching, history, computing.
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The need to check philosophy of mathematics against mathematical research doesn’t require explication. Many important philosophers of mathematics were mathematical researchers: Pascal, Descartes, Leibniz, d’Alembert, Hilbert, Brouwer, Poincaré, Rényi, and Bishop come to mind. Applied mathematics isn’t illegitimate or marginal. Advances in mathematics for science and technology often are inseparable from advances in pure mathematics. Examples: Newton on universal gravitation and the infinitesimal calculus; Gauss on electromagnetism, astronomy, and geodesy (the last inspired that beautiful pure subject—differential geometry); Poincare on celestial mechanics; and von Neumann on quantum mechanics, fluid dynamics, computer design, numerical analysis, and nuclear explosions.
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G. H. Hardy “famously” boasted: “I have never done anything ‘useful’. No discovery of mine has made, or is likely to make, directly or indirectly, for good or ill, the least difference to the amenity of the world.” Nevertheless, the Hardy-Weinberg law of genetics is better known than his profound contributions to analytic number theory. What’s worse, cryptology is making number theory applicable. Hardy’s contribution to that pure field may yet be useful. Twenty years after the war, mathematical purism was revived, influenced by the famous French group “Bourbaki.” That period is over. Today it’s difficult to find a mathematician who’ll say an unkind word about applied math.
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On the basis of this reduction, philosophers of mathematics generally limit their attention to set theory, logic, and arithmetic. What does this assumption, that all mathematics is fundamentally set theory, do to Euclid, Archimedes, Newton, Leibniz, and Euler? No one dares to say they were thinking in terms of sets, hundreds of years before the set-theoretic reduction was invented. The only way out (implicit, never explicit) is that their own understanding of what they did must be ignored! We know better than they how to explicate their work! That claim obscures history, and obscures the present, which is rooted in history.
An adequate philosophy of mathematics must be compatible with the history of mathematics. It should be capable of shedding light on that history. Why did the Greeks fail to develop mechanics, along the lines that they developed geometry? Why did mathematics lapse in Italy after Galileo, to leap ahead in England, France, and Germany? Why was non-Euclidean geometry not conceived until the nineteenth century, and then independently rediscovered three times? The philosopher of mathematics who is historically conscious can offer such questions to the historian. But if his philosophy makes these questions invisible, then instead of stimulating the history of mathematics, he stultifies it. Computing is a major part of mathematical practice. The use of computing machines in mathematical proof is controversial. An adequate philosophy of mathematics should shed some light on this controversy.
