Abstract
Mathematics in the first half of the twentieth century saw two striking tendencies. Dominant mathematical schools turned sharply away from interest in applications; and during the same years a “naïve platonism” became the dominant metaphysics among working mathematicians.
These two apparently related tendencies are so familiar we may forget how strange they are. The rejection of applications is singular in being so extreme, in comparison with parallel moves within other sciences. “Naïve platonism” is bizarre in that (i) most mathematicians insist on living by it but don’t really believe it; (ii) it followed an earnest examination of foundations which did not in any logical way lead to it; (iii) it is out of tune with the notions of scientific theory-building and theory-testing which developed during the same decades.
Explanation of such a peculiar phenomenon is called for. It should be sought in the social context of science and perhaps also in the state of physical theories. Only some very preliminary suggestions toward such an explanation are made in this paper. Specifically, two “folk explanations” are evaluated and found pertinent but insufficient; and it is ventured that a run of successes in applications, partly fortuitous, lengthened the sway of a paradigm internal to mathematics by making mathematics relatively exempt from criticism from outside.
The concluding section comments cursorily on the prospects for rapprochement between the prevailing mode of thought of academic mathematics and that of applied mathematics.
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I am perhaps referring to the same historical currents as Richard Courant in the introduction to Courant and Hilbert’s Methoden der Mathematischen Physik in 1924, which he still thought timely when the English edition appeared in 1953. ... mathematicians, turning away from the roots of mathematics in intuition, have concentrated on refinement and emphasized the postulational side of mathematics, and at times have overlooked the unity of their science with physics and other fields.... This rift is unquestionably a serious threat to science as a whole. Yet A. Douady writing just last year in the Gazette des Mathématiciens put the split between physicists and mathematicians as occurring about 1930. No later, surely.
Paul Halmos assembles in “Applied mathematics is bad mathematics” (in Mathematics Tomorrow, ed. L.A. Steen, Springer, New York, 1981) a number of polemical points, including this one.
An indication of wherein my informal statement of it is imprecise: All specific definitions comprise a countable set, perhaps, and the real numbers are said to be a more-than-countable set, yet all real numbers are said to exist.
Those who do not find it preposterous may find further polemic elsewhere. See for example Errett Bishop, Foundations of Constructive Analysis, McGraw-Hill, New York, 1967; New Directions in the Philosophy of Mathematics (ed. Thomas Tymoczko), Birkhäuser, 1985, especially Part I; and my own “Criticisms of the usual rationale for validity in mathematics”, in Physicalism in Mathematics (ed. A. Irvine), Kluwer, Dordrecht, 1989, pp. 343–356.
I am referring only to the first “crisis in foundations”, from Cantor and Weierstrass to Russell and Hilbert and Brouwer. The “crisis” to which Kurt Gödel was central came later,and did not reinforce naïve platonism so far as I can tell.
Hermann Weyl, to take the most striking example. See his Philosophy of Mathematics and Natural Science, Princeton 1949, especially p. 234. I discuss this in III of my “Materialist mathematics” in For Dirk Struik (ed. R.S. Cohen, J. Stachel, M. Wartofsky), Boston Studies in the Philosophy of Science XV, 1974.
Quoted by J.L. Kelley, “Once over lightly”, in A Century of Mathematics in America, Part III (ed. P. Düren, R.A. Askey, H.M. Edwards, U.C. Merzbach), Providence, American Mathematical Society, 1989, pp. 471–493, see p. 483.
In saying this I am asserting something about modernism. Medieval and Renaissance scientific mystique attached to other sciences as readily as to mathematics.
A theoretically ambitious attempt is Peter Bürger, Theorie der Avantgarde, Suhrkamp, Frankfurt, 1974; and explanatory suggestions are scattered through social histories like Carl Schorske, Fin-de-siécle Vienna, Knopf, New York, 1980.
Some mutual influence between different modernisms is seen in such movements as pata-physics. See Oulipo: A Primer of Potential Literature (ed. W.F. Motte), University of Nebraska Press, Lincoln & London, 1986. But this seems quite peripheral to me.
His article so titled is in Commun. Pure Appl. Math. 13(1960), 1–14. A disclaimer: some of the important queries made by Wigner fall outside the scope of my analysis here.
As Kurt Friedrichs said in irritation, “In analysis the phrase ‘everywhere dense’ is everywhere dense.”
“Criticisms of the usual rationale for validity in mathematics”, in Physicalism in Mathematics (ed. A. Irvine), Kluwer, Dordrecht, 1989, pp. 343–356; see Sec. 4(iii).
Gazette des Mathématiciens 32 (1987), 20–26.
I counted fewer. But he might have noted that of the five medals awarded at the Congress (four Fields medals and one Nevanlinna prize) at least three went to rather application-motivated work.
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Davis, C. (1994). Where Did Twentieth-Century Mathematics Go Wrong?. In: Sasaki, C., Sugiura, M., Dauben, J.W. (eds) The Intersection of History and Mathematics. Science Networks · Historical Studies, vol 15. Birkhäuser Basel. https://doi.org/10.1007/978-3-0348-7521-9_9
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