1. 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.
2. 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.
3. 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.
4. 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.
5. 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.