²"Accepting Scientific Ideas," The New York Times, April 28, 1982.

³The first was published; the second and third were prominently exhibited as concept art.

⁴There is a complete Bibliography of academic sources at the end.

⁵Again, see the Bibliography.

⁶E.g. Feyerabend wrote me a contemptuous, dismissive letter (October 23, 1978).

⁷A good example of mathematicians laughing at themselves is the spoof by "Phreilambud" in Reports of the Midwest Category Seminar IV. Such clowning does not redirect the field, nor does it become a firm rebuttal of any professional result.

⁸Yessenin-Volpin and several of his proteges were participants in prestigious conferences, whose proceedings were published by North-Holland and Springer, respectively. Yessenin-Volpin and Isles submitted a grant proposal to the U.S. N.S.F. in 1980. In rejecting the proposal, the judges essentially called Yessenin-Volpin a charlatan. (Copies in my possession.)

⁹In jurisprudence, mathematical proof is cited as the highest standard of proof--proof beyond the shadow of a doubt. Hilbert's Second Problem is the consistency of arithmetic.

¹⁰Mathematical Logic (1967), page 210.

¹¹Joseph Schoenfield, Mathematical Logic (1967), p. 4, p. 9, p. 107.

¹²E.T. Bell, in The Development of Mathematics, raged against the assumption that every culture must see the classical natural number series; or that our culture is more righteous because it does see it.

¹³Although these considerations may be linked to the declared subject-matter as its unadmitted preconditions, e.g. naive arithmetic competence, and the naive hermeneutic of geometry as positional relationships in the visual field.

¹⁴"Constructive mathematics as a philosophical problem," p. 137, in Logic and Foundations of Mathematics (1968).

¹⁵Again see the Bibliography.

¹⁶This will underlie some of the cases to follow, but is far from being the whole story.

¹⁷Blueprint for a Higher Civilization and the other publications.

¹⁸1948A in Collected Works, Volume I

¹⁹A published proof of the Diagonalization Lemma by Craig Smorynski, using a Gödel-type substitution function, would require major remediation not to be specious--at best. It's a question of professional courtesy: he is allowed to get away with it. Self-Reference and Modal Logic (1985), p. 6.

²⁰I.e. modern philosophy of mathematics--H.F.

²¹Philosophical Grammar, p. 322.

²²Taalen teken in de wiskunde Algemeen Nederlands Tijdschrift voor Wijsbegeerte 1947-8, pp. 121-31.

²³Not to mention neo-Platonism: a doctrine which is no longer studied, but which in the Middle Ages was regarded as the definitive science of mind.