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1Cosmology, History, and Theology, ed. Wolfgang Yourgrau and Allen Breck (1977), p. 99.

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.

24And cf. "A Draw That Is Really a Win," in James Gleick's article on computer chess, The New York Times, August 26, 1986, p. C1.