Back to H.F. Philosophy contents

^{2}"Accepting Scientific Ideas," *The New York Times*, April 28,
1982.

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

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

^{5}Again, see the Bibliography.

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

^{7}A 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.

^{8}Yessenin-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.)

^{9}In 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.

^{10}*Mathematical Logic* (1967), page 210.

^{11}Joseph Schoenfield, *Mathematical Logic* (1967), p. 4, p. 9,
p. 107.

^{12}E.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.

^{13}Although 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).

^{15}Again see the Bibliography.

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

^{17}*Blueprint for a Higher Civilization* and the other
publications.

^{18}1948A in *Collected Works*, Volume I

^{19}A 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.

^{20}I.e. modern philosophy of mathematics--H.F.

^{21}*Philosophical Grammar*, p. 322.

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

^{23}Not 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.

^{24}And 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.