Back to Is Mathematics a Scientific Discipline?

^{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.