Date: Jan 19, 2013 3:53 AM
Subject: Matheology § 193

Matheology § 193

{{In 1927 David Hilbert gave a talk at Hamburg university, where he
explained his opinions about the foundations of mathematics.}} It is a
great honour and at the same time a necessity for me to round out and
develop my thoughts on the foundations of mathematics, which was
expounded here one day five years ago {{compare Kalenderblatt 101212
to 101214
}} and which since then have constantly kept me most actively
occupied. With this new way of providing a foundation for mathematics,
which we may appropriately call a proof theory, I pursue a significant
goal, for I should like to eliminate once and for all the questions
regarding the foundations of mathematics [...]
I have already set forth the basic features of this proof theory of
mine on different occasions, in Copenhagen [1922], here in Hamburg
[1922], in Leipzig [1922], and in Münster [1925]; in the meantime much
fault has been found with it, and objections of all kinds have been
raised against it, all of which I consider just as unfair as it can
be. [...]
Poincaré already made various statements that conflict with my
views; above all, he denied from the outset the possibility of a
consistency proof for the arithmetic axioms, maintaining that the
consistency of the method of mathematical induction could never be
proved except through the inductive method itself. [...] Regrettably
Poincaré, the mathematician who in his generation was the richest in
ideas and the most fertile, had a decided prejudice against Cantor's
theory, which prevented him from forming a just opinion of Cantor's
magnificent conceptions. Under these circumstances Poincaré had to
reject my theory, which, incidentally, existed at that time only in
its completely inadequate early stages. Because of his authority,
Poincaré often exerted a one-sided influence on the younger
generation. {{Not to a sufficient degree, unfortunately. --- Then
Hilbert discusses the objections by Russell and Whitehead and finally
Brouwer. Hilbert concludes:}} I cannot for the most part agree with
their tendency; I feel, rather, that they are to a large extent behind
the times, as if they came from a period when Cantor's majestic world
of ideas had not yet been discovered. {{A world discovered by a man
who was behind his times, who did not recognize atoms in the late 19th
century, but rejected evolution, who believed in an infinite set of
angels and took the basis of his mathematics from the holy bible: "in
infinity and beyond".}}
[E. Artin et al. (eds.): "D. Hilbert: Die Grundlagen der
Mathematik" (1927). Abh. Math. Seminar Univ. Hamburg, vol. 6, Teubner,
Leipzig (1928) 65-85. English translation: J. van Heijenoort: "From
Frege to Gödel", Harvard Univ. Press, Cambridge, Mass. (1967) 464-479]

Regards, WM