In the "Preface for the Teacher", Landau remarks (p. 9): "I do
not, to be sure, prove the consistency of the five Peano axioms
(because that can not be done), ...."
Now we, today, know that this cannot be done (despite the fact
that Gerhard Gentzen actually did it!), but the preface is dated
December 28, 1929.
To be sure, it is not uncommon in the history of mathematics
for facts to be "known" before they are actually proved. No doubt the
ancient Greek mathematicians "knew" that angles could not be trisected
with straightedge and compasses (I believe that Pappus says this
somewhere) even though they did not have a rigorous proof. And I
guess that physicists have "known" for some time that the cubic
face-centered lattice gives the densest packing of spheres (or,
rather, "balls"). But this remark of Landau's seems rather strange
coming from a colleague of Hilbert's in Gottingen in 1929.
I read John Dawson's 1985 article "The Reception of Godel's
Incompleteness Theorems", reprinted in "Perspectives on the History of
Mathematical Logic", ed. by Thomas Drucker, Birkhauser, 1991,
pp. 84--100. Though the article is very interesting, it doesn't shed
much light on Landau's remark. Finsler had published an article
titled "Formal Proofs and Undecidability" in 1926 ---it's reprinted in
van Heijenoort's Source Book. Godel himself considered Finsler's
argument to be nonsense ---evidently with some justification, since
Finsler did not work with any formal system. And Post had done some
work which anticipated Godel's results, but this work had apparently
not been published.
On the other hand, Landau's words "weil man es nämlich nicht
kann" might just mean, "because I don't know how".
So I wonder what Landau had in mind.
Stacy Langton
University of San Diego
langton@acusd.edu