I looked up Huntington's paper, Trans. Amer. Math. Soc., 3(1902), 264-279,
which I think is based on the thesis. He claims originality only in proving
independence of his axioms. These are in fact for a type of ordered semigroup
S. He minimized their complexity to insure independence. Huntington builds
the real field from S more or less as modern logic texts do. This doesn't
quite amount to a categoricity proof for the axioms for a complete ordered
field: you need to show there's essentially only one way to construct the
reals from the positive integers. Who finally stated the theorem that way?
Huntington does cite Bettazzi, and notes that the book was reviewed in Bull.
Sci. Math. (2)15(1891), 76-101. But he says nothing further.
What led Huntington to this subject? He's one of the first American
postulate-theorists. I asked Michael Scanlon, who wrote a historical paper
on them in the Journal of Symbolic Logic. Scanlon replied that he knew
little, because Huntington's papers were destroyed at his request after
his death.
Huntington graduated from Harvard in 1895, received a master's degree there
in 1897, then taught at Williams for two years. He received the PhD in 1901
after two years' study at Strassburg. Why did he go there? The only
mathematician I've noticed there with such an interest was Reye. Huntington
is not listed as attending the meetings at the Paris exposition in 1900.
He's listed as attending some meetings in the years before that in the U.S.,
but I can't find anyone else with that interest simultaneously attending. I
haven't found any connection between Huntington then and the postulational
group forming at Chicago. Royce was at Harvard; but I checked his list of
courses and he doesn't seem to have taken an interest in logic until Huntington
returned from Europe. Osgood was at Harvard, had been at Goettingen, and, I
think, corresponded with the Goettinger mathematicians. Is that a possibility?
Has anyone heard of early correspondence between Huntington and anyone in
Europe? He can't have just arrived in Strassburg to go knocking at the gate!
Then there's the question of the relationship between what Bettazzi did and
what Hilbert did. Toepell doesn't seem to mention Bettazzi in his book on
Hilbert's Grundlagen. Fano spent a year in Goettingen around 1893. Perhaps
he provided a link there. That's part of a bigger inquiry about foundations
in general, but does anyone have more information that what's in Toepell?
--------------------------------
James T. Smith, Professor
Department of Mathematics
San Francisco State University
1600 Holloway Avenue
San Francisco, CA 94132
415-338-1368 (office, message)
415-338-1461 (fax)
http://math.sfsu.edu/smith
smith@math.sfsu.edu