Subject: categories: Eudoxus and the real numbers

From: "Stephen H. Schanuel" <[email-masked]>
Date: Thu, 20 Jan 2000 21:35:34 -0500

About 'disagreeable': As far as I can see, Mike Barr and I don't
disagree. I had trouble figuring out how to post something on the catnet,
so when my original note didn't appear, I added a couple of details and
reposted; I think that the second one was a bit clearer about what I was
crediting to Eudoxus, namely just the idea that cutting the measuring
stick into n equal parts to improve accuracy is unnecessary, and can be
replaced by multiplying the stick one wants to measure by n. (If I'm wrong
in crediting that to Eudoxus, somebody please enlighten me.) Now isn't it
reasonable to say that this simple observation ought to suggest that R can
be constructed without first constructing Q? (The only surprise is that it
doesn't require Z as ring, but only as unordered abelian group.)

    Whether Eudoxus was constructing the Cauchy reals or the Dedekind
reals (or the positive half of those) or 'constructing' anything at
all--which I don't think he was--is irrelevant to the point above, I
think. Of course I agree with Mike that the construction which I based on
E's observation gives the Cauchy reals, as I believe Mike pointed out many
years ago in Montreal when I neglected to make the distinction in a talk I
gave there. He was and is right to make this 'modern' Cauchy versus
Dedekind distinction, and if someone can show me that it isn't modern,
that would be even more interesting!