Subject: categories: re: Dedekind v Cantor

From: Jean-Pierre Marquis <[email-masked]>
Date: Mon, 24 Jan 2000 10:26:41 -0500 (EST)

>It looks more Dedekind than Cantor to me (why do people think that
>Cauchy had anything to do with this?).

In his "Analyse algebrique" (1821), Cauchy gives the first (still informal)
definition of a limit and says (without proof), in order to illustrate the
concept, that an irrational number is the limit of the various rational
sequences approximating it.  He also gives various criteria for
convergence.  Thus, although Cauchy certainly does not give a rigorous and
formal construction of the reels, people ascribe to him the basic idea.
But then, perhaps Bolzano, Weierstrass, Meray and Heine should also be
mentioned, no?

>But Steve is right when he says
>that the rational numbers don't appear: an incommensurable ratio is
>described as a partitioning of pairs of integers.

A relevant reference here is:
Stein, H., 1990, 'Eudoxus and Dedekind: On the ancient greek theory of
ratios and its relation to modern mathematics', Synthese, 84, 163-211.

Jean-Pierre Marquis