categories: Dedekind v Cantor

[Peter Freyd <pjf@saul.cis.upenn.edu>]

Folklore says that Book V of Euclid's Elements is the best extant
approximation to Eudoxus. The Joyce translation:


                          Euclid's Elements
                                      
                                Book V
Definition 5

       Magnitudes are said to be in the same ratio, the first to the
       second and the third to the fourth, when, if any equimultiples
       whatever are taken of the first and third, and any
       equimultiples whatever of the second and fourth, the former
       equimultiples alike exceed, are alike equal to, or alike fall
       short of, the latter equimultiples respectively taken in
       corresponding order.

(http://aleph0.clarku.edu/~djoyce/java/elements/bookV/bookV.html#defs)


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

According to Neugebauer the philosophical Greeks avoided the rationals:
they allowed _ratios_ named by pairs of integers, and they effectively
knew how to multiply ratios; but they considered the addition of ratios
as something allowed only by those entirely unphilosophical
calculators to be found in marketplaces.