>In Euclid there is no notion of a "system" of numbers of any kind, not
>even
>the "integers" or the "natural numbers". There are no "totalities" such
>as all positive integers
>or all real numbers. Dedekind is completely correct in stating that his
>pet equation \surd2\surd3=\surd6 cannot be found in Euclid's Book V.
>Euclid's book V is about magnitudes, that is to say, the geometric
>"objects" of the Elements such as lines and triangles, viewed not as
>figures per se but as measuring each other (see Defs 1 and 2 of Book V).
>Similar things can be said (mutatis mutandis) about Books VII-IX on
>"number". I cannot say whether this approach is "Greek" or particular to
>Euclid - I suspect the latter, but do not have sufficient evidence to
>commit.
Surely a system of magnitudes (lines, plane figures, solids, ETC.),
supplied with an addition for which x<y := \exists z [x+z = y] defines a
total ordering, is presupposed, e.g. by the common notions in Euclid and
the applications of the method of exhaustion. [In the latter, x=y is proved
by refuting x<y and y<x, and the refutation proceeds by inferring from x<y
that y-x exists.] Of course, = (isos) here is not identity---as you say
``the geometric
>"objects" of the Elements such as lines and triangles, viewed not as
>figures per se but as measuring each other''.
Equally surely, you are right to contrast the spirit of Euclid with that of
Dedekind. I can recommend a paper of my colleague, Howard Stein, on this
subject: ``Logos, Logic and Logistike'' in W. Asprey and P. Kitcher
*History and Philosophy of Modern Mathematics*, Minnesota Studies in the
Philosophy of Science XI, University of Minnesota Press 1988. pp. 238-259.
One aspect of the difference is Dedekind's willingness to consider
*sub*-totalities defined by some closure condition---the number fields,
modules, ideals, etc.
Regards from sunny Chicago,
Bill Tait