>
> Happy New Year!
>
> This is in reference to what you've written on HM regarding systems of
> magnitudes. I heard (perhaps mistakenly) the implication that an element
> of one of these systems couldn't be compared to one from another (as we
> can in today's mathematics using the real numbers).
>
> Suppose we have an m x n rectangle, m,n whole numbers (of course relative
> to some fixed unit length). Surely, Euclid would see no problem in
> constructing a line segment having the same length as the area of that
> rectangle. Now suppose we have incommensurability; Dedekind's example a
> sqrt(2)xsqrt(3) rectangle. Again the standard Euclidean product
> construction should work and give a line segment of length sqrt(6). Is
> this right? If not, why not?
>
> I'm posting this to you personally, but feel free to post it on HM if
> you think it useful.
>
> Best, Martin
Happy New Year to you, too---and to everyone on the list!
I did mean to imply that: the classical Greeks didn't compare magnitudes of
different species. But one could compare *ratios* A:B and c:d, where A and
B are from one magnitude system (e.g. plane figures) and c and d from
another (e.g. line segments). These ratios can carry the arithmetic of the
positive real numbers---indeed, both Newton and Euler defined numbers to be
ratios. Howard Stein suggested to me the following treatment: The
'multiplication' of ratios of a given species of magnitudes requires that
for every magnitude A of that species, every ratio of magnitudes of the
species can be put in the form A:X (the existence of so-called 'fourth
proportionals'). Then the compound ratio (multiplication)A:B x C:D of the
ratios A:B and C:D is defined as A:E, where C:D=B:E. (Euclid proves that
this is independent of the particular choices of the representatives of the
ratios.
Now for Dedekind's example, we have A:U x A:U = 2U:U and B:U x B:U = 3U:U.
So (A:U x B:U) X (A:U x B:U) = (by Euclid V, Proposition 23) (A:U x A:U) x
(B:U x B:U) = 2U:U x 3U:U = 6U:U.
We were not able to find this in Euclid---he does not even define the
notion of compound ratio explicitly (though he refers to it by that name
in Book VI); but he proves, as you see above, all the requisite
theorems---viz, that composition of ratios is a well-defined and
commutative operation; and so everything was there for proving Dedekind's
equation.
Note that stating Dedekind's equation in Greek terms (at least on the above
treatment), much less proving it, does involve the assumption of the
existence of fourth proportionals. But probably Dedekind's assertion that
it had never before been proved refers less to that than to his rejection
of geometric proofs in analysis in general. On the other hand, until a
definition of the real numbers was given---e.g. by him---what could sqrt(2)
have been other than a magnitude or ratio of magnitudes, hence depending on
geometry? So I wonder why he complained that the equation had never been
proved rather than that its terms had never been defined.
Stein has a good paper on this general topic: "Eudoxos and Dedekind"
*Frege's Philosophy of Mathematics* (edited by W. Demopoulos) Harvard U
Press, pp. 334-357. I don't know why it is in a collection of papers on
Frege's philosophy of mathematics.
Best, Bill