Subject: Re: [HM] Euclid and the unique factorization theorem
From: John Conway (conway@math.Princeton.EDU)
Date: Wed Feb 09 2000 - 13:35:05 EST
On Wed, 9 Feb 2000, Samuel S. Kutler wrote:
> As you know in IX 20, Euclid does not form the product of the primes, he
> takes the least common multiple. This is equal to the product, but I don't
> know why he did not ask us to multiply them all together. Probably it is
> because he has a procedure for finding the least common multiple. Do you
> have an opinion about this?
Well, I do. This is clearly connected with the business of defining
products of more than three terms. It's interesting to note that essentially
all of Euclid's number-theoretical investigations are phrased in terms of
the concept of one number's "measuring" some others (ie., being a common
multiple of them), rather than in terms of the notion of a "product" of
numbers.
To my mind, this is significant, but I wouldn't go so far as those who
confidently assert that the Greeks had no notion of the product of an
arbitrary number of numbers. What I do think, is that, in axiomatic mode
as Euclid clearly was, he wanted to deduce as much as possible from as
few assumptions as possible, so making sure his results would have extra
generality. The concept of one object's "measuring" another fits this bill,
because it can apply to things (like line-segments, or areas) other than
integers, and is so applied by Euclid.
My guess is that he did actually "know" the unique factorization
property, but refrained from stating it because he couldn't easily do
so in his geometrical language. What he did instead was state exactly
the "geometric" part of it - after all, he WAS writing a book on geometry.
However, there's little point in making such guesses, so it's only with
some reluctance that I've rejoined this discussion. Those who prefer to
make the contrary guess - that Euclid did NOT have the concept of unique
factorization - are obviously free to do so, since the only documentation
is his book (and a few commentaries that aren't very helpful about this
point).
I think some people go too far in the "no numbers" direction, though.
It's true that Euclid's book is thin on mentions of rational numbers,
but there are plenty of other Greek works that display an interest in
them. To my mind this supports my position - I'd guess that the rationals
are missing from his book not because he didn't know about them, but
rather because he wanted all his arguments to work for incommensurables
as well.
A very interesting point, to my mind, is Hero(n)'s famous formula
DELTA = root( s(s-a)(s-b)(s-c) ),
since this involves a "non-geometrical" product of four lines. Does anyone
know of any ancient discussion about this aspect of the formula?
John Conway
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