Subject: Re: [HM] ... and the unique factorization theorem
From: Jose Luis Garcia (jlgarcia@fcu.um.es)
Date: Fri Feb 11 2000 - 11:59:34 EST
Dear listmembers:
I am far from being a specialist in history of mathematics; so, please,
excuse me for being so uncounsciously bold to take part in this discussion.
Regarding Euclid and the fundamental theorem of Arithmetic, I think we
should distinguish two ways of looking at the question. From the purely
mathematical point of view, Euclid did have all the knowledge that is
needed in order to "essentially" knowing and proving the theorem. For the
existence of a decomposition into primes is trivial: Greeks knew very well
that a composite number is measured by a prime number and that the
factorization process can be repeated until only prime factors are left.
Regarding the uniqueness, Euclid also knew that if a number is written in
two different ways as a product, say x=ab=cd, then every prime factor of a
is also a prime factor either of c or of d; and that if you remove that
prime factor from x, then you still have a'b=c'd, say (a',c' being the
number of times each of the numbers a,c is exactly measured by the removed
prime) and you can continue in this way, until all prime factors have been
removed.
It remains the psychological point of view. This is what S.Kutler is
referring to when he writes
> 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?
This is precisely a psychological matter. For us, the notion of product is
simpler than least common multiple; but in Euclid's view, the lcm is the
simpler. In other words: product is simpler from the algorithmic viewpoint,
while lcm is simpler from the "essential" viewpoint. I think that for
Euclid and the ancient mathematicians, numbers had a meaning and an
essence: being the lcm of some numbers is a stronger and more defining
property than the somehow accidental fact of being their product.
But this cannot be more than a guess, and J.Conway already warned us
against this:
> 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.
[...]
> However, there's little point in making such guesses, so it's only with
> some reluctance that I've rejoined this discussion.
From what I said above it follows that I basically agree with J.Conway's
views above. Besides this, the actual statement of the theorem refers to
numbers that are equal to certain products of other numbers and thus does
not fit well in Euclid's way of thinking, which was strongly dependent of
meaning and far from the algorithmic way. But the only meaning of a long
product is the algorithmic one. I think this is why he was not much
interested in this kind of statements.
Best wishes,
J.L.Garcia.
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