Subject: Re: [HM] A question around zero
From: John Conway (conway@math.Princeton.edu)
Date: Tue Jan 25 2000 - 17:15:32 EST
On Tue, 25 Jan 2000, Don Cook wrote:
> Two questions:
> 1. Should we be having this debate in the year 2000 on whether zero is
> even or not.
We definitely should not! There are no two ways about it, as far as
mathematicians are concerned, and if common usage differs, we should be
trying to change common usage, rather than follow it.
There is some scope, of course, for discussions of what this or
that ancient author said about such things. But we shouldn't waste
many words on it. since the answers are unlikely to be important, and
very likely to be boring.
> 2. I need help with Euclid and zero. I thought, that for Euclid, all
> numbers were lengths of line segments. Since there are no line segments
> with either zero or negative length, these numbers do not exist. Will
> someone who has more knowlege than me expand on this?
What you say is essentially right, except for the fact (made much of by
some) that Euclid didn't actually make use of a length concept with number
values. His concept was, rather, a technical notion of proportion,
AB having the same proportion to CD as EF does to GH if and only if
for all positive integers p and q for which
p.AB >, = or < q.CD,
we have the same one of
p.EF >, = or < q.GH.
In our modern language, which is equivalent, this happens if and only
if CD/AB and GH/EF induce the same section of the rationals, in other
words, just if they determine the same real number.
But the difference in languages is unimportant for your question -
it IS true that the real numbers mentioned here were strictly positive.
To my mind, the "linguistic difference" here is certainly important,
as is the fact that Euclid wrote in Greek rather than English, but is
only a little bit more important than that fact!
John Conway
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