[FOM] Re: The Search for Mathematical Roots

[Timothy Y. Chow]

Marcus Schaefer wrote:

> I recently purchased Ivor Grattan-Guinness' "The Search for Mathematical
> Roots 1870-1940".

Your quotations do sound hair-raising.  This is a bit of a tangent, but 
one of the quotations raised a question in my mind.

> "He also proved here the existence of irrational numbers by a lovely
> reductio argument that has never gained the attention that it deserves:
> assume that the equation t^2 = Du^2 in integers (D not a square) has a
> solution, and let u be the smallest integer involved; then exhibit a
> smaller integer also to satisfy the equation, a contradiction which
> establishes sqrt(D) as irrational."

This sounds like a garbled version of the following proof that if D is not 
the square of an integer, then sqrt(D) is irrational.  Suppose to the 
contrary that sqrt(D) is rational but not an integer.  Then there exist
positive integers n such that n*sqrt(D) is an integer; pick the least
such n.  But now let m = n*(sqrt(D) - floor(sqrt(D))); then m is a 
positive integer less than n and m*sqrt(D) is an integer, which is a
contradiction.

In particular there is no appeal made to unique factorization.

Is this argument due to Dedekind?

Tim