I thought the Wikipedia article covered it well enough, if you have an eye for
history.
There are certainly many uses for restricting a solution set to integers, among
them that some crystal point groups can be scaled to have lattice points only at
integer locations on the plane, and people dealing with computer algorithms
(especially in the cryptographic community) are obsessed with countable sets.
I'd say it's "merely" semantics, but that restricting the solution set to very
specific nets is a remarkably fruitful realm of study. It's "semantic" in the
way that calling "Euclidean geometry" is "semantic": perhaps silly, if you know
differential geometry, or if you're only worried about topological questions,
but the way a large part of the world works.
The extension to when non-integer (or, more generally, rational) solutions are
allowed is called "algebra". :)
/**************************************
"The four points of the compass be logic, knowledge, wisdom and the unknown.
Some do bow in that final direction. Others advance upon it. To bow before the
one is to lose sight of the three. I may submit to the unknown, but never to the
unknowable." ~~Roger Zelazny, in "Lord of Light"
***************************************/
________________________________
From: Stefan Jeglinski <jeglin@4pi.com>
To: Forum for Physics Educators <phys-l@carnot.physics.buffalo.edu>
Sent: Sat, January 15, 2011 10:09:34 AM
Subject: [Phys-l] diophantine equations
Why are diophantine equations defined as having integer-only
solutions? Is it merely so that those solutions that are integer-only
have a name? It seems easy to find a "diophantine equation" whose
solutions are real or complex, as well as integer. Is it then no
longer a diophantine equation?
I guess I'm trying to get at whether this is just a semantic
distinction or whether there is a deeper reason for the naming
convention. Is there a name for the same equation when referring to
non-integer solutions?