The Greeks were able to prove by reductio that the square
root of two is "irrational" -- that is, it cannot be written
as the ratio of two whole numbers. This makes it what
we call a "surd," a good word we rarely use, which means
an "irrational root." We derive our word, "absurd," from
this word (or else, vice versa).
According to legend, when the Pythagoreans discovered this
fact about square roots, they were deeply disturbed because
it showed that numbers cannot be fully grasped by reason.
Plato, himself a Pythagorean, had his god (DEMIURGOS) in
Timaeus make atoms of the four elements out of two types
of triangles: 30-60-90 and 45-45-90. Each type has one
side that cannot be expressed "rationally" in terms of
the other sides. This is, in my view, wholly consistent with
his view that the physical world cannot be fully understood,
that it is a mere "shadow" or "imperfect copy" of the pure
forms, and thus we can have only "opinion" rather than
"knowledge" of it.
Ted Davis
Professor of the History of Science
Messiah College