16 Chapter 1
terse contradiction emerges if one assumes hypothetically that such a ratio could be found,
for if so, the “ numbers ” of that ratio would have to be simultaneously odd and even.^23 Our
symbol 2 does not resolve this problem but merely gestures symbolically toward a “ ratio ”
that is in fact no ratio of finite numbers but (as we would say) an infinite decimal,
1.41421356237 .... It is, indeed, both even and odd, or neither.
Thus, Greek mathematics could never speak of “ irrational numbers, ” as familiar as that
terminology became after the sixteenth century. In Greek mathematical texts, numbers
were by definition integers and did not include zero (a concept nowhere explicit in their
mathematical texts) or one. Indeed, some Pythagoreans considered two not a number but
a crucial intermediate, a mysterious dyad that bridges the utter solitude and uniqueness of
the One and the multitude of the Many.^24 In his own way, Plato treats the dyad as “ unlim-
ited ” by connecting pure Being — the One in its solitary splendor — and non-Being, leading
to the variable, ever-changing multiplicity of the cosmos, in which Being and non-Being
are not utterly separate but somehow interwoven into the structure of Becoming as we
experience it.^25 Throughout his dialogues, Socrates and his friends examine the strange
mixture of truth and story, logos and mythos , constituting the living stream of language
and thought. They keep looking to the realm outside our dark cave where, if only in
imaginative speech, we aspire to see the One and the other pure Forms or Ideas, each
distilling the ultimate essence of a number or concept. Socrates suggests that “ proceeding
to the major and more advanced part of geometry tends to make it easier to behold the
Idea of the Good, ” the highest Form, which many passages in Plato suggest may be identi-
fied with the One itself. As Socrates notes, everything in that realm of Forms tends to
compel “ soul to be turned round to that place in which the happiest of what is exists, which
soul must in every way behold. ”^26
Accordingly, Socrates pokes fun at geometers ’ use of phrases like “ squaring, ” “ apply-
ing, ” or “ constructing, ” “ as if all their words were for the sake of action ” rather than
“ undertaken for the sake of knowledge, ” meaning the philosophic contemplation “ of what
always is, not of what sometimes comes to be and passes away. ” Though he touches on
the practical uses of mathematics, which appeal to his companions, Socrates is much more
interested in its “ useless ” aspects because it awakens and sharpens “ an organ in the soul
of every man which is purified and rekindled in these studies when it has been destroyed
and blinded by other pursuits, an organ more worth saving than ten thousand eyes; for
truth is seen by it alone. ”^27
Given the exalted purity of what Socrates seeks to behold, he and the other Greek
mathematical writers would have been amused, if not dismayed, by the breezy nonchalance
with which later mathematicians speak of “ irrational numbers, ” which would have seemed
to them a sheer contradiction in terms, even nonsense: “ nonnumerical numbers, ” in their
terms, or “ uncountable counting numbers. ” Socrates teases his young friends for being
“ as irrational as lines, ” an ironic judgment he likely would have passed on the modern
mathematical proclivity to mix rational with irrational quantities. Socrates makes fun of