SOCRATES: This knowledge of military matters that you have, does it come from your being an
expert general or rhapsode?
ION: It doesn’t seem to me to make any difference.
SOCRATES: How can you say that it doesn’t make any difference? Do you mean that the
rhapsode and the general possess one skill, or two?
ION: One, it seems to me.
SOCRATES: Then whoever is an expert rhapsode is in fact also an expert general?
ION: Absolutely, Socrates.
SOCRATES: By the same token, then, whoever is in fact an expert general is also an expert
rhapsode.
ION: No, that doesn’t seem to me to be the case.
SOCRATES: But it does seem to you to be the case that whoever is an expert rhapsode is also an
expert general?
ION: Certainly.
SOCRATES: Well, you are the finest rhapsode in Greece, aren’t you?
ION: By far, Socrates!
SOCRATES: So are you also the finest general in Greece, Ion?
ION: Be assured that I am, Socrates! And I learned it all from Homer. (Plato, Ion 540e–541b)One of the areas of study that fascinated well-educated (and hence wealthy) Greeks in the time of
Socrates and Plato was geometry. In The Clouds, Socrates is presented as drawing geometric figures with
a pair of compasses (which he then uses to steal someone’s clothes) and the school that Plato founded, the
Academy, is supposed to have had the following inscribed over the entrance: “No admittance to anyone
ignorant of geometry.” Geometry had been a particular concern of the sixth-century philosopher
Pythagoras of Samos, who influenced Socrates and Plato in a number of important ways. Among other
things, Pythagoras had taught the doctrine of the transmigration of souls, namely that the soul outlives the
body and takes up residence in a succession of incarnations; he was also credited with the discovery of
the “Pythagorean Theorem,” although it was known to the Babylonians long before the time of Pythagoras.
According to the theorem, the sum of the squares on the sides of a right triangle is equal to the square on
the hypotenuse (figure 62). The appeal of a theorem like this is that it can be proved with absolute
certainty, and the appeal of geometry in general is that it deals with what is eternally the case. In contrast
to the messy world with which we come in contact every day, in which things are constantly changing and
about which intelligent people are often in violent disagreement, the objects of geometric investigation
enjoy a permanent, unchanging existence. What is more, the eternal truths about geometric figures like
triangle αβγ, for instance that α^2 + β^2 = γ^2 or that its area = αβ/2, can be known and can be proved to be
the case without our ever having seen triangle αβγ. In fact, it is quite impossible for geometric figures to
exist in the messy world with which we come in contact every day. And yet their existence cannot be
doubted: How could we know with absolute certainty what we know about them if they did not possess
existence at some level?