A friend today sent me this picture from the Boston Museum of Science. It’s especially funny to me, because I have formal education in both mathematics and philosophy. And anyone who’s read Plato’s Republic is bound to think it odd.

It’s odd because Plato’s good society, the republic, requires its citizens to study mathematics for ten years. So it’s not like Plato takes math lightly. This quotation arises during Socrates’ and Glaucon’s conversation about education (the character of Socrates speaks in the first person).

[Socrates:] Now, when all these studies reach the point of inter-communion and connection with one another, and come to be considered in their mutual affinities, then, I think, but not till then, will the pursuit of them have a value for our objects; otherwise there is no profit in them.

[Glaucon:] I suspect so; but you are speaking, Socrates, of a vast work.

What do you mean? I said; the prelude or what? Do you not know that all this is but the prelude to the actual strain which we have to learn? For you surely would not regard the skilled mathematician as a dialectician?

Assuredly not, he said; I have hardly ever known a mathematician who was capable of reasoning.

But do you imagine that men who are unable to give and take a reason will have the knowledge which we require of them?

Neither can this be supposed.

Source: Republic 531e, translated by B. Jowett

Jowett translated the Republic in the 19th Century, so having entered the public domain it’s well-distributed, if perhaps not as clear as it could be. Here are the same lines from a couple of other translations:

Surely you would not regard experts in mathematics as masters of dialectic.

Certainly not, except a very few of those I have met.

trans. F.M. Cornford

‘The subjects we’ve described are only a prelude to the main theme we have to learn. For you don’t think that people who are good at them are skilled in dialectic, do you?’

‘Heavens, no, though I have come across a few exceptions.’

trans. Desmond Lee

When Jowett’s translation talks about being “capable of reasoning,” it’s referring to “dialectic,” which for Plato is a special way of reasoning, not just reasoning in general. “Dialectic” is the capstone of Plato’s proposed education; after training in it for five years, the leaders of his republic should begin to understand philosophical issues.

So we can let mathematicians off the hook a little bit: Plato’s not saying they’re unreasonable, just that they generally haven’t achieved the heights of thinking made available by dialectic. Why this is so comes out a little bit later in the next book:

The detached studies in which they were educated as children will now be brought together in a comprehensive view of their connexions with one another and with reality.

Certainly that is the only kind of knowledge which takes firm root in the mind.

Yes, and the chief test of a natural gift for Dialectic, which is the same thing as the ability to see the connexions of things.
…
When they reach thirty they will be promoted to still higher privileges and tested by the power of Dialectic, to see which can dispense with sight and the other senses and follow truth into the region of pure reality.

Source: Republic 537, trans. Cornford

In other words, dialectic brings everything together. Referring to his famous allegory of the cave, Plato is saying that dialectic is like the last step, where the former prisoners are no longer seeing shadows or indirect images, but are looking at the “sun” of truth. The problem with mathematics by itself is that it doesn’t provide the big picture; it’s limited to a particular discipline, so it’s merely a shadow of the form of the Good.

I decided to track down this supposed Plato quotation and found the following in A History of Western Music:

Furthermore, the foundations of music once established must not be changed, for lawlessness in art and education inevitably leads to licence in manners and anarchy in society. For Plato the saying “Let me make the songs of a nation and I care not who makes its laws” would have expressed a political maxim; more than that, it would have been a pun, as the word nomos, with the general meaning of “custom” or “law,” was used also to designate the melodic patterns of a certain type of lyric song.

Grout, Donald J. A History of Western Music, 1973. p. 8

Notice the ambiguous “would have”: “For Plato the saying . . . would have expressed a political maxim.” It is possible that Grout could have used “would have” in the sense of “did,” but none of the footnotes near the quotation (there is none for the quotation itself) points to such a passage. In fact, the relevant passages in the footnotes—from Plato’s Laws—actually seem to support the converse: Plato wants law to control the music. But somewhere along the line an ambitious musical-political theorist made it through page eight of A History and decided to resolve the ambiguity in his own favor.

“Suppose a man was in charge of a large and powerful animal, and made a study of its moods and wants; he would learn when to approach and handle it, when and why it was especially savage or gentle, what the different noises it made meant, and what tone of voice to use to soothe or annoy it. All this he might learn by long experience and familiarity, and then call it a science, and reduce it to a system and set up to teach it. But he would not really know which of the creature’s tastes and desires was admirable or shameful, good or bad, right or wrong; he would simply use the terms on the basis of its reactions calling what pleased it good, what annoyed it bad. He would have no rational account to give of them, but would call the inevitable demands of the animal’s nature right and admirable, remaining quite blind to the real nature of and difference between inevitability and goodness, and quite unable to tell anyone else what it was. he would make a queer sort of teacher, wouldn’t he?”

“Very queer.”

“But is there really any difference between him and the man who thinks that the knowledge of the passions and pleasure of the mass of the common people is a science, whether he be painter, musician, or politician? If he keeps such company, and submits his poems or other productions, or his public services, to its judgment, he is going out of this way to make the public his master and to subject himself to the fatal necessity of producing only what it approves.”

Republic 493