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enthusiastically discovering the laws of geometrical objects to relegate them to mere human invention, the tics of certain minds in flight from the “real” world. When Euclid showed how you can prove the theorem of Pythagoras by constructing a set of parallelograms, with only a straight-edge and a compass—that is, with no numerals and no calibration —and by the rules of strict reason, he did not believe he was dissecting an unreality. More than that. He knew he had shown with absolute certainty that the theorem was correct, although neither Pythagoras nor he had ever seen or could ever see a line of infinitesimal thinness, or a circle exactly circular, or a right angle that was just right.
     
    After Socrates had needled his fellow Athenians for claiming to know what they had only heard, his pupil Plato strove to discover how we could come to certain truth, rather than accept convention or give up altogether. Naturally, then, he turned, as Immanuel Kant much later, to mathematics. But Plato did not make the tremendous error that has kept much of modern philosophy bottled up in symbolic logic and linguistic analysis. Plato didn’t assume that everything had to be demonstrated in the same way as the Pythagorean theorem was. Instead he asked about the nature of various kinds of objects, including mathematical objects, and about the various ways we have of knowing them, some more reliable than others. That’s why, according to one ancient account, he caused a sign to be hung over the door of his Academy: “No one ignorant of geometry may enter here.” 19
     
    Plato saw that the knowledge we have of a triangle was knowledge of a genuine thing, not a figment of the imagination. Such knowledge lay waiting for discovery. It was also knowledge of a universal. When we prove Pythagoras’ theorem, we know something about all right triangles, not just this one or that one. That led Plato to consider a mysterious property of language and the world. We say “cat” and “tree” and know that we are not talking necessarily about any particular cat or tree, or any cat or tree we have materially seen. We mean something other than “Tabby” or “the oak tree in front of my house.” But how can this be? What does the term “cat” denote? There are many cats, but what do I mean when I say “cat,” if I don’t intend any particular one, dead or alive?
     
    Plato concluded that knowledge could not be simply of matter, because we have knowledge of immaterial objects such as triangles, and because such words as “cat” are universal in their signifying, and not particular. Plato concluded, as had Anaximander, that material causes were not sufficient to explain the world or even to speak intelligibly about the things in it. So he developed his theory of the Forms or Ideas , universal, intelligible, immaterial, and immutable. We may see all the horses in the world, but unless we conceive of the idea of What It Is Essentially to be Horse, then we don’t know what a horse is. 20
     

The universal Good
     
    We can then, he argued, apply the same insight to morality, aesthetics, and politics. We may see this or that good deed. We agree that it is prudent for Themistocles to persuade the Athenians to use their new-found money to build a navy, and that it is courageous of Leonidas to stand with his small Spartan contingent to delay the Persians at Thermopylae. But what makes these actions good? What is the form of the good?
     
 
     
    Politics Before Polling
     
    But in other things [Pericles] did not comply with the giddy impulses of the citizens, nor quit his own resolutions to follow their fancies, when, carried away with the thought of their strength and great success, they were eager to interfere again in Egypt, and to disturb the King of Persia’s maritime dominions.
    Plutarch , Life of Pericles
     
     
    Pericles was more than a politician: he was a leader, a manly yet modest ruler of a people that had the power to overrule