[MUSIC] When Meno asks Socrates whether he thinks virtue is acquired by teaching, Socrates immediately insists on addressing a prior question. What is virtue? A question, to which Meno was confident he knows the answer. This is rather like the opening of the Eutifro, where the discussion if whether Eutifro's action is pious turns to the fundamental question, what is piety? Meno ends up being no more successful than Eutifro in answering this fundamental question. Even though just like Eutifro, he was confident that he knew the answer. As in the Eutifro, Socrates asked Meno to demonstrate his professed knowledge of virtue by stating what it is that is the same in all cases of virtue. What makes all these things cases of virtue? Meno makes a variety of proposals that upon questioning by Socrates he agrees do not fit the bill. Either he produces a list of different virtues rather than a single thing, or he proposes something like ruling others and acquiring wealth that he agrees isn't always virtuous. Since you can do these things unjustly. Or, he proposes something that is always virtuous, like ruling justly, but that still has part of virtue, justice, in the definition. By the time we get to page 80a, Meno is completely out of ideas, and he blames his predicament on Socrates. Just like Eutifro who says that it's Socrates fault that he can't say what piety is. Now this is a pattern that gets repeated many times in different Platonic dialogues. It turns out to be a lot harder than Socrates interlocutor expect to show that they know what they're talking about. And recall, Socrates claim in the apology that he never met anyone who did turn out upon examination to have the knowledge that they profess to have. Now one might wonder at this point whether Socrates philosophical method of cross examination, it gets called elenchos, is a purely negative method, capable only of debunking claims to knowledge. But unable to meet yield knowledge by finding suitable answers to questions like what is virtue? This is a worry that Plato takes up in the dialogue Meno, where he has Socrates explain how his method of inquiry can achieve positive results. Now, when Meno has run out of proposals about what virtue is, Socrates proposes that they inquire together to find out what it is. Meno then throws up his hands and asks, how on Earth such inquiry could ever be successful, giving that, according to Socrates, they don't know the least thing about virtue. Socrates reassures him with the momentous announcement that each person's soul has gone through many cycles of reincarnation, and during his travels outside the body it has learned the answer to all questions. So, inquiry is simply a matter of recalling what we already know, even if we've temporarily forgotten it. To illustrate what he means by saying that successful inquiry is simply a matter of recollecting what we already know, Socrates poses a geometry problem to a young slave in Meno's retinue. He invites Meno to observe the slave as he recollects the correct answer. I'm not teaching the boy, Socrates claims. All I am doing is asking him questions, and he is recollecting what he knew before. Now let's take a closer look at what happens here, and see whether we believe Socrates. Socrates starts by constructing a square that is 2 feet by 2 feet, and he asked the slave to calculate its area. He indicates a variety of strategies for doing this, and the slave correctly calculates that the area is 4 square feet. Socrates then asks what the area of a square would be if it was twice the size. The slave correctly answers that it would be 8 square feet. Socrates now opposes his main question. How long is the side of that 8 foot square? In effect, he is asking for the square root of 8. Not an easy question to answer. Do you know know what the square root of 8 is? The slave first thinks that the answer is easy. The 8 foot square, since it is double the area of the 4 foot square, must have a side double in length. Since the 4 foot square has a 2 foot side, the 8 foot square must have a 4 foot side. Are you sure, this is what you think? Socrates asks him. He constructs the square the slave has proposed with sides twice as long as those of the 4 foot square. And he gets the slave to see, that this square contains four squares the size of the original square. Thus, it is not twice as big as desired, but four times as large. The slave now recognizes that 4 feet is too long for the side of the 8 foot square, and he casts about for a different answer. Well, it's got to be bigger than two on the original square. And less than four, which was too big, so he proposes that three is the side of the 8 foot square. Socrates now constructs the square that is 3 feet by 3 feet, and he asked the slave to calculate its area. 3 times 3 is 9. So, once again, too big since the target area is eight. The slave is now completely stymied. He has no idea what answer to give. Just as Meno was when he ran out of proposals about what virtue is. Now the important part begins, because Socrates has promised to show Meno how one can arrive at a correct answer to a question by this method of inquiry. Not just refute wrong answers, so here goes a positive inquiry. Once again, Socrates insists that he is only asking questions, not teaching. He goes back to the result of the first wrong answer, the square with the 4 foot side, which turned out to be twice as big as the target square. He cuts each of the four constituent squares in half, and recall half of this over sized square is what they're looking for. He thereby constructs a square that is the target size. He gets the slave to realize this by asking him to count up the number of constituent triangles in the newly created square. There're four of them. And to compare them to the number of triangles in the original square, two. The slave now sees that they have constructed the square double an area to the original square. Even though he is not able to state how long the side is, it is after all, an irrational number. He is able to point to the line on which it is constructed. Socrates gives him a name or a label to attach to that line, he calls it the diagonal. It's what we would call the hypotenuse of a right angle triangle. With this label, the slave now can articulate the correct answer. A square doubled in area to a given square has its side the length of the diameter **diagonal** of the original square. The slave has, in effect, learned a special case of Pythagoras' theorem, that the square on the hypotenuse of a right angle triangle is the sum of the squares on the opposite two sides.
