[BLANK_AUDIO]. Let's read the darn dialogue, already. Some basic review, exactly what's going on just before we get to the geometry lesson. There's all that mythic stuff, right? But just before that, I talked about this last lesson too. Meno puts forward this silly argument when he goes into this sulk about having had his tongue paralyzed by Socrates, who he compares to a stingray. The silly argument goes more or less like this. It seems that concerning every possible subject of inquiry, you already know it or you don't. If you know it, you can't start inquiring, because you're already done. You know it. If you don't know it, you can't start inquiring, because, by hypothesis, you got nothing to start with. Conclusion: inquiry is impossible. That is, no one can learn nothing about nothing. You'd expect Socrates to respond in the obvious way. Presumably there is such a thing as sort of half knowing what you are talking about. You know enough to get started, but not enough that you're done before you're started. Maybe you know approximately what a bee is, to pick on that example again. You can tell a bee from a dog or an elephant, but you aren't quite sure what makes a bee different from an ant or a wasp, maybe. But you know Socrates, common sense is not his strong suit in these situations. Nope, he is going to say something weird. He says that he's heard things he thinks are true and beautiful from wise priests and priestesses. He quotes the weird Pindar bit, see last lesson, and then he summarizes like so. And as the soul is immortal, and has been reborn, time and again, and has seen both the things of this world and those of the underworld, and all matters, there is nothing it has not learned. So it is in no way surprising that is can recollect that which it knew before, both about virtue and other things. As everything in Nature is akin, and the soul has learned all, nothing prevents a man who has recalled one single thing, a process men call learning, from discovering everything else. Wow. That's pretty crazy. All knowledge is, in effect, innate. So it's true that, in a sense, we never learn anything. We can't learn anything, because we already know everything. So we can just recollect it. Meno says, this is very exciting, Socrates. But can you give me any evidence that it's actually, you know, true what they say? And Socrates proposes the geometry lesson as a sort of proof by example that what he just said is actually plausible. The boy is brought forward, and Meno testifies that the boy has never studied geometry before, and then Socrates proceeds to teach. But without teaching, supposedly. He's just going to ask questions, supposedly. The math is pretty simple, for those of us who've gone to school, so let me just talk you through it. Dropping the Socratic question form of the teaching and bringing up the bits that I think prove to be genuinely important for present purposes. The boy is shown a square that is two units on a side, thus it has an area of four square units. The question is, how long would the side of a figure be that was twice the area. That is, what is the side of a square that is eight square units in area? I already plot spoilered this one two videos back, so no harm in doing it again. It's the square root of eight, which, as I'm sure you know, is an irrational number. That is, it cannot be expressed as a simple fraction. 2.82842712475 and on, and on, and on, and on. In short, Socrates has asked the boy a seemingly simple question, but it has this incredibly weird answer. Do you remember when you first learned about this stuff in school? I said earlier that school tends to drill the wonder right out of math. But even so, maybe you remember learning about irrational numbers for the first time and feeling like, like the floor was falling out of the whole numbers business? There's a legend, that when the ancient Pythagoreans first discovered incommensurables, that is irrational numbers, they were so freaked out, they killed the guy who proved the stuff had to be real. Probably not true. But it makes sense that it might have freaked them out. The Pythagoreans were weird. They mixed up math and ethics in ways that seem strange to us today. For example, at least some of them associated numerical oddness with maleness and goodness, evenness with femaleness and badness. Sorry, ladies. Anyway, I imagine that it was thoroughly disturbing to them to discover that some numbers were neither even nor odd, just as many people today find it disturbing when gender assignments are uncertain. To say nothing of right and wrong getting mixed up. That really bothers people. Getting back to the dialogue. Socrates has not just asked a seemingly simple question that turns out to have a weird answer. He's asked an apparently concrete question, dirty anyway. He scratched this figure in the sand, or dirt. What could be more real than sand or dirt? You can see it, touch it, taste it, smell it. It's dirt. But it is going to turn out that this thing he's really pointing out by means of this figure is much more abstract, not the sort of thing that could be made out of dirt. Geometry, what is that name mean? It means earth measuring, dirt measuring, geo-meter. Very useful for establishing property lines and building houses, stuff like that. But what is it that we all know, who have taken geometry? The planes and figures we study in geometry, points and lines and circles and squares and all that, those aren't the same things as those illustrations in your geometry book, let alone the same as any crude thing you might scratch in the sand. You've seen circles, yeah. But you've never seen a real circle in the geometrical sense, every point on the line equidistant from the center, perfectly so, a perfect circle. Maybe when you do-, when you die, you'll get to see one. Let's think about this. It's not that unusual for sciences to sort of shift their subject matters, their domains as they mature and grow. People don't always know exactly what they're talking about when they start talking, do they? So maybe you set out to study fish, and you figure the whale is a fish. Swims in water, doesn't it? But maybe you decide after a while it's better to define fish in a way that excludes whales. Whales are mammals, because despite that whole swims in water thing, what whales have is more in common with humans and other mammals. Maybe it's more important than what they have in common with say, tuna fish. But geometry is a pretty striking case of a subject shift along these lines. It is the study of all and only entities that are, by nature, totally unlike the entities that people originally set out to study, namely, stuff scratched in the dirt. Let me underscore this point at the same time that I atone for those gender stereotypes in that graphic I showed at the beginning of the first video. When I learned geometry, I had a textbook in which there were two characters, Obtuse Ollie and Acute Alice. Per their names, Ollie got everything wrong and Alice got everything right. Girls can be good at math. Good going, textbook. But here's the thing, I don't know why this has stuck with me all these years, but Ollie was kind of an empiricist at heart. It's not obtuse to be an empiricist, is it? What am I talking about? Oftentimes the problem sets for this textbook would be framed like so, help Acute Alice correct Ollie's homework. And it turns out that Ollie said things like, it is obvious from looking at the picture or, I know it because I can see it or, because I measured the picture on the page, and none of that is allowed as valid proof procedures. But you know what? Seeing things, measuring things in the world are perfectly respectable scientific methods. Just not in geometry, but I'm getting ahead of the story. So far ahead in fact, that we won't really catch up until we get to Republic, the theory of the forms. Back to the boy. The question Socrates asks him, has this weird answer, so he's kind of doomed, isn't he? There is no way he has the intellectual resources to figure out about irrational numbers. But it works out pretty all right. Oh, it goes badly at first, don't get me wrong. The boy tries out the obvious sorts of bad answers you would expect. To double the area, he first tries doubling the side from two to four. I'm sure every kid would try this first. But that produces an area of 16, way too big. Then he tries three, halfway between two and four. What else are you going to try until you have more sophisticated ideas about number? Three is closer but still not right, gets you an area of nine. And Socrates emphasizes this, the boy can perfectly well see that his answers are wrong as soon as he offers them. Finally, Socrates helps him out a bit. What if we constructed a figure like this? It's actually the double the side square the boy himself tried first. That's not so important. This figure has twice the area we want, so it's not what we want, but is there some way we can construct a square that's half the size of this square? Well yes, it looks like there is. The big square is made of four little squares, if you like. If we cut each corner along the diagonal, we produce a square within the square that is clearly half the area. QED. It must be a square of area A, just what we're looking for. So the length of the side we want is that, isn't it? Point, point. Socrates tells the boy that wise men call that the diagonal, and he asks whether that is the line we want. And the boy can see that it is, and so he says yes. Lesson learned. But what lesson have we really learned here? We didn't actually get to all the fun stuff about irrational numbers, did we? But we're on the verge. By pointing at that, the diagonal, Socrates is pointing to a door which, if you open it and go through, contains truly amazing mathematical wonders. But what about the lesson about the boy, that is the lesson about learning? That's the point, remember? How do we acquire virtue? Well, how do we learn anything? That's what we want to know. Socrates: What do you think, Meno? In giving his answers, has he expressed any opinion that was not his own? Meno: no, they were all his own. And yet as we said, a short time ago, he did not know? Meno: that is true. So these opinions were in him all along, were they not? Yes. So the man who is ignorant about some subjects, whatever these things may be, nonetheless has within himself true opinions about these things he does not know? Meno agrees. But it hardly seems he should, and there are at least two problems. First, Socrates not only asks leading questions throughout the lesson, he more or less suggests the right answer in the end, drawing the diagram and asking the boy whether he thinks this is the answer. Second, what does this really mean, opinions that are his own or not his own? Let me attempt a quick salvage. Socrates can't seriously deny that he's been dropping hints, asking these questions he asked. He could plausibly saying that he is making what is sometimes called these days a poverty-of-the-stimulus style argument. That is, the hints he is dropping seriously under, underdetermined the boy's impressive output. But let's try a different strategy from making sense of how the opinions are the boy's own. Let's start in the next video with the idea that Socrates is a model educator.
