A Brief Guide to the Great Equations (5 page)

The paradox arises, as philosophers say today, from the mistaken assumption that knowledge comes in disconnected bits and pieces. In reality, we humans notice that something is unknown thanks to the whole matrix of things that we know already. We can extend this matrix – and fill in and flesh out gaps and thin areas – by
applying what we know to find what we don’t, bringing everything else to bear on it, inevitably uncovering new holes and weaknesses in the process. Acquiring knowledge is not like putting things someone else gives us in a mental warehouse, but a back-and-forth process in which we are constantly moving between parts and wholes, seeing and uncovering new things thanks to what we already know, acquiring a continually expanding base for understanding the world.

This isn’t the way Socrates puts it to Meno, of course. Meno cannot digest anything that subtle. Instead, Socrates couches it in a way the gullible lad can relate to, trying to entice him to move. Let me tell you an old legend believed by religious sages, Socrates says. They say souls are immortal, and thus have seen and learned everything under the sun. Deep within us, we already know everything, though during our earthly sojourn we’ve forgotten just about all. But if we are energetic enough, we can overcome this ignorance by recollecting it.

This legend is Socrates’ poetic way of telling Meno that learning is neither like getting something passively handed to you by someone else, nor like automatically following a rule. It’s an active and intensely personal process in which you motivate yourself to see something. You have to be on the move. And when you recognize something as true, you see that it belongs in that matrix as if it were a feature already there that you had overlooked. It feels so firmly nestled in your soul that it’s like you had it all along. It’s as if all the preparation and exercises and proofs you do in trying to learn something serves to help you to unforget it. This is the truth of the myth.

Meno likes the legend. But he still has not gotten the point, and asks for more explanation. Trying another tack, Socrates says he’ll show Meno the process in action. He asks Meno to summon one of his slaves – ‘whichever you like’ – and Meno complies. Socrates then coaxes this young slave, a naive boy innocent of any mathematical training, to go on a little journey, proving the geometrical theorem that the area of the square formed on the diagonal connecting the corners of another square is twice the area of that other square – thus, the Pythagorean
theorem involving an isosceles right-angled triangle. Socrates does so by drawing figures in the sand, step by step, asking Meno to keep him honest by listening carefully to make sure that Socrates does not smuggle any mathematical information into his questions and that the boy is ‘simply being reminded’ and not being spoon-fed.

Modern readers may see what follows as a fraud. They may think that Socrates is pulling the strings, playing with the slave’s head, getting the slave just to mouth the words. Modern readers are apt to find the idea of learning as recollection absurd, and think that real learning involves downloading new information into a person’s brain to be reinforced with homework and exercises. But if we read Plato carefully, we see that the slave is really learning – learning reduced to its elementals, as Socrates makes sure every new point emerges from the slave’s own experience. We see the slave boy going on a little journey in learning the Pythagorean theorem. Out of the infinite number of branching paths to follow, Socrates shows the boy which ones to choose, and provides him with some motivation to choose them.

You know what a square is? Socrates asks the boy, drawing a figure in the sand. A figure with four equal sides, like this? The youth says yes.

Do you know how to double its area? Socrates asks.

Of course, is the reply. You double the length of the sides. Obviously!

That’s wrong, of course, but Socrates doesn’t let on. A good teacher, he gets the student to spot his own mistake. When he extends the square, doubling the length of each side, the youth sees his error immediately – the new big square contains
four
squares of the original size, not two.

Try again, Socrates says. The boy proposes one and a half times the length of the first side. Socrates draws that square – and the boy sees that he’s overshot again.

Socrates asks the slave boy – dramatically, for Meno’s benefit – if he knows how to double the area of the square. No, I really don’t, says the youth.

This is the key moment! Socrates has, first, gotten the boy to see the limits of his knowledge – what he doesn’t know – and, second, dismantled the slave boy’s confidence. The boy had assumed that he knew, and now knows that he doesn’t know. It’s not true, of course, that the slave boy knows nothing. He knows a lot – that the answer to Socrates’ question lies in a narrow range, more than one but less than one and a half times the side. But the boy also knows more than he can say, and the knowing without being able to say doesn’t feel good to him now. The answer will come, but in a language the boy does not know yet. The slave boy has been made discomforted by encountering something he thinks he should understand and realizing that he does not. That bewilderment provokes a curiosity essential to learning. He is ready to let himself be led – ready to take a journey. It makes him want to
see
. He wants to
move
. We shall encounter the role of something triggering this desire – to move from where one is – again and again in the birth of equations. Sometimes the trigger is a chance event – perhaps the fall of an apple – while at other times it may be a passing remark, puzzling data, or an inconsistency between two theories. Here Socrates has bewildered the boy to induce the boy to want to follow him – a kind of seduction, which was one of the crimes Socrates would soon be accused of and for which he would be condemned to death.

Socrates capitalizes on the youth’s bewilderment. He rubs out the drawing and starts again with one of a square, 2 feet on a side, and then puts three other identical squares adjacent to it. Then he adds a new element to the diagram, a line crossing between two opposite corners: ‘the scholars call it a diagonal.’ The diagonal is not a totally new element. The slave has seen diagonals in floor mosaics and wall designs (an experience that has already given him an intuition of what is about to happen) and is merely being reminded what one is. But it’s new here. The diagonal suddenly casts the problem in a larger, richer context that makes the answer easier to see. It brings about a reformulation of the problem.

Resuming his coaxing, Socrates now easily gets the slave boy to see that a square built on that diagonal is equal to twice the area of the first square.

On the left is the first diagram drawn by Socrates. Socrates begins with a square 2 feet on a side and asks the slave boy how to double its size. The slave boy first suggests doubling the length of each side to 4 feet, but this quadruples the area of the square; then increasing the length of the side by one-and-a-half times (by 1 foot), but this, too, increases the area by too much, to 9 feet. In the second diagram on the right, Socrates introduces diagonals, and the slave boy then realizes that the area of the square enclosed by them is twice that of the original square.

Socrates turns to Meno, and tries to lead him on a journey of another kind. Has the boy gone from not knowing to knowing? Yes, admits Meno. Has Socrates fed him any information? No. He’s found the answers within himself? Yes. While these freshly stirred up opinions, being new, are ‘dreamlike’ now, Socrates continues, with more questioning – to make sure the learning is secure and does not slip away – the slave boy will carry this knowledge around inside him, and his ‘knowledge about these things would be as accurate as anyone’s.’ (We call such additional questioning ‘homework and exercises.’) And if we insist on sticking to the terms of Meno’s paradox and say that the boy either knew or didn’t know, then he must have known but forgotten, just as the legend said. Right, Meno admits. I wouldn’t swear to all of the legend, Socrates says, but I’m sure it’s got grains of truth.

Now that Meno is satisfied that learning is possible, the conversation reverts to the original question of virtue and how it might be
taught. Socrates and Meno begin discussing who the teachers of virtue might be. They quickly run out of candidates, for they determine that neither the good citizens nor even the esteemed rulers of the city are appropriate. At this point a wealthy and powerful Athenian named Anytus joins them. Anytus is angered by the conclusion that the good citizens don’t automatically make good teachers of virtue, and ominously warns Socrates not to ‘speak ill of people.’ A few years later, in fact, Anytus will be among the accusers who bring Socrates to the trial that will sentence him to death.

In the play-within-the-play, we see a lot of things; we readers go on a journey as well. We see the Pythagorean theorem taking shape before our eyes. We see the slave boy take a journey in learning the theorem. We see that Socrates is leading the boy, but also that a condition for being led is that the slave boy moves himself. We see Meno going on a journey, looking at the boy moving from ignorance to knowledge. We see what knowing is like: when we get stuck, we can go forward by adding elements to enrich our matrix of terms. The new line – the diagonal – is not present at first. Once introduced, it is as observable as any other line and enriches the matrix of elements to make the path clear. A more sophisticated and concise picture of education in action has never been penned.

But Plato is also showing us, the readers, something about our own situation. The play-within-the-play shows us that we are in the position of the slave boy without the benefit of Socrates to ask us the right questions and give us the right new terms. To some extent, human situations inspire their own implicit questions and create their own uncomfortable feelings, and chance sometimes drops in the diagonal for us; still, the answers often come in a language we don’t know yet, and we will have to forge ahead and create a denser language on our own. Like the young Pascal, we will have to learn how to add that next diagonal ourselves. Plato is also telling us to keep asking questions. Human beings are always tempted to turn what they know into something fixed and congealed, always exposed to the danger of having their deepest truths turn into illusions, reality
into dreams. That is why Socrates famously denounced books in the
Phaedrus
, calling them ‘orphaned remainders of living speech’, which don’t talk back. The only way out is to keep questioning, keep interrogating our experience, keep moving.

Plato has one final trick up his sleeve. He is using the episode to point out that, in our efforts, we will encounter two serious dangers. One is inertia from lazy academics, modern Menos, who will insist that we cannot really learn – that all we can do is add something that looks like what we already have and even if it looks new it’s only a projection, a construction. The second danger is from our politicians and their henchmen, modern Anytus’s, who will tell us that patriotism and the faith of the rulers takes precedence over scientific inquiry. Each group seeks to deny human cultural achievement in a different way. We will have to be patient with the first group; careful, even obsequious, with the second. In one of the most intricately plotted short pieces of literature extant, Plato uses the episode of the Pythagorean theorem in the
Meno
to show us that the journey of truth is much more difficult and perilous than the comfortable quest it is generally billed to be.

We all know the rule, but do we all know the proof? The Pythagorean theorem can be proven in many ways, sometimes even in ways that do not involve a single word. The Cité des Sciences et de l’Industrie in Paris, the largest science museum in Europe, has a visual wall display of the theorem in three dimensions. Three solid but hollow figures are built, one on each side of a right-angled triangle, partly filled with colored liquid that can flow from one solid into the others. When the display revolves, the liquid completely fills the solid built on the hypotenuse with no remainder – but then flows into the other two solids, filling them without remainder! And a nineteenth-century edition of Euclid’s
Elements
– known as ‘one of the oddest and most beautiful books of the century’, was exhibited at the famous Crystal Palace exhibition in London in 1851, and today regularly sells on eBay for thousands of dollars – cleverly used colored lines and figures to condense most of the text of the proofs, including that of the Pythagorean theorem, into almost purely visual presentations.
¹

Philosopher David Socher has a clever way to demonstrate the difference between the Pythagorean theorem, the rule, and the Pythagorean theorem, the proof, to his students of all ages.
²
Without telling them what he is up to, he hands each a large
white square and four colored triangles. ‘I simply explain that we’re going to do a little demonstration. I’m going to ask you to move the pieces in certain ways. It’s not any kind of trick. It’s not hard and it’s not a speed test. It’s a friendly little demonstration.’ He then asks them to arrange the four triangles (each of which happens to be 3 inches × 4 inches × 5 inches) on the square (with 7-inch sides) in two different arrangements. The students readily agree that, in each case, there is the same amount of white space left over. He asks what this says about triangles, and the students usually don’t say much. He asks what they know about triangles, and at least one person usually repeats the Pythagorean theorem, without realizing the connection to what’s in front of them. ‘
Pay dirt
’, Socher writes. For with a word or two more, the connection between rule and proof suddenly descends.

Four triangles in two different arrangements.

That is the unforgettable moment, the kind that we remember – and even long for – as adults. In
Quartered Safe Out Here: A Recollection of the War in Burma
, the British novelist George MacDonald Fraser tells of showing the Pythagorean theorem to his comrade Duke one night after their regiment had dug in along the road to Rangoon during World War II. Tired of conversation about cigarettes, the war, and the Japanese, Duke is on edge; later that night he will die horribly after a series of accidents and misunderstandings, cut almost in two
by a line of friendly machine gun fire as he stumbles in the dark. He asks Fraser to tell him ‘something educated’, wanting ‘a minute’s civilised conversation in which every other word isn’t ‘fook’.’ Fraser offers to ‘prove Pythagoras’, and Duke, delighted, promptly bets him he can’t.

I did it with a bayonet, on the earth beside my pit – which may have been how Pythagoras himself did it originally, for all I know. I went wrong once, having forgotten where to drop the perpendicular, but in the end there it was, and the Duke’s satisfaction was such that I went on, flown with success, to prove that an angle at the centre of a circle is twice an angle at the circumference. He followed it so intently that I felt slightly worried; after all, it’s hardly normal to be utterly absorbed in triangles and circles when the surrounding night may be stiff with Japanese.
³

And Albert Einstein wrote in an autobiographical essay of the ‘wonder’ and ‘indescribable impression’ left by his first encounter with Euclidean plane geometry as a child, when he proved the Pythagorean theorem for himself based on the similarity of triangles. ‘[F]or anyone who experiences [these feelings] for the first time’, Einstein wrote, ‘it is marvelous enough that man is capable at all to reach such a degree of certainty and purity in pure thinking.’
⁴

Einstein’s experience shows yet another kind of thrill that the Pythagorean theorem can teach. For those who do not merely learn how to prove it, but manage to come up with a new proof, the experience teaches the thrill of creativity itself. The person who does this is not merely watching the proof come into being as a spectator watches the unfolding of a little play – that person has become a playwright, doing what mathematicians do, practicing mathematics as a creative art, experiencing
the joy of creation, discovering that the true essence of mathematics is doing more mathematics. Such a person has discovered the power of discovery.

For Plato, Hobbes, Descartes, Hegel, Schopenhauer, Loomis, Einstein, Fraser, and countless others, the Pythagorean theorem was far more than a means to compute the length of hypotenuses. To someone who follows the reasoning, something more than the bare result becomes evident. In the experience of one thing – the content, the mathematics – there is a moment of manifestation in which something else, a structure of reasoning, also comes to appearance. It is a rugged, hardy, stubborn piece of knowledge that no religious conviction can dispel, no political ideology can disguise, no academic artifice can conceal.

In a similar way that 1 + 1 = 2 imparts the idea of addition, so the Pythagorean theorem imparts the idea of proof making. It makes possible what philosophers call categorical intuition: one can see in it more than bare content, but a structure of the understanding. It involves a journey short enough that its stages can be taken in at a glance to illustrate the journey of knowledge itself. It is a proof that demonstrates Proof.