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.
Interlude
RULES, PROOFS, AND
THE MAGIC OF MATHEMATICS
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 perpendi
cular, 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.
2
‘The Soul of Classical Mechanics’:
NEWTON’S SECOND LAW OF MOTION
F = ma
[Newton’s own] description: A change in motion is proportional to the motive force impressed and takes place along the straight line in which that force is impressed.
DISCOVERER: Isaac Newton
DATE: 1684–87
Newton’s second law of motion, F = ma, is the soul of classical mechanics.
– Frank Wilczek, Physics Today
The equation F = ma is shorthand for Newton’s second law of motion. It is the 1 + 1 = 2 of classical mechanics. It seems obvious and straightforward. The equation appears simply to translate an ordinary experience into measurable terms: push something and it either starts to move or moves differently.
Yet, like 1 + 1 = 2, F = ma erupts into mystery when looked at closely. It does not, in fact, refer to ordinary experience but to an abstract world of zero resistance: in the real world we have to continue pushing things like desks and carts to keep them moving at the same speed. The equation does not incorporate Einstein’s famous discovery of the interchangeability of mass and energy. It gives centre stage to force – a concept absent from most formulations of contemporary theories like relativity and quantum mechanics. Finally, the equation seems, in a contradictory way, to be both a name and a description. It seems both to define force, mass, and motion, and to state an empirically discovered and testable relationship among them.
How can such an elementary equation about something as ordinary as motion conceal so many complexities? The answer can be gleaned in the remarkable historical journey that led from ancient times to the equation’s formulation in the seventeenth century. To arrive at this equation, human beings had to train themselves to look at motion in new ways – to learn to look at different aspects of it, and to change how they thought about what they saw. In the course of this vast journey, new sights slowly and progressively came into view, occupied centre stage, and then vanished off the horizon, with each familiar landscape slowly yielding to another, until the travelers found themselves in an entirely new world.
Greek Notions of Motion and Change
The journey began in primitive times, when human beings saw the world as ruled by deities. This was natural and inevitable, perhaps the simplest and most straightforward way to make sense of things. All humans acquire a notion of force from individual experiences of pushes and pulls of daily life, in applying our muscles to lift, squeeze, or roll things. Generalizing that experience, early humans could readily conceive everything in nature – from nearby phenomena like thunder and rain to the movements of distant bodies like the sun and stars – as the result of spirits behaving and misbehaving, exerting their particular internal forces. Thus early ideas of force were closely connected with religious ideas of the direct presence of gods in the world.1
The earliest humans naturally tried to control nature by pleasing the spirits through ritual and prayer – the earliest forms of technology. But that did not succeed in bringing about the desired control. A far more effective way to predict and influence nature seemed to be to pay attention to the kinds and amounts of changes in nature – the recurrence of the seasons, the various movements of the planets and stars, the behaviour of fire and floods, and so forth. But nature is so varied! Sunlight and clouds, tides and storms, plants and animals, men and women, plans and ideas, houses and cities are constantly being born and dying, rising and falling, changing colors and forms, and moving about. How could one make sense of all these motions?
The Greek philosopher Aristotle (384–322 BC) was the earliest we know who drew up a systematic account of all kinds of motion or change – he used the same word, kinesis, for both. Kinesis is so important, he thought, that to understand it is tantamount to understanding nature itself, and he created a framework to include all varieties of kinesis: of animate and inanimate objects, with and without human intervention, on earth and in the heavens. He distinguished several kinds of kinesis: the substantial change of a thing being born or dying (fire consuming a log); the quantitative change of a thing growing or shrinking; the transformational change of one property changing into another (a green leaf turning brown); and local motion, or something changing its place.
Aristotle viewed these changes with biologically trained eyes. He regarded the world as a kind of cosmic ecosystem that contained many different levels of organization. Motion in this ecosystem is almost never random or chaotic, but a process of passing from one state to another in which something existing only in potential (a formal principle) is underway to being actualized. Many levels of organization are built on top of each other – human beings make up a state, organs make up a human being – so that any event is shaped by a complex network of different kinds of causes.
Aristotle understood this cosmic ecosystem in the framework of a set of key distinctions. He distinguished, for instance, between two kinds of motion: natural, or violent and forced. Natural motion was that of things moving themselves in their own habitats – acorns growing into oaks, or eggs into chickens – where the change actualizes some innate principle in the substance itself. Forced or violent motions occur when the change is imposed from the outside, as what happens to oak trees when humans fell them to build houses, or to chickens when humans slaughter them for food.
Aristotle also thought it mattered where a change happens. In the earthly realm below the moon, substances are composed of different mixtures of earth, air, fire, and water, and objects don’t move constantly but intermittently. In the heavenly realm, objects are made of an unchanging substance called ‘ether’, and mo
ve ceaselessly and circularly. If today we find this unjustified, it is a sign of how far we have traveled since Aristotle’s time and how our sight has changed, for his ideas were based on rational argument, logical deduction, and careful observation. For hundreds of years, astronomers in Greece and elsewhere had never witnessed any changes in celestial behaviour, nor seen anything but circular motion.2 Only circular movement can continue unceasingly, he thought, and only some special substance, not known on earth (hence the strange name ether), does not experience change.3 In the celestial realm, motion is initiated by a so-called unmoved mover, which drew the celestial spheres into motion. This was Aristotle’s analogue to God, though it was impersonal and not something with which one could have what we twenty-first-century humans call a ‘relationship.’ The celestial spheres, by various intermediaries, transmit the motion to the terrestrial sphere. Thus all motions in the cosmic ecosystem, however tiny, are connected, in a mediated way, with the first principle of the universe, and ultimately have to be understood in that context.
When Aristotle discussed what we call motion, then, we onlookers from 2,500 years later have to be careful not to read in our own assumptions. When he speaks of local motions, it is generally in the context of events such as a horse pulling a cart on the road or shipbuilders pushing a boat. Such events arise from a complicated network of purposes, plans, and designs that are being actualized, of which the local motion is only one aspect. And when Aristotle does discuss that aspect, he is not propounding and defending some hypothesis about local motion apart from the event itself, but rather speaking in general terms about the work required to accomplish such tasks to illustrate some other point. In these kinds of events, furthermore, the role of acceleration is almost nil, and rules of thumb such as ‘A force that moves a body against a certain resistance a certain distance in a certain time moves the same body half the distance in half the time’ work fine. Though at one point, in an assertion to become infamous two millennia later as the target of falling-body experiments, he remarked that ‘If half the weight moves the distance in a given time, its double (i.e., the whole weight) will take half the time.’4
A Brief Guide to the Great Equations Page 4
