In the fourth century ad, the Greek geometer Pappus of Alexandria discovered a theorem that extended Euclid’s. A few centuries later, Arabic mathematician Thābit Ibn Qurra (836–901), working in Baghdad, provided several new proofs in revising an earlier Arabic translation of the Elements. Two and a half centuries later, the Hindu mathematician Bhaskara (b. 1114) was so enamored of the visual simplicity of the Zhou Bi proof that he redid it in the form of a simple diagram, and instead of an explanation wrote a single word of instruction: ‘See.’
Later, Italian artist Leonardo da Vinci, Dutch scientist Christiaan Huygens, and German philosopher Gottfried Leibniz (1646–1716), all contributed new proofs. So did U.S. Congressman James Garfield, in 1876, before he became the twentieth U.S. president. Indeed, over a dozen collections of proofs of the Pythagorean theorem have appeared: in 1778, a list of thirty-eight was published in Paris, in 1880 a monograph appeared in Germany with forty-six proofs, while in 1914 a list of ninety-six proofs was published in Holland. The American Mathematical Monthly, the first general-interest mathematical magazine in the U.S., began publishing proofs in its first issues, starting in 1894. With some condescension, it stated that problem solving ‘is one of the lowest forms of mathematical research’, being applied and without scientific merit. Nevertheless, the magazine promised to devote ‘a due portion of its space to the solution of problems’ such as the Pythagorean theorem, to serve an educational purpose. ‘It [problem solving] is the ladder by which the mind ascends into the higher fields of original research and investigation. Many dormant minds have been aroused into activity through the mastery of a single problem.’12 In 1901, after publishing about a hundred proofs, its editor abandoned the effort, announcing that ‘there is no limit to the number of proofs – we just had to quit.’
Diagram on the basis of which U.S. President James A. Garfield invented a proof.
One who refused to quit was a schoolteacher and subscriber from Ohio named Elisha S. Loomis – a mason – who had contributed some of the proofs. Loomis continued to collect them, many passed on by teachers of bright youngsters who knew of his interest. In 1927 (by then a college professor) Loomis published The Pythagorean Proposition, a book containing 230 proofs; in 1940, the 87-year-old Loomis published a second edition containing 370 proofs.13 He dedicated both books to his Masonic lodge. Loomis divided the contents into geometric, algebraic, dynamic, and quaternion proofs. Most were geometric: number 31 was Huygens’s; 33, Euclid’s; 46, da Vinci’s; 225, Bhaskara’s; 231, Garfield’s; and the Zhou Bi’s was 243. Of the algebraic proofs, Leibniz’s was number 53. Loomis prized the way that the challenge of coming up with a new proof tested the mettle of students and, evidently fascinated by the process of proof, liked to signal interesting proofs, interesting people who had contributed proofs, or to commend youthful contributors.14 He was disapproving of those who, he thought, disrespected the subject. He chastised some American geometry textbooks that omitted Euclid’s proof – possibly to show ‘originality or independence’ – remarking wryly that ‘the leaving out of Euclid’s proof is like the play of Hamlet with Hamlet left out.’15 His final sentence of the second edition: ‘And the end is not yet.’16
Loomis was correct; it wasn’t. The Guinness Book of World Records Web site, under ‘Most Proofs of Pythagoras’s Theorem’, recently named a Greek who, it is claimed, has discovered 520 proofs. By the time you read this, more have surely appeared.
Whence the Magic?
All these proofs provoke two questions. The first is: Why isn’t one proof enough? We know why one application is not enough: the point of a rule is that it applies to many different circumstances. But proofs? A small number of proofs of the Pythagorean theorem generalize the theorem that Euclid proved, and thus extend what he did. Most in Loomis’s collections, however, are not of that type. Nor do they make the result more certain than it already is. Their fascination lies in the scientific desire not merely to discover, but to view a discovery from as many angles as possible – to convert implicit possibilities, or merely hypothesized or assumed results, into actualities. Science aims to enrich the world, to increase the variety of its forms, to let the reality of the things in the world show themselves. As science progresses, the landscape of the world develops with it.
The second question is: Why all the attention to this particular theorem, which has fascinated amateurs and professionals for thousands of years? A part of the answer is surely personal biography: the Pythagorean theorem tends to be the first deep proof that each of us encounters, the first proof where – as Hobbes’s experience shows – it is not obvious what it is we are setting out to prove. It is the first journey of mathematical discovery where we find something genuinely new at the other end. But that must be only a small part of the answer, for we also learn other beautiful proofs early on, such as of the irrationality of the square root of two or of the infinity of primes. We also learn proofs that are similar to the Pythagorean theorem (for instance, Proposition 31 of Book VI of Euclid’s Elements), or much more powerful and useful than the Pythagorean theorem, without these attracting anywhere near the same degree of attention. A striking example of the latter is the law of cosines – c2 = a2 + b2 – 2ab cos θ – which covers all triangles, not just right-angled triangles, and relates the lengths of the sides to the cosine of one of the angles; the Pythagorean theorem is but a special case of the law of cosines. Yet this law communicates no special magic – partly because one has to know trigonometry to prove it – and one can hardly imagine a Hobbes becoming as transformed.
The full answer as to why the Pythagorean theorem seems magical is threefold: the visibility of the hypotenuse rule’s applications, the accessibility of the proof, and the way that actually proving the theorem seems to elevate us to contemplate higher truths and thus acquaint us with the joy of knowing.
First, the theorem characterizes the space around us, and we thus encounter it not only in carpentry and architecture, physics and astronomy, but in nearly every application and profession. The Pythagorean rule for a distance in three-dimensional space – the diagonal of a shoebox, say – is the square root of x2 + y2 + z2; for four-dimensional Euclidean space it’s the square root of x2 + y2 + z2 + w2; in Minkowski’s interpretation of Einstein’s special theory of relativity, the four-dimensional space-time version is x2 + y2 + z2 1 (ct)2, where c is the speed of light. Suitably adapted, this formula enters into the equations of thermodynamics, in describing the three-dimensional motions of masses of molecules. It also enters into both the special and the general theory of relativity. (In the former, it is used to describe the path of light moving in one reference frame from the point of view of another, while in the latter, in a still more complex extension, it is used to describe the motion of light in curved, four-dimensional space-time.) And it is generalized still further in higher mathematics. In The Pythagorean Theorem: A 4,000-Year History, Eli Maor calls the Pythagorean theorem ‘the most frequently used theorem in all of mathematics.’17 This is not only because of its direct use but also due to what Maor calls ‘ghosts of the Pythagorean theorem’ – the host of other expressions that derive, directly or indirectly, from it. An example is Fermat’s famous ‘last theorem’, finally proven in 1994, which asserts that no integers satisfy the equation an = bn + cn (all variables stand for positive integers) for any n greater than two. Though, being the denial rather than the assertion of an equality, Fermat’s last theorem cannot be put in the form of an equation.
Second, as Hobbes’s experience indicates, even though the Pythagorean theorem involves a bit of knowledge whose proof seems implausible at the beginning, it can be proved simply and convincingly even without mathematical training. This is one reason why philosophers and scientists from Plato onward use it as an emblematic demonstration of reasoning itself. In On the World Systems, Galileo cited Pythagoras’s experience proving the theorem to illustrate the distinction between certainty and proof – what we now call the context of discovery and the context of jus
tification.18 In The Rules for the Direction of the Human Mind (Rule XVI), French philosopher and scientist René Descartes used the Pythagorean theorem to show the virtues of symbolic notation, which he was introducing into mathematics. G.W.F. Hegel viewed the proof as ‘superior to all others’ in the way it illustrates what it means for geometry to proceed scientifically, which for him meant showing how an identity contains differences.19 German philosopher Arthur Schopenhauer, one of the few critics of the way Euclid proved the Pythagorean theorem, viewed that proof as emblematic for another reason. Mocking it as a ‘mousetrap’ proof that lures readers in and then ‘springs’ a trap on them, Schopenhauer thought it logically true but overtly complicated, did not like the fact that not all its steps were intuitive (he much preferred proofs that appealed to intuition), and maintained that Euclid’s proof is a classical illustration of a misleading demonstration. Indeed, he saw Euclid’s proof of the Pythagorean theorem as emblematic of all that was wrong with the philosophy of his day, for it emphasized the triumph of sheer logic over insight and educated intuition. Hegel’s philosophical system, for Schopenhauer, was in effect no more than one huge conceptual mousetrap.20
Third, the Pythagorean theorem makes accessible the visceral thrill of discovery. Whenever we prove it, we can hardly be said to be ‘learning’ anything, for we learned the hypotenuse rule as schoolchildren. But as the proof proceeds – as we set the problem in a bigger context, and as the little pieces begin to snap together with an awesome inevitability – we seem to be taken out of the here and now to someplace else, a realm of truths far more ancient than we, a place we can reach with a little bit of effort no matter where we are. In that place, this particular right-angled triangle is nothing special; all are the same and we do not have to start the proof all over again to be certain of it. Something lies behind this particular triangle, of which it is but an instance. The experience is comforting, even thrilling, and you do not forget it. The proof arrives as the answer to a puzzle in a language that you did not have beforehand, a language that arrives in that instant yet which, paradoxically, you sense you already possessed. Without that moment of insight, the Pythagorean theorem remains a rule handed down authoritatively, rather than a proof gained insightfully.
The Pythagorean Theorem in Plato’s Meno
All three components of the magic of the Pythagorean theorem are evident in the earliest known, most celebrated, and most complexly described story of a journey to the Pythagorean theorem. That occurs in Plato’s dialogue Meno, written about 385 BC, or somewhat more than a century after Pythagoras and almost a century before Euclid’s Elements. It is the first extended illustration of the mathematical knowledge of ancient Greece that exists. In the Meno, Socrates coaxes a slave boy, ignorant of mathematics, to prove a particular instance of the theorem, one involving an isosceles right-angled triangle.
The principal participants are Socrates and Meno, a handsome youth from Thessaly. Meno is impatient, balks at difficult ideas, and likes impressive-sounding answers – a teacher’s nightmare. He’s been pestering Socrates about how it is possible to learn virtue. Socrates finds it difficult to get Meno’s mind going; his name, appropriately, means ‘stand fast’ or ‘stay put.’ The word ‘education’ means literally ‘to lead out.’ Socrates cannot lead Meno much of anywhere.
At one point, Meno throws up his hands and asks Socrates – in a famous query known as Meno’s paradox – how it is possible to learn anything at all. If you don’t know what you are looking for, you won’t be able to recognize it when you come across it – while if you do know you don’t bother to go looking for it. Meno is implying that it is fruitless even to try.
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.
A Brief Guide to the Great Equations Page 3
