by Ellen Kaplan
For these Russian mathematicians, this new conception inextricably mixed free will, set theory, redemption, and discontinuity, and out of this mixture came powerful results in the theory of functions, descriptive set theory (where naming became a creative force), probability, and many more fields of mathematics.be The great figures we know from the Moscow School, such as Alexandrov and Kolmogorov, were Luzin’s students—and name worshipping continues among the school’s descendants to this day.
It really isn’t surprising that a mind brimming with conjecture should have the analogies astir in it take their symbols as standing for external events as well as for one another, conjoining the mathematical and the metaphysical (the gnomon doesn’t just extend, it embraces the square it belongs with): that the childish and the childlike should seamlessly mingle, whether in Tarentum, Moscow, or Harvard. But Burkert’s claim that the Knowers would not have wanted to prove anything follows from what he takes to be obvious, that proofs diminish the high romance of cloudy symbols. He writes: “That which fills the naïve mind with amazement is seen [by alert mathematical analysis] as tautologous and therefore self-evident.” 14 This peculiar view belongs to what we’ll call Russell’s Syndrome, after Bertrand Russell, who wrote in his autobiography: “I set out with a more or less religious belief in a Platonic eternal world, in which mathematics shone with a beauty like that of the last Cantos of the Paradiso. I came to the conclusion that the eternal world is trivial, and that mathematics is only the art of saying the same thing in different ways.”15
This appalling misunderstanding of mathematics comes from having failed to notice that saying the same thing in different ways, like the slight difference in the angles of our eyes, gives stereoptical depth. The profundity of mathematics follows from its taking two different insights as equal. Why else would a great contemporary mathematician be overcome with appreciative joy every time he revisits a proof of the Pythagorean Theorem?16
So how did the Knowers in Tarentum prove the theorem that honors their founder’s name? Their lives were pervaded by music and their imagination by its cosmic power: not only did the stars circle harmoniously (we fail to hear their music only because it has been in our ears from birth),17 but all things are “locked together by harmony” (as the late Pythagorean, Philolaus, put it) “if they are to be held together in an ordered universe.”18 Their mathematics, accordingly, was steeped in proportions, since they had discovered that the chief notes of the scale came from the simplest whole number ratios. Pluck a string at its midpoint and it sounds the octave above (they expressed this half as the ratio 2:1). Stopping it in the ratio 3:2 produces the fifth, and 4:3 the fourth. There were those numbers again, 1, 2, 3, and 4, that made up 10,
Ratios were therefore on their minds when they looked at what had filtered through from Babylon, the 3-4-5 right triangles and their scaled-up versions. Since scaling preserves ratios (as the Babylonians, or anyone dealing with plans, would have known),bf it would have been natural to draw these similar triangles without any accompanying numbers, that one might stand for all. They let their thoughts loose on these generic figures: a momentous instance of less being more. Turning them this way and that, as their thoughts turned, a scatter of different sizes suddenly harmonized:
That vertical line, the join of two smaller right triangles within the larger, similar third, takes on a life of its own, since it looks just like the proportional stopping of a plucked string—and also like a mean proportional.bg But the use of two mean proportionals was in the air—whether because Hippocrates’s work on doubling the cube had preceded these thoughts, or followed like them from some prior inspiration. Watch how, instead of using these two means in series, they let them play together.
The two triangles, ADC and CDB, are similar: the second, that is, is the first simply scaled up; their corresponding angles are the same, and their sides are in proportion. The symbol for “similar” is, understandably, like a weakened “=”: ~. Since ADC ~ CDB,
and we say that x is the mean proportional between c − y and y. We have, from this equation, that
But also (because ACB ~ CDB) a is the mean proportional between c and y:
in right triangle CDB—and hence, similarly, in all right triangles.
The flash of an insight rarely has a single source (what is the light from one stick rubbing?), and Guido’s proof may have been involved, if it had indeed been invented and circulated by this time: for it is only a few hops of the imagination from moving four congruent right triangles around in a square box to fitting two similar right triangles into a triangular one (lengthening that square box to a rectangle and then cutting it along its diagonal would have been the first such hop).
In any case, the Knowers wouldn’t have put their proof algebraically as we did, but would likely have thought of it in terms of areas, drawing on the Old Babylonian manipulations that showed through the veils of distance and time. This is how it might have gone, pairing a different mean proportional involving the length b, rather than x, with a.
Drop the perpendicular CF from C to the far side, GE, of the square on side AB.
That is, the square on b has the same area as the rectangle AGFD.
Since the area of these two rectangles adds up to the area of the square on c, the theorem is proven.
A proof convinces us by showing that something apparently surprising or unlikely lies in a matrix of relations we recognize and truths we acknowledge. So Pythagoras’s proof of his numerous reincarnations tied them to objects we could see (the shield of Euphorbus), names we knew (the Argonauts’ herald) and figures from daily life (a peacock, a fisherman, a prostitute). By this criterion, Guido’s proof and that by proportions are on a par.
But from this criterion of harmonious connection a finer one soon emerges: not just the magic of seeing the astonishing object mirrored among its contemporaries (like Pythagoras appearing in several places at once), but seeing it derived from the foundation of things: deduction imitating descent. It is immeasurably more meaningful that Pythagoras reincarnated Apollo than Alco. By this higher standard, Guido’s proof is dazzling prestidigitation: now you don’t see it, now you do. But the proof by proportions strikes to the inner eye, which sees not appearance but structure.
Music and mathematics had combined to justify faith in the harmony of things. Only one step remained to attach this proof to the cosmic frame. Since for the Pythagoreans the universe was made of whole numbers and their ratios, and the ratios are manifest in this proof, just call up now their constituent numbers.bh
Ingenuity had taken these numbers away in order to gain the flexibility of generic figures, so that the numbers, having yielded this vital insight, could now be restored.
Just as the octave and all the musical intervals are built up from a unit tone, so here, the a, b, c, x, and y in a right triangle must have a common measure. If a and b are both 1, c must be a number which harmonizes with 1: a whole number—or, if not, a ratio of whole numbers, m/n, so that when we scale up the triangle’s sides by n, we get n, n, and m. But, if a and b are 1, by this very theorem c = . We therefore need to find the ratio—or, some would say, the fraction—, such that . This need ushers in the last act of our drama—tragic or comic, according to your lights. Travelers’ Internet would very likely have brought to Tarentum those brilliant approaches the Old Babylonians took to evaluating , like the one we saw reincarnated in Ptolemy. There just three iterations gave as
But if we recall how we got that value, it takes little sophistication to see that it might not be the final one. In fact, the amount of space left in the L-shaped room after putting in the first three gnomons makes it certain that more work remains. Worse: since successive gnomons clearly take up less and less area, the merest shadow of a doubt about ever being done flickers across the mind. If this mind is already disposed toward skepticism (as we suggested before that Greek minds were), doubt can take on a far more vital form and even raise the unthinkable question: might the problem not lie in th
e method but in itself: might it not be a ratio of whole numbers at all?
Perhaps this question wasn’t so thoroughly unthinkable. A rival musical theory existed at the time, which saw harmonic intervals not as ratios but as lengths of a plucked string: a viewpoint more congenial to awkward—even to continuous—divisions.19 Of course, it would have taken a mind open to ideas from a camp toward which the Pythagorean brotherhood was actively hostile. Something begins to materialize behind the veil—or someone, more shadowy still than Pythagoras: the hero or villain of the piece, Hippasus.
We hardly even know his dates—some time around 450 B.C., perhaps. That way the inner disruption would have neatly coincidedbi20 with the brotherhood’s sudden and surprising loss of political power in southern Italy—when rivals burned down their house at Croton and the only survivors, two young Pythagoreans, fled. We do know what emerged from Hippasus, or from someone taken to be him: we know the proof that couldn’t be a ratio of whole numbers. Again it took rubbing together two sticks, one of which came over the water: the Babylonian inheritance of the deep dichotomy between even and odd numbers. The interest the Pythagoreans took in these was primarily for their mathematical worth (such as square numbers being the sum of successive odds), but with value added on by other meanings: odd was male, light, good, square, and right, even was female, darkness, bad, oblong, and left.
The other stick traveled overland from Elea (modern Velia), a town a hundred miles west of Tarentum on the Tyrrhenian Sea. It was here that Parmenides devised the philosophy which, as we remarked earlier, the Pythagoreans would be ready for when it came: a philosophy devoted to the opposites of being and non-being.21 His message was as simple and profound as a tautology: being is, non-being isn’t. This recognition gave rise to the wholly new sort of proof by contradiction, based—like evens and odds—on opposition: if you assume something is, and following where your assumption leads lands you in something which can’t be, then your initial assumption must also not have been. An upbringing exposed to the numerous contradictions among the acusmata, as well as those in the life and teachings of their founder, would have helped prepare the mind for this way of using the power of contradiction.
Here is how Greek skepticism, possibly Oedipal rivalry, the even-odd distinction, and proof by contradiction combined into one of the great monuments to our power for distinguishing with the mind what the eye could never see, and for answering by a single act of reason what no amount of experiment could ever establish. We will decant its spirit into modern bottles.
THEOREM: isn’t a ratio of whole numbers.
PROOF:
Assume is such a ratio, i.e., that .
Had m and n a common factor, cancel it out before proceeding, so that is now in lowest terms (m and n, therefore, can’t both be even).
Squaring both sides,
Multiply both sides by n2: 2n2 = m2.
Hence m2 is even, and therefore (since even times even is even but odd times odd is odd), so is m. We therefore write m = 2a,
for some number a.
Substituting 2a for m, 2n2 = (2a)2, or
2n2 = 4a2.
Dividing both sides by 2, n2 = 2a2, so that n2 is even and therefore so is n.
Hence m and n are both even, contradicting our second step. We were therefore wrong in assuming that was a ratio of two whole numbers.
A proof as elegant, headlong, and devastating as anything in Sophocles. The incommensurable had torn the veil of numerical harmony, and with it the harmony of the soul, society, and music itself.22
A deeper understanding of what had been achieved by this proof would have to wait until almost the present: a wait exemplified by a young Guido of our acquaintance, who once drew on the blackboard an isosceles right triangle with legs of length 1. After having worked his way through Hippasus’s proof, however, he returned to his diagram and solemnly erased its hypotenuse, since it didn’t exist. He had yet to learn the deepest lesson of Parmenides: rather than not being rational, 2 is irrational. Being Is.bj
CHAPTER FOUR
Rebuilding the Cosmos
Euclid alone has looked on beauty bare.
—EDNA ST. VINCENT MILLAY
There is no excellent beauty that hath not some strangeness in the proportion.
—FRANCIS BACON
The eruption of the irrational shattered the Pythagorean cosmos and littered the ancient Greek landscape with its fragments. Whether or not Hippasus had hoped to take over the brotherhood, his banished the Pythagorean vision of harmony from the Theban landscape of thought. It took little further effort to see that wasn’t the sole exception to the rule of rationality: any rational multiple of it would be irrational too (for if k were such a multiple, and k could be expressed as a/b, then =a /kb, making rational—which we know to our sorrow it can’t be). And soon Theodorus of Cyrene showed that were irrational too, and then his student Theaetetus proved that irrationality was the rule: all square roots, save those of perfect squares, were irrational.
This was the mathematical chaos inherited by Greece at its acme: the serpent under the skin of that classical order you see in the thought of Plato, the statesmanship of Pericles, the works of the Athenian dramatists, architects, and sculptors. In each, however, discordant elements were reconciled: “harmonized opposites,” as Heraclitus put it, “as of the bow and the lyre.” After an epoch of exploration, with its triumphs, shocks, and confusions, mathematics needed time to make a stable foundation on which the shaken structure of number and shape could once again settle. Sense had to be made of the mathematical enterprise itself, and this in turn required a deeper understanding of proportion: one that would include irrational quantities too—not only to save the brilliant proof of the Pythagorean Theorem which you saw in the last chapter (and which depended on a theory of proportions dealing only with ratios of whole numbers), but to show how we may think through time about eternal things.
Eudoxus, who studied with Archytas and then briefly with Plato, came up with a way of handling proportion that answered these needs. It was as sophisticated as any of our own abstract devisings. Rather than asking what a proportion was, and whether there had to be a common measure among its terms, he laid down rules for how it must behave, whatever it was. To assert that , he said, meant that for any whole numbers m and n, if ma > nb, then mc > nd. If ma = nb, then mc = nd. But if ma < nb, then mc < nd. These rules of conduct were all we knew, and all we needed to know, about proportion.
If you find this unnerving, so would the contemporaries of Eudoxus. When Euclid, the Great Consolidator, put together his towering Elements toward the end of the turbulent fourth century, it took the whole of his Book V to work out the properties and consequences of this definition. His aim was to build mathematics up from self-evident truths, yet the Eudoxian view of proportion was anything but self-evident. Since Euclid needed the Pythagorean Theorem quite early on,1 in order to prove a number of fairly basic propositions, how could he do so, if the revised proof by proportions had to wait until the theory behind it stabilized, close to halfway through his work?
Perhaps it is only when you see human affairs in the middle distance that they make conventional sense. Farther back they become too hazy to decipher; closer to, they are utterly strange. In his steadily growing monument to mind, with its wealth of common notions, definitions, postulates, lemmas, porisms, and hundreds of propositions, Euclid proves the Pythagorean Theorem twice. The second time, in Book VI, is indeed by proportions—but at the end of Book I he gives a proof renowned for its ingenuity and notorious for its difficulty, which avoids proportions entirely.
Sir Thomas Heath long since pointed out that Euclid’s approach was nevertheless inspired by the Pythagorean. You might have expected instead that the proof would be like Guido’s, especially since Books I and II of the Elements are filled with the Babylonian Box. It isn’t—but we could see it as drawn from the spirit of that proof as well. Why hadn’t Euclid used such a simple, elegant, and intuitive approach, which needs no
more than a “Behold!” to convince you? Because his aim was to found mathematics forever—and in response to the Eleatic imperative, described in the last chapter, its pictures were to be of perfection, its objects grasped as unchanging. Triangles can’t be shunted around in a box, configured now this way, now that: these figures belonged to Being, and were therefore at rest. And just here, in our view, Euclid’s imagination triumphed: he let the figures remain still, but moved mind among them!
How? By comparing their areas.a If you want to show that the area of a given square, for example, is equal to that of a certain rectangle, find a pair of congruent triangles (whose areas would therefore be equal) and prove that one of them has half the area of the square, the other half that of the rectangle. This transitivity of area, through the triangle’s middle term, was in the spirit of a mean proportional, while its differently disposed triangles you could think of as derived from Guido-like manipulations; it was now, however, a legitimate act of thought, moving over the waters of the world.
Having understood this to be Euclid’s strategy, his proof of proposition I.47 becomes clear. Yet there is a veil of clarity that falls between it and the reader. In reaction to Pythagorean mysticism, the Greeks saw that a proof’s authenticity must follow not from rhetoric or its maker’s reputation, but from no more than mechanical inevitability (the ananke of their drama). They set what is still our style of passive imperative, virtually impersonal, mathematical narration, with all traces purged of the imaginative work and insights that went into a proof’s invention (the passive imperative becomes a perfect passive imperative when it comes to constructions: “Let this have been constructed,” thus solving once again the Eleatic problem that mathematical objects must be at rest; so they are, the constructing having occurred in some prior world, before the proving began). These formal proofs are held to clockwork standards, the gears seen to turn and mesh of themselves. Such are the proofs that seem so intimidating on the pages of most texts, where imagination is assumed to be at its 4 A.M. low ebb. Here is Euclid’s, rearranged organically.
