by Ellen Kaplan
t Number theory is that branch of mathematics which, like life, raises questions so easy to ask and so hard to answer. More specifically, the questions it asks are about the properties of whole numbers.
u The art of construing, as you’ve seen it practiced here, involves reworking mathematical sentences—i.e., equations. No wonder it might seem like hollow tautological play to the Hearers—but to the Knowers? It shifts massive concepts by moving the symbols that are their visible levers.
v Diophantus seems to have come up with a very similar algebraic method. Some think of him as a modern mathematician who just happened to live two thousand years ago; others, as one of those exemplars of pioneering, whose baffling writings are unexplored seas of their own.
w They might also set you wondering whether things are differently true on the whole than specifically because truth is a formal property, while imagination seems to grapple with and ultimately grasp the specific. Or is imagination a messenger, born of that interplay between example and exemplar, which has been a theme running through this book? What does the drawing of a triangle you see on the page stand for: its shaky self, with all the accidents of ink and position; triangularity per se; something beyond—or in between? Shakespeare’s prologue to Henry V asks you to
Think when we talk of horses, that you see them
Printing their proud hoofs i’ the receiving earth;
For ’tis your thoughts that now must deck our kings.
Imagination both bodies forth and disembodies.
x The only point with rational coordinates on the circle which our approach won’t capture is (− 1,0) since the line through it is vertical, and hence has an infinite slope m.
y We’d cast off on induction (in the appendix to Chapter Six) as showing us only that, not why, a statement was true. But it has this compensatory virtue: it takes our finite insights to infinity, where perhaps why and that converge.
z As you can imagine, we could pack our kit with complex numbers instead, or fancier sorts of multipliers—so long as each kit had a 0 in it, and the additive inverse of anything else it contained.
aa Much of this explanation is in a kind of shorthand that abbreviates centuries of engineering and numerous careful manoeuvres with inequalities, to ensure that such arrows of desire as in actually fall past any specified boundary for the one, and within any required ring from the bull’s-eye for the other.
ab If we look at the even more general measure
we see that the absolute value measure is just the special case p = 1. For just about all of these measures, it takes some fancy algebraic dancing to prove the triangle in equal ity.
ac Second order tactics, of course, such as stealth and deception, can modify the most Pythagorean of strategies. So the shortest distance in proof space is from insight to confirmation, but the richest voyages include those detours to sudden views that open up speculation on other and deeper vistas.
ad Stepping back, you may hear a taunting ring of tautology in all this: why do things tend to average out (or is it we who, defensively, do the averaging)? If we indeed live at the limit, why should the outlandish not be the rule rather than the exception—or would that make it the (absurdist) rule? What compels throws of two fair dice to distribute themselves better and better around the mean of 7—are these averages more than descriptive: predictive, or even prescriptive? Would it not be too curious were there no concord between the nature of ordering mind and nature itself, shown by such pointings as these? Or is asking such questions hopelessly wrong-headed, putting a cartload of results in front of the causal horse harnessed to pull them? Is there actually nowhere to step back to, from the play of phenomena? We tend to relegate these edgy concerns to the margins of our thoughts and the footnotes of our books.
ae Yet might this not be just a charming flaw in an otherwise resplendent fabric—a ‘spirit gate’, those errors some American Indians were said to have woven deliberately into their work? No: mathematics suffers from the “dram of eale” syndrome, so devastatingly clarified by Hamlet:
These men
Carrying I say the stamp of one defect
. . . be they as pure as grace,
As infinite as man may undergo,
Shall in the general censure take corruption
From that par ticular fault. The dram of eale
Doth all the noble substance of a doubt
To his own scandal.
Perhaps we can weave one true theorem after another into the tapestry of mathematics without ever completing it; but a single contradiction entangled in its tight threads unravels the whole.
af You can follow a brilliant exposition of this story in Michael Spivak’s Calculus, 2nd ed. (Berkeley, Calif.: Publish or Perish Press, 1980), 465–76. It is an exposition and an explanation that would have angered Luzin, since it is steeped in the juices of continuity in general and Weierstrass’s approach to it in particular—and these, for Luzin, were of the dev il, closing as they did those gaps of discontinuity through which such causeless darts as Grace might penetrate the world.
ag What right have we to interpret lines on the sphere as great circles, or to say that their two intersections are just one? What right have the French to call bread pain? That’s their interpretation of an object in the world, and it seems to work as consistently in their language as ‘bread’ does in ours. French and English are two incarnations of such axioms as language has. These axioms, like those for spherical geometry, don’t come with an ordained instantiation. Anything that faithfully represents them will do. You saw in Chapter Eight some of the widely different ways of interpreting the postulates for measuring distance. Here the choices for picturing lines on a sphere are more constrained, but what’s pleasing is that interpreting them as great circles neatly satisfies all the postulates of elliptic geometry without unnecessary distortions. After all, we could have interpreted lines on the Euclidean plane as cascades of ripples, or as solution sets to the equations y = mx + b, but the ornateness of the one, the abstraction of the other, might have blunted our intuition. The axioms are like regulative thoughts, expressed time and again in shapes as various as that of the hero Euphorbus and the herald Aethalides.
ah All are due to the nineteenth-century Italian geometer Eugenio Beltrami, although some have been named after their rediscoverers. Beltrami had variously been secretary to a railway engineer, a professor of rational mechanics, and a senator of the Kingdom of Italy (the contrast between these last two must have taken him aback).
ai We won’t let it trouble us that triangles on the hyperbolic plane have many peculiar features, such as that their area remains finite even when all their sides are infinite (although Lewis Carroll’s White Queen could believe six impossible things before breakfast, this was one that Reverend Dodgson couldn’t accept at any time of day, concluding that non-Euclidean geometry was non-sense). In our single-minded quest for the Pythagorean relation, all that matters is that we’ve managed to cook up some way of measuring distance here, and so can speak of the side-lengths of triangles and the ratios among them.
aj What follows as well from the power series for sin, cos, and e is that for complex numbers z, sin2z + cos2z = 1.
ak The easiest of the various ways to carry this out would be on a different model of the hyperbolic plane—the so-called Poincaré disk, which is the interior of a circle C, where the ‘points’ are Euclidean points but ‘lines’ are the arcs of other circles that meet C at right angles (along with the diameters of C), not including the points on C’s circumference.
al Having first chosen the same units for measuring time and space—units that con veniently make the speed of light 1.
am An analogy would be to song form, ABA: one melody is followed by a second, then the first repeats. This simple and universal template evolves into sonata form, where ABA has become exposition, development, recapitulation: a format within which the most far-reaching musical adventures happen. It is as if form had passed from facilitating to underly
ing—or is the formal cause neither efficient nor final but—formal?
an Should the madness seize you, you can read more in Richard Guy, Unsolved Problems in Number Theory, 3rd ed. (Berlin: Springer, 2004), section D18; and see Oliver Knill’s “Trea sure Hunting for Perfect Euler Bricks”, http://www.math.harvard .edu/~knill/various/eulercuboid/lecture.pdf. That informed opinion is Robin Hartshorne and Ronald van Luijk’s, in their “Non-Euclidean Pythagorean Triples, A Problem of Euler’s, and Rational Points on K3 Surfaces”, remark 5.4, http://www.math.leidenuniv.nl/~rvl/ps/noneuc.pdf.
ao In every case, these triples are ancestral, as were the Pythagorean triples we found: each begets an infinite family of multiples that work too.
ap A similar problem in this papyrus has the solution (12, 16, 20), which is the triple (3, 4, 5) multiplied by 4.
aq George Sarton, A History of Science: Ancient Science Through the Golden Age of Greece (Cambridge: Harvard University Press, 1952, 39), thinks that the Pythagorean interpretation of rope-stretching is a red herring, being a misunderstanding of an astronomically-slanted ceremony (“Stretching the cord”) at the founding of a temple.
ar You will see in Chapter Seven this beautiful play of untrammeled thought.
as 159, according to Robson, “Neither Sherlock Holmes nor Babylon” 177–78, reduced from the possible pairs of generators available in the 44 numbers of a standard table of reciprocals, if we assume that this already astounding scribe was also familiar with the idea of numbers being relatively prime.
at An error in it, easily explained as a mistake in copying, and another recording the square of a number rather than the number itself, suggest to some that this was a copy, which in turn suggests a teacher’s duplicate from a master.
au It will not have escaped you that YBC 7289 is a special case of Guido’s proof (making all of his four boxes congruent)—and you marvel again at how narrow is the gap between what should and what did generalize, and between the truths of mathematics, which are a priori, and their discovery, which is synthetic.
av How neat it would be were this Ishkur-Mansum, or Ku-Ningal, servant of the moon-god Sin, the first mathematicians whose names we know.
aw A third method was reconstructed by Christopher Walker (who discovered that the two tablets belonged together) and Eleanor Robson, from what Robson describes (in “Three Old Babylonian Methods”, 56–57, 63–64) as “tiny fragments only”. This third method is supposedly the Pythagorean Theorem itself. Since the theorem can’t give a way of approximating, and since, as Robson comments, “it is impossible [from the condition of the fragments] to determine which approximation to the diagonal was used here”, this reconstruction doesn’t enter into our thinking—especially since it makes up out of whole clay 48 percent of the text (in translation) of one problem, 72 percent of the second, and 100 percent of the third. Unlike Miss Froy’s name, no breathing on these fragments will, it seems, make their message reappear.
ax We do the same thing in effect when finding the decimal form of, say, 3/7: 7 doesn’t go into 3, so we make it 7 into 3.0 and divide as if it were 7 into 30.
ay A similar benign neglect comes up for us in calculus, when we show with such boxes what the derivative of a product of functions is. Here again we get a little square, where two rectangles overlap, and cavalierly ignore its area because it contributes negligibly little to the whole—how long the legs of ancient invention.
az So a Russian linguist, Alexander Lipson, playfully concocted “hypothetical roots” of irregular Russian verbs, to make learning the language easier.
ba A companion distinction may have been at work here as well, between the spoken and the written word. The former plays a key role in the wonders of bardic recitation, incantatory magic, and secret doctrine; the latter in the greater complexity and abstractness of argument that it bears and promotes. These Pythagorean events took place at a time when traditions were shifting from the former to the latter: the spoken word was indeed in the beginning, but the written in the end.
bb This law, proposed by Stephen Stigler in his 1980 “Stigler’s Law of Epo nymy” (in Science and Social Structure: A Festschrift for Robert K. Merton, ed. T. F. Gieryn [New York: New York Academy of Sciences, 1980], 147–57), was named by Stigler after himself because, he says, it isn’t originally his.
bc There are four lives of Pythagoras (as befits a man of many incarnations), each more fantastic than its prede ces sor (see www.completepythagoras.net for Guthrie’s translation of these four lives, and all fragments). You will find a paragraph-brief life in Burkert, 165, valuable as much for what it tells us about the character of modern revisionism as for what it says about its subject.
bd That the problem was then differently solved by Archytas’s pupil Eudoxus, who came from Cnidos, a thousand miles away in the Anatolian Peninsula; and after, differently still, by Eudoxus’s pupil Menaechmus, from another town in Asia Minor five hundred miles farther from Cnidos, shows how broad was the reach of information, and how cosmopolitan the Pythagorean community.
be You may rightly wonder how discontinuity, free will, set theory, and redemption could possibly combine. Pavel Florenskii, the mathematician and mystic behind the Name Worshippers, attributed the ethical decline of the nineteenth century to its infatuation with calculus, built as it was on a faith in continuity, which in turn, he saw, established the world as deterministic (differential equations showed what must follow from initial conditions). But Cantorian set theory had demoted continuous sets to but one of many varieties. With discontinuity restored to the worldview, determinism fell, free will was now possible, therefore religious autonomy and, beyond causality, redemption. For the fuller story, see Loren Graham and Jean-Michel Kantor’s Naming Infinity (Cambridge: Belknap Press of Harvard University Press, 2009).
bf It was certainly known to Thales, some fifty years before these events, who determined the height of the Egyptian pyramids by comparing the length of their shadows to that of a measured stick he had set up. The significance of this rang through the isles of Greece: the world afar behaves like the world at hand.
bg It seems clear (Burkert, 440–42) that the Pythagoreans were familiar with the arithmetic and harmonic as well as the geometric means. Only the first two arise in divisions of a one-dimensional string (“the cutting of the canon”), the third bringing us, as you see, into the second and third dimensions.
bh Was there a harbinger to this Pythagorean summoning of number by form in the Old Babylonian play between shapes—their geometrical gnomons—and the numbers that came from filling them up with pebbles? We certainly recognize, in this discarding of a means once the end it leads to is in sight, a process that plays out again and again in mathematics (think of the epsilon-delta approaches to a limit in calculus)—along with its opposite, the means replacing the importance of the original end.
bi But was even the internal crisis wholly due to Hippasus? The stories about him are cloudy. Was he drowned at sea by the gods for his impiety in having discovered (or just disclosed?) the irrationality of 2? Was it instead the irrationality of the Golden Section within the sacred pentagon that he revealed? Or did he more thoroughly subvert the teachings of Pythagoras or, Oedipus-like, betray and displace him? Did he lead the Knowers against the Hearers, only to be abandoned even by them and called a plagiarist? Or was it he who came to understand through theory and experiment the mathematics of musical concord? The utter discord among ancient accounts of Hippasus is well presented in Burkert, 455–65.
bj But what it is becomes less, not more, comprehensible. Not only are the creatures making it up more various and complex than we had thought, but our ways of coming to know them grow ever more subtle. So, in this story, we have passed from diagrams with specific lengths to generic diagrams—and now, with proof by contradiction, to no diagrams at all, where we are on the brink of algebra, with saying replacing showing. Our architectural instinct carries us toward transparent structure—toward structure per se, because (as the Pythagorea
ns’ contemporary Heraclitus wrote, and you read at the outset), a hidden connection is stronger than an apparent one. This is the ultimate meaning of the veil through which we see.
Selected Bibliography
Beauregard, R. A., and E. R. Suryanarayan. “Arithmetic Triangles.” Mathematics Magazine 70, no. 2 (April 1997): 105–15.
———. “Proof Without Words: Parametric Representation of Primitive Pythagorean Triangles.” Mathematics Magazine 69, no. 3 (June 1996): 189.
Borzacchini, Luigi, David Fowler, and David Reed. “Music and Incommensurability.” E-mail exchange on http://mathforum.org/kb/thread.jspa?threadID=384376&messageID=1186550, July 1999.
Bogomolny, Alexander. “Pythagorean Theorem.” http://www.cut-the-knot.org/pythagoras/index.shtml.
Burkert, Walter. Lore and Science in Ancient Pythagoreanism. Trans. Edwin L. Minar Jr. Cambridge: Harvard University Press, 1972.
Fowler, D. H. The Mathematics of Plato’s Academy: A New Reconstruction. New York: Oxford University Press, 1987.
Fowler, David, and Eleanor Robson. “Square Root Approximations in Old Babylonian Mathematics: YBC 7289 in Context.” Historia Mathematica 25 (1998): 366–78.
Friberg, Jöran. Amazing Traces of a Babylonian Origin in Greek Mathematics. Hackensack, N.J.: World Scientific, 2007.
———. A Remarkable Collection of Babylonian Mathematical Texts. New York: Springer, 2007.
———. Unexpected Links Between Egyptian and Babylonian Mathematics. Hackensack, N.J.: World Scientific, 2005.