No part of this immense logistical tail by itself compels belief or elicits assent. It was Euclid’s genius to grasp the whole and to trust in the reader to follow what he had grasped.
1. Euclid’s twenty-seventh proposition may suggest Euclid’s original parallel postulate. This is incorrect. Euclid’s twenty-seventh proposition is inverse to Euclid’s parallel postulate and so logically equivalent to its converse.
2. Euclid’s twenty-seventh proposition is logically equivalent to Euclid’s sixteenth proposition, something that the logician August De Morgan observed in the nineteenth century. Euclid might well have begun with his sixteenth proposition and after demonstrating it, arrived at his twenty-seventh proposition by contraposition. The resulting proof would have been impeccable, but it would not have directly mentioned those parallel lines bound for far places that figure in his twenty-seventh proposition. To get them to come forward, he would have had to reverse logical steps and restore the original proposition.
1. James Joyce, Finnegan’s Wake, Library of the University of Adelaide, South Australia, 2005, p. 213.
Chapter VIII
THE DEVIL’S OFFER
Algebra is the offer made by the devil to the mathematician. The devil says: “I will give you this powerful machine, it will answer any question you like. All you need to do is give me your soul: give up geometry and you will have this marvelous machine.”
—MICHAEL ATIYAH
IF THERE ARE numbers and there are shapes in nature, which comes first?
Common sense: neither.
First in what?
Although Euclid’s Elements is a treatise about geometry, some idea of the natural numbers 1, 2, 3, . . . is present as background. It could not be otherwise. Euclid talks of triangles, after all, meaning more than one, and there is one and only one line parallel to a given line specified by Playfair’s axiom. The natural numbers are among the common notions “on which all men base their proofs.” The reverse is true as well. No mathematician could study arithmetic if the numerals did not have stable geometrical properties. Imagine trying to prove that 2 plus 2 is 4 and seeing the numeral “2” undertake a sinuous deformation on the blackboard, or in the mathematician’s mind, vanishing, perhaps, at the very moment of intellectual triumph.
The oil of compromise having been spread, the question, of course, remains. Which does comes first, geometry or arithmetic—first in the sense of being more fundamental, as bread is more fundamental than butter, and thus first in the sense that geometry may be derived from arithmetic, or arithmetic from geometry?
THE NATURAL NUMBERS: 1, 2, 3, . . . Although the number 1 is smaller than all the rest, there is no number greater than any of the others. If there were such a number n, the number n + 1 would be greater still. Does it follow that the natural numbers are infinite? The great Gauss offered a warning. “I protest,” Gauss remarks, “against the use of infinite magnitude as something completed, which is never permissible in mathematics. Infinity is merely a way of speaking.” The proper way to speak is to speak guardedly. The natural numbers are potentially infinite. The mathematician ascends from 1 to 2, from 2 to 3, and from n to n + 1. But unless counting can go on forever, this analysis can hardly do justice to the natural numbers; and if it can go on forever, why not admit the infinite once and for all and be done with it? Georg Cantor, the creator of set theory, argued in the late nineteenth century that the set of natural numbers comprises something infinite all at once, a great thing, complete in its luxuriance. What Cantor could not say is just how the human mind gains access to the infinite if not by climbing the staircase of the numbers, one step at a time. The natural numbers comprise an infinite set. Point to Cantor. Access to the infinite is incremental. Point to Gauss.
This is not Euclid’s way.
IN BOOKS V, VII, and X of the Elements, Euclid talks about the numbers. “A unit,” he says, “is that by virtue of which each of the things that exist is called one.” A number is a “multitude composed of units.” Ancient commentators, writing before and after Euclid, suggested that a unit was the least or smallest answer to the question how many? The answer: just one. They were well aware, of course, that whatever one might be, it, too, could be divided, but this, they argued, would only return the mathematician to whatever multiplicities the number one was supposed to resolve in favor of the least of them.
Although Book VII of the Elements addresses the numbers explicitly, the book is logically subordinate to Book V. The numbers of Book VII are, in fact, the magnitudes of Book V. Book X of the Elements is in an act of double deference subordinate to both. A rare lapse in organization is present throughout. Book V introduces Euclid’s theory of magnitudes. It is widely considered one of Euclid’s masterpieces, although whether by Euclid or whether a masterpiece are other questions. Historians of Greek mathematics now suggest that Euclid’s theory is due largely to the Greek mathematician Eudoxus of Cnidus. Having anticipated the calculus with his method of exhaustion, Eudoxus also appears to have anticipated the real numbers with his theory of proportions. In his treatise Elementary Geometry from an Advanced Standpoint, the mathematician Edwin Moise argued that with respect to the real numbers, modern mathematicians had no need to create what Eudoxus had known all along.
With a unit fixed—a unit chosen—Euclid gains access to a geometrical simulacrum of the natural numbers and to the rational numbers or fractions, such as ½ or ⅓. The number 7 corresponds to 7 units laid tail to head, and the number to the ratio of 1 unit to 7 of them.
In the progression of the Elements, a magnitude represents a new idea, but the easy familiarity with which Euclid handles it suggests that he thought of magnitudes as natural aspects of his system—friends of the family. Euclid never quite says what a magnitude is, but the general idea is of extent, an occupation of space, an expanse. Whatever the expanse encompassed by a unit, it corresponds in the plane to a straight line fixed between two points. This is how Euclid illustrates every one of his arithmetic ideas. It follows that the natural numbers, since they are without end, must correspond to the indefinite production of a straight line.
“It seems that the aging Plato,” René Thom remarked (with some indifference to the English language), “considered this type of generativity to be of the type [of] discrete generativity [that is characteristic] of the sequence of the natural integers.” The very old Plato may have murmured a word into the very young Euclid’s ear; if not, his words were still within hearing distance.
A striking simile is at work in the Elements, one that is today anachronistic. Euclid had found the source of arithmetical generation in a geometrical object. The shadows play on the Euclidean plane. Beyond, there is the play of real things in the real world. Whence the simile. The Euclidean line moves in Euclidean space as a physical object moves in physical space. The idea occurred to many Greek mathematicians. It is the basis for the Greek scheme of geometrical algebra. As it works its way through Euclid’s system, it introduces a degree of contrivance into Euclid’s thoughts. The Euclidean line flows through the points that it touches, but it can be divided into discrete segments only by means of the geometer’s retrospective artifice. From the outset, the Euclidean idea of number conflicts with the simile by which it is explained. In the end, what may seem nothing more than a conceptual conflict grew until it threatened the integrity of the Euclidean system itself.
IT IS NOT disrespectful—is it?—to say that geometrical algebra has in Euclid’s hands all of the elegance of bears chained and taught to dance. In his proof of the Pythagorean theorem, Euclid ignores the algebraic equation in which the facts are so easily expressed—a2 + b2 = h2—and occupies himself with the construction of those rather oafish squares, seeing in their area the secret to the theorem’s meaning. It is a clumsy business. The first proposition of Book II of the Elements affirms that the area of a given rectangle is equal to the sum of its subrectangles. This is in algebraic terms, the distributive law a(b + c + d) = ab + ac + ad, where a, b, c, and d are
numbers. The rectangles are illustrations; they get in the way. Euclid takes geometrical algebra as far as he can go, but by the time he gets to where he is going, the tide must already have begun to turn. And while it took a long time to flow out, in the end it flew out, until mathematicians universally acknowledged the imperatives of analytic geometry, the countercurrent.
Writing in the seventeenth century, René Descartes created analytic geometry in a work titled La Géométrie. Descartes was not quite sure what he was doing. His great work he left almost as an afterthought. In analytic geometry, the Euclidean plane is made accessible, and so it is opened up, by means of a coordinate system. A point is chosen arbitrarily, the origin. Since all points are in the end the same, it hardly matters which point is chosen. Whatever the point, it corresponds to the number zero. Thereafter, the point is bisected by two straight and perpendicular lines, the coordinate axes of the system. The positive natural numbers run from the origin out to infinity, the negative numbers run out the other way, back-street boys to the end, and precisely the same scheme is repeated for the vertical axis, making four line segments starting at zero and proceeding inexorably to the edge of the blackboard and the space beyond.
Any point in the plane may now be identified by a pair of numbers (Figure VIII.1). Hidden previously in the sameness of space, a point acquires a vivid arithmetical identity. It is the point corresponding to two numbers, where before it was some drab or other, anonymous. Once points have acquired their numerical identity, the mathematician can deploy the magnificent machinery of algebraic analysis to endow Euclidean geometry with a second and incomparably more vivid form of life.
FIGURE VIII.1. Euclidean coordinate system
ARITHMETIC IS THE place where the numbers are found, and algebra, the place where they are treated in their most general aspects. The points and straight lines of Euclidean geometry—make of them what you will. They are undefined. Now that a geometrical point has been identified with a pair of numbers, a straight line can be defined by the equation Ax + By + C = 0, where A, B, and C are numerical parameters, fixed place markers, and x and y variables denoting points resident on the line.
Did Euclid have circles to command? He did. A circle whose center is at the point (a, b), and whose radius is R, is perfectly and completely described by the formula (x–a)2 + (y–b)2 = R2. The identity of the circle has been dominated by a numerical regime: its center is a pair of numbers; its radius, another number, and its circumference, an endless succession of numbers.
Analytic geometry has the power to depict a great many familiar geometrical shapes, such as the parabola, the ellipse, and the hyperbola. There is also the cardioid, its penciled heart emerging from the billet-doux of (x2 + y2 + ax)2 = a2 (x2 + y2) (Figure VIII.2).
FIGURE VIII.2. The cardioid
There are curves that look like a woman’s smile, or the valley between hills, or the exuberant petals of some tropical flower.
There is an abundance.
IN A LITTLE book titled The Coordinate Method, a troika of Russian mathematicians (I. M. Gelfand, E. G. Glagoleva, and A. A. Kirillov) offers this account of analytic geometry: “By introducing coordinates, we establish a correspondence between numbers and points of a straight line.” They then add: “In doing so, we exploit the following remarkable fact: There is a unique number corresponding to each point of the line and a unique point of the line corresponding to each number” (emphasis added). The remarkable fact to which they appeal is often described as the Cantor-Dedekind axiom, although how a fact could be an axiom, they do not say.
It hardly matters. There is no such fact, and neither is there any such axiom. There are some numbers that no accessible magnitude can express. In Book X of the Elements, Euclid offers a proof that this is so, one based on an earlier proof attributed to the Pythagorean school. He fails only to notice that what he has done constitutes an act of immolation.
The hypotenuse of a right triangle whose two sides are both 1 is, by the Pythagorean theorem, the square root of 2. The square root of 2 is neither a natural number nor the ratio of natural numbers. A proof simpler than Euclid’s own proceeds by contradiction. Suppose that the square root of 2 could be represented as the ratio of two integers so that √2 = a/b. Squaring both sides of this little equation: 2 = a2/b2. Cross-cutting: a2 = 2b2.
Now the fundamental theorem of arithmetic affirms that every positive integer can be represented as a unique product of positive prime numbers. A prime number is a number divisible only by itself and the number 1. Known to the Greeks, this theorem was known to Euclid. It was widely known; it had gotten around.
The little equation a2 = 2b2 is shortly to undergo a bad accident. Whatever the number of prime factors in a, there must be an even number of them in a2. There are twice as many. Ditto for b2. But the number 2b2 has an odd number of prime factors. The number 2 is, after all, prime. Either the square root of 2 is not a number, or some numbers cannot be expressed as natural numbers or as the ratio of natural numbers.
This is the bad accident.
The consequences are obvious. If the Euclidean line does not contain a point corresponding to the square root of 2, how can the Cantor-Dedekind axiom be true, and if it does, how can the line be Euclidean?
THE NATURAL NUMBERS 1, 2, 3, . . . constitute the smallest set of numbers whose existence cannot be denied without commonly being thought insane. Whatever their nature, nineteenth-century mathematicians discovered how numbers beyond them might be defined and so made useful. A single analytical tool is at work. New numbers arise as they are needed to solve equations that cannot be solved using numbers that are old. Zero is the number that results when any positive number is taken from itself and so appears as the solution to equations of the form x − x = z. The negative numbers provide solutions to equations of the form x–y = z, where y is greater than x. The common fractions, numerator riding shotgun on top, denominator ridden below, are solutions in style to any equation of the form x ÷ y = z.
There remained equations such as x2 = 2. The equation is there in plain sight. What, then, is x? The answer proved difficult to contrive. The Greeks endeavored to find a sense suitable to the square root of 2, but they did not entirely succeed, and beyond the imperative of solving this equation, mathematicians had no common currency with which they could easily pay for its solution. They had nothing in experience.
In the late nineteenth century, Richard Dedekind defined the irrational numbers—how did numbers that are not rational come to be irrational?—in terms of cuts, a partitioning of the integers into two classes, A and B. Every number in A, Dedekind affirmed, is less than any number in B, and what is more, there is no greatest number in A. The cuts themselves he counted as new numbers, the enigmatic square root of 2 corresponding to the cuts A and B in which all numbers less than the square root of 2 are in A, and all those greater than the square root of 2 are in B. Dedekind’s cuts are not the sort of animals one is apt to find in an ordinary zoo. Dedekind’s cuts are, it must be admitted, transgendered, their identity as numbers at odds with their appearance as classes. Dedekind demonstrated, nevertheless, that they were what they did not seem to be, and that is number-like in nature. They could be added and multiplied together; they could be divided and subtracted from one another. They took a lot of abuse. They were fine. They were, in any event, more appealing than the supposition that where there really should be a number answering to the square root of 2, there was no number at all.
The formal introduction of the real numbers in the nineteenth century brought to a close an arithmetical saga, one in which numbers that had once inspired unease acquired their own, their sovereign, identity. The positive integers, zero, the negative integers, the fractions, and the real numbers were all in place. They had acquired an indubitable existence in the minds of mathematicians. The system had a kind of abstract integrity. It held together under scrutiny. It was not adventitious.
THE REAL NUMBER system represented the confluence of two triumphs: the triumph
of arithmetic and the triumph of algebra. The triumph of arithmetic is obvious. The real number system is a system of real numbers. The triumph of algebra, less so. The real numbers satisfy the axioms for an identifiable algebraic structure, what mathematicians call a field. The great achievement of nineteenth- and early twentieth-century mathematics was by a python-like compression of concepts to detach the structure from its examples. Writing in 1910, the German mathematician Ernst Steinitz proposed to make use of fields in an “abstrakten und algmeinen Weise”—in an abstract and general sense. A field, he wrote, is a system of elements with two operations: addition and multiplication. That is all that it is. Steinitz then introduced the distinctively new, entirely modern note, the one that marks a decisive promotion of an interesting idea into an independent idea. Never mind the question, the field of what? The abstract concept of a field is itself at the mittelpunkt of his interests. The examples dwindle away and disappear. The field remains. It becomes itself.1
THE AXIOMS FOR a field bind its various far-flung properties together.2 Their exposition calls to mind the lawyers in Bleak House rising to make a point.
—A field is a set of elements, M’Lud . . .
—Elements, M’lud, anything really.
—Feel it my duty to add, M’lud, that there are two operations on these elements . . .
The King of Infinite Space Page 7