The King of Infinite Space

by David Berlinski

  —Beg pardon? Any two distinct operations, M’lud.

  —Feel it my duty to add that there is 0 somewhere, M’lud. Yes, here it is.

  —Do? It does nothing M’lud: a + 0 is always a.

  —There is a 1, too, M’lud. Yes, I have it here. Beg pardon? Nothing. It does nothing M’lud: 1a is always a.

  —Feel it my duty to add a word about inverses, M’lud. I have them here.

  —Beg pardon? Do? They invert, M’lud. Any element plus its inverse is 0, and any element times its inverse is 1.

  There is no need to pursue this particular courtroom drama beyond the judge’s demand that his attorneys sit down. A field is an abstract object, and so above it all. Still, it is an abstract object whose most compelling example is the ordinary real numbers. An associative law holds force: a + (b + c) = (a + b) + c. And so does a distributive law: a(b + c) = ab + ac. Identities in 0 and 1, and inverses in the negative numbers and fractions, make possible the recovery of subtraction and division. It is, as lawyers say, familiar fare. A last lawyer rises to remind the judge that the real numbers are ordered. It is always one number before the other, or after the other; it is always, as the judge mutters, one thing or another.

  No matter the lawyers, this idea has been a triumph, the second, after the definition of the real numbers themselves. This prompts the obvious question: a triumph over what?

  IN 1899, DAVID Hilbert published a slender treatise titled Grundlagen der Geometrie (The foundations of geometry). Having for many years lost himself in abstractions, a great mathematician had chosen to revisit his roots. Over the next thirty years, Hilbert would revise his book, changing its emphasis slightly, fiddling, never perfectly satisfied. The Grundlagen—the German word has an earthiness lacking in English—is a moving book, at once a gesture of historical respect and an achievement in self-consciousness. In writing about Euclidean geometry, Hilbert was sensitive to the anxieties running through nineteenth-century thought. Well hidden beneath the exuberant development of various non-Euclidean geometries, the anxieties could often seem arcane. But what mathematicians had suppressed was a concern, sometimes amounting to a doubt, that in geometry, the monumental aspect of Euclid’s system might all along have disguised the fact that none of it made any sense.

  Were the axioms of Euclidean geometry consistent? Or was there buried in the dark flood of their consequences propositions that together with their negations could both be demonstrated? To imagine that Euclidean geometry might be inconsistent would be to place in doubt more than an axiomatic system, but the way of life that it engendered. Hilbert’s Grundlagen did not answer this question completely because it cannot be completely answered. Hilbert showed that geometry is consistent if arithmetic is consistent, an achievement a little like demonstrating that one building is tall if another is taller, but an achievement nonetheless.

  Hilbert undertook the reformation of Euclidean geometry by expanding to twenty Euclid’s original list of five axioms. In a remark of some cheekiness, Thom described Hilbert’s system as a work of “tedious complexity.” The details are onerous. Hilbert had found and then corrected a number of logical lapses in Euclid; he was fastidious. Hilbert accepted, as Euclid did, points, lines, and planes as fundamental, bringing them explicitly into existence by assumption. He had already outlined his method in an essay titled “Uber den Zahlbegriff” (On the concept of number): “One begins by assuming the existence of all elements (that is one assumes at the beginning three different systems of things: points, lines and planes) and one puts these elements into certain relations to one another by means of certain axioms, in particular the axioms of connection, order, congruence and continuity.”

  Having set out twenty axioms, Hilbert then steps back to cast a cool, appraising eye on what he has done. There is a change in emphasis, a heightened sense of explicitness. Euclid’s analysis is directed toward the world of shapes, but Hilbert has begun to think about the analysis itself, his patient, as so often happens, left droning on the leather couch.

  The subtle distinctions needed to make these issues immediate did not exist at the beginning of the twentieth century. Logicians required time to develop them. Hilbert was careful; he made no mistakes in his treatise, but he was not up to date.

  A theory, logicians now say, consists of a set of axioms together with its logical consequences. Euclidean geometry is a theory, the first in human history. A model of a theory consists of the structures in which it is satisfied, a mathematical world, a place in which a theory is at home. Euclidean geometry is satisfied in the Euclidean plane. The simple idea in which theories are juxtaposed to their models makes it possible to ask what models make theories true and whether one theory could be expressed within the alembic of another. It is this idea of re-expression or reinterpretation that Hilbert advanced in his treatise, the tool that he developed.

  HILBERT’S GRUNDLAGEN IS a work with divided purposes. It is, among other things, a defense of classical analytical geometry.

  In thinking about the numbers, Hilbert considered two axioms, the so-called Archimedean axiom, and the so- called completeness axiom. No two axioms have ever been so many times so-called. The first axiom may be found in Eudoxus; it is an implicit aspect of his theory of proportions. The axiom has a simple, powerful intuitive meaning.2 When it comes to certain numbers, there is no greatest among them and no least among them either. The axiom is satisfied by the rational numbers. The axiom went far in the ancient world, but it did not go far enough. It did not suffice to characterize the real numbers, and for this, the completeness axiom was required.

  “To a system of points, straight lines, and planes,” Hilbert wrote, “it is impossible to add other elements in such a manner that the system thus generalized shall form a new geometry obeying all of the five groups of axioms.” There are as many points on the line as there are real numbers. There are enough to go around. This is not the Cantor-Dedekind axiom, which speaks to a correspondence between points and numbers. The completeness axiom is of the enforcement variety. It establishes the existence of those points. It brings them about. It guarantees them. The guarantee makes possible, if not plausible, the techniques of analytic geometry.

  But Hilbert’s completeness axiom is not an axiom of geometry. The objects that the axiom introduces to complete the points on the Euclidean line are not Euclidean: they are not geometric. They belong to arithmetic and they come from afar.

  HAVING OFFERED AN aggressive defense of analytical geometry—here they are, the real numbers, take them or leave them—Hilbert at once revised his tone and tome in order to argue peacefully in favor of a version of Euclidean geometry requiring no direct concourse whatsoever with the arithmetical side of things.

  In Books V, VII, and X of the Elements, Euclid attempted to see in the lights and shadows of a purely geometrical world the stable figures of arithmetic. He looked very hard, but what he discerned, he never discerned completely. Too many lights and, of course, too many shadows. In the Grundlagen, Hilbert justified Euclid and made him whole, son frère du silence éternel.

  The device that Hilbert employed, he called the calculus of segments. Segments are line segments, as Euclid had supposed. There is no form of arithmetical generation at work. Geometry is paramount. With the kind of patience that prompted Thom to complain that he was bored, Hilbert endowed Euclid’s line segments with arithmetic powers all their own. They could be added together, subtracted from one another, multiplied and divided, arranged in continuous proportional arrays, and they could encompass some square roots.

  So it could be carried out—yes, the old haunting, incomplete, Euclidean arithmetic scheme. It could be carried out, as Hilbert understood perfectly, but not carried out completely. Line segments are not numbers. They may be used to illustrate numbers or to form a picture by which the numbers are understood, but they are not numbers.

  THE FRENCH VERB engloutir means to swallow something without chewing—swallowing it whole, an annihilation. Having given up brawlin
g in favor of good works, still another Hilbert is prepared to give up good works in favor of the revolution. This Hilbert is prepared to show that Euclidean geometry may be swallowed by the field of the real numbers. His work complete, Hilbert the Red demonstrated that the points, lines, and planes of Euclidean geometry are actors in an algebraic world not of their making and indifferent to their nature, their impression to the contrary entirely a matter of false consciousness.

  THE GEOMETRICAL THEORY that Hilbert presented in the Grundlagen is very much Euclid’s theory. Hilbert’s axioms are more precisely expressed. There are many more of them than may be found in Euclid. The spirit is the same. Euclid had, of course, intended his theory to be interpreted in the Euclidean plane, geometrical axioms satisfied in a purely geometrical model. This will not do because that did not serve Hilbert’s radical agenda. For his own model, Hilbert chose an arithmetic object, one composed of a set of numbers Ω. These numbers begin with 1, and they include all the numbers that may be made from 1 by the operations of addition, subtraction, multiplication, and division, and the numbers . This structure generates the real numbers, but it does not generate all of them. It is limited. This spare structure also satisfies the axioms for a field. Hilbert made his choice of Ω for the sake of convenience; just a few pages later, he recognized that having introduced some of the real numbers but not all of them, he might well have introduced all of the real numbers and not just some of them.

  What follows is swallowing in steps. “Let us regard,” Hilbert writes, “a pair of numbers (x, y) . . . as defining a point.” What is undefined in geometry has just met what is perfectly defined in arithmetic. In the ensuing friction, any sense of a correlation between points and numbers is lost. There is an identity: a point is a pair of numbers. The Euclidean plane gives way. All is dark. The mathematician’s attention has swept out from one theory in geometry to another theory in arithmetic.

  Euclidean straight lines follow Euclidean points into the void, only to emerge again, reborn in arithmetic. A straight line, Hilbert writes, is a ratio of three numbers (a: b: c). An insistent arithmetical identity now imposes itself on the old-fashioned and rapidly receding Euclidean shape. A straight line is the ratio of three numbers. The choice of three numbers suggests, of course, the equation for the straight line, the symbols Ax + By + C = 0, bringing all of them under the control of a single symbolic form. The numbers are expressed as a ratio because two equations a0x + b0y + c0 = 0 and a1x + b1y + c1 = 0 define—they are—one and the same straight line if the numbers a, b, and c are proportional. In this way, Hilbert has offered an interpretation of the undefined terms of Euclidean geometry in terms of the defined elements of a real ordered field. He has gotten one theory to speak in another theory’s voice.

  It is like hearing a cat bark.

  THE EUCLIDEAN POINT has vanished in favor of pairs of numbers; the Euclidean straight line in favor of triplets of numbers—their ratios. Hilbert is now free to provide an interpretation of Euclid’s axioms in arithmetic. Following Euclid, Hilbert affirms that two distinct points A and B always determine a straight line a. This axiom is assumed in geometry, but in arithmetic, it is not assumed at all. It is demonstrable. Hilbert had already gained the power to say what it means for a point to lie on a straight line without ever mentioning points or straight lines at all. “The equation ax + by + c = 0,” he writes, “expresses the condition that the point (x, y) lies on the straight line (a: b: c).”

  Euclid’s first axiom talks of two points A and B, and one straight line a. The point A is equal to a pair of numbers (x1, y1). Ditto the point B, like A also equal to a pair of numbers (x2, y2). The straight line a is equal to a ratio of numbers (a, b, c).

  Euclid’s first axiom is true in its arithmetical model, Ω, if some equation may be found satisfying both A and B. And, of course, there is. The point (x1, y1) lies on the straight line ax1 + by1 + c = 0. The point (x2, y2) lies on the straight line ax2 + by2 + c = 0. Subtracting the second equation from the first yields a(x2 − x1) + b(y2 – y1) = 0. The parameter c has vanished; a and b may now be banished in the equation (y1 − y2)x + (x2 − x1)y + (x1y2 − x2y1) = 0. Two points: but one straight line.

  What has come forward has come back. The circle is closed. Step by patient step, Hilbert shows how every one of the axioms of Euclidean geometry can be interpreted within a purely arithmetic model. But Hilbert, of course, does more. The fact that every two points determine a straight line is not only true in the real ordered field, it is demonstrable. Geometrical axioms have become arithmetical theorems. By this maneuver, Euclidean geometry has been swallowed by arithmetic, the swallowing giving rise to what are today called Euclidean vector spaces, new structures, ubiquitous throughout mathematics, their sleek compact lines concealing all traces of the annihilation by which they were created.

  There is no coordination, no counterpart, no mapping, no scheme of correlation between points and pairs of number. This is because under Hilbert’s analysis, there are no Euclidean points of old.

  “Analytical geometry,” the French mathematician Jean Dieudonné observed happily in remarks now famous, “has never existed. There are only people who do linear geometry badly, by taking coordinates, and they call this analytical geometry. Out with them!”

  “Down with Euclid,” he added, just to be sure.

  1. Ernst Steinitz, “Algebraische Theorie der Körper,” Crelles Journal (1910) (my own translation).

  2. For which, see my own One, Two, Three (New York: Pantheon Books, 2011).

  2. I discuss that simple, powerful intuitive meaning in David Berlinski, A Tour of the Calculus (New York: Pantheon, 1995).

  Chapter IX

  THE EUCLIDEAN JOINT STOCK COMPANY

  A tradition is kept alive only by something being added to it.

  —HENRY JAMES

  AT THE BEGINNING of the nineteenth century, the Euclidean Joint Stock Company was wholly owned by Euclid and his Elements. If there was among mathematicians a residual uneasiness about Euclid’s parallel postulate, it did little to diminish their sense that Euclidean geometry had, over the course of 2,300 years, been valued at its true price. The philosophers agreed. Euclidean geometry, Immanuel Kant argued in The Critique of Pure Reason, was not only true—of course it was—but necessarily true, an aspect of the human mind, the expression of the way the mind confronted the sensuous world of shapes.

  Within sixty years, the famous old company had undergone a dilution of ownership. Kant had doubled his investment at the very moment his investment had been halved in value. And not only Kant. The philosophers had all missed something; theirs was an insufficiency of daring.

  By the end of the nineteenth century, it had become clear that there were geometrical schemes in which Euclid’s parallel postulate might be replaced by its denial. The long- expected shambles did not emerge. When in 1915, Albert Einstein published the field equations for his theory of general relativity, non-Euclidean geometries acquired a dignified physical standing. Mathematicians such as Carl Friedrich Gauss, Nicolai Lobachevsky, János Bolyai, and Bernhard Riemann, the men who had made non-Euclidean geometry, demanded substantial ownership positions in the Euclidean company, and they received them.

  EUCLID’S FIFTH AXIOM is one of the few statements in mathematics to have achieved a stable sort of notoriety. It is shady; this everyone understands. And controversial. This, too, is understood. Judging from the popular literature, no one is quite sure why.

  Euclid’s parallel postulate has today been enveloped by the cloak of its flamboyant history. Euclid made no use of his postulate until he came to his twenty-ninth proposition. It would seem that he had brooded suspiciously. He had put things off. Common sense might suggest that Euclid did not use the parallel postulate before he did, because he had not required it before he had. This is a prosaic view and for this reason not widely entertained.

  Ancient commentators wondered whether the parallel postulate might be deduced from Euclid’s other four axioms. They sensed its ano
malous character. It made them uneasy. They could not entirely say why, and sometimes they were betrayed by their scruples. Proclus rejected a fallacious proof of the parallel postulate by Ptolemy and at once advanced a fallacious proof of his own. Ancient mathematicians often assumed what they intended to show, a circumstance that enlarged their frustrations without ever resolving their anxieties.

  Mathematicians uninterested in proving the parallel postulate were often interested in demonstrating that it was equivalent to something else, perhaps in the expectation that swapping things around would reveal something simpler and more compelling. The parallel postulate is a proposition too powerful to be exhausted by a single identity. The Pythagorean theorem and Euclid’s parallel postulate are the same. The first leads logically to the second; the second leads logically to the first. In Book I of the Elements, Euclid demonstrated that A + B + C = π, where A, B, and C are the interior angles of any triangle. It is his thirty-second proposition. But A + B + C = π is logically equivalent to the parallel postulate. Euclid had demonstrated what he had already assumed, the parallel postulate exerting a deforming force on the very structure of Euclidean deduction.

  Attempts to prove the parallel postulate continued sporadically over the next two thousand years. Some mathematicians gave the matter a look and, after a few desultory attempts, turned away. The parallel postulate seemed a hard little knot, a twisted root. Mathematicians of the long and brilliant Arab renaissance were as intrigued by the parallel postulate as the Greeks before them had been. Writing in the tenth century, Ibn al Haytham thought that the postulate might require an indirect proof. Euclid had often demonstrated theorems in the Elements by assuming that they were false and searching for the contradiction that ensued. Ibn al Haytham did the same; he searched for the shambles. He found nothing, his assumption, for the sake of argument, that the parallel postulate was false leaving everything the same, glassy and undisturbed.