It is difficult to imagine an objection being framed. Did anyone in the fifth century BC propose that what is, is not, or that what is not, is? Yet from the premise that nothing is, after all, nothing, Parmenides drew the conclusion that there is no void between atoms, because it makes no sense to say of a void that it is.
It then follows that space is just one thing, and not many things. What beyond spatial separation could mark the distinction between atoms, the more so if like Euclidean points, they have no parts? If space is filled, then motion and, indeed, change, are impossible. There is no place to go, and if no place to go, no place to have come from either.
These strange ideas belong to the pre-Socratic world, one that in the popular imagination contrasts unfavorably with our own. But Euclid lived and worked within historical memory of the pre-Socratic philosophers. Parmenides was as close to his consciousness as Abraham Lincoln is to ours. Those bony Parmenidean fingers were poking into Euclid’s shoulder.
If there are points in the plane, then Euclidean space is replete with them, for between two Euclidean points along any straight line, there is always another Euclidean point. The inference is almost immediate. Euclid’s third definition identifies the ends of a line with two points, and his twenty-third definition establishes that a straight line may be produced indefinitely. Suppose that there is no point between the points P and Q lying on the straight line L. Then starting at P, L could not fall short of Q. Lacking any other point by assumption, one of its ends would dangle uselessly. In that case, how could L be produced from P?
This downward descent by which points lead to points must, so one might imagine, end either with nothing between points or with something still further. It is an inference at odds with the geometer, eager to get from one point to another.
If nothing, how? If something, what?
IN THE COMPETITION between contending ghosts, Parmenides has made his influence felt. Democritus, too. Euclidean points may well be like atoms, but there is no void anywhere in the Elements, no suggestion that there is nothing between points. For Euclid, it is points all the way down.
The discussion is hardly at an end. In his little book Das Kontinuum, the twentieth-century mathematician Hermann Weyl found himself interrogating the pre-Socratics all over again. It is quite a crowd in Euclid’s study. Between any two points, there is a third. Yet time flows, and things change, and there is a distinction between the flow of time and the points used to mark that flow. The points are like diamonds in a skein of silk: attend to them, and they catch. But as time flows, it does not catch. “The view of a flow,” Weyl wrote, “consisting of points and, therefore, also dissolving into points turns out to be mistaken: precisely what eludes us is the nature of the continuity, the flowing from point to point; in other words, the secret of how the continually enduring present can continually slip away into the receding past.”
About these issues, Euclid said nothing at all.
Chapter V
THE AXIOMS
Nempe nullas vias hominibus patere ad cognitionem certam veritatis praeter evidentem intuitum, et necassariam deductionem (There are only two routes open to human beings to arrive at sound knowledge of the truth, evident intuition and necessary deduction).
—RENÉ DESCARTES
The dull mind, once arriving at an inference that flatters the desire, is rarely able to retain the impression that the notion from which the inference started was purely problematic.
—GEORGE ELIOT
EUCLID PROPOSED FIVE axioms for geometry. These axioms cannot, of course, be themselves derived from still further assumptions. Or from anything else. “No science,” Aristotle dryly remarks, “proves its own principles.” It is possible, of course, that if some theorems were made axioms, then some axioms could be made theorems. The American logician Harvey Friedman has for this reason studied the extent to which something standing on its feet could be made to stand on its head. This does not mean that Euclid’s axioms are unjustified or arbitrary. If that were so, what would be their interest? Euclid accepted self-evidence as the justification for his axioms, and he was troubled to discover that not all of his assumptions were evident, not even to himself.
The first three of Euclid’s axioms are commonly grouped together. “Let the following be postulated,” Euclid writes:
1. To draw a straight line from any point to any point.
2. To produce a finite straight line continuously in a straight line.
3. To describe a circle with any center and distance.
These assertions are hardly controversial. They seem to make perfect sense. Two points, one straight line. What could be simpler? But if intellectually disarming, these axioms are also disconcerting. They cede to the reader powers properly the mathematician’s, or if not the mathematician’s, then obviously not the reader’s: to draw, to produce, and to describe.
What if that reader, unwilling to do anything, is unwilling to draw, produce, or describe? Or if he does not know how? What then? “Geometry does not teach us to draw these lines,” Isaac Newton remarked in the Principia, “but requires them to be drawn.”
Euclidean geometers have traditionally explained the first three of Euclid’s axioms by reference to a straight-edge and compass. In his wonderful companion to Euclidean geometry, Geometry, Euclid and Beyond, the contemporary mathematician Robin Hartshorne remarks that Euclid’s proofs are “carried out with specific tools, the ruler (or straightedge) and compass.” Faithful to his policy of saying as little as possible, Euclid himself never once mentions either a straightedge or a compass in the Elements. Nor does Hartshorne. When at last he defines a geometrical construction, Hartshorne abjures both ruler and compass and writes instead about “constructible numbers.”
Having been introduced at some moment after Euclid put down his stylus, the straight-edge and compass proved a very considerable success. Students enjoyed stabbing paper with a compass point and drawing aimless circles. Some things could be done with just these two instruments, and some things not. This made for a nice series of discoveries. It is impossible to square a circle using only straight-edge and compass, and impossible again to trisect an arbitrary angle. In a celebrated theorem, Gauss demonstrated that a polygon with seventeen sides could be constructed using a straight-edge and compass.
The introduction of a straight-edge and compass does very little to discharge the unease conveyed by the first three of Euclid’s axioms, a sense of their uselessness.
Between any two points, it is possible to draw a straight line. This is Euclid straight up, the Euclid of the Elements.
Then there is Euclid revised: between any two points, it is possible to draw a straight line using a straight-edge.
Now a reminder: a straight-edge is an edge ending in a straight line. What else could it be?
Whereupon the conclusion that it is possible to draw a straight line with a straight line.
Uh-huh.
IN THE NINETEENTH and twentieth century, mathematicians with briskness and brusqueness in mind, offered Euclid their retrospective assistance in saying what he meant. That business of drawing, producing, and describing? Gone. Euclid’s axioms they recast as assertions of existence and uniqueness. There is something, and, by God, there is only one of them.
1a. Between any two distinct points, there exists a unique straight line.
2a. For any straight line segment, there exists a unique extension.
3a. For every point, there exists a unique circle of fixed radius.
These axioms control the way that the Euclidean universe is filled. They are very powerful: they provide an implicit definition of shape itself. A Euclidean shape is whatever exists by means of Euclid’s first three axioms or by repeated application of his first three axioms. The Euclidean constructions were an attempt to capture in physical movement a logical power of the mind. They are gone. It is the arrow of inference that moves. Nothing else.
IN ALL THIS, something is missing, or if not missing, then amiss.
Euclid’s axioms assume the existence of points. Where else would those straight lines go if not between them? Yet Euclid never once affirms that there exist any points at all, let alone a universe of them.
To Euclid’s first three axioms must be added an axiom still more fundamental: that there are points. What is more, there are infinitely many of them, an infinite set of points in modern geometries, a collection or gathering of them, or even a single point allowed Tantric powers of multiplication. Whatever the image, such points exist before anything else does, and in Euclidean geometry, they must exist if anything else does.
A universe of points does not by itself make everything clear where before some things were dark. It is surely false that any two points can be joined by a straight line, for unless one thinks of a point as the shrunken head of a straight line, no straight line can join a point to itself. Should one say instead that any two distinct points may be joined by a straight line? What makes points distinct? It can be nothing about their internal properties. They have none. To say that two points are distinct only if they are separated in space is to invite the question what separates them? If the answer is a straight line, nothing has been gained.
Euclid’s first three axioms lack the sparkle of logical impeccability, spic, but not, perhaps, span. They are doing the work of creation. It would have been a miracle had they done anything more.
EUCLID’S FOURTH AXIOM asserts that
4. All right angles are equal.
This axiom is noticeably different from Euclid’s first three axioms. It does not say that anything exists, let alone the right angles. The first three of Euclid’s axioms are concerned to get things under way. The fourth is intended to establish a companionable identity among right angles, a brotherhood. Still, whatever the identity of the right angles, their nature must be encompassed by the first three of Euclid’s axioms, together with the decorative ancilla of his definitions.
How might this have worked? Ancient geometers were divided. A right angle?
Geometer A: A right angle is the angle formed when two straight lines are crossed at the perpendicular.
Geometer B: Two straight lines are crossed at the perpendicular when they form two right angles.
Geometer C: Two right angles arise when two straight lines are crossed at the perpendicular.
Geometer D: Gentlemen, gentlemen.
Before right angles are declared equal it would be immensely helpful to know what an angle is in the first place. In this respect, Euclid’s axiom is rather like the declaration that all close siblings are competitive. What is a sibling? But then again, what is an angle?
Euclid does say in his eighth definition that “a plane angle is the inclination to one another of two lines in a plane which meet one another and do not lie in a straight line.” In his very next definition, Euclid seems to suggest that an angle is what, by his previous definition, an angle contains. It is better not to go there. Revising Euclidean geometry early in the twentieth century, David Hilbert considered Euclid’s eighth definition and thought that with a bit of polish, it would do nicely, the brass showing through the smudge (see Chapter VIII for Hilbert’s system). “Let α be any arbitrary plane,” Hilbert writes, “and h, k any two distinct half-rays lying in α and emanating from the point O so as to form a part of two different straight lines. We call the system formed by these two half-rays h, k an angle.” An angle is thus a matter of two straight lines suavely exiting a common point.
But Hilbert’s definition invites the question when these systems are the same, and when they are different.
Euclid and Hilbert both required some general principle under which angles of any size are judged equal or unequal. A principle is easy enough to contrive. Consider two angles separated widely in space. Two angles and so two systems. Two systems and so four straight lines. Four straight lines and so two points. Two such systems, and so two such angles, are equal if they coincide.
Euclid and Hilbert appear well satisfied.
But to determine whether two angles separated in space coincide, both Euclid and Hilbert must suppose that one of them is moved so that it is imposed on the other. But if moved, then moved in such a way that its own angle remains unchanged. This requires a commitment to the homogeneity of space, the idea that as they are moved in space, Euclidean figures do not change in shape. How might this be established without an antecedent account of the identity of their angles?
It is not so much that Euclid’s definition is smudged. There seems to be no brass underneath the definition, no matter how much polish is applied to its surface.
THE FIFTH AND final axiom of Euclid’s system is more famous than the other four. It is said to have troubled Euclid, who squirmed and turned, wheezed and whistled, before accepting it:
5. If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on the side on which are the angles less than two right angles.
The axiom is troubling because it seems to assess the property of parallelism by an appeal to what it is not. The theorem’s two straight lines converge at a point; they are not parallel. The subject of Euclid’s fifth axiom happens to be lines that are parallel. What about them?
An eighteenth-century form of the axiom credited to the Scottish mathematician Francis Playfair is far more intuitive than Euclid’s own, and as mathematicians almost at once realized, both versions are logically equivalent:
5a. One and only one straight line may be drawn through any point P in the plane parallel to a given straight line AB.
The phrase “may be drawn” is permissive when permission is not needed. The axiom affirms that in addition to being unique—“one and only one”—a line parallel to AB and passing through P exists.
Point taken, motion adopted. Playfair’s axiom says that through a point outside a given line, there exists one and only one line parallel to the given line.
Playfair’s axiom completes the axiomatic structure of Euclidean geometry.
It is the last.
EUCLID WAS SAID to be troubled by this axiom because it seemed more complicated than the others. On other accounts, it seemed to him less evident. And still other accounts ascribe to Euclid the contrary conviction that the parallel postulate is simple enough to be a theorem. Those doubts of his are today taken as evidence of Euclid’s superb logical intuition. He knew something was wrong, or if not wrong, then not right.
It would be fine to have a Euclidean double willing to join the discussion and enter into the record a few doubts about these doubts. What might he say—this Euclid scrabbling along the path the real Euclid never chose? For one thing, he would, I hope, reject the idea that the fifth axiom is more complicated than the other axioms. Complexity requires a measure or metric, and neither is in the case of these axioms forthcoming. I am sure that a geometer might be found championing the first of Euclid’s axioms as more complicated than all the others, just for the heck of it. The fact that Playfair’s axiom is simpler than the axiom Euclid introduced is evidence that one and the same axiom may have both a simple and a complicated formulation.
An imaginary Euclid might be equally inclined to disabuse the real Euclid of his petit soucis that his fifth axiom might not be evident. Mais non! If an axiom is not self-evident, then somehow it must encourage the suspicion, however long deferred, that it might be false. For this reason, no one supposes that the statement that snow is white is self-evident. True, yes; evident on inspection, that, too; self-evident, no. The denial of self-evidence requires some imaginative contingency of the snow is white but it might have been black variety.
But Euclid’s parallel postulate is true under the circumstances that Euclid sketched on a brimming dust board, and there is no obvious way in which it might be false. The parallel axiom is obviously not provable; an axiom is an assumption. But neither is the axiom obviously deniable. If it were obviously deniable, it would be possible obviously to de
ny it.
How would that proceed, that imaginative exercise?
Still, modern mathematicians have seen better and seen further than either of our Euclids. The parallel postulate is anomalous. It is not necessary. It can lapse.
But in every single world in which the parallel postulate fails, it fails either because the underlying space has changed, or because certain common geometrical terms such as distance have been given a new meaning. In the contrived universe that Euclid limned, it does not fail at all.
Chapter VI
THE GREATER EUCLID
Si les triangles faisaient un dieu, ils lui donneraient trois côtés (If triangles had a god, they would give him three sides).
—VOLTAIRE
EUCLID’S ELEMENTS BELONGS to a curious tradition, one that it created and now incarnates—the mountain- climbing pastoral. Mathematicians regard themselves as men of ascent. “Mathematical study and research are very suggestive of mountaineering,” the English mathematician Louis Joel Mordell remarked, recalling with satisfaction that when Edward Whymper made the first ascent of the Matterhorn, four of his colleagues perished on the climb. The genre is pastoral because the Elements expresses Euclid’s intense demand for an idealized world, one in which things are free of friction and inferences smooth as ice. In his influential study, Some Versions of Pastoral, William Empson identified the pastoral with the imperative to “put the complex into the simple.” What could be more Euclidean? Euclid’s Elements is that rare thing: its own best example.
If the theorems of Euclid’s Elements are its peaks, the proofs are a record of his climbs. In some, Euclid gets to the top quickly; in others, he is obliged to grunt and slog, and in these he is like some grizzled old climber recalling how once he was threatened by congelation of the anus. No matter the proofs that he offers, Euclid expects the reader to grasp the drama that they encompass. The proofs communicate tension, release, triumph. They allow the reader to experience the author’s discomfort at a distance.
The King of Infinite Space Page 4
