The King of Infinite Space (5 page)

Surfaces as well as curves may depart from straightness. If the plane were balanced on the top of a sphere, like a book balanced on an apple, then one might say that the sphere is curved at its apex, by virtue of the increasing distances between the plane and the surface of the sphere. The apple has undertaken its own departure.

To see this requires of an observer a complicated maneuver in which apple and book, plane and sphere, are somehow embedded in a three-dimensional space, the extra dimension required to place both objects in juxtaposition. The result is a standard measure of curvature and so of flatness—
extrinsic
curvature, to use the suggestive name given it by mathematicians—with curvature now a relative property, one space curved when measured by the standards of another, almost as if what is crooked could be understood only against what is straight. It is a principle known to be useful in the criminal law as well as in mathematical physics.

Still, there is no ultimate decisiveness to extrinsic curvature. The sphere is curved when measured against the plane. The first has positive curvature—it swells—and the second, no curvature at all. It is flat. But wherein flatness itself?

I
S THERE A
measure of flatness accessible to an observer within a two-dimensional space, to an ant, say? Could that
ant, bound forever to wander the blackboard,
discover
that the blackboard is flat? The answer was provided by Carl Friedrich Gauss in a remarkable theorem that he published under the title
Theorema Egregium
. The intrinsic curvature of a surface, Gauss demonstrated, may be deduced entirely by using local clues such as angles and distances and the way that they change. No appeal to spaces beyond a surface is necessary, and what is more, intrinsic and extrinsic curvature coincide and they coincide perfectly.

In reaching these conclusions, Gauss went considerably beyond anything in Euclidean geometry itself. His
Theorema Egregium
is an exquisite achievement, but it is an exquisite achievement in differential geometry, one of the innumerable mixed marriages in mathematics, this one between the analytic apparatus of the differential calculus and the classical concerns of Euclidean geometry. Euclid did not discuss differential geometry and could not have foreseen its development.

S
O FAR
,
SO
pretty good.

What lies between two points in the Euclidean plane? One answer is nothing. This is the answer suggested by Democritus in the fifth century
BC
. There are in nature only atoms and the void, Democritus argued, the atomic theory of matter just budding at his fingertips. Ancient atoms were
both indivisible and indestructible. In the twenty-first century, those atoms have given way to elementary particles, but the idea of a radical dissection of material objects into their parts remains as imperishable as the atoms it countenances.

There is a very considerable difference between a physical atom and a Euclidean point, if only because one is physical, the other not, but Euclid in his study may well have felt Democritus behind his back, a gray ghost hanging over his shoulder, as ghosts so often do, one man's point an idealization of the other man's atom. Nothing between atoms; nothing between points; and, so, nothing all around.

By whatever means he found himself in Euclid's study, Democritus was not alone. Parmenides, his predecessor, was there, too, muttering. At some time in the fifth century, Parmenides had composed a long poem titled
On Nature
. Surviving in fragments, his voice comes to us over an immense distance, sun-baked, half-mad, delirious. It is not at all modern.

“What is, is,” Parmenides says, and as for what is not, “it is not.”

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?

I
N 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.

E
UCLID 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:
t
o
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.

I
N 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.

I
N 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.

E
UCLID
'
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.