The King of Infinite Space

by David Berlinski

  The distinction between the concrete and the abstract models of Euclidean geometry offers a nice place in which to watch this uneasiness emerge and then separate itself into a destructive dilemma.

  Does the idea of coincidence apply to the concrete or the abstract models of Euclidean geometry? Or neither, or both? Not the concrete models, surely, for physical triangles are never completely coincident, no matter how they are moved. Something is always left out, or something always left over. How on earth can two physical objects coincide perfectly?

  Not on earth is the correct answer; it is the only answer. If it is true that concrete triangles are never coincident, it is equally true that abstract triangles cannot be moved. They are beyond space and time. Moving about is not among the things that they do, because they do not do anything.

  Sensitive to just this point, Russell dismissed the idea that in Euclidean geometry, anything is moving or being moved. Writing in the supplement to the 1902 edition of the Encyclopedia Britannica, Russell remarked that “what in geometry is called a motion is merely the transference of our attention from one figure to another.”

  But the geometer’s attention is like the wind: it goeth where it listeth. Where it goeth is of little interest unless it goeth from one figure to another equal figure.

  Coincidence is a condition that the concrete models of Euclidean geometry cannot satisfy: they are never the same. And it is a condition that the abstract models of Euclidean do not meet: they cannot be moved.

  THERE IS FINALLY the last of Euclid’s common notions, the principle that the whole is greater than the part. Far from expressing a belief on which “all men base their proofs,” the proposition is either trivially true or false.

  If the whole of something is by definition greater than its parts, Euclid has not advanced his cause or his case; but if the very idea of a part standing to a whole is left undefined, it is easy enough to construct examples in which the whole is less than its parts or equal to them.

  The number 6, to take an example, has its own internal structure. It may make sense to say that 0 and 1 are simple numbers, quite without parts, but the number 6 is the sum and product of various numbers and thus has a richness in its identity, an otherwise hidden complexity. Is the number 6 greater than its parts? Is it greater than the sum of its parts? Not if the parts of the number are composed of its divisors, 1, 2, and 3. Their sum is equal to 6.

  The number 12, on the other hand, is less than the sum of its parts, 1, 2, 3, 4, and 6.

  The relationship between wholes and parts is exquisitely sensitive, then, to the way in which the underlying ideas are specified. If this is so, then it is difficult to ascribe Euclid’s fifth common notion to those beliefs “on which all men base their proofs.” Too much by way of circumstantial dependency is involved for this to be a common notion at all.

  Infinitely large objects present problems all their own. Is the assertion that the whole is greater than its parts true of the natural numbers? Skepticism arises because the natural numbers 1, 2, 3, . . . may be put into a tight correspondence with the even numbers 2, 4, 6. . . . The correspondence is tight enough so that for every natural number, there is an even number, and vice versa. The set of natural numbers and the set of even numbers, as logicians say, have the same cardinality. They are the same size.

  But surely, the even numbers are a part of the natural numbers? If they are not, what residual meaning can be assigned to the now-vagrant terms part and whole?

  THE GOAL OF listing once and for all those ideas on which “all men base their proofs” is profoundly compelling. A list is something explicit and thus open to inspection; once open to inspection, a list of common notions satisfies the desire to have all the cards on the table. Hidden assumptions, like hidden cards, suggest that what is hidden is somehow disreputable.

  The explicitness with which Euclid affirms certain common notions is, of course, no reason by itself to think his common notions any good. Euclid never suggests otherwise. His common notions are what they seem. They express assumptions that are more general than his axioms but no less undefended.

  If Euclid’s common assumptions cannot be derived from anything further, they make their claim by means of their inescapability. Without them, Euclid believes, there could be no proof at all. Whatever their inescapability, Euclid’s common notions suggest a question that neither he nor Aristotle ever considered. Can these common notions be faulted because they are incomplete? Whenever an explicit list of common assumptions is offered, after all, it is easy enough to step back and with some assurance point to the assumptions on which the assumptions themselves depend.

  Like any other mathematician, Euclid took a good deal for granted that he never noticed. In order to say anything at all, we must suppose the world stable enough so that some things stay the same, even as other things change. This idea of general stability is self-referential. In order to express what it says, one must assume what it means.

  Euclid expressed himself in Greek; I am writing in English. Neither Euclid’s Greek nor my English says of itself that it is Greek or English. It is hardly helpful to be told that a book is written in English if one must also be told that written in English is written in English. Whatever the language, its identification is a part of the background. This particular background must necessarily remain in the back, any effort to move it forward leading to an infinite regress, assurances requiring assurances in turn.

  These examples suggest what is at work in any attempt to describe once and for all the beliefs “on which all men base their proofs.” It suggests something about the ever-receding landscape of demonstration and so ratifies the fact that even the most impeccable of proofs is an artifact.

  Chapter IV

  DARKER BY DEFINITION

  Sometimes things may be made darker by definition. I see a cow. I define her, Animal quadrupes ruminans cornutum. Cow is plainer.

  —SAMUEL JOHNSON

  THE ELEMENTS CONTAINS twenty-three definitions. Of these, the first seven, and the twenty-third, are fundamental:

  1. A point is that which has no part.

  2. A line is length without breadth.

  3. The extremities of a line are points.

  4. A straight line is a line which lies evenly with the points on itself.

  5. A surface is that which has length and breadth only.

  6. The extremities of a surface are lines.

  7. A plane surface is a surface which lies evenly with the straight lines on itself.

  23. Parallel straight lines are straight lines which, being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction.

  Nineteenth- and twentieth-century mathematicians have almost to a man objected to these definitions. Both Moritz Pasch and David Hilbert criticized Euclid because in struggling to say what he meant, Euclid rejected what he knew: things come to an end. If axioms must be accepted without proof, then some terms must be accepted without definition. In his ninth through twenty-second definitions, Euclid is almost impeccable, defining terms that are new by an appeal to terms that are old. A triangle is a figure contained by three straight lines. This is Euclid’s nineteenth definition. Not perfect. What is a figure? But not bad. There remain his initial definitions. A point, Euclid affirms, has no parts. It is the first thing that he says, circumstances suggesting that he meant to say it. And since it is the first thing Euclid says, it is the first definition to which critics object. “This [definition] means little,” Morris Kline argues in Mathematical Thought from Ancient to Modern Times, “for what is the meaning of parts?”

  Yet if Kline intended to rebuke Euclid for a logical mistake, he has done so by making a mistake all his own. The haunch of a cow is one of its parts, but only the word haunch carries meaning. The haunches have nothing to do with it. They are busy supporting cows. When physicists say that an electron has no parts, they are talking about electrons and not the meaning of the words that might denote th
em. So, too, Euclid. His first definition seems much less a proper definition than a fact about points: that they have no parts.

  In his twenty-three definitions, Euclid blurs the line between the claims that he makes and the terms that he defines. He is not fully the master of distinctions. His definitions are, for this reason, moving. They reveal a great mind entering uncertainly into a space that logicians would not fully command for two thousand years. The definitions are what they seem: an instruction, a guide to Euclid’s thoughts, a way into the labyrinth.

  GEOMETRY IS THE study of shapes in space. There is not one without the other. The enveloping plane, Euclid makes clear in his fifth definition, has in length and breadth two dimensions. What is a dimension, and why are there two of them? Length and breadth are terms of ordinary experience—hands outstretched or one above the other, as if measuring a flounder. The third dimension of space is often represented by extending a flat palm forward—in—and then retracting it—out. This reversion to experience might suggest that the Euclidean plane is simply whatever is left over when one dimension of space is subtracted from the original three. It is a position impossible to fault.

  Indifferent to sturdy common sense, textbook authors often say that the plane has two dimensions because two numbers are sufficient to identify any point. It is by no means clear that this is the improvement commonly supposed. Two numbers are sufficient to identify any point in space if the space has two dimensions. Not otherwise. If we had some analytic understanding of just how points comprise a space of two dimensions, there would be no need to appeal to two numbers, and if not, of what use is the appeal?

  Euclid’s introduction of length and breadth is nevertheless not entirely misplaced. An appeal to the power of geometrical objects to move, or to be moved, is latent throughout the Elements. It is the power behind proposition four, and thus the point of coordination to Euclid’s system of equality, the one in which he considers shapes the same or different. It hardly matters whether a geometrical object, having taken it in its head to vacate its premises, moves on its own or is moved by the geometer. With motion assigned geometrical figures, in how many ways could any one of them move? The behavior of a marble on a plate-glass table suggests that there is no end to the possibilities. And this is true enough. There is no end. What the question demands, however, is not an overall count but a kind of classification, a reduction to essentials by which what the moving marble does may be collapsed into a finite scheme.

  There are three ways to move in the plane: by translation, moving straight ahead on any convenient straight line: by rotation, turning in an arc at a point; and by reflection, as when a flounder’s horrible two eyes, having looked out from the plane, are persuaded out of common decency to look into the plane.

  This is the finite scheme.

  THERE ARE THREE degrees of freedom in the Euclidean plane, geometers say—a nice phrase and a reminder that even in mathematics, there are ties between ideas that are austere and abstract—degrees—and ideas that appeal to human agency—freedom. If there are three degrees of freedom, then two dimensions. A little formula coordinates the degrees of freedom and dimensionality: n (n + 1) / 2, where n is the dimensions of space, and n (n + 1) / 2, the degrees of freedom.

  One thing, the dimension of space, has been defined in terms of another, its degree of freedom, but there is a long, glistening rodent trail leading backward from these ideas to the far more primitive idea of some mental movement by which the geometer shuttles between dimensions, undertaking observations and seeing things where in real life no observations could be made and nothing could be seen.

  EUCLIDEAN SPACE HAS a dash of the distant in its veins; this much is clear from Euclid’s twenty-third definition. The word infinite does not appear in the definition itself. What Euclid does say is that straight lines may be produced indefinitely in both directions. There is as much of space, it follows, as might be needed to encompass an ever-expanding straight line. Still, the definition is peremptory, failing to make distinctions that are dying to be made. There is a difference, after all, between a space that is unbounded and one that is infinite. The surface of a sphere is unbounded but not infinite, and a line of fractions narrowing to zero is infinite but not unbounded. Just a century before Euclid wrote, Zeno the Eleatic provided a discussion of infinity and its paradoxes that is to this day matchless in its subtlety. It is possible that Euclid said as little as he did because he understood that he stood to gain nothing by saying more.

  Why make trouble?

  ABOUT FLATNESS, EUCLID has two things to say. A straight line, he affirms in his fourth definition, lies evenly with the points on itself, and in his seventh definition, Euclid makes much the same claim for the plane itself. The plane lies evenly with its embedded straight lines. The contrast is between straight lines and curves, and between the flat and level plane and other surfaces such as the surface of a sphere.

  The idea of flatness has a certain emotional valence well beyond geometry. Ideas, champagne, and chests may all be flat; this is rarely said to be a good thing. Both falling flat and flattery share a common Old Norse root in flatr, which designates a leveling down, a featurelessness. The Euclidean plane is everywhere the same. This invites the question: The same with respect to what? In the concept of superposition, or coincidence of shape, Euclidean geometry makes a concession to the idea that Euclidean figures may be moved. The triangles of proposition four are congruent to themselves no matter how they shuffle or are shuffled across the plane. Euclidean figures are indifferent to the fallacy that distance makes a difference. Caelum non animum mutant qui trans mare currunt, as Horace observed. They change the sky but not their souls who flee across the sea.

  But if the Euclidean plane is homogeneous, it hardly follows that it is flat. The sphere is everywhere the same, and so is the geometry of the earth, as jaded travelers well know. It is not flat.

  IN THE CALCULUS, the curvature of a line is defined by an appeal to the straightness of straight lines; they have no curvature at all. In his treatise Relativity and Geometry, the physicist Roberto Torretti writes that “the curvature of a plane curve at a point measures the rate at which the curve is changing direction.” Curvature is a falling away. Torretti then adds something wonderfully vivid. What curvature really measures at a point is the extent to which a curve is “departing from straightness.”

  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?

  IS 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.

  SO 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.”