Like the pyramids, an axiomatic system is a public work.
EUCLIDEAN GEOMETRY IS the study of shapes in space. Shapes are not bound to the wheel of time. There is no place where the Euclidean triangle resides, and no time at which it arrived there. Plato argued that the shapes are a part of the Kingdom of Forms, caves and cavemen, shadows and the ecstatic sun. No philosopher since Plato has been entirely satisfied with the kingdom. Existing in the great beyond, the Platonic forms have no obvious causal powers. Yet if they have no obvious causal powers, they have obvious causal effects. Euclid reached his conclusions about triangles by reasoning about the form of the triangle, the essential thing.
If the Platonic forms are difficult to accept, they are impossible to avoid. There is no escaping them. Mathematicians often draw a distinction between concrete and abstract models of Euclidean geometry. In the abstract models of Euclidean geometry, shapes enjoy a pure Platonic existence. The concrete models are in the physical world. Freeways masquerade as straight lines, ink drops as points, amphitheaters as circles, and planetary orbits as ellipses. Mathematicians have often supposed that the concrete models of Euclidean geometry have a degree of vitality denied the Platonic models. “One must be able to say at all times,” the German mathematician David Hilbert remarked, “instead of points, straight lines, and planes—tables, beer mugs, and chairs.” These words convey a reassuring impression of ordinary life. Tables, beer mugs, and chairs! What could be more down-to-earth? But the phrase instead of prompts a reservation. The shortest distance between two beers is a straight line in time or in space. Yes, that is certainly true. But the shortest distance between two beers is a straight line because the shortest distance between two points is a straight line. Nothing is instead of anything.
Without the Platonic models, the concrete models would have no interest. Euclid does not, after all, invite his readers to consider more or less straight lines. How much more, how much less? And if there are no purely straight lines, what would the comparison come to?
The concrete models of Euclidean geometry include the tables, chairs, and beer mugs. They are where they have always been, and that is in the barroom.
The Platonic models of Euclidean geometry include the points, lines, and planes. They are where they have always been as well, and that is God knows where.
IF THE THEOREMS of an axiomatic system follow from its axioms, it is reasonable to ask what following from might mean. What does it mean? The image is physical, as when a bruise follows a blow, but the connection is metaphorical. The relationship between the axioms and the theorems of an axiomatic system is, when metaphors are discarded, remarkably recondite, invisible for this reason to all of the ancient civilizations but the Greek.
The men of the ancient Near East no doubt knew what arguments were. They had so many of them. What they knew, they knew imperfectly. They lacked words to make clear the distinctions that they sensed. Why assess an argument when it was so much easier to end it by either violence or indifference? This point of view has never completely lost favor. It was the Greeks who did the assessment and forced the very idea of an inference into consciousness, asking patiently for an account of its nature, the way it controlled the movement of the mind, and where in the catalog of human powers it belonged.
At roughly the same time that Euclid composed the Elements, Aristotle provided a subtle and refined analysis of syllogistic inference, the pattern in argument that takes Socrates—and the rest of us, alas—to his death by virtue of the fact that he is a man and we are mortal. Born in 384 BC and dying in 322 BC (another victim of his own syllogism), Aristotle might conceivably have known Euclid when Euclid was still a young man, perhaps even palpating his togaed shoulder. Far too little is known about the circumstances of Euclid’s life to say just whose hand he might have shaken. The two men worked hand in hand all the same.
ARGUMENTS, ARISTOTLE ARGUED, may be divided into those that are good and those that are not. In the syllogism, two premises resolve themselves in one conclusion:
All dogs are mammals.
All mammals are animals.
All dogs are animals.
Good.
No fish are dogs.
No dogs can fly.
All fish can fly.
Bad.
Any dog who has not lost something still has it.
No dog has lost a fifth foot.
All dogs have five feet.
Shame.
As these examples might indicate, good arguments are good by virtue of their form and not their content. The logician is indifferent to the distinction between all dogs are mammals and all men are mortal; both cases are swallowed whole by all A’s are B. This is the Aristotelian insight, and logicians have accepted it ever since. The conclusion of a valid argument is entrained by its premises. Truth plays an ancillary role. If the premises of a valid argument are true, then their conclusion must be true, but whether they are true is a matter on which the logician has little to say; an argument may be good even though its premises are false, and bad even though its premises are true.
It is tempting to imagine a fraternal give-and-take between Euclid and Aristotle, Euclid taking, Aristotle giving, with Euclid advancing proofs and arguments that Aristotle had antecedently assessed and classified. This is not quite so. The Elements is a work of great logical sophistication, but it is not a work of logical self-consciousness. Euclid’s subject is geometry, his business is proof, and Euclid was not a mathematician disposed to step back to catch himself in the act of stepping back. That his arguments were valid, he had no doubt, but in the question of what made them valid, he had no interest.
Euclid often made use of arguments that Aristotle had not analyzed properly or analyzed at all. If the natural numbers progress by 1, then there is no natural number between 3 and 4. The natural numbers do progress by 1. So there is no natural number between 3 and 4. The inference proceeds by the stately music of modus ponens. No syllogism is involved, just the straightforward play between propositions and their particles—if, then, and. Euclid is especially fond of reaching his conclusions by demonstrating that a given proposition leads to a contradiction, and so must be rejected. In Euclid’s hands, this style of reasoning often becomes a torpedo.
There remains the matter of the distinction between an axiomatic system and an argument.
There is none.
An argument is a small axiomatic system, and an axiomatic system is a large argument.
Chapter III
COMMON BELIEFS
La dernière démarche de la raison, c’est de connaître qu’il y a une infinité de choses qui la surpassent. (The last step of reason is to grasp that there are infinitely many things beyond reason).
—PASCAL
“EUCLID ALONE,” EDNA St. Vincent Millay once wrote, “has looked on beauty bare.” It is the first line in a sonnet of the same name. Literary critics are often embarrassed by the sonnet, and mathematicians by Edna St. Vincent Millay. Euclid alone? Still, the idea that “Euclid alone looked on beauty bare” elegantly draws attention to the nakedness of inference exhibited by every Euclidean proof. It is something rarely seen beyond mathematics—this hidden, if somewhat lurid, power of a Euclidean proof to compel fascination. Up go the axioms on the blackboard; down come the theorems. Students and readers alike are encouraged to think of the display as something stirring. And it is. So much of ordinary argument and inference is fully clothed.
But this way of presenting Euclid and the Elements imposes a gross distortion on Euclid’s thoughts: it allows the staged drama of his proofs to stand for the grandeur of his system as a whole. Euclid meant his proofs to be grasped against the background of his common notions and definitions. In almost every proof, he appeals to his own common notions and, in many proofs, either to his definitions or to ideas that follow naturally from his definitions. Beyond any of this, there are Euclid’s ideas about space and human agency and the exaltation of geometry that is so conspicuous a feature of his thought
s. Focus, control, and tension—they are there in Euclid’s proofs, but these moments, as any athlete knows, do not appear as isolated, brief, bursting miracles. They are not isolated at all, and they are not miracles either. They are grounded in Euclid’s meditations about what may be supposed and what not, and how difficult ideas may be defined or, at least, exposed. In all this, the master, unbending to explain himself, remains entirely in character, his severity undiminished, no word wasted, as prudent, compact, and tight as the stretched skins on which he wrote.
EUCLID’S COMMON NOTIONS represent the “beliefs on which all men base their proofs.” The words are Aristotle’s, but the idea that there are beliefs on which all men base their proofs must itself have been one of them, for Euclid appropriated the idea without hesitation and without argument.
There are five common notions in all:
1. Things that are equal to the same thing are also equal to one another.
2. If equals be added to equals, the wholes are equal.
3. If equals be subtracted from equals, the remainders are equal.
4. Things that coincide with one another are equal to one another.
5. The whole is greater than the part.
These principles convey an air of what is obvious. They have authority. No one either in Euclid’s time or our own is proposing that if equals are added to equals, the result might be unequal. A surprising delicacy is nonetheless required to say just what these principles mean. It is a delicacy that Euclid did not possess. This might suggest that Euclid’s conviction that these beliefs are common represented on his part a willingness to repose his confidence in things he could neither explain nor justify. To say as much involves no rebuke. If Euclid could neither explain nor justify the common beliefs that he invoked, we can do as little, or as much, with respect to our own. It was Euclid’s genius to grasp that whatever the powers of his geometrical system, it did rest on certain common beliefs.
It was Euclid’s business to say what those beliefs were.
And our business to say what they mean.
EQUALITY IS AN indispensable idea. It is like water to the fish—everywhere at once, but easy to ignore and difficult to define. To say of two things that they are equal is always false, and to say of one thing that it is equal to itself is always trivial. This is an uncommonly stern conceptual rebuke.
Euclid’s first common notion is often illustrated by three straight lines labeled A, B, and C and an insouciant appeal to intuition. If A is equal to B, and B is equal to C, then A is equal to C.
The appeal is not misplaced, but it is misleading. For one thing, neither illustration nor intuition says much about the concept of equality. For another thing, what Euclid says of equality is also true of size: if A is greater than B, and B greater than C, then A is greater than C.
Euclid’s statement of his first common notion covers up a chamboulement, a disorder. The illustration, those lines—this is starting well. But two equal lines? With the long history of Euclidean geometry at our back, we can say easily enough that two lines are equal if they are equal in length. A one-foot line in Moscow is the same length as a one-foot line in Seattle. But equality in length is a far narrower concept than equality itself, and it is not a concept that Euclid made accessible to himself. Euclidean geometry contains no scheme under which numbers are directly associated with distances.
Euclid’s fourth common notion expresses the Euclidean concept of geometrical equality. Having been in the grapple, Euclid has, we may suppose, gotten the better of things. Two things are equal if they coincide. This principle of superposition Euclid puts to work throughout the Elements. In the case of those straight lines, it admits of immediate application. Two lines are equal if they coincide. A question having been posed about equality, a very similar question now arises about coincidence: just when do things coincide? To say that two things coincide when they coincide equally is not obviously an improvement. Having fastened on coincidence as crucial, Euclid may well have remembered that in his definitions, he affirms that a line, although it has length, has no width. What investigation might justify the conclusion that two lines without width coincide? If no investigation, how could we say that two lines coincide even in length if we cannot say whether they coincide at all?
The wheel of time required twenty-three centuries before George Boole and C. S. Peirce assessed equality in its proper, its logical, context. Mathematicians today take it all in stride. Aristotle and Euclid were more strode upon than striding.
THE PROPOSITION THAT Euclid is wise says of Euclid that he is wise. His wisdom is something that he has, an aspect of the man. Euclid is wiser than Aristotle says of Euclid and Aristotle that one man is wiser than the other. It puts them both in their places—two men, but one relationship.
Equality is a relationship and, as such, a member of a great, worldwide fraternity: things bigger, taller, slighter, smaller, greater, grander, fathers and sons, daughters and mothers, before and after. To them, the logic of relationships, a general account of just how an A might be related to a B, the rules of the road.
Equality is in the first place reflexive. A = A. No relationship could be closer. Or more universally enjoyed.
And symmetric. If A = B, then B = A.
And transitive. If A = B, and B = C, then A = C.
Euclid saw the transitivity of equality. It is the first of his common notions. But symmetry and reflexivity he missed or did not mention.
In his second and third common notions, Euclid juxtaposes the relationship of equality and the operations of addition and subtraction. Things are added to one another or subtracted from one another. Inasmuch as subtraction is a way of undoing addition, Euclid’s second and third common notions might be collapsed into one encompassing declaration: If A = B and C = D, then A ± C = B ± D.
There is no reason, one might think, to restrict this principle to arithmetical operations; there is no reason to restrict it at all. A = B if and only if whatever is true of A is true of B. This is sometimes thought a definition of equality, and so a way of eliminating a troublesome concept altogether. It is not clear that this maneuver confers any great benefits. Among the things true of A is surely that A is equal to itself. The concept destined to be disappeared has just been reappeared. This might suggest that equality cannot easily be eliminated in favor of the truth because it cannot be eliminated at all.
Just so.
EUCLID’S FOURTH COMMON notion expresses a criterion of identity, a principle by which triangles, circles, or straight lines may be judged the same. The idea is implicit in every theorem that Euclid demonstrates. It is of the essence. If the geometer cannot tell when two shapes are the same, he cannot tell when they are different, and if he cannot tell whether shapes are the same or different, of what use is he?
Suppose now that two triangles are separated in space. They become coincident when one of them is moved so that it covers the other in such a way that the two figures are perfectly aligned. Nothing is left over, extrudes, or sticks out.
Coincidence or superposition offers the geometer a rough-and-ready measure of sameness in shape. What is not entirely obvious in all this rough-and-readiness is just how figures separated in space—a triangle here, another one there—can be moved through space so that their coincidence may be tested.
The point emerges early in the Elements; it emerges in Euclid’s fourth proposition:
If two triangles have the two sides equal to the two sides respectively, and have the angles contained by the equal straight lines equal, they will also have the base equal to the base, the triangle will be equal to the triangle, and the remaining angles will be equal to the remaining angles respectively, namely those which the equal sides subtend.
Two triangles are equal, Euclid has affirmed, if they are congruent, and they are congruent if two of their sides are equal, along with the angles the sides subtend.
The proof is simple in its notoriety, for Euclid deviates at once into the swamp of concepts that he
has not analyzed and cannot justify: “If,” he says, “the triangle ABC be applied to the triangle DEF, and if the point A be placed on the point D and the straight line AB on DE, then the point B will coincide with the point E, because AB is equal to DE.”
Euclid is at the podium. He has just pointed to his dust board with the tip of an outstretched finger. Beaming with satisfaction, he is about to say . . .
When he is interrupted.
—Applied by whom, Sir?
One question.
—Placed how, Professor?
Another.
—Coincide when, Maître?
A third.
BOTH BERTRAND RUSSELL and David Hilbert thought that Euclid would have been better served had he accepted proposition four as an axiom instead of claiming it as a theorem. It is a policy, as Russell remarked in another context, that has all the advantages of theft over honest toil. Designating Euclid’s fourth proposition an axiom does not do much to diminish the sense that in moving things around on the blackboard, the geometer has undertaken something at odds with the rigor of Euclidean geometry. In a little book titled Leçons de géométrie élémentaire (Lessons of elementary geometry), the French mathematician Jacques Hadamard proposed that coincidence be subordinated to some catalog of the ways in which shapes in Euclidean space might move. If the Euclidean idea of coincidence is a theorem, it depends on assumptions that Euclid did not make; if an axiom, it makes those assumptions without defending them; and if based on some antecedent assessment of motions allowed Euclidean figures, then it is both.
The King of Infinite Space Page 2