The King of Infinite Space Read Free Page B

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 conceptthan 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.
    T HE 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