The King of Infinite Space Read Online Free Page A

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.
    T HERE 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 menbase 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 ?
    T HE 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