By Philip J. Davis, William G. Chinn (auth.)
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As another example, from Postulates 4 and 5 we get II. Each line contains exactly three points. Postulate 7 guarantees at least one point of intersection for any two lines, and Postulate 3 implies that two or more intersections are impossible since each pair of points can lie on at most one line; hence, III. Any two lines intersect in one and only one point. Postulate 6 implies that for each line in this set there exists at least one point of the set not lying on the line. Consequently, this outside point together with two points which lie on the line guarantee that IV.
It is then that we must face up to the complexities of addition, subtraction, multiplication, division, square roots and the more elaborate dealings with numbers. The complexity of a civilization is mirrored in the complexity of its numbers. Twenty-five hundred years ago the Babylonians used simple integers to deal with the ownership of a few sheep, and simple arithmeti9 to record the motions of the planets. Today mathematical economists use matrix algebra to describe the interconnections of hundreds of industries, and physicists use transformations in "Hilbert space"-a number concept seven levels of abstraction higher than the positive integers-to predict quantum phenomena.
Who, if not the mathematician, is the custodian of the odd numbers and the even numbers, the square numbers and the round numbers? To what other authority shall we look for information and help on Fibonacci numbers, Liouville numbers, hypercomplex numbers and transfinite numbers? Let us make no mistake about it: mathematics is and always has been the numbers game par excellence. The great American mathematician G. D. BirkhofI once remarked that simple conundrums raised about the integers have been a source of revitalization for mathematics over the centuries.