To Infinity and Beyond: A Cultural History of the Infinite
Mind Tools: The Five Levels of Mathematical Reality
“The world is colors and motion, feelings and thought…and what does math have to do with it? Not much, if “math” means being bored in high school, but in truth mathematics is the one universal science. Mathematics is the study of pure pattern, and everything in the cosmos is a kind of pattern.”
In the above quotation, the first paragraph of Rudy Rucker’s latest book, observe the word “pure.” Mathematical patterns are pure, timeless concepts, uncontaminated by reality. Yet the outside world is so structured that these patterns in the mind apply to it with eerie accuracy. Nothing has more radically altered human history than this uncanny, to some inexplicable, interplay of pure math and the structure of whatever is “out there.” The interplay is responsible for all science and technology.
Perhaps it is a dim awareness of the explosive role of mathematics in altering the world, together with the low quality of math teaching in this country, that accounts for the growing number of books intended to teach mathematics to those who hated it in school. The two books here under review are general surveys, in the tradition of such popular classics as Edward Kasner and James Newman’s Mathematics and the Imagination. Unlike most such surveys, each book is organized around a unifying concept.
For Eli Maor, an Israeli mathematician now at Oakland University in Rochester, Michigan, the unifying concept is infinity. “Finite mathematics,” a term that has come into recent use, is precalculus math in which infinity is avoided as much as possible, yet even in the most elementary math there is no way to escape completely from the concept. As Maor points out, counting numbers go on forever, and straight lines are endless in both directions. Textbooks on finite math have chapters on probability, but what is meant when you say the odds are equal that a flipped coin will fall heads or tails? “We tacitly assume,” writes Maor, “that an infinite number of tosses would produce an equal outcome.”
Maor begins his admirable survey with the concept of limit. In one of Zeno’s notorious paradoxes, a runner can’t get from A to B until he first goes half the distance. Now he must run half the remaining distance, then half the still remaining distance, and so on into an infinite regress. Because at any time the number of distances yet to be traversed is infinite, how can he reach B? Worse than that, how can he begin? If the distance is sixteen miles, he must first run eight miles. To go eight he must go four. Again, the halves form an infinite sequence. How does he get started? Of course mathematicians are no longer troubled by such paradoxes of motion, but it is impossible to resolve them without a clear notion of the limits of infinite sequences of magnitudes in both time and space.
Maor’s well-chosen examples are wide-ranging. Archimedes determined the value of pi (the ratio of a circle’s circumference to its…
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