In his article about transcendental numbers for New Scientist, mathematician Richard Elwes suggests that Cantor’s work on the topic had profound consequences: “It means that the range of numbers that human brains and computers are equipped to handle - essentially those easily derived from the integers - are actually just an infinitesimal sliver of the numerical universe. Swarming around the integers and fractions is an infinitely larger collection of transcendental numbers. They are the "dark matter" of mathematics: they constitute the overwhelming majority of numbers, yet known examples are rare.” (e: the mystery number, New Scientist, 21 Jul 2007, p. 38)
Transcendentals are numbers that cannot be related to the integers by any sequence of ordinary arithmetical operations like addition, multiplication, or raising to powers. The square root of two can’t be written as a fraction, and the Pythagoreans found that outrageous enough; but it just multiplying it by itself gets you back to 2, an integer. The number e was proved to be transcendental in 1873, pi less than a decade later. However, finding more examples has proved remarkably difficult.
The article implies that being able to call on a geometric intuition – that is, one that can be referred back to the concrete world – is important even to mathematicians. It mentions that work by Alain Connes that offers deep insights but “defies all usual geometric intuition and is disconcertingly difficult to get to grips with”; however, some more recent work promises to “demystify Connes's extreme levels of abstraction” and provide “a raft of techniques for understanding Connes's abstract geometry in a more intuitive way.” It’s reassuring that not only cognitive scientists like Lakoff and Nunez feel that the practice of even mathematics has to be grounded in the physical.
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