by Rishidev Chaudhuri
Monsters lurk in the artificial paradises that mathematicians construct for themselves, swimming, slouching and fluttering into view from the corners of the imagination. These strange beasts – odd constructions or unexpected consequences of axioms – trouble the mathematical mind and threaten to expel the mathematician into chaos. By embodying the limits and the peculiarities of the concepts we use they force us to clarify, to draw distinctions where none had been drawn before, and to interrogate which objects we may meaningfully speak of and which properties we can assume. They also serve as a fascinating and wondrous gallery of the oddities of the mathematical universe, reflecting elegant and treacherous subtleties in their construction. Many mathematicians collect menageries of counterexamples and unusual cases, much as a naturalist might display upon returning from some distant continent.
A number of these emerge from the delicate structure of the number line, and the ideas of infinity and continuity that follow. One of the first and best-known is the square root of 2, along with the (probably apocryphal) story of the trouble it caused the Pythagoreans. Unlike most of the numbers we normally encounter, the square root of 2 cannot be written as a fraction (the ratio of two integers – what a mathematician would call a rational number)1. Allegedly the Pythagoreans, driven by their number mysticism and belief in a cosmos governed by whole numbers and their ratios, tried to keep this discovery a secret, going so far as to execute a member for revealing it to the outside world.
In fact, the situation is much worse. Not just most, but effectively all of the real numbers (the numbers on the number line; the numbers that can be written with possibly infinite decimal expansions) cannot be written as fractions. The numbers that we deal with everyday, numbers like ½, ¼, 5 and so on, are an infinitely small proportion of the total, lost in a host of numbers that we never dream of. This realization is the entry into the strangeness that is the real number line, and is a glimpse at the long attempt to grapple with and understand infinity and continuity.