A Brief History of Infinity; April 1995; Scientific American Magazine; by Moore; 5 Page(s)
For more than two millennia, mathematicians, like most people, were unsure what to make of the infi- nite. Several paradoxes devised by Greek and medieval thinkers had convinced them that the infinite could not be pondered with impunity. Then, in the 1870s, the German mathematician Georg Cantor unveiled transfinite mathematics, a branch of mathematics that seemingly resolved all the puzzles the infinite had posed. In his work Cantor showed that infinite numbers existed, that they came in different sizes and that they could be used to measure the extent of infinite sets. But did he really dispel all doubt about mathematical dealings with in- finity? Most people now believe he did, but I shall suggest that in fact he may have reinforced that doubt.
The hostility of mathematicians toward infinity began in the fifth century B. C., when Zeno of Elea, a student of Parmenides, formulated the well-known paradox of Achilles and the tortoise [see "Resolving Zeno's Paradoxes," by William I. McLaughlin; SCIENTIFIC AMERICAN, November 1994]. In this conundrum the swift demigod challenges the slow tortoise to a race and grants her a head start. Before he can overtake her, he must reach the point at which she began, by which time she will have advanced a little. Achilles must now make up the new distance separating them, but by the time he does so, she will have advanced again. And so on, ad infinitum. It seems that Achilles can never overtake the tortoise. In like manner Zeno argued that it is impossible to complete a racecourse. To do so, it is necessary to reach the halfway point, then the three-quarters point, then the seven-eighths point, and so on. Zeno concluded not only that motion is impossible but that we do best not to think in terms of the infinite.