Infinity as a numerical concept is a description of unbounded growth rather than any attained quantity. In 1831, the German mathematician Gauss wrote that "the infinite is but a figure of speech; an abridged form for the statement that limits exist which certain ratios may approach as close as we desire, while other magnitudes may be permitted to grow beyond all bounds." When mathematicians say that something goes to infinity, they mean that it will eventually have a value larger than any prescribed integer. Around 250 b.c. Archimedes devised a method for approximating the value of pi to any desired accuracy by computing the perimeters of polygons with more and more sides. In modern language mathematicians would say that pi is the limit of the polygonal approximations to a circle of unit diameter as the number of sides goes to infinity.

For centuries mathematicians were able to work with...