Mathematical Digest

Short Summaries of Articles about Mathematics
in the Popular Press

"Zero, Zilch and Zip," by Ian Stewart. New Scientist, 25 April 1998, pages 40-44.

What's at the heart of mathematics? Nothing!

That's cheating a bit--in math, "nothing" means a total void, an emptiness. What lies at the heart of math is zero--definitely something (it's a number) which has given mathematicians more joy, and more heartaches, than any other number.

The number zero came into existence implicitly as a placeholder. On an ancient abacus, the absence of beads in a row made it possible to differentiate between 15 and 105, for example. Still, zero wasn't thought of as a number in its own right until around 800 A.D., when it was incorporated into arithmetic. The Indian mathematician Mahavira explained that multiplying a number by 0 gives 0 and subtracting 0 from a number gives the number back. Suddenly, nothing was something to study carefully.

Immediately, nothing generated problems, like what happens when we divide by zero. Some mathematicians thought that a non-zero number divided by zero should be "infinity," but that led to even more arithmetic problems. Today we say it's "undefined." Zero divided by zero is also considered undefined, but in the late 1600s, Newton and Leibnitz each ran into this quotient when they invented the calculus. Enter Bishop Berkeley, who saw their inconsistencies and cried foul. It wasn't until 120 years later that Karl Weierstrauss settled the issue by rigorously defining the concept of a "limit." After this development, the whole debate turned out to be about nothing.

In the twentieth century, nothing played a key role in the development of set theory. John von Neumann, trying to rigorously answer the question of what a number is, realized that once you introduced the number 0, you could "bootstrap" all the rest in sets like this: 1 = {0}, 2 = {0,1}, and so on. What about 0, then? This must be empty set--the set with no elements-- introduced by Georg Cantor in the 19th century.

Nothing is still being considered in mathematics today. In graph theory, where a graph is defined as a set of points connected by lines, there was recently an argument about whether the "null graph"--with no points and no lines--is worth considering. Once again, nothing was something to be reckoned with.

--- Ben Stein