Bunch-Hellemans, The limits of mathematics

From its inception in the 17th century, the calculus worked but its foundation was nearly inexplicable. In the 19th century, however, mathematics purged itself of the murkiness of many of the basic concepts of the calculus by providing a logical framework based on simple arithmetic and geometry.

Mathematicians called the new foundation rigor. It seemed possible that by the end of the century the process of rigorization could reach completion. To accomplish this mathematics would need to be both consistent within itself and complete in the sense that everything provable could be proved. In a famous speech in 1900, the mathematician David Hilbert proposed 23 unsolved problems for 20th-century mathematics to handle. Proving the consistency of arithmetic with integers was problem number two on his list. A few years earlier, Hilbert himself had set geometry on a new foundation that is internally consistent provided that arithmetic is consistent.

Georg Cantor, the founder of set theory, had discovered 17 years earlier some unsettling problems that suggested that there were roadblocks in the way of Hilbert’s goal, but Hilbert did not think they would be insurmountable. For one thing, Cantor’s discoveries concerned contradictions — generally called paradoxes by mathematicians — in his own set theory. One of these contradictions came from Cantor’s proof that for every set, there is a set with more members. The paradox occurs when one considers the set of all sets. How can there be another set with more members than that?

Two years after Hilbert’s speech, Bertrand Russell discovered an even more devastating paradox: Is the set of all sets that are not elements of themselves an element of itself? Russell’s paradox was so close to the heart of set theory that he, and others, made major revisions and restrictions in the theory in an effort to avoid it. Still, as long as the paradoxes were confined to the newly created set theory, they did not seem to affect arithmetic, now used as the foundation for both geometry and calculus.

In the 1920s Hilbert launched a major effort in collaboration with other mathematicians to prove that mathematics is consistent. His group, the Formalists, seemingly achieved large parts of their goals, but the price was too high. They had to assume ideas that were far more complicated than the ones they were trying to prove.

In 1931 Kurt Gödel ended many of the arguments and effectively halted the efforts of the Formalists. That year he proved what is now known as Gödel’s incompleteness theorem: For any consistent axiomatic theory that is strong enough to produce the properties of natural numbers (1, 2, 3, …), there exists a statement whose truth or falsity cannot be proved (or else the system is inconsistent). In other words, mathematics is either inconsistent or incomplete so long as it is based on arithmetic.

Since Gödel’s incompleteness theorem, other mathematicians have used methods similar to his to show specific examples of statements that cannot be proved from widely accepted sets of axioms that are supposed to underlie all of mathematics. Despite incompleteness, however, mathematics continues to progress. Some mathematicians think mathematics is even more interesting now that they know that it is not unlimited in scope.