The limits of mathematics |
| In the nineteenth century mathematics both purged itself
of the murkiness of many of the basic concepts of the calculus and liberated
itself with the discovery of non-Euclidean geometry. It seemed possible
by the end of the century that the process could reach completion by showing
that mathematics was 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 twentieth-century
mathematics to handle. Proving the consistency of arithmetic with integers
was problem number two in his list. 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? A year 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. In the 1920s, Hilbert launched a major effort in collaboration with other mathematicians to prove that mathematics is consistent. His group, the formallsts, 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 formallsts. 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. 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. |
Alexander Hellemans and Bryan Bunch
"Timetables of Science" p462
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