The reticent and relentlessly abstract logician Kurt Gödel might seem an unlikely candidate for popular appreciation. But that's what Rebecca Goldstein aims for in her new book Incompleteness, an account of Gödel's most famous theorem, which was announced 75 years ago this October. Goldstein calls Gödel's incompleteness theorem "the third leg, together with Heisenberg's uncertainty principle and Einstein's relativity, of that tripod of theoretical cataclysms that have been felt to force disturbances deep down in the foundations of the 'exact sciences.' "
What is this great theorem? And what difference does it really make?
Mathematicians, like other scientists, strive for simplicity; we want to boil messy phenomena down to some short list of first principles called axioms, akin to basic physical laws, from which everything we see can be derived. This tendency goes back as far as Euclid, who used just five postulates to deduce his geometrical theorems.
But plane geometry isn't all of mathematics, and other fields proved surprisingly resistant to axiomatization; irritating paradoxes kept springing up, to be knocked down again by more refined axiomatic systems. The so-called "formalist program" aimed to find a master list of axioms, from which all of mathematics could be derived by rigid logical deduction. Goldstein cleverly compares this objective to a "Communist takeover of mathematics" in which individuality and intuition would be subjugated, for the common good, to logical rules. By the early 20th century, this outcome was understood to be the condition toward which mathematics must strive.
Then Gödel kicked the whole thing over.
Gödel's incompleteness theorem says:
Given any system of axioms that produces no paradoxes, there exist statements about numbers which are true, but which cannot be proved using the given axioms.
In other words, there is no hope of reducing even mere arithmetic, the starting point of mathematics, to axioms; any such system will miss out on some truths. And Gödel not only shows that true-but-unprovable statements exist—he produces one! His method is a marvel of ingenuity; he encodes the notion of "provability" itself into arithmetic and thereby devises an arithmetic statement P that, when decoded, reads:
P is not provable using the given axioms.
So a proof of P would imply that P was false—in other words, the proof of P would itself constitute a disproof of P, and we have found a paradox. So we're forced to concede that P is not provable—which is precisely what P claims. So P is a true statement that cannot be proved with the given axioms. (The dizzy-making self-reference inherent in this argument is the subject of Douglas Hofstadter's Pulitzer Prize-winning Gödel, Escher, Bach, a mathematical exposition of clarity, liveliness, and scope unequalled since its publication in 1979.)