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.)
One way to understand Gödel’s theorem (in combination with his 1929 “completeness theorem”) is that no system of logical axioms can produce all truths about numbers because no system of logical axioms can pin down exactly what numbers are. My fourth-grade teacher used to ask the class to define a peanut butter sandwich, with comic results. Whatever definition you propose (say, “two slices of bread with peanut butter in between”), there are still lots of non-peanut-butter-sandwiches that fall within its scope (say, two pieces of bread laid side by side with a stripe of peanut butter spread on the table between them). Mathematics, post-Gödel, is very similar: There are many different things we could mean by the word “number,” all of which will be perfectly compatible with our axioms. Now Gödel’s undecidable statement P doesn’t seem so paradoxical. Under some interpretations of the word “number,” it is true; under others, it is false.
In his recent New York Times review of Incompleteness, Edward Rothstein wrote that it’s “difficult to overstate the impact of Gödel’s theorem.” But actually, it’s easy to overstate it: Goldstein does it when she likens the impact of Gödel’s incompleteness theorem to that of relativity and quantum mechanics and calls him “the most famous mathematician that you have most likely never heard of.” But what’s most startling about Gödel’s theorem, given its conceptual importance, is not how much it’s changed mathematics, but how little. No theoretical physicist could start a career today without a thorough understanding of Einstein’s and Heisenberg’s contributions. But most pure mathematicians can easily go through life with only a vague acquaintance with Gödel’s work. So far, I’ve done it myself.
How can this be, when Gödel cuts the very definition of “number” out from under us? Well, don’t forget that just as there are some statements that are true under any definition of “peanut butter sandwich”—for instance, “peanut butter sandwiches contain peanut butter”—there are some statements that are true under any definition of “number”—for instance, “2 + 2 = 4.” It turns out that, at least so far, interesting statements about number theory are much more likely to resemble “2 + 2 = 4” than Gödel’s vexing “P.” Gödel’s theorem, for most working mathematicians, is like a sign warning us away from logical terrain we’d never visit anyway.
What is it about Gödel’s theorem that so captures the imagination? Probably that its oversimplified plain-English form—”There are true things which cannot be proved”—is naturally appealing to anyone with a remotely romantic sensibility. Call it “the curse of the slogan”: Any scientific result that can be approximated by an aphorism is ripe for misappropriation. The precise mathematical formulation that is Gödel’s theorem doesn’t really say “there are true things which cannot be proved” any more than Einstein’s theory means “everything is relative, dude, it just depends on your point of view.” And it certainly doesn’t say anything directly about the world outside mathematics, though the physicist Roger Penrose does use the incompleteness theorem in making his controversial case for the role of quantum mechanics in human consciousness. Yet, Gödel is routinely deployed by people with antirationalist agendas as a stick to whack any offending piece of science that happens by. A typical recent article, “Why Evolutionary Theories Are Unbelievable,” claims, “Basically, Gödel’s theorems prove the Doctrine of Original Sin, the need for the sacrament of penance, and that there is a future eternity.” If Gödel’s theorems could prove that, he’d be even more important than Einstein and Heisenberg!
One person who would not have been surprised about the relative inconsequence of Gödel’s theorem is Gödel himself. He believed that mathematical objects, like numbers, were not human constructions but real things, as real as peanut butter sandwiches. Goldstein, whose training is in philosophy, is at her strongest when tracing the relation between Gödel’s mathematical results and his philosophical commitments. If numbers are real things, independent of our minds, they don’t care whether or not we can define them; we apprehend them through some intuitive faculty whose nature remains a mystery. From this point of view, it’s not at all strange that the mathematics we do today is very much like the mathematics we’d be doing if Gödel had never knocked out the possibility of axiomatic foundations. For Gödel, axiomatic foundations, however useful, were never truly necessary in the first place. His work was revolutionary, yes, but it was a revolution of the most unusual kind: one that abolished the constitution while leaving the material circumstances of the citizens more or less unchanged.