Visions of Infinity
A mathematical voyage brought Kurt Gödel to the shores of madness.
Jun 19, 2006, Vol. 11, No. 38 • By DAVID GUASPARI
A theory is formal if a computer can be programmed to recognize whether or not any alleged proof is valid. Insight may be needed to devise proofs but, if they are formal, not to check them: A computer can scan a sequence of symbols and mindlessly apply mechanical rules to determine whether it represents valid logical steps starting from axioms.
Any "useful" theory can be made formal (because all mathematics can), must be consistent (not allow the deduction of contradictory results), and will incorporate at least the ability to perform elementary arithmetic (with rules for addition, multiplication, and simple algebra, such as x+y=y+x).
The First Incompleteness Theorem, slightly cleaned up, says that any such "useful" theory is incomplete; that is, there are propositions of elementary arithmetic that the theory can neither prove nor disprove. In fact, we can find such a proposition that is essentially the statement of an elaborate algebraic rule. And someone who understands what that proposition means will recognize that it is true--because, under a clever interpretation, it means that it cannot be proven.
Were we to add this proposition as a new axiom, the Incompleteness Theorem would immediately provide another truth unprovable in this beefed-up theory. There is no escape.
This is head-spinning. How, for example, can we "recognize" that some arithmetical rule is true if we cannot prove it from beloved axioms that, hitherto, seemed to say all we knew about numbers? And just how does one reason to a conclusion about the limits of reasoning?
It is also an affront. David Hilbert, one of history's greatest mathematicians, testified to an ancient faith when he said that "In mathematics, there is no ignorabimus." When we seek an answer, "we must know. We will know."
Goldstein presents a sketch of Gödel's reasoning that should be accessible to a general reader. Some small technical miscues are not seriously misleading. (For the cognoscenti: The fixed-point lemma is misstated; and the demonstration that the "Gödel sentence" is unprovable appeals unnecessarily to its meaning, and thereby relies on a stronger hypothesis than consistency.)
Gödel announced his results in one terse paragraph at the end of a three-day technical meeting--and made not a ripple. Only the great scientific polymath John von Neumann realized that something important had happened. But what?
Gödel's interpretation is startling. It follows, he says, that at least one of the following two things must be true--and he, in fact, believed both: "Either the human mind cannot be reduced to the working of the brain" or "mathematical objects and facts . . . exist objectively and independently of our mental acts and decisions." (These quotations come from Logical Dilemmas, John Dawson's biography of Gödel. Goldstein cites opinions a bit less provocative.)
Here's my guess at what he meant: If my mind is only my brain, it's some kind of computer, so all its purely deductive workings are captured within some "useful" theory. But my mind can also prove the First Incompleteness Theorem, allowing it to recognize some mathematical truths that the theory does not imply. That act of recognition can't be a deduction, so must be a perception--of an independently existing mathematical realm.
Imagine Gödel's dismay, if not despair, when many drew the opposite conclusion: that mathematics had lost its certainty and been revealed as a social practice based on arbitrary decisions about what axioms to choose. Or when, as Goldstein says, "[Ludwig] Wittgenstein never accepted that Gödel had proved what he provably did prove" since it would have contradicted Wittgenstein's deep belief that logic was necessarily empty, and could contain no surprises.
Gödel's last years became a perverse reductio ad absurdum of his sustaining belief that everything has a rational explanation. What had been intermittent episodes of paranoia and depression became chronic. He feared that his food might be poisoned, and that dangerous gases were seeping from his radiators and ice box. He starved to death.