David Peat

42 From Certainty to UncertaintyMathematics is also inconsistent because it is possible for a statementand its negation to exist simultaneously within the same system.Kurt Gödel’s result staggered the world of mathematics. His proofappears irrefutable. The final refuge of certainty had been mathematics,and now Gödel had kicked away its last prop. But, as with somethingas revolutionary as Heisenberg’s uncertainty principle, mathematiciansand philosophers continue to ask about the deeper significanceof Gödel’s theorem. How is it to be interpreted? What are its implications?To take one example, what exactly does it mean that there are truemathematical statements that cannot be proved? What would suchtruths look like? How would we recognize one if we saw it?Unprovable TruthsOne example of an unprovable mathematical statement may be“Goldbach’s conjecture.” It states that “every even number is the sumof two primes” ( A “prime,” or “prime number,” is a number that canonly be divided by itself and 1 without leaving a remainder.)It certainly appears to work in practice, as the following examplesshow:20 = 17 + 310 = 7 + 38 = 7 + 1No mathematician has ever found an exception to this conjecture,and it has been tested on enormously large numbers using computers,though it has not been tested on every number there is—after all thereare an infinite number of numbers. Mathematicians are quite certainthat Goldbach’s conjecture is true, but no one has ever been able toprove it. Is this the sort of unprovable truth to which Gödel was referring?Or will it turn out one day, as with Fermat’s last theorem, thatingenious mathematicians will figure out a proof?Suppose Goldbach’s conjecture is a basic truth about numbers, a