Gödel’s Proof
January 24, 2007 – 3:35 amby Ernest Nagel and James R. Newman
The author’s present an exegesis (for laymen) of Kurt Gödel’s 1931 paper: “On Formally Undecidable Propositions of Principia Mathematica and Related Systems”
Gödel basically showed that mathematics is not finite. (as far as I can tell… you shouldn’t quote me on that :P)
Gödel’s conclusions bear on the question wether a calculating machine can be constructed that would match the human brain in mathematical intelligence. Today’s calculating machines have a fixed set of directives built into them; these directives correspond to the fixed rules of inference of formalized axiomatic procedure. The machines thus supply answers to problems by operating in a step-by-step manner, each step being controlled by the built-in directives. But, as Gödel showed in his incompleteness theorem, there are innumerable problems in elementary number theory that fall outside the scope of a fixed axiomatic method, and that such engines are incapable of answering, however intricate and ingenious their built-in mechanisms may be and however rapid their operations. Given a definite problem, a machine of this type might be built for solving it; but no one such machine can be built for solving every problem.
Basically the principals behind the operation of our minds is beyond any formalized system of rules we have come up with. Our minds work on a separate order from formalized rules (i.e. it isn’t that we haven’t figured out all the rules for how the mind works)
