David Peat

On Incompleteness 43truth that can never be proved. Why not incorporate it as one of theunderlying axioms of mathematics? All we have to do is increase theaxioms of arithmetic by one and we begin a whole new ball game.Does this get us around Gödel’s theorem? No, for Gödel’s theoremstates that once you add a new axiom, further unprovable truths willarise. No matter how you look at it, there is no avoiding Gödel’s proofthat mathematics is inherently incomplete.The meaning of Gödel’s result continues to be debated. For someit is a major headache, a failure to find ultimate security in logic andmathematics. Others see it in a more positive light. After all, Hilbert’sgreat program was to reduce all mathematics to symbolic manipulationsthat could, in principle, be performed on a computer. A proof,Hilbert said, can be achieved through a series of algorithms, and suchsteps could be automated. But now Gödel is telling us that such anapproach has limits and cannot encompass the whole of mathematics.There are things that human mathematicians do that can never beachieved by computers.Limits to AlgorithmsTake, for example, the idea of algorithms. 5 An algorithm is a simplerule, or elementary task, that is repeated over and over again. In thisway algorithms can produce structures of astounding complexity. Theycan be used with a computer to produce fractals, for example. Mathematicalfractals are generated by repeating the same simple steps atever decreasing scales. In this way an apparently complex shape, containingendless detail, can be generated by the repeated application of asimple algorithm. In turn these fractals mimic some of the complexforms found in nature. After all, many organisms and colonies alsogrow though the repetition of elementary processes such as, for5An algorithm is “a set of well defined rules for the solution of a problem in afinite number of steps” (McGraw-Hill Dictionary of Physics and Mathematics. NewYork: McGraw-Hill, 1978); or, in other words, a recipe for solving a mathematicalproblem.