The Dependence of Mathematics on Reality

Abstract

MOST mathematicians today regard mathematics as an abstract structure built up from axioms by the use of logical principles. By altering these axioms, different types of mathematics can be developed; for example, Euclidean and non-Euclidean geometries differ by only one axiom—the parallel postulate—but this difference is quite sufficient to lead to very different conclusions in each system. This modern concept of mathematics stresses the idea that man invents his mathematics (and hence controls it to some extent.) rather than discovers it (the assumption here being that mathematics exists apart from and independent of man). Mathematics becomes an intellectual game. You select your axioms according to taste and work out the corresponding mathematics, using logical principles. If the results are aesthetically pleasing or elegant, this new system becomes a part of mathematics. Notice here that for the modern mathematician there is no mention of applicability to reality. The system stands alone on its consistency and beauty. If, for some reason, the system can be used in explaining certain properties of the real world, this is of no import to the mathematician. G. H. Hardy, in his book A Mathematician's Apology, expresses this view quite forcefully. It has been said by Huxley that six monkeys pecking unintelligently on typewriters would, after millions of millions of years, eventually write all the books in the British Museum (Sir James Jeans quotes Huxley's statement on page 4 of The Mysterious Universe). Similarly, mathematicians randomly selecting various axioms would eventually produce significant mathematical systems, but undoubtedly only after an enormous amount of time had been wasted.