Mathematics in the real world

This paper explores aspects of an experimental approach to mathematical proof, most notably number crunching, or the verification of subsequent particular cases of universal propositions. Since the rise of the computer age, this technique has indeed conquered practice, even if it implies the abandonment of the ideal of absolute certainty. Thus, it seems that also in mathematical research, the qualitative criterion of effectiveness, i.e. to reach one's goals, gets increasingly weighed against the quantitative one of efficiency, i.e. to minimize one's means/ends ratio. We probe for mathematical reasons and philosophical justifications for this rising popularity of `going inductive'. Our story will lead to the consideration of limit cases, opening up the possibility of proofs of infinite length being surveyed in a finite time. This should show that mathematical practice in crucial aspects depends upon what the actual world is (or is not) like. Note that this does not at all entail a rejection of the notion of a purely formal or deductive proof, even in cases where the latter should have actually ceased to essentially contribute to establishing the correctness of underlying mathematical claims.

In the proposed scenarios it remains perfectly possible to be a Platonist, thus claim that mathematical knowledge is necessary, and nevertheless accept that, depending on the world you live in, some mathematical statements are either trivial or extremely difficult to answer (e.g., if one happens to live in our universe). What should become clear however, is that an isolationist strategy, whereby in order to preserve the purity of mathematics, one has the mathematical domain shrunk until all external influences are excluded, will be of no avail. Indeed, this rather cynical procedure, which has helped create the miracle of the effectiveness of mathematics (Wigner), cannot do much work here, since only pure mathematical statements will be talked about to begin with. Moreover, working mathematicians absolutely do not shun away from all inductive techniques, methods, and ideas described in this paper.

Unfortunately, there is (still) not much willingness among mathematicians to `out' themselves on this philosophically laden topic, to the effect that, in terms of the metaphor introduced by Reuben Hersh, most empirical or experimental elements currently remain relegated to the back stage, while only formal proofs are held to occupy the front stage and confront the public. If it is however accepted that the front stage cannot exist without the back stage, then it is realized that no theatre can function as a whole without taking into account the economical necessities also. Already today, mathematicians amply rely on computers to warrant mathematical results, and work with conjectures that are only probable to a certain degree. Every so often, we get a glimpse of what is happening back stage, but what seems to be really required is not merely the idea that the front can only work if the whole of the theatre is taken into account, but also that, in order to actually understand what is happening front stage, an insight and understanding of the whole is required. If not, a \emph{deus ex machina} will be permanently needed.