ABSTRACT
Some people argue that because all the sets we encounter in everyday life are finite, we should exclude mathematics in which infinite sets appear. This is not a popular view! The main reason for this is that however finite the material universe appears, its mathematics inevitably leads one to the consideration of infinite sets and approximation processes involving algorithmic repetition. One can, with great difficulty, reconstruct a large part of classical mathematics on a finitary basis. But in doing this we just mimic in our mathematics the concrete processes by which nature itself is constrained. And because of this, such mathematics does nothing to divert us from the overwhelming impression that we do need the mathematics of the infinite to fully understand our own Universe. In practice, of course, it isn’t philosophy which widens our mathematical horizons. From Newton onwards, it has been the pure explanatory power of infinitary mathematics which has left us little alternative. In the same way, there is much we will never understand about the world we
live in without a mathematics which takes due account of how incomputable phenomena arise and interrelate. We have already seen how the iteration of very simple rules can lead to incomputable sets — even diophantine functions, using basic arithmetical operations, can have incomputable ranges. Such iterations are abundant in nature, where we find a level of incomputability associated with chaotic phenomena built on computable local events (e.g., problems in predicting the weather). Fractals provide a more graphic expression of the mathematics. For instance, Roger Penrose has asked about the computability of familiar objects like the Mandelbrot and Julia sets. In this chapter we set out to answer such basic questions as: Are there
different sorts of incomputable sets? How do we mathematically model situations involving interactions between incomputable phenomena?
