Many should find familiar the notions of materialism and reductionism, and should recognize that these doctrines enjoy large numbers of adherents. Fewer need have heard of finitism because of its presently smaller number of adherents, though many should recognize some of its aspects in current scientific and technological trends. This chapter tries to collect and address some of these issues as they relate to a broadened mechanics.

What is finitism?

I use the term finitism to refer to the thesis that the spatial and material world and its behavior are finite, not just finitely axiomatizable (as are the infinity of natural and real numbers) but actually finite in the sense of being composed of a finite number of bits of stuff that may undergo finite numbers of possible changes at each of a set of discrete temporal instants. The finitistic picture of the world in some locality thus resembles an enormous, possibly nondeterministic or probabilistic finite automaton, or more naturally, as a cellular automaton.

One can consider strengthenings of this local notion of finiteness to finiteness of space and time as well. Finiteness of space means that at each instant there are only finitely many places at which events may occur, so that the entire universe looks instantaneously like a cellular automaton. Finiteness of time means that the event world contains only finitely many temporal instants. Thus the strongest notion of finitism, involving both spatial and temporal finiteness, views the entire universe as a gigantic finite automaton.