A Computational Mathematics View of Space, Time and Complexity

Abstract

Modern computational mathematics requires a philosophical perspective largely at odds with that of traditional mathematics, since current computational mathematics (as distinct from computer science) is by its very nature is discrete, not continuous, and tied to the real world in ways that the more theoretical branches of mathematics (and computer science) often are not. Indeed, computational mathematics provides a means to escape the trap feared by John von Neumann when he wrote,

[T]here is a grave danger that the subject [of mathematics] will develop along the line of least resistance, that the stream so far from its source [in empirical reality] will separate into a multitude of insignificant branches, and that the discipline will become a disorganized mass of details and complexities.

But even a computational approach to mathematics has limits, not the least of which are the uncertainties of errors in hardware, software and algorithms that inevitably are part-and-parcel with computation, although there are ways to limit these uncertainties. In our chapter, bulwarked by concrete examples, we will try to situate past, present and future mathematical views of space, time, infinity and certainty within a computational context in which, for example, error due to quantum effects begins to compete with traditional sources of logical and numerical inaccuracy. We shall also argue that traditional taxonomies of complexity and completeness are not only outmoded but actually destructive of progress.

$^{\dagger }$ Borwein sadly passed away on 2nd August 2016