Unrealistic models for realistic computations: how idealisations help represent mathematical structures and found scientific computing

Abstract

We examine two very different approaches to formalising real computation, commonly referred to as "Computable Analysis" and "the BSS approach". The main models of computation underlying these approaches -bit computation (or Type-2 Effectivity) and BSS, respectively-have also been put forward as appropriate foundations for scientific computing. The two frameworks offer useful computability and complexity results about problems whose underlying domain is an uncountable space (such as R or C). Since typically the problems dealt with in physical sciences, applied mathematics, economics, and engineering are also defined in uncountable domains, it is fitting that we choose between these two approaches a foundational framework for scientific computing. However, the models are incompatible as to their results. What is more, the BSS model is highly idealised and unrealistic; yet, it is the de facto implicit model in various areas of computational mathematics, with virtually no problems for the everyday practice.This paper serves three purposes. First, we attempt to delineate what the goal of developing foundations for scientific computing exactly is. We distinguish between two very different interpretations of that goal, and on the separate basis of each one, we put forward answers about the appropriateness of each framework. Second, we provide an account of the fruitfulness and wide use of BSS, despite its unrealistic assumptions. Third, according to one of our proposed interpretations of the scope of foundations, the target domain of both models is a certain mathematical structure (namely, floating-point arithmetic). In a clear sense, then, we are using idealised models to study a purely mathematical structure (actually a class of such structures). The third purpose is to point out and explain this intriguing (perhaps unique) phenomenon and attempt to make connections with the typical case of idealised models of empirical domains.

The Nature of Physical Computation

“…) notes that this approach is also founded on Turing's work (Turing 1948). See also Feferman (2013) and Papayannopoulos (2020a), who discuss the relationship between the two approaches. 48 The exposition here is adopted from Earman (1986).…”