It is often indeterminate what function a given computational system computes. This phenomenon has been referred to as “computational indeterminacy” or “multiplicity of computations.” In this paper, we argue that what has typically been considered and referred to as the (unique) challenge of computational indeterminacy in fact subsumes two distinct phenomena, which are typically bundled together and should be teased apart. One kind of indeterminacy concerns a functional (or formal) characterization of the system’s relevant behavior (briefly: how its physical states are grouped together and corresponded to abstract states). Another kind concerns the manner in which the abstract (or computational) states are interpreted (briefly: what function the system computes). We discuss the similarities and differences between the two kinds of computational indeterminacy, their implications for certain accounts of “computational individuation” in the literature, and their relevance to different levels of description within the computational system. We also examine the inter-relationships between our proposed accounts of the two kinds of indeterminacy and the main accounts of “computational implementation.”

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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.

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I examine the classical idea of ‘algorithm’ as a sequential, step-by-step, deterministic procedure (i.e., the idea of ‘algorithm’ that was already in use by the 1930s), with respect to three themes, its relation to the notion of an ‘effective procedure’, its different roles and uses in logic, computer science, and mathematics (focused on numerical analysis), and its different formal definitions proposed by practitioners in these areas. I argue that ‘algorithm’ has been conceptualized and used in contrasting ways in the above areas, and discuss challenges and prospects for adopting a final foundational theory of (classical) ‘algorithms’.