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 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 characterization of the system’s relevant behavior. Another kind concerns the manner in which the (...) abstract states are interpreted. 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.”. (shrink)

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’. (shrink)

We examine the interrelationships between analog computational modelling and analogue (physical) modelling. To this end, we attempt a regimentation of the informal distinction between analog and digital, which turns on the consideration of computing in a broader context. We argue that in doing so one comes to see that (scientific) computation is better conceptualised as an epistemic process relative to agents, wherein representations play a key role. We distinguish between two, conceptually distinct, kinds of representation that, we argue, are both (...) involved in each case of computing. Based on the semantic and syntactic properties of each of these representations, we put forward a new account of the distinction between analog and digital computing. We discuss how the developed account is able to explain various properties of different models of computation, and we finish by conceptually comparing analog computational modelling to analogue (scale) modelling. It is concluded that, contrary to the standard view, the two practices are orthogonal, differing both in their foundations and in the epistemic functions they fulfil. (shrink)

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 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. 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. In a clear sense, then, we are using idealised models to study a purely mathematical structure. The third purpose is to point out and explain this intriguing phenomenon and attempt to make connections with the typical case of idealised models of empirical domains. (shrink)

This dissertation examines aspects of the interplay between computing and scientific practice. The appropriate foundational framework for such an endeavour is rather real computability than the classical computability theory. This is so because physical sciences, engineering, and applied mathematics mostly employ functions defined in continuous domains. But, contrary to the case of computation over natural numbers, there is no universally accepted framework for real computation; rather, there are two incompatible approaches --computable analysis and BSS model--, both claiming to formalise algorithmic (...) computation and to offer foundations for scientific computing. -/- The dissertation consists of three parts. In the first part, we examine what notion of 'algorithmic computation' underlies each approach and how it is respectively formalised. It is argued that the very existence of the two rival frameworks indicates that 'algorithm' is not one unique concept in mathematics, but it is used in more than one way. We test this hypothesis for consistency with mathematical practice as well as with key foundational works that aim to define the term. As a result, new connections between certain subfields of mathematics and computer science are drawn, and a distinction between 'algorithms' and 'effective procedures' is proposed. -/- In the second part, we focus on the second goal of the two rival approaches to real computation; namely, to provide foundations for scientific computing. We examine both frameworks in detail, what idealisations they employ, and how they relate to floating-point arithmetic systems used in real computers. We explore limitations and advantages of both frameworks, and answer questions about which one is preferable for computational modelling and which one for addressing general computability issues. -/- In the third part, analog computing and its relation to analogue (physical) modelling in science are investigated. Based on some paradigmatic cases of the former, a certain view about the nature of computation is defended, and the indispensable role of representation in it is emphasized and accounted for. We also propose a novel account of the distinction between analog and digital computation and, based on it, we compare analog computational modelling to physical modelling. It is concluded that the two practices, despite their apparent similarities, are orthogonal. (shrink)

Computational physical systems may exhibit indeterminacy of computation (IC). Their identified physical dynamics may not suffice to select a unique computational profile. We consider this phenomenon from the point of view of cognitive science and examine how computational profiles of cognitive systems are identified and justified in practice, in the light of IC. To that end, we look at the literature on the underdetermination of theory by evidence and argue that the same devices that can be successfully employed to confirm (...) physical hypotheses can also be used to rationally single out computational profiles, notwithstanding IC. (shrink)

On the basis of two neglected testimonia, this short note argues that the terminus ante quem for Philippos of Amphipolis should be moved forward to the third century or to the early fourth century c.e. if not earlier.

Sisenna Historiarum lib. iii: Lucium Memmium, socerum Gai Scriboni, tribunum plebis, quern Marci Livi consiliarium fuisse callebant et tune Gurionis oratorem … . Erat Hortensius in bello primo anno miles, altero tribunus militum, Sulpicius legatus; aberat etiam M. Antonius; exercebatur una lege iudicium Varia, ceteris propter bellum intermissis; cui frequens aderam, quamquam pro se ipsi dicebant oratores non illi quidem principes, L. Memmius et Q. Pompeius, sed oratores tamen, teste diserto utique [Jahn: MSS. uterque] Philippo, cuius in testimonio contentio et (...) vim accusatoris habebat et copiam. Reliqui qui turn principes numerabantur in magistratibus erant cotidieque fere a nobis in contionibus audiebantur …. (shrink)

Vers la fin du XVI e siècle, le ramisme se répandit en Allemagne, gagna un grand nombre de prosélytes et fit preuve d'une remarquable vitalité. L'histoire de ses effets constitue un intéressant, néanmoins peu recherché chapitre de l'historiographie de la philosophie, et la question portant sur les ‹dettes› éventuelles de Leibniz envers le ramisme se tiend au coeur de cet essai. En premier lieu, quelques caractères théoriques du ramisme allemand et surtout du philippo-ramisme sont mis en évidence, après cela on (...) analyse l'emploi de Leibniz des notions ‹dialectic›, ‹topica› et ‹ars inveniendi› et l'on découvre une syntonisation conceptuelle entre cet emploi et la manière de penser des ramistes. Sans en tirer des conclusions hasardeuses, on peut comprendre le ramisme comme un ingrédient essentiel de l'univers leibnizien complexe et comme un thème de plus en plus important pour Leibniz. (shrink)

Of the four surviving fragments of Varius' De Morte1 perhaps the most widely discussed has been the first: Vendidit hie Latium populis agrosque Quiritum eripuit, fixit leges pretio atque refixit This is imitated by Virgil, whose Sibyl says of a soul in Tartarus: Vendidit hie auro patriam dominumque potentem imposuit; fixit leges pretio atque refixit Most commentators, quoting Cic. Phil. 12.5.12, connect both passages exclusively with Antony, and rightly point to Servius' words on v. 622, ‘possumus Antonium accipere’. What should (...) be stressed, however, is that Servius also thinks the words ‘vendidit hie auro patriam’ have a general reference, but are at the same time designed to recall historical individuals, of whom he names two: etiam haec licet generaliter dicantur, habent tamen specialitatem: nam Lasthenes Olynthum Philippo vendidit, Curio Caesari XXVII. S. Romam: de quo Lucanus Gallorum captus spoliis et Caesaris auro. (shrink)

Dans cet article, composé en hommage aux travaux de Peter Sharratt, l’auteur examine l’attitude de Pierre de La Ramée (Ramus) et d’Omer Talon à l’égard de l’Éthique à Nicomaque. En tant qu’humanistes, ces derniers s’occupent intensément des arts du discours. Tout en reconnaissant la nécessité d’un fondement éthique dans tout discours, ils se montrent violemment hostiles au traité aristotélicien. Le traité posthume de théologie composé par Ramus propose le Décalogue comme source exclusive de la morale chrétienne. Ramus et Talon refusent (...) de la sorte de s’engager dans la voie ouverte quelques décennies plus tôt par Ph. Melanchthon, qui avait intégré de manière originale la morale antique, notamment celle d’Aristote, dans la théologie protestante. Toutefois, vers la fin du xvie siècle, le philosophe hollandais Rodolphe Snellius, père adoptif du théologien Arminius, composera un traité de morale dans lequel il tente de concilier la morale aristotélicienne avec la méthode développée par Ramus et son équipe. (shrink)