An important task in mathematical sciences is to make quantitative predictions, which is often done via the solution of differential equations. In this paper, we investigate why, to perform this task, scientists sometimes choose to use numerical methods instead of analytical solutions. Via several examples, we argue that the choice for numerical methods can be explained by the fact that, while making quantitative predictions seems at first glance to be facilitated by analytical solutions, this is actually often much easier with (...) numerical methods. Thus we challenge the widely presumed superiority of analytical solutions over numerical methods. (shrink)

The transference theory reduces causation to the transmission of physical conserved quantities, like energy or momenta. Although this theory aims at applying to all felds of physics, we claim that it fails to account for a quantum electrodynamic effect, viz. the Aharonov-Bohm effect. After having argued that the Aharonov-Bohm effect is a genuine counter-example for the transference theory, we offer a new physicalist approach of causation, ontic and modal, in which this effect is embedded.

Uffink and Valente claim that there is no time-asymmetric ingredient that, added to the Hamiltonian equations of motion, allows to obtain the Boltzmann equation within the Lanford’s derivation. This paper is a discussion and a reply to that analysis. More specifically, I focus on two mathematical tools used in this derivation, viz. the Boltzmann–Grad limit and the incoming configurations. Although none of them are time-asymmetric ingredients, by themselves, I claim that the use of incoming configurations, as taken within the B–G (...) limit, is such a time-asymmetric ingredient. Accordingly, this leads to reconsider a kind of Stoßzahlansatz within Lanford’s derivation. (shrink)

In this paper, I present a discrete solution for the paradox of Achilles and the tortoise. I argue that Achilles overtakes the tortoise after a finite number of steps of Zeno’s argument if time is represented as discrete. I then answer two objections that could be made against this solution. First, I argue that the discrete solution is not an ad hoc solution. It is embedded in a discrete formulation of classical mechanics. Second, I show that the discrete solution cannot (...) be falsified experimentally. (shrink)

Je propose d’analyser une pratique de la recherche en mathématiques constructives, celle qui consiste à reformuler constructivement les théories physiques. Je discute plus précisément trois aspects de cette pratique. Je montre d’abord que celle-ci a la particularité d’être motivée par des considérations philosophiques et comment la physique est utilisée pour arbitrer un débat de philosophie des mathématiques entre constructivisme et classicisme. Ensuite, j’identifie la méthodologie de la recherche en mathématiques que cette pratique implique et montre qu’il s’agit, selon une terminologie (...) empruntée à Poincaré, d’une méthodologie de « logiciens ». Enfin, je montre que dans cette pratique, les théories physiques ont un rôle heuristique sur le développement des mathématiques constructives. Elles permettent avec succès de stimuler la recherche d’énoncés constructifs et d’orienter la recherche en mathématiques constructives vers de nouveaux domaines. (shrink)

Various errors can affect scientific code and detecting them is a central concern within computational science. Could formal verification methods, which are now available tools, be widely adopted to guarantee the general reliability of scientific code? After discussing their benefits and drawbacks, we claim that, absent significant changes as regards features like their user-friendliness and versatility, these methods are unlikely to be adopted throughout computational science, beyond certain specific contexts for which they are well-suited. This issue exemplifies the epistemological heterogeneity (...) of computational science: profoundly different practices can be appropriate to meet the reliability challenge that rises for scientific code. (shrink)

It has been argued that the Duhem problem is renewed with computational models since model assumptions having a representational aim and computational assumptions cannot be tested in isolation. In particular, while the Verification and Validation methodology is supposed to prevent such holism, Winsberg argues that verification and validation cannot be separated in practice. Morrison replies that Winsberg overstates the entanglement between the steps. The paper aims at arbitrating these two positions, by stressing their respective validity in relation to domains of (...) application. It importantly argues for an increasing use of formal methods in verification, that makes disentanglement possible. (shrink)

Scientific models need to be investigated if they are to provide valuable information about the systems they represent. Surprisingly, the epistemological question of what enables this investigation has hardly been investigated. Even authors who consider the inferential role of models as central, like Hughes or Bueno and Colyvan, content themselves with claiming that models contain mathematical resources that provide inferential power. We claim that these notions require further analysis and argue that mathematical formalisms contribute to this inferential role. We characterize (...) formalisms, illustrate how they extend our mathematical resources, and highlight how distinct formalisms offer various inferential affordances. (shrink)

Je propose d’analyser une pratique de la recherche en mathématiques constructives, celle qui consiste à reformuler constructivement les théories physiques. Je discute plus précisément trois aspects de cette pratique. Je montre d’abord que celle-ci a la particularité d’être motivée par des considérations philosophiques et comment la physique est utilisée pour arbitrer un débat de philosophie des mathématiques entre constructivisme et classicisme. Ensuite, j’identifie la méthodologie de la recherche en mathématiques que cette pratique implique et montre qu’il s’agit, selon une terminologie (...) empruntée à Poincaré, d’une méthodologie de « logiciens ». Enfin, je montre que dans cette pratique, les théories physiques ont un rôle heuristique sur le développement des mathématiques constructives. Elles permettent avec succès de stimuler la recherche d’énoncés constructifs et d’orienter la recherche en mathématiques constructives vers de nouveaux domaines. (shrink)

In philosophical studies regarding mathematical models of dynamical systems, instability due to sensitive dependence on initial conditions, on the one side, and instability due to sensitive dependence on model structure, on the other, have by now been extensively discussed. Yet there is a third kind of instability, which by contrast has thus far been rather overlooked, that is also a challenge for model predictions about dynamical systems. This is the numerical instability due to the employment of numerical methods involving a (...) discretization process, where discretization is required to solve the differential equations of dynamical systems on a computer. We argue that the criteria for numerical stability, as usually provided by numerical analysis textbooks, are insufficient, and, after mentioning the promising development of backward analysis, we discuss to what extent, in practice, numerical instability can be controlled or avoided. (shrink)

Various errors can affect scientific code and detecting them is a central concern within computational science. Could formal verification methods, which are now available tools, be widely adopted to guarantee the general reliability of scientific code? After discussing their benefits and drawbacks, we claim that, absent significant changes as regards features like their user-friendliness and versatility, these methods are unlikely to be adopted throughout computational science, beyond certain specific contexts for which they are well-suited. This issue exemplifies the epistemological heterogeneity (...) of computational science: profoundly different practices can be appropriate to meet the reliability challenge that rises for scientific code. (shrink)

Can Brownian motion arise from a deterministic system of particles? This paper addresses this question by analysing the derivation of Brownian motion as the limit of a deterministic hard-spheres gas with Lanford’s theorem. In particular, we examine the role of the Boltzmann-Grad limit in the loss of memory of the deterministic system and compare this derivation and the derivation of Brownian motion with the Langevin equation.

In his book, The Structure of Time, William H. Newton-Smith claims that we cannot decide whether the structure of time is continuous or merely dense. I discuss here the extent to which he is correct. I argue that he fails to provide strong enough arguments to defend his claim, but I then suggest how he might overcome this issue. I also argue that we must reconsider the consequences that he draws from his “undecidability result”.

Vincent Ardourel discusses the eliminability of infinite limits in the explanations of phase transitions—an important point in the debate on the reducibility of thermodynamics to statistical mechanics. To this end, he examines alternative physical theories that deal with phase transitions in finite systems.