Vincent Ardourel

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.

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.

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.

The finite-size scaling (FSS) theory is a relatively new and important attempt to study critical phenomena; this paper aims to contribute to clarifying the philosophical significance of this theory. We maintain that, contrary to initial appearances and to some recent claims in the literature, the FSS theory cannot arbitrate the debate between the reductionists and anti-reductionists about phase transitions. Although the theory allows scientists to provide predictions for finite systems, the analysis we carry on here shows that it involves the intertwinement of both finite and infinite systems. But, we argue, the FSS theory has another virtue, as it provides quantitative predictions and explanations for finite systems close to the critical point; it thus complements in a distinctive manner the standard Renormalization Group qualitative approach relying on infinite systems.

The starting point of the chapter is a twofold observation: (i) most current physical theories make use of a continuous parameter t that plays the role of time; (ii) the current practice of modeling, namely, of computational modeling, makes use of a discrete parameter t k (k = 1, 2, …, n) that plays the role of time, because computers cannot handle continuous quantities. A “parameter” here is a symbol that does not play the same role as the other symbols in the physical theories or models: whereas the other symbols represent physical quantities, t (or t k ) just plays the role of that within which other quantities evolve. It is “time”, but in a rather shallow sense.

Le modèle d’Ising décrit les interactions locales entre des flèches fixées aux nœuds d’un réseau et orientées vers le haut ou vers le bas. Ce modèle mathématique, très abstrait, mais aussi très simple, est l’un des plus étudiés en physique moderne. Cet article s’intéresse à la nature de ce modèle et à ses fonctions épistémiques. Nous montrons qu’il a d’abord rempli une fonction représentationnelle en tant que « modèle approché » du magnétisme. Ensuite, comme « modèle-jouet » il a permis la compréhension qualitative des transitions de phase. C’est enfin comme « modèle minimal » pour l’étude des phénomènes critiques qu’il remplit désormais une fonction explicative.

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.

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.

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.

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.

I focus on a practice of mathematical research which consists in reformulating constructively physical theories. More precisely, I discuss three features of this practice. First, I show that it is stimulated by philosophical views and discuss how physics is used in the debate between classicism and constructivism in the philosophy of mathematics. Secondly, I characterise the methodology of the research in mathematics involved in this practice and show this is a methodology of "logicians" as Poincaré said. Finally, I show that in this practice, physical theories play a heuristic role for the progress of constructive mathematics. Successfully, they stimulate researches on constructive results and lead to the study of new provinces of constructive mathematics.

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.