Deductive nomological model and mathematics: making dissatisfaction more satisfactory

El debate sobre la explicación matemática ha heredado el mismo sentimiento de insatisfacción que los filósofos de la ciencia expresaron, en el contexto de la explicación científica, hacia el modelo nomológico deductivo. Este modelo se considera incapaz de cubrir casos de explicaciones matemáticas genuinas y, además, continúa siendo ignorado en gran medida en la literatura relevante. Sorprendentemente, las razones de este ostracismo no son suficientemente manifiestas. En este artículo exploro una posible extensión del modelo al caso de las explicaciones matemáticas y sostengo que hay por lo menos dos razones para juzgar la imagen nomológico deductiva de la explicación como inadecuada en este contexto

Direct and converse applications: Two sides of the same coin?

In this paper I present two cases, taken from the history of science, in which mathematics and physics successfully interplay. These cases provide, respectively, an example of the successful application of mathematics in astronomy and an example of the successful application of mechanics in mathematics. I claim that an illustration of these cases has a twofold value in the context of the applicability debate. First, it enriches the debate with an historical perspective which is largely omitted in the contemporary discussion. Second, it reveals a component of the applicability problem that has received little attention. This component concerns the successful application of physical principles in mathematical practice. With the help of the two examples, in the final part of the paper I address the following question: are successful applications of mathematics to physics (direct applications) and successful applications of physics to mathematics (converse applications) two sides of the same problem

The Weak Objectivity of Mathematics and Its Reasonable Effectiveness in Science

Philosophical analysis of mathematical knowledge are commonly conducted within the realist/antirealist dichotomy. Nevertheless, philosophers working within this dichotomy pay little attention to the way in which mathematics evolves and structures itself. Focusing on mathematical practice, I propose a weak notion of objectivity of mathematical knowledge that preserves the intersubjective character of mathematical knowledge but does not bear on a view of mathematics as a body of mind-independent necessary truths. Furthermore, I show how that the successful application of mathematics in science is an important trigger for the objectivity of mathematical knowledge

Intended and Unintended Mathematics: The Case of the Lagrange Multipliers

We can distinguish between two different ways in which mathematics is applied in science: when mathematics is introduced and developed in the context of a particular scientific application; when mathematics is used in the context of a particular scientific application but it has been developed independently from that application. Nevertheless, there might also exist intermediate cases in which mathematics is developed independently from an application but it is nonetheless introduced in the context of that particular application. In this paper I present a case study, that of the Lagrange multipliers, which concerns such type of intermediate application. I offer a reconstruction of how Lagrange developed the method of multipliers and I argue that the philosophical significance of this case-study analysis is twofold. In the context of the applicability debate, my historically-driven considerations pull towards the reasonable effectiveness of mathematics in science. Secondly, I maintain that the practice of applying the same mathematical result in different scientific settings can be regarded as a form of crosschecking that contributes to the objectivity of a mathematical result

Deductive Nomological Model and Mathematics: Making Dissatisfaction more Satisfactory

The discussion on mathematical explanation has inherited the same sense of dissatisfaction that philosophers of science expressed, in the context of scientific explanation, towards the deductive-nomological model. This model is regarded as unable to cover cases of bona fide mathematical explanations and, furthermore, it is largely ignored in the relevant literature. Surprisingly, the reasons for this ostracism are not sufficiently manifest. In this paper I explore a possible extension of the model to the case of mathematical explanations and I claim that there are at least two reasons to judge the deductive-nomological picture of explanation as inadequate in that context

Evidence, explanation and enhanced indispensability

International audienceIn this paper I shall adopt a possible reading of the notions of ‘explanatory indispensability’ and ‘genuine mathematical explanation in science’ on which the Enhanced Indispensability Argument (EIA) proposed by Alan Baker is based. Furthermore, I shall propose two examples of mathematical explanation in science and I shall show that, whether the EIA-partisans accept the reading I suggest, they are easily caught in a dilemma. To escape this dilemma they need to adopt some account of explanation and offer a plausible answer to the following ‘question of evidence’: What is a genuine mathematical explanation in empirical science and on what basis do we consider it as such? Finally, I shall suggest how a possible answer to the question of evidence might be given through a specific account of mathematical explanation in science. Nevertheless, the price of adopting this standpoint is that the genuineness of mathematical explanations of scientific facts turns out to be dependent on pragmatic constraints and therefore cannot be plugged in EIA and used to establish existential claims about mathematical objects

What’s the Matter with the Deductive Nomological Model?

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The Epistemological Import of Euclidean Diagrams

International audienceIn this paper I concentrate on Euclidean diagrams, namely on those diagrams that are licensed by the rules of Euclid’s plane geometry. I shall overview some philosophical stances that have recently been proposed in philosophy of mathematics to account for the role of such diagrams in mathematics, and more particularly in Euclid’s Elements. Furthermore, I shall provide an original analysis of the epistemic role that Euclidean diagrams may (and, indeed) have in empirical sciences, more specifically in physics. I shall claim that, although the world we live in is not Euclidean, Euclidean diagrams permit to obtain knowledge of the world through a specific mechanism of inference I shall call inheritance

Learning from Euler. From Mathematical Practice to Mathematical Explanation

International audienceIn his "Découverte d'un nouveau principe de mécanique" (1750) Euler offered, for the first time, a proof of the so-called Euler's Theorem. In this paper I will focus on Euler's original proof and I will show how a look at Euler's practice as a mathematician can inform the philosophical debate about the notion of explanatory proofs in mathematics. In particular, I will show how one of the major accounts of mathematical explanation, the one proposed by Mark Steiner in his paper "Mathematical explanation" (1978), is not able to account for the explanatory character of Euler's proof. This contradicts the original intuitions of the mathematician Euler, who attributed to his proof a particular explanatory character