Michele Ginammi

In this paper, I will do preparatory work for a generalized account of applicability, that is, for an account which works for math-to-physics, math-to-math, and physics-to-math application. I am going to present and discuss some examples of these three kinds of application, and I will confront them in order to see whether it is possible to find analogies among them and whether they can be ultimately considered as instantiations of a unique pattern. I will argue that these analogies can be exploited in order to get a better understanding of the applicability of mathematics to physics and of the complex relationship between physics and mathematics.

In this paper I will take into examination the relevance of the main indispensability arguments for the comprehension of the applicability of mathematics. I will conclude not only that none of these indispensability arguments are of any help for understanding mathematical applicability, but also that these arguments rather require a preliminary analysis of the problems raised by the applicability of mathematics in order to avoid some tricky difficulties in their formulations. As a consequence, we cannot any longer consider the applicability problems as subordinate to ontological ones: no ontological stance on mathematical entities can offer an easy road to the comprehension of the applicability of mathematics.

In this article I will sketch a general introduction to the problem of the applicability of mathematics as it is conceived in contemporary philosophy of mathematics. After some brief considerations concerning the historical reason why this problem was dismissed by the philosophical analysis during the first half of 20th century, I will expound Wigner’s puzzle, and I will make some theoretical considerations about it. In particular, I will show that the problem of applicability is independent from the ontological framework we can eventually assume and that it depends rather on the existence of an epistemological gap between mathematics and physics. Finally, I will try to sketch a possible line of inquiry for understanding the applicability of mathematics in physics.

According to Steiner (1998), in contemporary physics new important discoveries are often obtained by means of strategies which rely on purely formal mathematical considerations. In such discoveries, mathematics seems to have a peculiar and controversial role, which apparently cannot be accounted for by means of standard methodological criteria. M. Gell-Mann and Y. Ne׳eman׳s prediction of the Ω− particle is usually considered a typical example of application of this kind of strategy. According to Bangu (2008), this prediction is apparently based on the employment of a highly controversial principle—what he calls the “reification principle”. Bangu himself takes this principle to be methodologically unjustifiable, but still indispensable to make the prediction logically sound. In the present paper I will offer a new reconstruction of the reasoning that led to this prediction. By means of this reconstruction, I will show that we do not need to postulate any “reificatory” role of mathematics in contemporary physics and I will contextually clarify the representative and heuristic role of mathematics in science.