According to a standpoint that is widely shared by philosophers, mathematics deals with abstract entities that have no spatio-temporal location and lack causality. The empirical sciences, on the other hand, deal with concrete entities that have spatio-temporal location and causal efficacy. This characterization is very effective to render the different nature of the subject matter of mathematics and that of the empirical sciences. Nevertheless, surprisingly, the objects of mathematics and those of the empirical sciences successfully interact. This is especially evident if we consider how many results obtained in the empirical sciences through mathematics receive successful empirical confirmations and allow to make successful empirical predictions. When this happens, we speak of ‘successful applications of mathematics in the empirical sciences’ and the relevant philosophical question connected to this phenomenon is the following: How can we account for the successful applicability of mathematics in the empirical sciences?
Many efforts have been recently made by philosophers to provide an answer to this question and account for the success of applications of mathematics in science. The present research project pursues a different, although connected, goal. It aims to investigate the converse question that arises from the successful interplay between mathematics and the empirical sciences, namely: How can we account for the successful applicability of the empirical sciences in mathematics?
This question has not yet been addressed by philosophers of science and mathematics, and the motivation for addressing it lies in the following remarks. An analysis of mathematical practice reveals that some mathematicians do use results from the empirical sciences in their mathematical arguments. Moreover, the mathematical arguments constructed in this way sometimes lead to correct mathematical results (i.e., results that are correct from a purely mathematical point of view). Therefore, there is an interesting philosophical question as to how these successful applications of the empirical sciences to mathematics can be accounted for. The main goal of this project is to provide a philosophical investigation of such applications through a twofold strategy.
First, together with my team, we will examine the hypothesis that the structural mapping account of applicability, namely the most influential view that has been proposed by philosophers to account for the success of applications of mathematics in science (‘direct applications’), can be used to account for successful applications of science in mathematics (‘converse applications’). The idea behind this strategy is to take into consideration those examples of converse applications that are already available in the context of mathematical practice and see if the structuralist standpoint can be used (in its original form or with proper modifications and extensions) to account for them.
Secondly, we will examine the hypothesis that the reason why the empirical sciences apply in mathematics is due to the particular modal character of the scientific laws that are applied in mathematics. The idea behind this second strategy is that, since in many cases of successful applications of science in mathematics what is applied in mathematics are conservation laws and since mathematical statements have a specific necessary status (they are metaphysically necessary), it may be the case that the successful applicability of science in mathematics essentially depends on the fact that conservation laws and mathematical statements share the same necessary status. Conservation laws are, in fact, regarded as metaphysically necessary by some philosophers of science. And mathematical truths are usually considered by philosophers as paradigmatic metaphysical necessities. If conservation laws and mathematical statements successfully interact in converse applications and if they share the same modal status, there is an interesting research path that concerns the import of their modal status for this successful interplay.
The philosophical analysis of converse applications has not yet been addressed by philosophers and the two research strategies that I am proposing are totally novel. Thus, the main significance of this exploratory research project will be to provide, for the first time, a philosophical analysis of converse
applications. Furthermore, by assessing the viability of adopting a specific philosophical stance to account for cases of applications of the empirical sciences in mathematics, the present investigation will disclose new directions of analysis and favor a better understanding of the successful interplay between mathematics and science.