Structure and Applicability : an Analysis of the Problem of the Applicability of Mathematics

From the introduction: [...] In the first part (Historical Considerations), I will deal with the historical problem of understanding why the applicability problem has been dismissed after Frege’s and logicists’ analysis (Chapter 1: A Neglected Problem). I will show that their answer is no longer satisfying and that such a dismissal was not due to a real overcoming of the problem. The second part (Philosophical Problems) will be devoted to the analysis of the specific philosophical problems lying behind the applicability of mathematics. First I will discuss Wigner’s (1960) famous analysis (Chapter 2: Do Miracles Occur? ), and then I will deal with Steiner’s (1998) fundamental work on the topic (Chapter 3: Applicabilities of Mathematics). As we will see, there are many philosophical problems concerning the applicability of mathematics. A further chapter (Chapter 4: Applicability and Ontological Issues) will be devoted to an analysis of the relations between ontological questions in mathematics and its applicability and effectiveness in science, in order to remove any misunderstanding about the possibility that the problems of mathematical applicability are nothing but a consequence of a certain ontological choice. Finally, in the third part (An Account for Mathematical Representativeness) I will offer an original account for one of the main roles played by mathematics in science: the representative role, which is at the very base of so many scientific discoveries and improvements in contemporary physics. First, (Chapter 5: Structures and Applicability) I will present my account in a purely theoretical way, and then I will offer some concrete examples in support of such an account (Chapter 6: Some Concrete Examples).