Some foundational problems in mathematics suggested by physics

Abstract

It is largely recognized that physics has represented an important source of suggestions in the historical development of mathematics and, in more recent times, even of logic. At the same time, if we look to the foundational investigations about both sciences in our century, one cannot help but notice that these researches have very rarely inter acted. Further, a number of discussions about the foundations of mathematics seem to go along with a somewhat naive and old fashioned image of physics, whereas foundational investigations about physics often propose an oversimplified image of the world of mathe matics (stressing for instance the characterization of mathematics as a merely analytical science). I would like to discuss in what sense contemporary physics might provide some interesting arguments and theoretical results, which could have a bearing on the foundational studies about mathematics. Let me start with a traditional philosophical question. It is well known that many discussions of the golden period of the foundational studies in mathematics in our century are essentially founded on a somewhat schematic contraposition between a platonistic view according to which, roughly, mathematical objects are described) and a concep tualista view (according to which mathematical objects are invented or constructed). Now, such a contraposition is clearly based on an extrapolation from the theoretical constructs of empirical sciences. Namely, the starting point of the traditional platonist in mathematics seems to be founded on a very uncritical concept of physical object, to which the platonist intends to assimilate even the concept of mathema tical object. In other words (according to the classical platonistic view), the concept of mathematical object should share the same logical and gnoseological characters as the concept of physical object. But what are the relevant properties of the concept of physical object? It is well known that contemporary physics has submitted to a strong criticism the traditional concept of object, developed by classical macrophysics [10]. I would like to recall only four main reasons, which in my opinion confirm the thesis according to which the concept of