ON SOBER PLATONISM: NEW PERSPECTIVES IN MATHEMATICAL PLATONISM BEYOND STRONG ONTOLOGICAL ASSUMPTIONS

This work aims at analyzing a trend which in recent years has been developed in mathematical Platonism. I have identified four theories which seem to me paradigmatic of this new trend: Full-Blooded Platonism by Mark Balaguer, ante rem Structuralism by Stewart Shapiro, Abstract Object Theory by Edward Zalta and Trivialism by Agustìn Rayo. These four theories share a platonist attitude towards mathematical objects, assuming that mathematical objects, as the reference of the terms in mathematical statements, actually exist. But contrary to classical mathematical Platonism, their ontological assumptions are so moderate, or sober, as to give the impression that these theories aren’t even genuinely platonist. I therefore propose to call ‘Sober Platonism' those approaches that support Platonism, without endorsing strong ontological commitment. The key feature of this trend is that the assumption of the existence of mathematical objects is no longer considered the starting-point of a theory of mathematical objects, but becomes a necessary condition to the occurrence of a fact: the human mind accesses to mathematical knowledge. Consequently, mathematical objects must exist and be such as to make possible a connection between mathematical objects and the human mind. Hence, the ultimate aim of Sober Platonism is to obtain a description of mathematics as practiced, which does not impose any philosophical constrain, but is able to answer philosophical questions. The first chapter of this work is devoted to the analysis of classical mathematical Platonism. I propose to consider this line of thought as the sum of three major theses: Independence (mathematical objects are independent of human thought and practices), Existence (mathematical objects exist) and Epistemology (mathematical objects are knowable). The latter thesis is further divided into three sub-theses: Theory of Knowledge, Reference and Truth. In the second, third, fourth and fifth chapter I discussed the proposals of the four aforementioned authors, matched together by their implicit, or sober, ontological commitment towards mathematical objects. These four theories take into account the existence of mathematical objects, the possibility to access to mathematical knowledge, the meaning of mathematical statements and the reference of their terms as philosophically relevant questions. Their main objective, however, is rather the development of an accurate description of mathematics in its autonomy. In the last chapter I have defined Sober Platonism through its adherence to the same theses to which classical Platonism adheres, Independence, Existence and Epistemology (again analyzed as Theory of Knowledge, Reference and Truth). After a comparative evaluation, it becomes clear that Sober Platonism assumes largely what is assumed by classical Platonism. The real element of distinction lies in the relationship between philosophy and mathematics, since in Sober Platonism the autonomy and dignity of mathematics are clearly established. The proper role of Philosophy is then to deliver a methodological description of how mathematics is performed, rather than a normative prescription of how mathematics should be performed. Beyond the results that may be achieved until today, Sober Platonism promises to have what it takes to reduce the importance of at least some of those issues that seems to be relevant to the philosophy of mathematics, but are not relevant for mathematics as practiced. In conclusion, Sober Platonism offers both an innovative approach in the philosophy of mathematics, and a fruitful contribution in providing both philosophy and mathematics with a genuine domain of inquiry.