Mathematical Explanation: Problems and Prospects

Since this issue is devoted to the interaction between philosophy of mathematics and mathematical practice, I would like to begin with an introductory reflection on this topic, before I enter the specifics of my contribution. In the past thirty years there has been a marked shift in philosophy of mathematics, due to the appearance of research on aspects of mathematics that were previously ignored by philosophers of mathematics. A short, and very incomplete, list includes work on the dynamics of mathematical growth, the debate on computer proofs, the role of diagrammatic reasoning in mathematics, induction and conjecture in mathematics, problems at the interface of theoretical physics and mathematics. In some cases, these contributions have been accompanied by much fanfare about the need to pay attention to mathematical practice and by an attack on philosophy of mathematics as “foundations of mathematics”, variously called, “formalism”, “foundationalism”, “justificationism.” Whereas some of the polemical tone might have served to bring attention to new and exciting developments, I find that overall it is unwarranted and tends to muddle the issues. First of all, the characterization of the foundational programs, which are being attacked, is often one-sided at best and patently false in the worst cases. But even leaving questions of historical accuracy aside, all the programs in foundations of mathematics in this century have, in my opinion, been concerned with mathematical practice. In the grand foundational programs, say Hilbert’s, attention to practice was necessary to insure that the consistency program be able to account for all of mathematics, as opposed to a small part of it. And setting up the formalisms does require a very good sense of how much you need for various parts of mathematical practice. In this sense, many programs in contemporary logical foundations, such as reverse mathematics or predicative mathematics, are extremely sensitive to issues of mathematical practice. Moreover, the distinction between elementary and non-elementary methods, which was one of the cornerstones of Hilbert’s program, is a typical issue emerging from mathematical practice. However, it is true that many of the classical foundational programs “filter out” many aspects of mathematical practice which are irrelevant to their goals. Hence, there is a kernel of truth in the above mentioned criticisms. There are many aspects of mathematical practice that are irrelevant for some of the classical foundational programs but nonetheless worthy of philosophical attention. Thus, for instance, while a study of mathematical heuristics is not relevant to Hilbert’s program, it has much to offer to the philosophers and mathematicians who are interested in aspects of mathematics which go beyond the specific aims set by Hilbert for his task. But this, contrary to some of the polemical claims I referred to above, in itself does not invalidate Hilbert’s program (other considerations do!). It only calls for a liberalization concerning what aspects of mathematics should be objects of philosophical interest. I think that much of the alleged opposition between these developments can be deflated if one keeps in mind that the aims of both traditions are legitimate and all provide essential information about the complex reality we are interested in, i.e. mathematics. The topic of my paper, mathematical explanation, also escapes traditional foundational work. Part of the reason is that the subject area is admittedly vague, and consequently difficult to treat with precise mathematical or logical tools. Moreover, it does not bear directly upon some of the traditional foundational concerns, such as certainty, which have dominated much of philosophy of mathematics. It is nonetheless a subject of great philosophical interest. Consider, for instance, the situation in philosophy of science. There the topic of scientific explanation has received much attention. In this Mathematical Explanation: Problems and Prospects Paolo Mancosu