Introduction

Understood as a philosophical reflection about mathematics, the philosophy of mathematics has a long history that is intertwined, in intriguing and complex ways, with the development of philosophy itself.

In this essay, rather than providing a historical overview of the philosophy of mathematics, I will focus on some of the central issues in the field, giving the reader a sense of where some of the current debates are. In particular, I will examine three interrelated issues. First, I will consider an ontological issue: the issue of the existence and nature of mathematical objects. Do they exist? And what kinds of things are mathematical objects? Are they mind-independent abstract entities (that is, entities that exist independently of us, but which are not located in space-time), or are mathematical objects particular mental constructions (the result of certain operations per­formed in the mathematicians’ minds)? Moreover, should mathematical objects be thought of as individual entities (such as particular numbers), or as overall structures (such as various relations among numbers)? There are significant disagreements about how these questions should be answered, and distinctive proposals have been advanced to support the corresponding answers.

Second, I will examine an epistemological issue pertaining to the nature of our mathematical knowledge. How do we know the mathematics that we know? Do we have some form of access to mathematical objects akin to the perception of physical objects? And how is mathematical knowledge possible if mathematical objects are taken to be abstract entities that exist independently of us? Once again, distinctive views have been developed to address these problems.

Finally, I will discuss the problem of the application of mathematics.

As will become clear, these three issues are closely connected, and answers to one of them will constrain and, in part, shape answers to the others. Typically, to develop a particular philosophy of mathematics involves, at a minimum, the development of views about these issues. At least those who have approached the philosophy of mathematics systematically have attempted to do that. Thus, instead of examining these issues separately, I will consider them together, as part of the development of different conceptions of mathematics. I will start by describing four different versions of Platonism in mathematics - according to which mathematical objects exist and they are abstract - and how these various proposals have addressed the issues. I will then consider three versions of nominalism in mathematics - according to which mathematical objects do not exist - and discuss how these issues have been re-conceptualized by nominalists.

Like most philosophical proposals, those discussed in this essay face challenges. Nevertheless, I have decided to focus on the positive contribu­tions they have made, leaving the assessment of their drawbacks for another occasion. Even if none of these proposals has settled the issues under con­sideration, each has made significant contributions to our understanding of the philosophical issues about mathematics. For this reason alone, they deserve close study.

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