
Stewart Shapiro
Thinking about mathematics
Plato thought that numbers existed 'out there' they had a reality independent of everything else, but Aristotle's view was that they were much more tied to the ways we use them in our lives. The book goes on to consider the views of Kant and Mill, and then looks at the three points of view of the early 20^{th} century, Logicism, Formalism and Intuitionism. Shapiro then discusses more recent arguments about whether numbers actually exist or not, finishing with a chapter on structualism, where he puts forward his own point of view.
The book does not require too much prior experience in either mathematics or philosophy, but it's scope seemed considerably narrower than the title suggests  it is essentially just the ontology of numbers. Those wanting a wider view of the philosophy of mathematics  for instance the struggles with infinity, the incompleteness of arithmetic, and more recently the status of computers in mathematics  would be advised to look elsewhere. So who might want to read this book? Well those wanting to study the philosophy of mathematics will find it useful as a starting point for the ontology of numbers. If you're interested Shapiro's views then the final chapter gives a good indication. The book is also useful as an introduction to the works of other authors  I have added Hartry Field's Science without Numbers to my 'Books to Read' list.