The Philosophy of Mathematics: An Introductory Essay

When you first pick it up, this book looks really boring. Lend me your ear, and I'll explain why it's not:Mathematical systems are formalized, unambiguous, mechanical processes for manipulating symbols on a piece of paper. These systems are useful to the extent that they are models of things in the real world, or of thought processes in our minds. Specifically, mathematical systems make great models because, unlike words or thoughts or pictures, they are formalized and unambiguous.People who use mathematics often begin to believe that the symbolic models are "real". That is, they start to think that mathematical principles are undeniable truths about things in the real world, or at least about thought processes in our minds. (For example, before Einstein many people believed that Euclidean geometry accurately modeled triangles and squares in physical space. Before Gödel, many people believed that set theory could be used to prove any logical statement which is true.)The philosophy of mathematics is an investigation of the correspondence between symbols on paper and the realities observed by thinking people. That is, it questions the basic assumptions people use when they interpret mathematical models. It also addresses the ironic circularity introduced when the very description of a mathematical model assumes the concepts of "truth" that it defines.The philosophy of mathematics has a long and interesting history, most of which has been forgotten by today's mathematicians and flimsy philosophy departments. This book is a great introduction to this history, written at a level that doesn't require too much previous knowledge. It does require a bit of patience for people who have not yet mastered the vocabulary of classical philosophers, or the concepts of higher mathematics.After some diligence, I found the essay to be quite fascinating. Although the author's diction could be simplified, his ideas are well organized and thoughtfully presented.