Introduction to Mathematical Philosophy [ペーパーバック]

I have devoted substantial time and effort to this 200 page book. Unless you are a student of logic, this book may not be for you. I suggest alternatives below.

I stayed the course and worked my way through each chapter, sometimes backing up, and often repeating several chapters before advancing again. Bertrand Russell is admired for his eloquence and style. Nonetheless, I can assure you that a methodical reading will require much effort.

I was forewarned. At one point a friend and colleague, a previous professor of mathematics at Texas A&M, expressed surprise that I was tackling this particular book. He considered Russell's work to be dated and not particularly easy going. I continued plodding along.

Russell begins with familiar ground, Peano's effort to derive the entire theory of natural numbers from five premises and three undefined terms (primitives). Russell demonstrates why Peano's approach fails to serve as an adequate basis for arithmetic.

In chapter 2 Russell introduces the work of Frege, who first succeeded in logicising arithmetic. We are led to a definition of number: the number of a class is the class of all those classes that are similar to it, or more simply, a number is anything which is the number of some class.

The third chapter introduces properties termed hereditary, posterity, and inductive. After some effort, we define the natural numbers as those to which proofs by mathematical induction can be applied. We also learn that mathematical induction is not valid for infinite numbers.

Russell now addresses the serial character of natural numbers, a characteristic involving finding or construction of an asymmetrical transitive connected relation.

In Chapters 5 and 6 Russell distinguished between cardinal numbers (the earlier definition of number) and relation numbers (also called ordinal numbers). I had difficulty with the interplay between the relations aliorelative, transitive, asymmetrical, square, and connected. For example, an asymmetrical relation is the same thing as a relation whose square is an aliorelative.

In chapter 7 I was initially surprised by Russell's assertion that the common belief that the complex numbers include the real numbers, the real numbers include the rational numbers, and the rational numbers include the natural numbers is erroneous and must be discarded.

The next thee chapters - infinite cardinal numbers, infinite series and ordinals, and limits and continuity - were more difficult. Eight more chapters follow.

Introduction to Mathematical Philosophy is philosophy, logic, and mathematics. It investigates the logical foundations of mathematics. It requires very careful reading.

I can suggest alternatives. Howard Eves in his delightful Foundations and Fundamental Concepts of Mathematics offers an excellent chapter titled Logic and Philosophy that compares three approaches - Logicism (Russell and Whitehead), Intuitionism (Brouwer and Heyting), and Formalism (Hilbert's Grundlagen der Geometrie). He also provides in an appendix a short overview of Godel's theorems (1931) which demonstrated that no complete or consistent axiomatic development of mathematics is attainable.

I also highly recommend Godel's Proof, a short book by Ernest Nagel and James R. Newman. Godel's Proof demonstrates that Russell and Whitehead's Principia Mathematica must necessarily be incomplete and inconsistent.

Like anything Russell wrote, it is a pleasure to read - his writing style is wonderful, and quite extraordinary when one realises how quickly he wrote this book (in prison, too!), but I suspect that for many readers the mathematical content will prove a little tricky to grasp.

As a historical document, it is fascinating; as an introduction to mathematical philosophy it is too narrow-minded for 1999.