Introduction to Mathematical Philosophy [Paperback]

Bertrand Russell (Author)

Book Description

Publication Date: September 14, 1993

Seminal work by great modern philosopher and mathematician focuses on certain issues of mathematical logic that Russell believed invalidated much traditional and contemporary philosophy. Topics include number, order, relations, limits and continuity, propositional functions, descriptions and classes, more. Clear, accessible excursion into the realm where mathematics and philosophy meet.

Product Details

• Paperback: 224 pages
• Publisher: Dover Publications (September 14, 1993)
• Language: English
• ISBN-10: 0486277240
• ISBN-13: 978-0486277240
• Product Dimensions: 8.4 x 5.4 x 0.5 inches
• Shipping Weight: 8 ounces (View shipping rates and policies)
• Average Customer Review: 4.2 out of 5 stars  See all reviews (10 customer reviews)
• Amazon Best Sellers Rank: #243,962 in Books (See Top 100 in Books)

More About the Author

Bertrand Russell (1872 - 1970). Philosopher, mathematician, educational and sexual reformer, pacifist, prolific letter writer, author and columnist, Bertrand Russell was one of the most influential and widely known intellectual figures of the twentieth century. In 1950 he was awarded the Noble Prize for Literature in 1950 for his extensive contributions to world literature and for his "rationality and humanity, as a fearless champion of free speech and free thought in the West."

Customer Reviews

Most Helpful Customer Reviews

107 of 109 people found the following review helpful:

3.0 out of 5 stars Substantial effort required. Careful reading necessary., October 1, 2003

By 

Michael Wischmeyer (Houston, Texas) - See all my reviews
(VINE VOICE)    (REAL NAME)   

This review is from: Introduction to Mathematical Philosophy (Paperback)

Bertrand Russell and Alfred North Whitehead created the monumental work Principia Mathematica (1910-1913), the ambitious and comprehensive effort to provide a detailed reduction of the whole of mathematics to logic. In 1919 Russell was jailed for antiwar protests and while in prison he wrote Introduction to Mathematical Philosophy, a seminal work in the field for more than 70 years.

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.

67 of 74 people found the following review helpful:

3.0 out of 5 stars A very dated and one-sided introduction to the subject, July 11, 1999

By A Customer

This review is from: Introduction to Mathematical Philosophy (Paperback)

This book is important for revealing Russell's views, at a certain point in his career, on the philosphies of mathematics and logic. But it says little on other philosophical viewpoints (even if only to criticise them). It might be better titled now 'Introduction to a Mathematical Philosophy (Called Logicism)'. We can hardly blame Russell for not knowing about the later developments of the subject (especially Godel), but it is worth bearing in mind that the book was written before some very important discoveries.

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.

28 of 29 people found the following review helpful:

5.0 out of 5 stars A very accessible mathematical classic, September 29, 1998

By A Customer

This review is from: Introduction to Mathematical Philosophy (Paperback)

An excellent and lucid exposition of what we really mean when we talk about 2 houses, or 1/2 an hour, or square root of 2 meters, or that the counting numbers are infinite. It does not require any prior mathematical knowledge beyond the basics, although it probably will be of interest only to those that care about math at its most abstract. It is fascinating to realize how much we take for granted when we do math and how much ingenuity it takes to pin down the concept of number. Highly recommended.