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Introduction to Mathematical Philosophy

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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.

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.

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.

Bertand Russell's "Introduction to Mathematical Philosophy" provides the reader with a great understanding of mathematical philosophy in a very simple and straightforward manner. Though this is an introductory work it may not be casual reading to all who endeavor to read it. Beginning with definition of numbers and sets it expands to provide definitions of simple and complex and builds to provide a good understanding of the logic behind mathematics. While much of what is spoken about may seem very elementary the logic behind certainly is not. While the book is not nearly as expansive ad "Principia Mathematica" it is a good distillation of the bigger work and provides a great introduction to anyone wishing to explore that work. I recommend this book to anyone interested in formal logic and believe that it should be in the required reading for any formal logic introductory class. Further anyone interested in reading Goedel's work's which expand on Russell's work needs at least to read this work prior to Goedel. I find this book to be very succinct and readable and ultimately very worthy of the effort it takes to read.

-- Ted Murena

Two of the very first courses I took in graduate school were in the foundations of mathematics, a decision that I have repeatedly praised myself for since. By learning the basic structure of mathematics, it was much easier to understand what came later. In this book, Bertrand Russell, one of the giants of mathematical philosophy, writes about the subject for a general audience.
Russell, known in mathematical circles more for his giant work "Principia Mathematica" co-authored by Alfred North Whitehead, does an excellent job in describing the foundations of mathematics for the non-mathematician. It is a difficult task, as it is hard to describe mathematics without using mathematics. While there are some sections where Russell has no choice but to mention some higher-level mathematics, he does so only when necessary and explains it well. Most people with at least some exposure to mathematics will be able to understand it. There are no proofs in the book.
As a primer on many of the basic ideas of mathematics, this book is one of the best. Russell was also a great expository writer and he demonstrates that trait here.

Once upon a time, long long ago there was a group of people that believed that mathematics could be completely reduced to just a study of logic. One of the principal members of this group was Bertrand Russell (who along with Alfred North Whitehead wrote the almost incomprehendable Principia Mathematica). Jump ahead 20 years when there entered men like Godel who showed that the entire endevour was doomed for failure.

This is a text written before that fateful discovery, and as such does not have the benefit of the Incompleteness Theorem to flesh out the ideas. As such, most of the material is wanting, at best, to the contemporary reader of mathematics. Adding to this the fact that the communication of mathematical ideas has tremendously changed in the intervening years, and the result is a text that, though one day had great significance, today seems like a much faded phtotgraph from a by-gone era.

Maybe this makes the text interesting in itself. However, those readers that wish for a current look at mathematical thought, and an introduction to the philosophy of mathematics may be best served by looking elsewhere.

A great book by a great philosopher. Of course, much of the material was for its time advanced and revolutionary now it is more of a classic introductory text given a basic preparation in critical reading and basic mathematics to sufficiently appreciate the nuance of his thought.

Okay, I have to be honest- I was a little intrepid picking up this book, and it had nothing to do with Russell's math. I had this really dogmatic atheist friend who used to endlessly quote "Why I am not a Christian," and it put me off of Bertrand Russell.

This book is a joy. It's easy to read, interesting to think about, and inexpensive. Three virtues of math books that are hard to find in combination!

First, we have the mathematical text only, second, this text was written by a mortal to mortals although not for all mortals, maybe only for some hundreds, and I can begin to name them rereading first chapter the Series of Natural Numbers, pp.1-10: Euclid, Pythagoras, Peano, Frege, Russell, me (or you, or anybody else reading the same pages but existing for our design now nowhere).

In the first part of the book Veblen, Whitehead, Dedekind, and Cantor join us, further Leibnits, Weierstrass, Zermelo, Kant, Sheffer, Nicod, Occam, Laplace, and Wittgenstein in the last XVlll chapter Mathematics and Logic. Usual themes look like Definition of Number, The Definition of Order, Kinds of Relations, Rational, Real, And Complex Numbers, Limits And Continuity, Functions, Descriptions, Classes, and Index pp. 207-208. Guess how many people gave their ideas to write this book?--Not more than twenty.

One question of my child's curiosity led me some weeks through all chapters: how it could help to make intelligible my own problems of explaining of 80 (!!!) interlacing triangles inside of one circumference (people didn't see it before at all during last three millennium) of more difficult star polygons than Sri Yantra (43 triangles), and now I have a strange feeling that to my taste numbers cannot be only symbols coexisting in a text and in a mind but must also have the co-ordinates in the spaces (agreement with progression) and directions where they move or not either like matter, particles, ideas, or energy according to rhythms including or not category of time. The true god is geometry.

For those who have read this note to the end and think it was pure fun I have to say jazz, Chinese green tea, and chocolate Nestle helped me a lot:)

The highest point of mathematical philosophy is a total lacking of numbers but a total sense of humour instead. Maybe, a bit of Zen too ... a lot of

A magnificent, fantastic and very readable introduction to the highly abstract world of formal logic and the foundations of mathematics. Lord Russell is not only one of the greatest logicians of all time, but he is also an astonishing writer, blessed with an elegant, heavenly precise concision of style and sharpness of argumentation.