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Title: Introduction to Mathematical Philosophy by Bertrand Arthur Russell ISBN: 0-671-20927-2 Publisher: Simon & Schuster Books Pub. Date: May, 1971 Format: Paperback Volumes: 1 List Price(USD): $9.95 |
Average Customer Rating: 3.8 (5 reviews)
Rating: 3
Summary: Substantial effort required. Careful reading necessary.
Comment: 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.
Rating: 5
Summary: As all Russell's writings, a masterwork.
Comment: 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.
Rating: 3
Summary: A postcard from the past
Comment: 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.
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Title: The Principles of Mathematics by Bertrand Arthur Russell ISBN: 0393314049 Publisher: W.W. Norton & Company Pub. Date: February, 1996 List Price(USD): $17.95 |
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Title: The Problems of Philosophy by Bertrand Arthur Russell, John Perry ISBN: 019511552X Publisher: Oxford Press Pub. Date: September, 1998 List Price(USD): $10.95 |
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Title: On Formally Undecidable Propositions of Principia Mathematica and Related Systems by Kurt Gödel ISBN: 0486669807 Publisher: Dover Pubns Pub. Date: 01 April, 1992 List Price(USD): $6.95 |
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Title: Godel's Proof by Ernest Nagel, James Roy Newman, Douglas R. Hofstadter ISBN: 0814758169 Publisher: New York University Press Pub. Date: February, 2002 List Price(USD): $18.95 |
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Title: The History of Western Philosophy by Bertrand Russell ISBN: 0671201581 Publisher: Touchstone Books List Price(USD): $24.00 |
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