Customer Reviews


15 Reviews
5 star:
 (8)
4 star:
 (3)
3 star:
 (3)
2 star:    (0)
1 star:
 (1)
 
 
 
 
 
Average Customer Review
Share your thoughts with other customers
Create your own review
 
 

The most helpful favorable review
The most helpful critical review


36 of 37 people found the following review helpful
5.0 out of 5 stars A very accessible mathematical classic
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...
Published on September 29, 1998

versus
131 of 133 people found the following review helpful
3.0 out of 5 stars Substantial effort required. Careful reading necessary.
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...
Published on October 1, 2003 by Michael Wischmeyer


‹ Previous | 1 2 | Next ›
Most Helpful First | Newest First

131 of 133 people found the following review helpful
3.0 out of 5 stars Substantial effort required. Careful reading necessary., October 1, 2003
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.
Help other customers find the most helpful reviews 
Was this review helpful to you? Yes No


36 of 37 people found the following review helpful
5.0 out of 5 stars A very accessible mathematical classic, September 29, 1998
By A Customer
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.
Help other customers find the most helpful reviews 
Was this review helpful to you? Yes No


71 of 78 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 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.
Help other customers find the most helpful reviews 
Was this review helpful to you? Yes No


12 of 12 people found the following review helpful
4.0 out of 5 stars Good introduction To Mathematical Logic, July 8, 2005
By 
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
Help other customers find the most helpful reviews 
Was this review helpful to you? Yes No


7 of 7 people found the following review helpful
5.0 out of 5 stars Russell does an excellent job in describing the foundations of mathematics for the non-mathematician, November 29, 2008
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.
Help other customers find the most helpful reviews 
Was this review helpful to you? Yes No


27 of 34 people found the following review helpful
3.0 out of 5 stars A postcard from the past, April 12, 2002
By 
A. Fischer (Toronto, Ontario Canada) - See all my reviews
(REAL NAME)   
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.
Help other customers find the most helpful reviews 
Was this review helpful to you? Yes No


5 of 6 people found the following review helpful
4.0 out of 5 stars A Philosophy Reading Classic, September 10, 2005
By 
Erik G. (New York, NY) - See all my reviews
Verified Purchase(What's this?)
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.
Help other customers find the most helpful reviews 
Was this review helpful to you? Yes No


2 of 2 people found the following review helpful
4.0 out of 5 stars Dense made clear, December 30, 2013
"For the moment, I do not know how to define "tautology"....It would be easy to offer a definition which might seem satisfactory for a while; but I know of none that I feel to be satisfactory, in spite of feeling thoroughly familiar with the characteristic of which a definition is wanted. At this point, therefore, for the moment, we reach the frontier of knowledge on our backward journey into the logical foundations of mathematics." -Bertrand Russell (p.204)

So it is with modern math's struggle for ever-increasing rigor. I started reading this book in high school and quickly realized that I didn't have the math background to make sense of it.

Now after I've taken logic and math classes at the University of Chicago, I revisited the book. Published in 1919, Russell offers what is, by any account, an excellent attempt at building a working system of mathematics from the bare bones of logical deduction. This alone is fascinating, as I've heard that it was possible but didn't know what it would look like (there's some things they don't teach you, even at college).

Russell is an endless well of information on the fields of logic, math, and the gray areas between the two. He never fails to let you know where the current questions or problems in set theory or definitions or propositions are. A perfect example is in the statement and discussion surrounding two hypotheses of (I think) purely deductive set theory: multiplication and infinity. Both of them are not logically necessary, and yet these processes appear to function as if they are true. Intuitively, they should be true, yet we just don't know if they are!. This is the problem of proving things that have absolutely no grounding in the world as we know it, and Russell makes you aware of how pure mathematicians deal with these problems.

In 1919, Russell skirts around the edges of a key aspect of logic--one that Godel later proved in his incompleteness theorems in 1931--that is:

1) In a logically deduced system that is complex enough to perform math as we know it,
2) There are certain hypotheses that are true and can never be proved or disproved.

This is astounding, and Russell is right on the edge of predicting what I'm assuming is a long-conjectured, impossibly-strange aspect of logic. This particular concept blows my mind in a way a symbolic logic class never can. Russell covers these difficulties perfectly; he exemplifies intellectual honesty and objectivity.

This book is essentially an extended essay that condenses Russell's own Principles of Mathematics--no short work at 576-pages--and the gigantic, 3-volume, nearly 2,000 page Principia Mathematica co-authored with Whitehead. He does this in plain english, which is perhaps the book's greatest merit. One can see how reading this slim book in order to better understand Russell's philosophy is a huge savings of time and difficulty.

Russell is an excellent writer, especially when it comes to making impossibly dense subjects appear clear. That being said, this book is considerably dry at points. It is also way over my head but I think it's interesting and I learned a lot about the philosophy of higher mathematics.
Help other customers find the most helpful reviews 
Was this review helpful to you? Yes No


1 of 1 people found the following review helpful
1.0 out of 5 stars Not worth a penny, May 1, 2014
Verified Purchase(What's this?)
Like other math books I have tried from the kindle store, this one has egregious errors in transcribing any mathematical expressions. I am certain that nobody was paying any attention; all done by OCR and assumed to be correct. Amazon should be ashamed of offering such garbage for sale. For some of these books I would be willing to pay quite a bit more, if they were edited a little more. C
an you tell I am disgusted with this travesty?
Help other customers find the most helpful reviews 
Was this review helpful to you? Yes No


1 of 1 people found the following review helpful
5.0 out of 5 stars One Step Back, January 4, 2013
Verified Purchase(What's this?)
This book takes you back one or two steps in the understanding of math when definitions in the modern age just doesn't seem to define with clarity most concepts of math. This book takes you to a world of propositional logic that is of utter importance to grow as a mathematician. Bertrand Russell is one of a kind, you can easily see why he got a Nobel Prize in Literature. This book must be read slowly as to digest the concepts so that you can internalize them and put them to use in proofs in the modern way. Enjoy!!
Help other customers find the most helpful reviews 
Was this review helpful to you? Yes No


‹ Previous | 1 2 | Next ›
Most Helpful First | Newest First

Details

Introduction to Mathematical Philosophy (Classic Reprint)
Introduction to Mathematical Philosophy (Classic Reprint) by Bertrand Russell (Paperback - June 9, 2012)
$8.64 $7.78
In Stock
Add to cart Add to wishlist
Search these reviews only
Rate and Discover Movies
Send us feedback How can we make Amazon Customer Reviews better for you? Let us know here.