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The Principles Of Mathematics | 
 enlarge | Author: Bertrand Russell
Publisher: Merchant Books
Category: Book
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Sales Rank: 660283
Media: Paperback
Number Of Items: 1
Pages: 564
Shipping Weight (lbs): 2.1
Dimensions (in): 9.2 x 7.3 x 1
ISBN: 160386119X
Dewey Decimal Number: 511
EAN: 9781603861199
ASIN: 160386119X
Publication Date: June 21, 2008
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Product Description
An unabridged, digitally enlarged printing, with a comprehensive index.
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| Customer Reviews: Read 1 more reviews...
 Excellent Introduction to Mathematics and its Conceptual Structure June 23, 2008
This is an excellent introduction to the fundamental principles and the core concepts of mathematics. There is no need to be mathematically inclined or a mathematical specialist to gain significantly from reading this book. Serious students of mathematics, logic, intellectual history, or philosophy will also gain significantly from its lucid and sharp explanations, and Bertrand's ability to question and challenge and manipulate even the most presumed unchangeable fundamental categories of mathematics.
This book is cogently written and is for the serious student and reader (yet there is no new mathematical or logical symbol system that needs to be learned, like in his and A.N. Whitehead's Principia Mathematica). A consistent theme throughout is on the philosophical nature of mathematical knowledge.
Some of the topics and concepts covered:
Part I - The Indefinables of Mathematics
Pure Mathematics
Symbolic Logic [includes propositional logic, calculus of classes, calculus of relations, and Peano's symbolic logic]
Implication and Formal Implication
Proper Names, Adjectives and Verbs
Denoting
Classes
Propositional Functions
The Variable
Relations
The Contradiction
Part II - Number
Definition of Cardinal Numbers
Addition and Multiplication
Finite and Infinite
Theory of Finite Numbers
Addition of Terms and Addition of Classes
Whole and Part
Infinite Wholes
Ratios and Fractions
Part III - Quantity
The Meaning of Magnitude
The Range of Quantity
Numbers as Expressing Magnitude: Measurement
Zero
Infinity, the Infinitesimal, and Continuity
Part IV - Order
The Genesis of Series
The Meaning of Order
Asymmetrical Relations
Difference of Sense and Difference of Sign
On the Difference between Open and Closed Series
Progressions and Ordinal Numbers
Dedekind's Theory of Number
Distance
Part V - Infinity and Continuity
The Correlation of Series
Real Numbers
Limits and Irrational Numbers [includes Weiserstrass's theory and Cantor's theory]
Cantor's First Definition of Continuity
Ordinal Continuity
Transfinite Cardinals
Transfinite Ordinals
The Infinitesimal Calculus
The Infinitesimal and the Improper Infinite
Philosophical Arguments Concerning the Infinitesimal
The Philosophy of the Continuum
The Philosophy of the Infinite
Part VI - Space
Dimensions and Complex Numbers
Projective Geometry
Descriptive Geometry
Metrical Geometry
Relation of Metrical to Projective and Descriptive Geometry
Definitions of Various Spaces
The Continuity of Space
Logical Arguments Against Points
Kant's Theory of Space
Part VII - Matter and Motion
Motion
Causality
Definition of a Dynamical World
Newton's Laws of Motion [discusses also causality in dynamics]
Absolute and Relative Motion
Hertz's Dynamics
Appendix A
The Logical and Arithmetical Doctrines of Frege
Appendix B
The Doctrine of Types
 An interesting read after the Principia November 5, 2006
1 out of 3 found this review helpful
I don't have much to say beyond what I would say about Russell: a clear writer but nothing sweeping philisophically appears here.
Classic March 1, 2006
5 out of 7 found this review helpful
Russell was a keen and original thinker. He and Whitehead wrote the Principia in an attempt to explain mathematics in terms of logic and put it on a firm logical basis. This was proved impossible by Godel later in the century. This book gives Russell's definitions and thinking on the subject, and discusses Frege and Cantor and Dekind and Hilbert and their approaches to mathematics and number system. I find the book historically
interesting, but I am not qualified to criticize the mathematics
or axioms proposed in the volume.
Russell's Magnum Opus October 10, 2005
27 out of 29 found this review helpful
Bertrand Russell's greatest pieces of philosophical writing could probably be said to be "The Principles of Mathematics", "On Denoting" and with Alfred North Whitehead "Principia Mathematica", there is however one sense in which it could be said that the russellian magnum opus is The Principles of Mathematics, from here on TPM.
TPM is, arguably, the culmination in print of a long process of thought and concern, philosophically speaking, of Russell's intellectual preoccupations from his adolescence, youth and maturity with questions relating to the foundations of mathematics. Ever since Russell read Mill in his adolescence he had thought there was something suspect with the millian view that mathematical knowledge is in some sense empirical. Though he lacked the sophistication at the time to propose a different view of the foundations in mathematics his concerns with these topics remained with him well into the completion of Principia Mathematica. Logic and Mathematics were, by that time, seen as separate subjects dealing with distinct subject-matters, it came to be, however, the intuition of Russell (an intuition shared, and indeed, anticipated by Frege) that mathematics was nothing more than the later stages of logic. He did not came into this view easily, after a long period of hegelianism and kantianism in philosophy, in which Russell sought to overcome the so called antinomies of the infinite and the infinitesimal, etc; Russell saw light coming, not from the works of philosophers, but from the work of mathematicians working to introduce rigour in mathematics. Through the developments introduced by such mathematicians as Cantor and Dedekind Russell saw, or indeed thought he saw, that the difficulties in the notion of infinite and infinitesimal could be dealt with by solely mathematical methods, and it was through the continued development of formal logic by Peano and his followers that Russell saw the possibility of defining the notions of zero, number & successor in purely logical terms. Before all of these developments and ideas were put together by Russell and developed into the philosophy of mathematics known as logicism he made several sophisticated though unsuccesful attempts at questions having to do with the foundations of mathematics, one such attempt is his "An Analysis of Mathematical Reasoning". In TPM all of these developments are taken together with the formal logic Russell was developing following the steps of Peano, indeed the TRUE foundations of mathematics are for Russell: the calculus of classes (Set Theory), the propositional calculus and the predicate calculus (first-order classical logic).
And indeed the book not only presents these developments, argues for them and introduces the reader to the whole theoretical and philosophical edifice of formal logic, but also with these tools Russell delves in an exploration of all or most concepts relevant in the mathematics of the day. He shows that with the methods he proposes he can construct the whole of the real numbers, and that the concept of infinity can be dealt with through the set-theory of Cantor. Russell's theory of relations, a theory which made possible to deal with relations in formal logic as well as to refute the metaphysical views of Bradley and others, appears in the book. The chapter on "The Philosophy of the Infinite" is a tour de force for anyone interested in the philosophy of mathematics. This book is quite long, but it is absolutely breathtaking in its depth, its subtle arguments, its sophistication and originality (for its time). The book already contains a philosophy of language and reference not very different from that of Frege in "Sense and Reference". As I said, it is thorough in its philosophical examination and explanation of mathematical concepts, and it delves into physics through the russellian investigation of space and time, as well as his incorporation of logicism into physics through rational dynamics. Russell's paradox makes its first appeareance in this book, it has a chapter to itself. And indeed, Russell's theory of types also makes an appearance in an earlier form in one of its appendix. It is well known that Russell and Frege each came to his views independently, and indeed Russell had just read Frege by the time his book had been finished and so added another appendix discussing and commending Frege's work. All in all, this book is worth every penny, it is one of the masterpieces of XX century philosophy by any standards. One professor of mine once remarked that if Russell had developed his famous theory of descriptions by the time he wrote TPM and had included it in the book, the already master piece would then be wholly perfect, I am inclined to agree.
Spliting Hairs Infinitesimally May 7, 2003
25 out of 27 found this review helpful
He doesn't do much theorem proving, but he tackles
head on all the basic problem of mathematics that were known
a hundred years ago. It was how well he did everything
that makes this still a must read if you love mathematics.
There is actually only one equation in his book that I can think of:
and it is of a Clifford geometry measure! This man was a mathematician's
mathematician and a metamathematics master in the language of
philosophy as well! The pages are falling out and I still
go to this and Sommerville when I want inspiration or understanding of really hard issues.
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