Product Description

INTRODUCTION TO THE SECOND EDITION THE Principles of Mathematics " was published in 1903, and most of it was written in 1900. In the subsequent years the subjects of which it treats have been widely discussed, and the technique of mathematical logic has been greatly improved ; while some new problems have arisen, some old ones have been solved, and others, though, they remain in a controversial condition, have taken on completely new forms. In these circumstances, it seemed useless to attempt to amend this or that, in the book, which no longer expresses my present views. Such interest as the book now possesses is historical, and consists in the fact that it represents a certain stage in the development of its subject. I have therefore altered nothing, but shall endeavour, in this Introduction, to say in what respects 1 adhere to the opinions which it expresses, and in what other respects subsequent research seems to me to have shown them to be erroneous. The fundamental thesis of th

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Classic        Rating:

If you are looking for a book to curl-up with for a few years, this is it.

Excellent Introduction to Mathematics and its Conceptual Structure        Rating:

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.

Since you cannot really get a sense of this book because there is no listing of table of contents or excerpt, etc. I though I would post 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        Rating:

I don't have much to say beyond what I would say about Russell: a clear writer but nothing sweeping philisophically appears here.

Classic        Rating:

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.

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