Principles of Mathematics

by Bertrand Russell

Book Overview

His ideas have had a profound influence on twentieth-century work on logic and the foundations of mathematics.

• Format: Paperback
• Language: English
• ISBN: 0393314049
• ISBN13: 9780393314045
• Release Date: February 1996
• Publisher: W. W. Norton & Company
• Length: 576 Pages
• Weight: 1.23 lbs.
• Dimensions: 1.1" x 5.5" x 8.3"

Customer Reviews

Excellent Introduction to Mathematics and its Conceptual Structure

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

Classic

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

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 pro

Spliting Hairs Infinitesimally

He doesn't do much theorem proving, but he tackleshead on all the basic problem of mathematics that were knowna hundred years ago. It was how well he did everythingthat 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'smathematician and a metamathematics master in the language ofphilosophy as well! The pages are falling out and I stillgo to this and Sommerville when I want inspiration or understanding of really hard issues.

Dated, but still a gold mine.

10-Point Rating: (8.75)One of the claims of the analytical school of western philosophy is that math is reducible to logic, specifically the logic of groups, classes, or sets. In this vein, I can think of no better introduction than Russell's Principles of Mathematics. Although many of the ideas he proposes are intellectually outdated, Russell's method is rigorous and his presentation is lucid. While this book is not for everyone, no serious student of mathematical foundations should be without it. The chapters on zero and the concept of continuity are especially insightful.