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The Principles of Mathematics

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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 & that mathematics is, so to speak, the most abstract of empirical sciences, but empirical nonetheless. Though he lacked the sophistication at the time to propose a different philosophy of 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 come 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 into 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" (now in the Collected Papers). 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, the propositional calculus and the predicate calculus. Of course, for Russell, the notion of class is a purely logical notion which is defined intensionally, by the comprehension axiom, rather than extensionally, by the enumeration of its members. This means that a class can be determined solely by a property which all of its members share. For example, the property of being blue determines the class of all blue things. The view that every property determines a class is what leads to Russell's paradox (more on this below).

Indeed, the book not only presents these developments, argues for them and introduces the reader to the 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. As promised, he shows that Peano's primitives in the Peano-Dedekind axioms: zero, number & succesor, can be defined in purely logical terms (according to his view of logic which is not philosophically neutral). He gives a definition of cardinal number in terms of one-one relations betwen classes. Indeed, a cardinal number is just the number of a class of similar classes, that is, the number of a class is the class of all classes similar to the given class & two classes are similar if and only if there is a one-one bijection beween their members. For instance, the number '2' is the class of all couples and the term 'couple' can be further analysed through quantification & identity (thus the definition is not circular). With this, & Peano's axioms, he gets the natural numbers & 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 problem of the unity of the proposition, as well as perennial difficulties in the philosophy of language, rear their heads. The chapter on "The Philosophy of the Infinite" is a tour de force for anyone interested in the philosophy of mathematics. Zeno's paradoxes are discussed with the new methods, yielding valuable insights. Russell even engages in a brief, yet sophisticated, discussion of the philosophy of matter towards the end of the book.

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 all that different from that of Frege in "Sense and Reference", though less sophisticated, still capable on its own of dealing with definite & indefinite descriptions via the use of the denoting concept (this gives the theory enough resources to deal w both impossible & non-existent objects). 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. Given Russell's assumption that every property determines a class, one might ask, what of the property of not being a member of itself, a property which some classes have, like the class of humans, it is not a human & therefore not a member of itself. But then, what of the class of all those classes which are not members of themselves? If the class is a member of itself, then it is not. But if it is not, then it is, the class is a member of itself if and only if it is not a member of itself. This paradox puts the entire philosophical project at risk, Frege would respond to it by saying that the only possible foundation of mathematics has been shattered. Indeed, a sketch of Russell's theory of types, his eventual solution to the paradox, also makes an appearance in one of the books appendix's.

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. Probably the first philosophical discussion of Frege's work in the English-speaking world.

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 masterpiece would then be wholly perfect, I am inclined to agree.

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.

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.

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

When an editor simply republish something out of copyright protection, the only thing he has to do is to copy the pages, all the pages. If he just do that the client will be happy. That editor was incapable of doing that simple thing. Some pages from Bertrand Russell's book are *missing*. This is simply infuriating and absolutely inexcusable. To add insult to injury, the font is so degraded as to make the reading difficult. A double rip-off.

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

Russell is undoubtedly a brilliant mind and this sometimes goes with a cryptic expression... But here, the "sometimes" turns into "a lot".

If you are both patient and ready to skip the convoluted, if not indigestive sections, then you might safely reach the last page of this monumental work.

If not, then a better route is to use Russell's later "Introduction to Mathematical Philosophy", where he tried and succeeded in clarifying and correcting his thoughts in just 200 pages.

This is not to say that PM doesn't contain illuminating sections, it does but they are gems, lost in dense magma.

Dear reader,

The Principles of Mathematics is a classic of the literature of mathematical logic that should be read by those who nurture an interest in mathematics. The Forgotten Books Collection is readable and accessible.
Luiz Roberto Rosa
Itatiba - SP - Brazil

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.

200 pages missing on the book from 134 to 253 - Chapters XVI to XXX. Ashame! I would like to return and the policy said that it will be only paid half of the price + I have to assume shipping costs.
With the money I spend buying books in amazon, I think this is a poor solution. Janet