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ByMoisés Macías Bustoson October 10, 2005

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 XXth 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.

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 XXth 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.

8 people found this helpful

ByAndré Gargouraon October 24, 2011

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 indigestible 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.

If you are both patient and ready to skip the convoluted, if not indigestible 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.

ByMoisés Macías Bustoson October 10, 2005

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 XXth 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.

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 XXth 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.

Byjohn warrenon August 8, 2000

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.

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.

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ByWilliam J. Romanoson 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.

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

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

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ByRoger Bagulaon May 7, 2003

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.

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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ByAndré Gargouraon October 24, 2011

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 indigestible 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.

If you are both patient and ready to skip the convoluted, if not indigestible 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.

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Byadamon February 11, 2014

there are tons of typos, looks like the book was scanned and OCR'd and not further edited. it is super annoying, kinda like how typing on a kindle paper white is super annoying. jesus. None of the equations are readable at all... This review is for the kindle edition.

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ByRead and thinkon July 18, 2010

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.

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ByJonathan Carteron November 3, 2013

Was going to give this one star, but the content is good. This copy is based on a scanned book and looks like a photo copy with binding. Not very good quality.

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ByAmazon Customeron January 4, 2013

Only when you love math can you appreciate Bertrand Russell's insights on math. He has a particular way of thinking and writing it down in a way that makes his ideas come to life. This book is slow reading and void of much symbolism. It is very retorical, but it is overrided by the exquisite syntax. For those students, professionals, and aficionados of Pure Mathematics this is the way to master the proofs and definitions in a whole new light, even though Bertrand's book go back to the beggining of the 20th Century, it is a world of wisdom that won't fail you if you have a true desire to better your understanding of Math in its most needed depth.

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Byimmanuelon March 15, 2013

I was a liitle upset after reading theese comments, they all complained about the font and chapters missing, but i just recived my copy and its perfect everything in there and the font is perfect. If you want this book buy it...

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