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Principles of Mathematics
A precursor to Russell’s famous Principa Mathematica, this is one of the most original and comprehensive treatises on the logical foundation of mathematics available today.
602 pages, Kindle Edition
First published January 1, 1903
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2676 people want to read
About the author
Bertrand Russell
1,152 books7,214 followersBertrand Arthur William Russell, 3rd Earl Russell, OM, FRS, was a Welsh philosopher, historian, logician, mathematician, advocate for social reform, pacifist, and prominent rationalist. Although he was usually regarded as English, as he spent the majority of his life in England, he was born in Wales, where he also died.
He was awarded the Nobel Prize in Literature in 1950 "in recognition of his varied and significant writings in which he champions humanitarian ideals and freedom of thought."
He was awarded the Nobel Prize in Literature in 1950 "in recognition of his varied and significant writings in which he champions humanitarian ideals and freedom of thought."
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Displaying 1 - 15 of 15 reviews
June 18, 2009
Despite its title, this is NOT a math book, at least in the conventional definition of the term. It is indeed true that the subject matter of the book is indeed mathematics, but it neither teaches the reader any math nor assumes that the reader knows much math. At first glance, it seems to explore the question "What is mathematical knowledge?". At a deeper level, however, this is a book about philosophy, specifically epistemology. What is knowledge and how is it different from mere belief, and what is the nature of truth? Bertrand Russell, the famed philosopher, tries to argue that mathematics is objectively true, and that by extension it is possible for a belief to be objectively true.
April 26, 2016
One of the most comprehensive works on logic ever written. Synthesizes many key principles of classical logic and adds new ones that are extremely innovative and useful. Successfully demonstrates the logical nature of language and the universe and how it translates into the symbolic realm of mathematics. Very useful for humanities students looking to ensure soundness in their rhetoric and provides a good framework to critique arguments.
February 28, 2017
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.
Want to read
August 14, 2007has anybody read this??
July 7, 2022
Excellent book
July 14, 2021
This is Russell's attempt to reach a wider audience to explain his and A.N. Whitehead's work "Principia Mathematica" a three-volume work trying to ground mathematics in symbolic logic as a foundation. I own a used set of those three volumes and it requires patience that I don't have to go through the detailed logic and reams of propositions. I think this is true of most people so Russell wrote a more popular work which if you know some college math isn't too bad and it sketches in under six hundred pages in one volume the program behind "Principia Mathematica". I recommend this to other, close to normal, people like myself if someone wants to get at Russell and Whitehead were both trying to accomplish.
November 12, 2024
"Principia Mathematica" is a three-volume work on the foundations of mathematics, written by Alfred North Whitehead and Bertrand Russell, and published between 1910 and 1913.
It is one of the most important works in mathematical logic and philosophy.
Key Points:
Purpose: The primary aim of Principia Mathematica is to derive all mathematical truths from a set of axioms and inference rules using symbolic logic. The authors sought to show that mathematics could be reduced to logic, a viewpoint known as logicism.
Structure: The work is divided into three volumes:
Volume I covers the basics of logic, including propositional logic and the theory of classes.
Volume II deals with cardinal numbers, ordinal numbers, and relations.
Volume III focuses on more advanced topics, such as real numbers and measure theory.
Notation and Symbols: The book uses a complex system of notation to express logical and mathematical concepts. This notation has influenced the development of symbolic logic.
Influence: Principia Mathematica has had a profound impact on the fields of logic, mathematics, and the philosophy of mathematics. It laid the groundwork for subsequent developments in formal logic and computer science.
The work is notable for its rigor and precision, though it is also famously difficult to read due to its dense and technical nature. Despite this, it remains a landmark achievement in the field of mathematical logic.
By Inti, Lake Titicaca Perú
It is one of the most important works in mathematical logic and philosophy.
Key Points:
Purpose: The primary aim of Principia Mathematica is to derive all mathematical truths from a set of axioms and inference rules using symbolic logic. The authors sought to show that mathematics could be reduced to logic, a viewpoint known as logicism.
Structure: The work is divided into three volumes:
Volume I covers the basics of logic, including propositional logic and the theory of classes.
Volume II deals with cardinal numbers, ordinal numbers, and relations.
Volume III focuses on more advanced topics, such as real numbers and measure theory.
Notation and Symbols: The book uses a complex system of notation to express logical and mathematical concepts. This notation has influenced the development of symbolic logic.
Influence: Principia Mathematica has had a profound impact on the fields of logic, mathematics, and the philosophy of mathematics. It laid the groundwork for subsequent developments in formal logic and computer science.
The work is notable for its rigor and precision, though it is also famously difficult to read due to its dense and technical nature. Despite this, it remains a landmark achievement in the field of mathematical logic.
By Inti, Lake Titicaca Perú
October 13, 2024
ARE MATHEMATICS AND LOGIC “IDENTICAL”?
Bertrand Arthur William Russell (1872-1970) was an influential British philosopher, logician, mathematician, and political activist. In 1950, he was awarded the Nobel Prize in Literature, in recognition of his many books.
He wrote in the Introduction to the second edition, “The fundamental thesis of the following pages, that mathematics and logic are identical, is one which I have never since seen any reason to modify. This thesis was, at first, unpopular, because logic is traditionally associated with philosophy and Aristotle, so that mathematicians felt it to be none of their business, and those who considered themselves logicians resented being asked to master a new and rather difficult mathematical technique. But such feelings would have had no lasting influence if they had been unable to find support in more serious reasons for doubt. These reasons are… first, that there are certain unsolved difficulties in mathematical logic, which make it appear less certain than mathematics is believed to be; and secondly that, if the logical basis of mathematics is accepted, it … tends to justify, much work, such as that of Georg Cantor, which is viewed with suspicion by many mathematicians on account of the unsolved paradoxes which it shares with logic.”
He continues, “logic aims at independence of empirical fact, and the existence of the universe is an empirical fact… In practice, a good deal of mathematics is possible without assuming the existence of anything… The doctrines of Pythagoras… influenced all subsequent philosophy and mathematics more profoundly than is generally realized. Numbers were immutable and eternal, like the heavenly bodies… the science of numbers was the key to the universe. The last of these beliefs has misled mathematicians and the Board of Education down to the present day. Consequently, to say that numbers are symbols which mean nothing appears as a horrible form of atheism.”
He concludes this Introduction, “There are still many controversial questions in mathematical logic which, in the above pages, I have made no attempt to solve… Broadly speaking, I still think this book is in the right where it disagrees with what had been previously held, but where it agrees with older theories it is apt to be wrong… the technical advances of mathematical logic… have simplified the apparatus of primitive ideas and propositions, and have swept away many apparent entities, such as classes, points, and instants. Broadly, the result is an outlook which is less Platonic, or less realist in the mediaeval sense of the word. How far it remains to go in the direction of nonrealism remains, to my mind, an unsolved question, but one which, whether completely soluble of not, can only be adequately investigated by means of mathematical logic.”
He outlines the famous contradiction he discovered: “the class of all classes is a class… Do all the classes that have this property form a class? If so, is it as a member of itself as many or not?” (§101) He continues, “A natural suggestion for escaping the contradiction would be to demur to the notion of ALL terms or of ALL classes. It might be urged that no such sum-total is conceivable… But… if this view were maintained against ANY term, all formal truth would be impossible, and Mathematics…would be abolished at one stroke. Thus the correct statement of formal truths requires the notion of ANY term or EVERY term, but not the collective notion of ALL terms.
"It should be observed… that no peculiar philosophy is involved in the above contradiction, which … can only be solved by abandoning some common-sense assumption. Only the Hegelian philosophy, which nourishes itself on contradictions can remain indifferent, because it finds similar problems everywhere.” (§105)
Much later, he makes the interesting statement, “As regards proof of mathematical induction, it is to be observed that it makes the practically equivalent assumption that numbers form the chain of one of them. Either can be deduced from the other, and the choice as to which is to be an axiom, which a theorem, is mainly a matter of taste.” (§241)
He begins Chapter XLI with the statement, “We have now completed our summary review of what mathematics has to say concerning the continuous, the infinite, and the infinitesimal. And here, if no previous philosophers had treated of these topics, we might leave the discussion, and apply our doctrine to space and time. For I hold the paradoxical opinion that what can be mathematically demonstrated is true. As, however, almost all philosophers disagree with this opinion, and as many have written elaborate arguments in favor of views different from those above expounded, it will be necessary to examine controversially the principle types of opposing theories, and to defend, so far as possible, the points in which I differ from standard writers.” (§315)
He concludes chapter LI with the statement, “I conclude… that absolute position is not logically inadmissible, and that a space composed of points is not self-contradictory. The difficulties which used to be found in the nature or infinity depended upon adherence to one definite axiom, namely, that a whole must have more terms than a part; those in the nature of space, on the other hand, seem to have been derived almost exclusively from general logic. With a subject-predicate theory of judgment, space necessarily appears to involve contradictions; but when once the irreducible nature of relational propositions is admitted, all the supposed difficulties vanish like smoke. There is no reason, therefore… to deny the ultimate and absolute philosophical validity of a theory of geometry which regards space as composed of points, and not as a mere assemblage of relations between non-spatial terms.” (§431)
For those interested in advanced mathematical theory, this is one of the “key” books to study.
Bertrand Arthur William Russell (1872-1970) was an influential British philosopher, logician, mathematician, and political activist. In 1950, he was awarded the Nobel Prize in Literature, in recognition of his many books.
He wrote in the Introduction to the second edition, “The fundamental thesis of the following pages, that mathematics and logic are identical, is one which I have never since seen any reason to modify. This thesis was, at first, unpopular, because logic is traditionally associated with philosophy and Aristotle, so that mathematicians felt it to be none of their business, and those who considered themselves logicians resented being asked to master a new and rather difficult mathematical technique. But such feelings would have had no lasting influence if they had been unable to find support in more serious reasons for doubt. These reasons are… first, that there are certain unsolved difficulties in mathematical logic, which make it appear less certain than mathematics is believed to be; and secondly that, if the logical basis of mathematics is accepted, it … tends to justify, much work, such as that of Georg Cantor, which is viewed with suspicion by many mathematicians on account of the unsolved paradoxes which it shares with logic.”
He continues, “logic aims at independence of empirical fact, and the existence of the universe is an empirical fact… In practice, a good deal of mathematics is possible without assuming the existence of anything… The doctrines of Pythagoras… influenced all subsequent philosophy and mathematics more profoundly than is generally realized. Numbers were immutable and eternal, like the heavenly bodies… the science of numbers was the key to the universe. The last of these beliefs has misled mathematicians and the Board of Education down to the present day. Consequently, to say that numbers are symbols which mean nothing appears as a horrible form of atheism.”
He concludes this Introduction, “There are still many controversial questions in mathematical logic which, in the above pages, I have made no attempt to solve… Broadly speaking, I still think this book is in the right where it disagrees with what had been previously held, but where it agrees with older theories it is apt to be wrong… the technical advances of mathematical logic… have simplified the apparatus of primitive ideas and propositions, and have swept away many apparent entities, such as classes, points, and instants. Broadly, the result is an outlook which is less Platonic, or less realist in the mediaeval sense of the word. How far it remains to go in the direction of nonrealism remains, to my mind, an unsolved question, but one which, whether completely soluble of not, can only be adequately investigated by means of mathematical logic.”
He outlines the famous contradiction he discovered: “the class of all classes is a class… Do all the classes that have this property form a class? If so, is it as a member of itself as many or not?” (§101) He continues, “A natural suggestion for escaping the contradiction would be to demur to the notion of ALL terms or of ALL classes. It might be urged that no such sum-total is conceivable… But… if this view were maintained against ANY term, all formal truth would be impossible, and Mathematics…would be abolished at one stroke. Thus the correct statement of formal truths requires the notion of ANY term or EVERY term, but not the collective notion of ALL terms.
"It should be observed… that no peculiar philosophy is involved in the above contradiction, which … can only be solved by abandoning some common-sense assumption. Only the Hegelian philosophy, which nourishes itself on contradictions can remain indifferent, because it finds similar problems everywhere.” (§105)
Much later, he makes the interesting statement, “As regards proof of mathematical induction, it is to be observed that it makes the practically equivalent assumption that numbers form the chain of one of them. Either can be deduced from the other, and the choice as to which is to be an axiom, which a theorem, is mainly a matter of taste.” (§241)
He begins Chapter XLI with the statement, “We have now completed our summary review of what mathematics has to say concerning the continuous, the infinite, and the infinitesimal. And here, if no previous philosophers had treated of these topics, we might leave the discussion, and apply our doctrine to space and time. For I hold the paradoxical opinion that what can be mathematically demonstrated is true. As, however, almost all philosophers disagree with this opinion, and as many have written elaborate arguments in favor of views different from those above expounded, it will be necessary to examine controversially the principle types of opposing theories, and to defend, so far as possible, the points in which I differ from standard writers.” (§315)
He concludes chapter LI with the statement, “I conclude… that absolute position is not logically inadmissible, and that a space composed of points is not self-contradictory. The difficulties which used to be found in the nature or infinity depended upon adherence to one definite axiom, namely, that a whole must have more terms than a part; those in the nature of space, on the other hand, seem to have been derived almost exclusively from general logic. With a subject-predicate theory of judgment, space necessarily appears to involve contradictions; but when once the irreducible nature of relational propositions is admitted, all the supposed difficulties vanish like smoke. There is no reason, therefore… to deny the ultimate and absolute philosophical validity of a theory of geometry which regards space as composed of points, and not as a mere assemblage of relations between non-spatial terms.” (§431)
For those interested in advanced mathematical theory, this is one of the “key” books to study.
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February 11, 2020There are more suggestions
Currently reading
October 7, 2024Did I read this whole 500-page book at one point?… hrm.
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February 7, 2017Meu conhecimento científico é quase nenhum. Mas lí, claro, a Lógica da Pesquisa Científica, de Karl Popper, quando entendi o que esses cabras querem. Para quem quer um começo apenas, recomendo o prefácio do Novum Organum, de Francis Bacon, que quer dizer, o título, novo instrumento, e Bacon explica o método científico e o que objetiva a ciência. E para complementá-lo leia o prefácio dos Os Princípios Matemáticos da Filosofia Natural, de Isaac Newton, e o prefácio de Bertrand Russell e Alfred North Whitehead de seus Princípios da Matemática. Também vale a pena ler a História da Filosofia Ocidental, de Bertrand Russell, e o capítulo sobre Positivismo Lógico, que é a filosofia calcada no conhecimento científico. Em resumo, tudo que pode ser provado lógica e matematicamente, é filosofia. O resto não é. Acho isso perfeitamente aceitável. Dispenso o resto.
This entire review has been hidden because of spoilers.
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September 7, 2015This method is, to define as the number of a class the class of all classes similar to the given class. Membership of this class of classes (considered as a predicate) is a common property of all the similar classes and of no others; moreover every class of the set of similar classes has to the set of a relation which it has to nothing else, and which every class has to its own set. Thus the conditions are completely fulfilled by this class of classes, and it has the merit of being determinate when a class is given, and of being different for two classes which are not similar. This, then, is an irreproachable definition of the number of a class in purely logical terms.
to-keep-reference
October 18, 2016Ampliamente discutido en Gödel, Escher, Bach: Un eterno y grácil bucle
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April 11, 2017This book is very interesting. However, there are places where it is confused and uneven. It was written extremely quickly, and Russell changed his ideas over the course of the writing, and afterwards when he read Frege.
An important book in the history of philosophy and well worth reading, although it is much better with a competent guide in the field.
An important book in the history of philosophy and well worth reading, although it is much better with a competent guide in the field.
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