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Rating: 
Summary: A Hallmark in the History of Mathematics and Philosophy.
Review:
Much nonsense has been said on the subject of the importance of Principia Mathematica by people ignorant of the history of mathematics and logic. Principia Mathematica together with Frege's Grundgesetze der Arithmetik is the book which gives birth to modern logic. It is absurd to assume that Russell and Whitehead intended their axiomatization of mathematics as a guide to learn the subject, as one reviewer thinks, in fact what they tried to show was that the whole of mathematics could be deduced from a small stock of premises and inference rules and using only notions of first order logic and set theory. In doing this they were following a trend in mathematical thought in the late XIX century, that of introducing more rigour to the subject, they intended to do this by demonstrating that the derivation of mathematics needed only logic (think of Weierstrass, Dedekind, Cantor, Frege). From a philosophical standpoint they also did it to rebut the intuitionist views of Kant and Poincare as well as certain opinions regarding truth coming from British Idealism (think of Bradley). Of course there are much more rigurous treatises on logic, but they would have been impossible without PM because PM was the first thorough treatment of this subject-matter and, indeed, the first book to use the modern day notation. As another reviewer pointed out, Godel's proof would've been impossible without Principia; someone first needed to show that you could reduce mathematics to logic to a great extent (Russell and Whitehead were aware that their treatment used certain axioms unprovable within the system, like the axiom of infinity, but were hopeful a solution would be found, Godel found it, it was a negative solution, there could be no complete system PM like). This book together with Frege's gave birth to modern logic, it gave a tremendous boost to research in set theory, it influenced the presentation of modern mathematics to the extent that every student has to learn about sets at the beginning of a mathematics course, it showed also the scope of the deductive powers of logic and axiomatic systems which made possible the revolution in computers and AI. It developed an influential and responsive philosophy of mathematics, perhaps the most influential of the XX century. In it Russell's superb theory of descriptions, a cornerstone in logic and philosophy, is applied with success. This theory is tremendously important in logic through its use of quantification to break up much more complex expressions revealing their true logical form. In philosophy it provided a theory which would prove immensely useful and important in epistemology, metaphysics and the philosophy of language. Russell's paradox ( regarding those sets of sets which are not members of themselves) is disposed through ramified type-theory, now obsolete in logic (though not in computer science), because, thanks to it, other ways to avoid the paradox were developed, think of Zermelo-Fraenkl or Ramsey's simple type theory. Carnap, Hilbert, Weiner, Ramsey, Quine, Wittgenstein, Turing, Tarski, Godel etc were, as thinkers, tremendously influenced by it. In short, this work is one of the greatest achievements in the history of thought, its importance for mathematics, logic, philosophy (linguistics also) and computer science is first rate, suffice to say that none of these studies would be as advanced as they are now, or as complex, or in the same direction were it not for Russell and Whitehead's groundbreaking scientific work. Of course, like Newton's Philosophia Naturalis Principia Mathematica it is now, because the subjects it initiated are today tremendously advanced, mostly of historical interest, however, for the philosophers at least, Russell's introduction still holds great philosophical interest and rigourous arguments helpful in the contemporary debate. For more details check out Ivor Grattan Guiness's great works on the history of mathematics, logic and set theory.
Rating:
Summary: If you don't know know this book then you don't need it
Review: Let me try to give a balanced review. First this is a monumental work and one of the most influential works of the 20th century. I am not giving it five stars: this book earned them. With that said I don't think is the most influential book of the 20th century because such a book doesn't exist. In my opinion that kind of debate is totally misleading. However the five stars do not suggest that you should buy this book. With the exception of libraries and scholars specializing in Russell or related subjects, I can't see anybody else spending [this amount] on a copy of this work. That is unless they like to collect books. For a math or philosophy student the paperback copy to *56 is all you need. Unless you are a mathematician, a logician or a philosopher with a strong background in logic and philosophy of mathematics and aware of the issues surrounding the problems in the foundations of mathematics at the beginning of the 20th century then you are not going to benefit from STUDYING this book. The emphasis in studying is important because this book needs to be studied not just read like some reviewers may suggest. If you are not an expert in this area and you want to learn about the subject then you may want to start with Bertrand Russell's "Introduction to Mathematical Philosophy". It summarizes the major points of this work for the layman and is Russell at its best (he won a Nobel prize mostly due to this book). Read it with a critical mind and then you can continue reading Quine, Putnam, Brower, Heyting and the rest. You can get a good bibliography from Benacerraf and Putnam's "Philosophy of Mathematics". Finally if you are a mathematician, a logician or a philosopher you already know about this book and you don't need this review. Moreover you know you can borrow a copy from the university library for study...that is unless you like to collect books.
Rating:
Summary: A spoiler!
Review: The denouement in which we discover that the Vicar was murdered by the Butler, in the Conservatory, with a Candlestick was weak. But the sex scenes, on pages 183 - 879 were the most sensitive yet erotic that I have ever read (except for page 1334 of the "Catalogue of Insects, Arachnids and Marsupials vol XXIV"). Top work, Whitehead and Russell! I eagerly await volume 4.
Rating:
Summary: A spoiler!
Review: The denouement in which we discover that the Vicar was murdered by the Butler, in the Conservatory, with a Candlestick was weak. But the sex scenes, on pages 183 - 879 were the most sensitive yet erotic that I have ever read (except for page 1334 of the "Catalogue of Insects, Arachnids and Marsupials vol XXIV"). Top work, Whitehead and Russell! I eagerly await volume 4.
Rating: 
Summary: Mostly of historical interest
Review: The notation of PM is hard to read by anyone who learned logic post 1960, say. The typesetting is archaic. Hundreds of theorems are proved, but it is not clear where
they all lead. Russell and Whitehead are guilty of a number of major philosophical confusions, such as use and mention, between meta- and object language, and their confused notion of "propositional function." Their choice of axioms can be much improved upon. The PM theory of types and orders is a complicated horror; Chwistek, Ramsey, and others later showed that it could be radically simplified. R & W think they can substitute the intensional for the extensional, and ultimately define sets and relations in logical terms. PM does not have a clue about model theory or metatheory. There is no hint of proofs of consistency, completeness, categoricity, and Loewenheim-Skolem. In this sense, the fathers of modern logic are Skolem, Goedel, Tarski, and Church. And Goedel did indeed prove that there must exist mathematical truths that cannot be proved true using the axioms of PM, or any other finite set of axioms. But this is still one of the greatest works of mathematics and philosophy of all time. The long prose introduction is a philosophical masterpiece. The collaboration between Russell and Whitehead may be the greatest scientific collaboration in British history. Whitehead, who was trained as a mathematician, went on to become one of the shrewder philosophers of the 20th century, and supervised Quine's PhD thesis. PM's treatment of the algebra of relations (a brilliant generalisation of Boolean algebra that
has not received the study it deserves) is perhaps the most thorough ever. Mathematical logic is indeed the abstract structure that underlies the digital electronics revolution. And PM is still perhaps the greatest work of math logic ever penned.
Rating:
Summary: A monument of mathematical logic
Review: This book is the ultimate attempt to derive all of mathematics from logic while avoiding paradoxes of the sort that Russell himself sprang on Frege--and in passing, it gives in rigorous symbolic form Russell's "theory of descriptions." Just as Bach took the baroque style of music about as far as it could go, Russell and Whitehead took this attempt to put mathematics on a firm logical footing about as far as it could go (and Goedel's incompleteness theorem killed off the hopes that mathematicians such as Hilbert had for the goal). Nevertheless, like any really good problem, it turned up worthwhile byproducts. Alas, my exposure to the full three-volume set is confined to time spent at a university library; I could only afford the paperback volume of the first fifty-six chapters. I hope to eventually buy a copy of this classic work in its entirety.
Rating: 
Summary: Useless book
Review: This is a terse review, but quite literally this is the greatest achievement of twentieth century logic and mathematics. Only reading it can compel one to understand it.
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