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	Principia Mathematica to *56 (Cambridge Mathematical Library) Alfred North Whitehead, Bertrand Russell Cambridge University Press, 1997 - 460 pages average customer review:based on 13 reviews view larger image
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highly recommended

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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.

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A Hallmark in the History of Mathematics and Philosophy.


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 in the philosophy of mathematics. For more details, historical background and a well-documented account check out Ivor Grattan Guiness's great works on the history of mathematics, logic and set theory. For an appropiate and easy-going understanding of the scope and purpose of this work read Russell's brilliant "The Principles of Mathematics", his "Introduction to Mathematical Philosophy", or Frank Ramsey's papers on the "Foundations of Mathematics". Even easier is Penrose's account of it in his "The Emperor's New Mind" or his "Shadows of the Mind." If you want to see the direct influence of Russell and Whitehead's work check the works of Quine, Wittgenstein, Godel, Tarski or some of the papers of Turing in Mind (some are available online); van Heijenoort's "From Frege to Godel" is a superb sourcebook on papers which detail the development of mathematical logic.

Considering some statements from mathematicians arguing for the thesis of the irrelevance of the book based on the fact that probably no mathematician of notice has read the work in the last fifty or so years shows the misunderstandings to which people who dislike history are prone, and shows some contempt for the history of mathematics and logic. I am reminded of the comment I heard once, that the theories of the Milesians (all is water, etc) are absurd, a view which I am convinced would only be put forward by someone wholly indifferent to historical context, and who does not consider those theories as the first step towards the current scientific worldview. It is like saying that Bacon's methodology of science is irrelevant because we now have a deeper understanding of how science works, or even like saying that the study of the work of Adam Smith is worthless since for free-market economies we can now consider Hayek's or Milton Friedman's work. This analogy will, hopefully, show the preposterousness of views which do not consider the historical context of such major works. Indeed one does not need to review the proofs in PM (poor by modern standards) that 1 plus 1 equals 2, to understand the important place of this book in contemporary thought. It is only necessary to glance at any contemporary book on logic or set theory, most of the ideas there, the notation, and most developments in both disciplines in the past fifty or so years. Developments which are in debt of the work done by Zermelo, Hilbert, Quine, Turing, Weiner, Tarski, Godel etc, who, as anyone who has studied a bit of their works (as in authored by them) will know, owe their own ideas, developments and work to the study of Principia Mathematica during the first fifty years of the twentieth century. Indeed I would be the first to suggest that no one should read this book from cover to cover if one wants to learn logic (even Russell used to joke he only knew of a couple of poles who had read it and had then perished in WWII), just as I wouldn't suggest anyone interested in contemporary calculus and advanced mathematics to read Newton's Principia, or anyone interested in Set Theory to read Cantor's papers, or again, anyone interested in Einstein's special relativity to read his 1905 papers. In fact I cannot believe anyone would have to stress this point, but I am forced to on account of the various misunderstandings I see here, and by mathematicians, which one would presume would be the most rigurous of thinkers. These days the value of the book is mostly historical (with the introduction, mostly chapters II and III, having philosophical value), but, and I must once again stress this strongly, its tremendously influential and important place in the DEVELOPMENT of logic and set theory (and metatheory with the discovery of Russell's paradox) cannot be doubted, it can indeed be traced, if one takes the time to do so, to the various seminal thinkers it influenced strongly. Its value should be doubted even less by those academics ignorant of the history of their own disciplines not because they disagree with me (I could hardly be that vain) but rather because their misunderstandings are on par with disminishing Darwin's importance to contemporary biology on the grounds that his works are not cited in the bibliography of the most important papers written on the subject nowadays.

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Mostly of historical interest

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.

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Principia

I decided to write a review, because, when reading the existing ones,- I realized their incorrectness. Leaving out the "Customer from Christchurch New Zealand", the rest shows an evident shallowness of mind. The reader "La-la land" utilizes an enormous mass of epithets discrediting Russell and Whitehead, which could be valuable in a form, but instead,- he shows a stupid prejudice that must have learned in his Mathematical-logic "polytechnic" course. I will only refute his last thought( which is the base of his "thesis"), because the others refute themselves. He presents Russell as a "Fruitless Mathematician", and even more stupid, compares him with Hilbert, saying: " at least he proved himself worthy.....". Throughout all Mathematics history we have individuals with enormous logic-constructive aptitudes, who although creating fundamentals results, were unable to understand their significance. Two perfect examples are Newton and Leibniz, both creators of the "infinitesimal calculus". One went on to construct the modern mechanistic view of physics in his "Principia". The other, with a much more profound understanding of logic, a superficial "monadic-substantial" and teleological ontology. Newtonian physics was a major episode in modern science, and Leibniz "subject-predicate" logic is the first glance at mathematical-logic.But their incorrect understanding of the infinitesimal calculus made them see, in it, the proof of an omnipotent god: they both conceived a universe with its first cause as god, and the human aptitude is, within it, merely an "algorithmic" one, which could never fully calculate god's creation. Hilbert, also providing fundamental results in constructive knowledge, went on to expose a somewhat "Hegelian" conception of mathematics, giving an almost silly definition of numbers. Both of this errors cause enormous damage, which I don't have space to describe now. Russell's "Principia Mathematica", although written with the wrong "motivation"( that is: to reduce the whole of mathematics into axiomatic form, finding the "universal method"), achieved unquestionable logic-mathematical results: The most valuable and original, the "theory of descriptions". in an abridged explanation, these theory comprehends the next: "algorithmic" function in logic and mathematics. when you say, " this is black", the theory of descriptions shows that you are only saying something about "this", which is a subject-variable(x), and black is an element-predicate, calculable within the conjunct "this". The theory permits mathematical-logic understand algorithmic functions, and is, also, what makes possible via your computer processor to read codified information. The result is more than a "fruit". it gives you the possibility of grasping that, like any other mathematical fruit, men is able of creating it,- and of reading it(calculate it). these means: Mathematical creations are only valuable as a source of human power, not as mystic ontological formulae,- that stupid motivation in all pseudo "Mathematicians".
In terms of actuality, the axiomatic system, the method, has been perfected, simplified, and transcended. If I had to recommend some books on the matter, I would say Tarski's: "Introduction to logic and to the methodology of the deductive sciences", Patrick Suppes:"axiomatic set theory", continued by the reading of the: "Gödel proofs" by Raymond Smullyan, some other text dealing whith "boolean algebra" such as: "logic as algebra" by Halmos. This would give any self-educated person, the basic models he needs to comprehend math-logic, the "method" with which he can possibly contribute to this "powerful trend of modern thought" as described by tarski. Remember that Russell and whitehead say in the introduction that they not claim having the most perfect axiomatic reduction, only that the one presented was enough to reduce mathematics into that form, which was, until godel, true, or at least "thought possible"(completely). Is important to undersatnd that "principia mathematica" made "possible" the incompleteness proofs of Godel: his original paper was named "on formally undecidable propositions of principia mathematica and related systems"(see dover edition), and although he uses mostly the axioms of peano in his system, if someone as Russel had not attempted successfully such axiomatic construction of math, godel would have never found or seen the incompleteness of arithmetic's. Something similar could be said of the later notions of completeness of first order logic, metamathematics, etc. The few works (few only in number) independent from principia may be the ones of: 1) the polish masters: Lukasiewicz, Lesniewski, and the last king Tarski. 2) the forgotten Richard Martin's and Rudolf Carnap's logic-syntaxic-semantic conception of math-logic. The rest walked, continued walking the path of principia. Individual example: Quine. ...

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reviews: page 1, 2, 3

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