Principia Mathematica to *56 (Cambridge Mathematical Library) | 
 enlarge | Authors: Alfred North Whitehead, Bertrand Russell
Publisher: Cambridge University Press
Category: Book
List Price: $80.00
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Rating:  13 reviews
Sales Rank: 872970
Media: Paperback
Edition: 2
Pages: 456
Number Of Items: 1
Shipping Weight (lbs): 1.5
Dimensions (in): 8.8 x 6 x 1.3
ISBN: 0521626064
Dewey Decimal Number: 511.3
EAN: 9780521626064
Publication Date: October 13, 1997
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Could it be true that Whitehead and Russell's Principia Mathematica is the most influential book written in the 20th century? Ask any mathematician or philosopher--or anyone who understands the impact these fields have had on modern thinking--and you'll get a short answer: yes. Their goal, to set mathematics on a firm logical foundation, was revolutionary, and their tools and rigor continue to influence modern professionals. Using Peano's symbolic logic, they formalized axioms and produced theorems (including the famous "1 + 1 = 2") in orderings, continuous functions, and other areas of mathematics. Although the Principia is far from comprehensive, Whitehead and Russell's method and program captivate their readers. The audacity to hope to formalize all of mathematics logically was inspirational and helped to give great boosts to math and logical philosophy. Though Goedel proved in 1931 that any such program is doomed to incompleteness, the tools found in and developed from the three volumes helped build the atomic bomb and the Internet. It may not be summer vacation reading (for most), but Principia Mathematica will reward the dedicated student with a deeper understanding of how we got here. --Rob Lightner
Product Description
The great three-volume Principia Mathematica (CUP 1927) is deservedly the most famous work ever written on the foundations of mathematics. Its aim is to deduce all the fundamental propositions of logic and mathematics from a small number of logical premises and primitive ideas, establishing that mathematics is a development of logic. This abridged text of Volume I contains the material that is most relevant to an introductory study of logic and the philosophy of mathematics (more advanced students will of course wish to refer to the complete edition). It contains the whole of the preliminary sections (which present the authors' justification of the philosophical standpoint adopted at the outset of their work); the whole of Part I (in which the logical properties of propositions, propositional functions, classes and relations are established); section A of Part II (dealing with unit classes and couples); and Appendices A and C (which give further developments of the argument on the theory of deduction and truth functions).
Book Description
The great three-volume Principia Mathematica is deservedly the most famous work ever written on the foundations of mathematics. Its aim is to deduce all the fundamental propositions of logic and mathematics from a small number of logical premisses and primitive ideas, and so to prove that mathematics is a development of logic.This abridged text of Volume I contains the material that is most relevant to an introductory study of logic and the philosophy of mathematics (more advanced students will of course wish to refer to the complete edition).
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| Customer Reviews: Read 8 more reviews...
 A Hallmark in the History of Mathematics and Philosophy. July 22, 2004
Moises Macias (Mexico City, Mexico)
30 out of 30 found this review helpful
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.
A monument of mathematical logic September 3, 1999
12 out of 20 found this review helpful
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.
Nonpareil April 20, 2001
4 out of 12 found this review helpful
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.
This book is a masterpiece of Mathamatics. Extreamly good. January 26, 1999
Jason R. McDiarmid (jasonmc@cqnet.com.au) (Queensland, Australia)
7 out of 16 found this review helpful
This is an extreamly good book with more information than you could poke a stick at. I got left behind a few times and had to re-read a chapter, but as long as you love ADVANCED math and like to think logicaly, this book is for you. It goes into many areas of math in general as well as dozens of spacific areas and provides more than enough information. It is a very "deep" book and can sometimes be a bit diffucult to understand, and i recoment this book to people who love deep, deep thought. It is not for everybody, in fact it is probibly for very few, but thouse who read and understand it, will come to love and value it...
Godel and The Principia August 12, 2001
15 out of 34 found this review helpful
To the person who wrote that Godel proved the Principia wrong is incorrect. He only proved that not every math truth can be proved logicaly. He did not disprove the notion that all truths can be expressed logicaly, and therefore a truth of logic.
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