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Principia Mathematica to *56 (Cambridge Mathematical Library) 2nd Edition

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ISBN-13: 978-0521626064
ISBN-10: 0521626064
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Editorial Reviews Review

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 Gödel 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

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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Product Details

  • Series: Cambridge Mathematical Library
  • Paperback: 460 pages
  • Publisher: Cambridge University Press; 2 edition (October 13, 1997)
  • Language: English
  • ISBN-10: 0521626064
  • ISBN-13: 978-0521626064
  • Product Dimensions: 6 x 1 x 9 inches
  • Shipping Weight: 1.5 pounds (View shipping rates and policies)
  • Average Customer Review: 4.4 out of 5 stars  See all reviews (8 customer reviews)
  • Amazon Best Sellers Rank: #2,478,018 in Books (See Top 100 in Books)

Customer Reviews

Most Helpful Customer Reviews

60 of 63 people found the following review helpful By Moisés Macías Bustos on July 22, 2004
Format: Paperback
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 & Begriffsschrift are the books which give 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 logic and set theory (the former they also conceived as logic, equating it with the theory of properties). 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, succint & elegant treatises on logic and the foundations of mathematics, 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 something like the modern day notation. As another reviewer pointed out, Godel's proof would've been extremely hard to come up with in the absence Principia or a systems such as PM; someone first needed to show that you could axiomatize mathematics to a great extent for there to be possible to reflect on the metalogic of such systems.
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48 of 56 people found the following review helpful By galloamericanus on October 19, 2002
Format: Paperback
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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24 of 30 people found the following review helpful By A Customer on July 30, 2003
Format: Paperback
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.Read more ›
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