Principia Mathematica 3 Volume Set [Hardcover]

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.

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. ...

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.