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Principia Mathematica - Volume One Paperback – October 26, 2011
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It is difficult to think of any other work in the mathematical foundations literature which has been so significant or influential in the development of the modern approach. Although the Whitehead/Russell theory of types has been abandoned in favour of the type-free Zermelo-Fraenkel and Neumann-Bernays-Gödel approaches, most aspects of the Whitehead/Russell approach have been incorporated into the modern literature.
The first half of Volume I presents mathematical logic, both propositional calculus and predicate calculus, and a theory of classes which is essentially equivalent to a set theory. The second half of Volume I presents some introductory concepts which are used in the first half of Volume II for cardinal arithmetic.
The notation of the Principia Mathematica is unfamiliar in some ways, but a large proportion of the modern conventions for notating logic and set theory have their historical origins in this book, which in turn obtained many notations from the works of Peano and Frege. The main obstacle for the modern reader might be the dot-notation. Unfortunately, this annoying dot-notation (which was based on Peano's 1889 dot-notation) was adopted by many authors during the 20th century. With a bit of effort, though, one starts to be able to quickly convert the logical formulas into modern parenthesis-notation.
After the Russell/Whitehead Principia Mathematica, no other work attempted to fully write out a derivation of the foundations of mathematics from the lowest levels of logic up to the theories of real numbers and series. Their bold claim that mathematics can be securely based on pure logic issued a challenge to later authors to continue and develop the logicism research program, which effectively achieved success by about 1970.
The first volume is the rosetta stone of what might look like some arcane language, but it is actually both a practical and philosphical discussion of how to logically approach mathematics.
While their text is not casual, it is conversational as they talk through the process. Their discussion illustrates common fallacies and serve as instruction to the student. It is by no means pleasure reading, but if you really read the text, you will learn alot. At least I am.
Some of the criticisms that have been aimed towards the text actually refer to the wrong edition, as I have heard. This is the correct edition, and has a lot more genuine content than any of the other editions that I know about.
This book is heavy on the logic, and I guess, heavy on the math as well (from a logician's point of view). But it is not an introductory text, and may require undo specialism to support its relevance to mathematics. Some of the criticisms are valid, whereas others fail to realize the book's very actual complexity.