well that site has only geometry, but it did provide the following link:
[I will say however that without PM probably Godels great work would never have been done, so in a sense PM did something wonderful by inspiring the refutation of its own thesis. sort of like watson inspiring holmes by accident.]
http://plato.stanford.edu/entries/principia-mathematica/#COPM
from which the next excerpt is taken:
Significance of Principia Mathematica
Achieving Principia's main goal proved to be controversial. Primarily at issue were the kinds of assumptions that Whitehead and Russell needed to complete their project. Although Principia succeeded in providing detailed derivations of many major theorems in set theory, finite and transfinite arithmetic, and elementary measure theory, two axioms in particular were arguably non-logical in character: the axiom of infinity and the axiom of reducibility. The axiom of infinity in effect stated that there exists an infinite number of objects. Thus, it made the kind of assumption that is generally thought to be empirical rather than logical in nature. The axiom of reducibility was introduced as a means of overcoming the not completely satisfactory effects of the theory of types, the theory that Russell and Whitehead used to restrict the notion of a well-formed expression, and so to avoid paradoxes such as Russell's paradox. Although technically feasible, many critics concluded that the axiom of reducibility was simply too ad hoc to be justified philosophically. As a result, the question of whether mathematics could be reduced to logic, or whether it could be reduced only to set theory, remained open.
Despite these criticisms, Principia Mathematica proved to be remarkably influential in at least three other ways. First, it popularized modern mathematical logic to an extent undreamt of by its authors. By using a notation superior in many ways to that of Frege, Whitehead and Russell managed to convey the remarkable expressive power of modern predicate logic in a way that previous writers had been unable to achieve. Second, by exhibiting so clearly the deductive power of the new logic, Whitehead and Russell were able to show how powerful the modern idea of a formal system could be, thus opening up new work in what was soon to be called metalogic. Third, Principia Mathematica reaffirmed clear and interesting connections between logicism and two main branches of traditional philosophy, namely metaphysics and epistemology, thus initiating new and interesting work in both these and other areas.
Thus, not only did Principia introduce a wide range of philosophically rich notions (such as propositional function, logical construction, and type theory), it also set the stage for the discovery of classical metatheoretic results (such as those of Kurt Gödel and others) and initiated a tradition of common technical work in fields as diverse as philosophy, mathematics, linguistics, economics and computer science.
Today there remains controversy over the ultimate substantive contribution of Principia, with some authors holding that, with the appropriate modifications, logicism remains a feasible project. Others hold that the philosophical and technical underpinnings of the Whitehead/Russell project simply remain too weak or confused to be of much use to the logicist. Interested readers are encouraged to consult Hale and Wright (2001), Quine (1966a), Quine (1966b), Landini (1998) and Linsky (1999).
Contents of
"Contents of Principia Mathematica
Principia Mathematica originally appeared in three volumes. Together these three volumes are divided into six parts. Volume 1 begins with a lengthy Introduction containing sections entitled "Preliminary Explanations of Ideas and Notations," "The Theory of Logical Types," and "Incomplete Symbols." It also contains Part I, entitled "Mathematical Logic," which contains sections on "The Theory of Deduction," "Theory of Apparent Variables," "Classes and Relations," "Logic of Relations," and "Products and Sums of Classes"; and Part II, entitled "Prolegomena to Cardinal Arithmetic," which contains sections on "Unit Classes and Couples," "Sub-Classes, Sub-Relations, and Relative Types," "One-Many, Many-One and One-One Relations," "Selections," and "Inductive Relations."
Volume 2 begins with a "Prefatory Statement of Symbolic Conventions." It then continues with Part III, entitled "Cardinal Arithmetic," which itself contains sections on "Definition and Logical Properties of Cardinal Numbers," "Addition, Multiplication and Exponentiation," and "Finite and Infinite"; Part IV, entitled Relation-Arithmetic," which contains sections on "Ordinal Similarity and Relation-Numbers," "Addition of Relations, and the Product of Two Relations," "The Principle of First Differences, and the Multiplication and Exponentiation of Relations," and "Arithmetic of Relation-Numbers"; and the first half of Part V, entitled "Series," which contains sections on "General Theory of Series," "On Sections, Segments, Stretches, and Derivatives," and "On Convergence, and the Limits of Functions."
Volume 3 continues Part V with sections on "Well-Ordered Series," "Finite and Infinite Series and Ordinals," and "Compact Series, Rational Series, and Continuous Series." It also contains Part VI, entitled "Quantity," which itself contains sections on "Generalization of Number," "Vector-Families," "Measurement," and "Cyclic Families."
A fourth volume was planned but never completed.
Contemporary readers (i.e., those who have learned logic in the second half of the twentieth century or later) will find the book's notation somewhat antiquated and clumsy. Even so, the book remains one of the great scientific documents of the twentieth century."
