is Principia Mathematica ("Mathematical Principles"
by Alfred North Whitehead and Bertrand Russell -- two names which philosophy students should recognize. It was an attempt to describe a set of axioms and rules in symbolic logic from which all mathematics could be proven.
There is a famous quote on about page 400 of the first volume, following a considerable amount of symbolic logic, "From this proposition it will follow, when arithmetical addition has been defined, that 1+1=2." With the additional comment, "The above proposition is occasionally useful."
Unfortunately for Whitehead and Russell, there are a number of mathematical objections which can be raised. By the rules of the Principia Mathematica, the Axiom of Infinity (that there exists at least one infinite set) and the Axiom of Choice (that the product of a collection of non-empty sets is non-empty) are conditionals. However, the Axiom of Reducibility (required by Russell's Paradox -- the question about whether the set of sets which are not a members of themselves is a member of itself) says that they are not conditionals. Kurt Goedel's Incompleteness Theorem showed that it is impossible to derive all mathematics from a finite set of axioms. (When I was taking Introduction to Metaphysics, I had an argument with the professor about Goedel's Incompleteness Theorem and the Heisenberg Uncertainty Principle, which I maintained were philosophically significant, because they demonstrated that certain questions are unanswerable and certain facts are unknowable; he disagreed. I still say he was wrong.)
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