Logicism

Logicism is the theory that mathematics is an extension of logic and therefore all mathematics are reducible to logic. Modern philosophers believed that proof of this theory was the means of extricating the befuddlement of ordinary language and metaphysics from philosophical argumentation. Bertrand Russell and Alfred North Whitehead championed this theory fathered by Gottlob Frege. Frege gave up on the project after Russell recognized a paradox exposing an inconsistency in naïve set theory. Russell and Whitehead continued on with the project in their Principia Mathematica with success except for the paradox of trying to formulate a logical definition of natural numbers in terms of classes. Kurt Gödel's incompleteness theorem ultimately undermined the purpose of the project. The attempted resurrection of this theory is styled neo-logicism.

What both began and ended the original project of Logicism was the notion that Logic, as it was reconcieved in the 19th century, was essentially free from contradiction, where as mathematics was not. Mathematicians, being deeply intertwined with physics, has always been more pragmatic in nature. For even though, for example, Calculus seemed to introduce many paradoxes into the normal discourse of mathematics, its value as a tool entirely vindicated it.

What ended up being so devestating about Gödel's theorem to the project of Logicism was that, even though Logic was still far more internally consistent as a whole then mathematics, the rationalization for it being a foundational theory, its sort of "Philosophic purity", had been undermined.

This isn't to say that the Logicalist project was a failure, for it is still true that Set Theory is the most widely accepted (though nevertheless, far from the only) foundational theory of mathematics, and many important discoveries have been a direct result. However, while mathematics has always been understood to largely consist of the formal manipulation of axioms according to the rules of logic, it was the failure of the original Logicalist position which allowed logicians to realize that, even though the principals of logic are extremely self evident, they are not perfect, and Logic, in the end, was still axiomatic.

In a sense, what has happened since the failure of the original project has been a return to the original question concerning the philosophy of mathematics, which is, essentially, "How and why is it that our capacity for Mathematics can both be so productive yet seem to have so little to do with the empirical world". However, in light of modern developments, the question has been broadened to logic as well. For now, one of the central questions of the philosophy of Logic is, "What relationship does it have with our faculty of reason, and what relationship does our faculty of reason have with the outside world?"