A new philosophical movement began to develop in late 19th century and blossomed through the entire 20th century. It became known as analytic philosophy and focused on clarity, argument, and a logical, linguistic, and mathematical formulation of philosophical ideas. Although this new school dominated the philosophy of English-speaking countries, its forerunner is a German mathematician and philosopher named Gottlob Frege.
A Revolutionary Aristotelian
Gottlob Frege (1848–1925) was a German philosopher, logician, and mathematician, generally regarded as the father of modern philosophy of language. Working on the borderline between philosophy and mathematics, Frege discovered the fundamental ideas and invented the entire discipline of mathematical logic. Frege has been called “the greatest logician since Aristotle”, and one of the most profound philosophers of mathematics ever.
Frege believed that one could prove all the truths of arithmetic deductively from a limited number of logical axioms and, in fact, he demonstrated all the basic laws of arithmetic within his own system of logic. He set out to learn if proofs in arithmetic rest on pure logic, or do they need support from empirical facts. The answer which he gave was that arithmetic is a branch of logic and can be formalized without the use of non-logical axioms.
Frege opposed Kant’s idea that the truths of mathematics were synthetic a priori, and that our knowledge of them depended on intuition. He postulated that arithmetic is analytic and can be proved from purely logical principles.
One of Frege’s concerns was to remove intuition from the structure of logical inference. If there was an intuitive element in the logical structure, he isolated it and represented it as an axiom. This brings up an interesting question: is intuition a set of undeclared axioms?
In his two-volume book Foundations of Arithmetic, Frege presented a system which derived all arithmetic from the rules of logic. This was to be a crowning achievement of Frege’s work until mathematician and philosopher Bertrand Russell proved that Frege’s reliance on…
