Frege among the Formalists

This project will help us better understand Gottlob Freges logic and theory of meaning by investigating the context in which they arose: the movement for greater rigor in mathematics in the nineteenth century. In the late nineteenth century, mathematics was developing rapidly. New methods were producing exciting new results, but unintuitive and paradoxical results were also starting to appear. Mathematicians began calling for closer scrutiny of their new methods and greater rigor in their proofs, to ensure that these paradoxes did not lead to contradictions and that mathematical knowledge remained trustworthy. As part of this effort, they began to pay close attention to the question of what signs in mathematics mean. Frege was one of those mathematicians. Frege invented modern formal logic, a symbolic language and a system for proving statements expressed in this language. Frege wanted to show that much of mathematics could be proven from just the basic principles of this logic. As part of this effort, he developed a theory about what the symbols of a logical language mean: what kinds of symbols there are, what kinds of meanings they can have, and how these meanings combine to express complete thoughts in the language. Freges logic sparked a revolution in mathematics, computing, and philosophy, where it plays a fundamental role today. But there are still many open questions about the theory of meaning he developed to go with it. Other mathematicians also wanted to put mathematics on secure foundations, and developed their own methods for achieving rigor. Mathematical formalism was another approach which attracted many mathematicians at the time, including important figures like Karl Weierstrass and David Hilbert. Though formalism was similar to Freges program in some ways, Frege saw it as a rival and argued frequently against it. The main disagreement between them concerned the meaning of signs. Whereas Frege thought a sign must stand for or designate something, for a formalist, a sign in mathematics is like a wooden piece in a chess game: its meaning lies not in what it stands for, but in how it is manipulated according to rules. This project examines the debates between Frege and the formalists. These debates are excellent sources for Freges views on meaning, but have so far not received close attention from researchers. By investigating those views in their historical and mathematical context, the project will shed light on fundamental issues in Freges logic and theory of meaning and their role in mathematics and philosophy today.