The Foundations of Arithmetic

"Yes, although it wasn’t written in the 20th century, it became very influential in the 20th century. Frege wrote it in 1884 in the middle of a large project in the philosophy of mathematics. The first part of the project was to develop a new system of logic capable of formalising the notion of proof in mathematics. The result was a system that replaced the classical Aristotelian logic of the syllogism, and represented the most important advance in logic in 2,000 years. The second part of his project was to demonstrate that the axioms of arithmetic can be derived from his system of logic plus his logical definitions of all arithmetical concepts. He outlines the strategy for doing this in The Foundations of Arithmetic . Simply put, the idea is to show first that arithmetic is at bottom nothing but an elaboration of pure logic, and second that higher mathematics is at bottom nothing more than an elaboration of arithmetic. So the goal, as explained in the book, is to show how all of mathematics can be established with the unchallengeable a priori certainty of pure logic. It is the best, most accessible work ever in the philosophy of mathematics. It is also beautifully conceived and executed. For those who want to know what philosophical analysis is, this is among the best example ever produced. Its vision, though complicated in details, is simple and compelling. In the end, Frege didn’t accomplish everything he hoped for. But he did succeed in laying the foundation for the stunning advances in mathematical logic in the 20th century that themselves provided frameworks for modern theories both of computation and of linguistically encoded information."