Title: Introduction to Mathematical Thinking: The Formation of Concepts in Modern Mathematics
Author: Friedrich Waismann
Short Description: This book, first published in 1936, gives an introduction to the philosophy of mathematics and the foundations of analysis. It shows how to construct number systems, some very elementary set theory and how to make differential calculus precise. The focus is on concepts. It includes a discussion of things like continuuous, nowhere differentiable functions and discusses why R and R2 are isomorphic as sets but not topologically. It even gives an very short and intuitive proof that R and R2 are not homeomorphic.
The philosophical discussion is, naturally, a litlle bit dated. It discusses formalism, logicism and intuitivism. The philosophy still makes for exciting historical reading. It is clear that this discussion was very lively when this book was written.
The level should youalify as ++, the book is demanding but can certainly be read before going to college.