The contents of this article result from the perplexities and questions which assail one in teaching or expounding, in talking and writing about, mathematics. What do we mean by “thinking mathematically”? Whence comes the “certainty” attached to the result of a mathematical argument—a certainty not shared by theology, artistic criticism, politics, history, or even by biology, chemistry or physics? How comes it that the abstractions of mathematics enable us to deal better with the problems of the real world around us? How does mathematics grow—is it created, evolved, or discovered? And so on! I cannot deal completely or comprehensively with all these questions, and I doubt whether final answers to most of them are yet known. I will therefore confine myself to four topics: proof, classification (including recognition), contradiction (including consistency), and discovery