Presenting an outline of mathematics and its history, with particular emphasis on events that shook up its philosophy, this work ranges from the discovery of irrational numbers in ancient Greece to the 19th- and 20th-century discoveries on the nature of infinity and proof. Recurring themes are intuition and logic, meaning and existence, and the discrete and the continuous. These themes have evolved under the influence of new mathematical discoveries and the story of their evolution is, to a large extent, the story of philosophy of mathematics.