The aim of this book is to explain carefully but not technically the differences between advanced research-level mathematics & the sort of mathematics we learn at school. The most fundamental differences are philosophical & readers of this book will emerge with a clearer understanding of paradoxical-sounding concepts such as infinity curved space & imaginary numbers. The first few chapters are about general aspects of mathematical thought. These are followed by discussions of more specific topics & the book closes with a chapter answering common sociological questions about the mathematical community (such as Is it true that mathematicians burn out at the age of 25?")"