You are at the right place in your mathematical career to be reading this book if you liked Trigonometry and Calculus, were able to solve all the problems, but felt mildly annoyed with the text when it put in these verbose, incomprehensible things called “proofs.” Those things probably bugged you because a whole lot of verbiage (not to mention a sprinkling of epsilons and deltas) was wasted on showing that a thing was true, which was obviously true! Your physical intuition is sufficient to convince you that a statement like the Intermediate Value Theorem just has to be true – how can a function move from one value at a to a different value at b without passing through all the values in between?

Mathematicians discovered something fundamental hundreds of years before other scientists – physical intuition is worthless in certain extreme situations. Probably you’ve heard of some of the odd behavior of particles in Quantum Mechanics or General Relativity. Physicists have learned, the hard way, not to trust their intuitions. At least, not until those intuitions have been retrained to fit reality! Go back to your Calculus textbook and look up the Intermediate Value Theorem. You’ll probably be surprised to find that it doesn’t say anything about all functions, only those that are continuous. So what, you say, aren’t most functions continuous? Actually, the number of functions that aren’t continuous represents an infinity so huge that it outweighs the infinity of the real numbers!

The point of this book is to help you with the transition from doing math at an elementary level (which is concerned mostly with solving problems) to doing math at an advanced level (which is much more concerned with axiomatic systems and proving statements within those systems).