Beyond A-Level
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What Is Mathematics? by Courand/Robbins
An excellent book where the authors try to cover all of modern math. It's not the sort of book that you can read from cover-to-cover, it's more of a reference book. I find myself coming back to this book on a regular basis.
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The Infinite by A W Moore
This gem of a book tackles one of my favourite mathematical topics - The Infinite. It assumes no prior knowledge, so might be suitable to a younger audience at first but it will get tough. Mathematical philosophy at its finest.
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Godel, Esher and Bach by Douglas Hofstadter
I still haven't managed to read more that 3/4 of this book because it gets pretty tough, but what I have read is BRILLIANT! Anyone interested in mathematical philosophy should at least give this book a try. It will take a long time to read because you end up having to think for half-an-hour about each page. Then you discover things that you would never imagine like the whole passage you just read was the same if you read the lines from bottom to top, but you didn't realise this until the end!
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To Mock A Mockingbird by Raymond Smullyan
The Master of Logic gives us a superb book full of logical puzzles. "Why is this in this section?" I hear you ask. The answer is that this book gets very tough later on and really does demand a lot of effort from the reader. If you do get to the end, you will have an excellent background into one of my fave math topics - Logic (it will be obvious which bits of math I like best from the topics tackled in the books in this section). But if you only manage 40 pages you will still have opened your mind to some very interesting questions.