For a small buffet of abstract math topics (with lots of exposition putting ideas into historical context and an emphasis on building geometric intuition) try Courant & Robbins "What is Mathematics?". Covers number theory, number systems, geometry, topology, optimization, algebra, and knots, among others. You could also play around with branches of math like Knot and Braid Theory, which are interesting in their own right, give insight into lots of different branches of math, and many interesting problems are still accessible to lay people. Number theory is also like this I think it can also…
I had the same problem with math. There are two books which changed my mindset forever: http://en.wikipedia.org/wiki/What_Is_Mathematics%3F http://en.wikipedia.org/wiki/Concrete_Mathematics The first one is the general book about math. It's a classical book. The second one is Donald Knuth's book written specifically for computer science guys.
Here is a great one: http://www.amazon.com/Introduction-Graph-Theory-Dover-Mathem... "A stimulating excursion into pure mathematics aimed at "the mathematically traumatized," but great fun for mathematical hobbyists and serious mathematicians as well. Requiring only high school algebra as mathematical background, the book leads the reader from simple graphs through planar graphs, Euler's formula, Platonic graphs, coloring, the genus of a graph, Euler walks, Hamilton walks, and a discussion of The Seven Bridges of Konigsberg."
> I’ve been asked over and over for good books about mathematics for a layperson, someone who hasn’t taken advanced courses in university and is more simply interested in learning about what math is, and some of the more interesting historical figures and results from mathematics. The book you want is "What is Mathematics?" [0] by Courant and Robbins. [0] http://books.google.no/books?id=_kYBqLc5QoQC&printsec=frontc...
The book What is Mathematics by Courant et. al. is a terrific, read, and highly recommended from generations of mathematicians. It's one of the few math books good enough to compel you to read through the whole thing. http://www.amazon.com/Mathematics-Elementary-Approach-Ideas-... There's also the Princeton Companion for Mathematics, which isn't out yet but is available online. It's a wonderful book. http://pcm.tandtproductions.com/ User: Guest Pass: PCM
If you want fundmantal, intuitive understanding I suggest What is Mathematics? by Courant and Robbins. I got this after pg recommended it in an essay, and it's wonderful.
"What is Mathematics" by Courant and Robbins is quite good and respected, but it will challenge you: http://www.amazon.com/Mathematics-Elementary-Approach-Ideas-... It may be more advanced then what you're looking for though.
Based on what you wrote you would hate Courant and Robbins What is Mathematics?, too rigorous. I think you would enjoy Mathematics from the Birth of Numbers by Jan Gullberg and Calculus Made Easy by Sylvanus Thompson. You might also find Unknown Quantity interesting. I think Gullberg would be my #1 req for you.
I would recommend the book "What is Mathematics" to you. It covers a wide range of math topics from first principles. The beginning of each chapter is light reading, but gets more rigorous as the chapter progresses. It talks about mathematical notation a good amount, which is surprisingly helpful because many math ideas are intuitive but hidden behind extremely inaccessible notation. The Kindle version has bad formatting, but the Google Books version works well on all devices supported by Google Books.