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Before digging into the various books others have suggested, you would do well to read "Where Mathematics Comes From" by George Lakoff and Rafael Nunez: http://www.amazon.com/Where-Mathematics-Comes-Embodied-Bring...

That book explains the origins and understanding of the basic items of mathematical analysis: infinity, sets, classes, limits, the epsilon-delta of calculus and alternatives, infinitesimals, etc. The explanation is from the viewpoint of psychological understanding. It details how we build up a scaffolding of tools (starting with basic counting) sufficient to slay the dragons of modern physics and mathematics.

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