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Can you recommend a popular science book that deals with axiomatic foundation of mathematical areas and consequences? The areas could be geometry, algebra/numbers, set theory, etc. It should be exact and not too hand-wavy, but by "popular" I mean a book that just gives a nice overview. Basically I'd like to find out about these approaches, but I don't have the time to go into all details.

EDIT: I do prefer mathematical notation and reasoning. I just want to avoid many lemmas and proofs :)

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Try snipurl.com/23f8po6 (Another interesting book to have) –  Kirthi Raman May 8 '12 at 17:34
    
I'm not exactly what you're looking for, but Godel, Escher, Bach is a classic. It's a long read but does get into many of the issues of axiomatic mathematics in a rather deep way. –  Michael Joyce May 9 '12 at 6:02
    
Not a popular science, but how about the colorful version of Six Books of Euclid? –  Sniper Clown May 9 '12 at 6:15
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You might try "Naive Set Theory" by Paul Halmos. It would seem to meet your requirements in that it strives to be exact while remaining "popular" by using ordinary language and avoiding the deep logic foundations of mathematics. As set theory can generally be used to formulate all other mathematics, this would seem to be a good place to start.

Note that it does not meet the popular definition of "popular" in the sense that a book by Greene or Hawking would. Halmos' book still requires a certain mathematical mind.

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Sounds pretty good :) Thanks! Actually I also didn't mean the common meaning of "popular" and I do want the mathematical mind way. –  Gerenuk May 8 '12 at 17:03
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I first learned about these things as a teenager by reading the third volume of James R. Newman's anthology, World of Mathematics. It contains essays by many of the original workers in the field, as well as commentary by Newman. This may be at a lower level than what you're looking for, but I highly recommend it nonetheless.

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