Few books to consider:
1)Godel, Escher, Bach: An Eternal Golden Braid, By Douglas R Hofstadter : This Pulitzer prize winning book will be good for your needs. Has short and to great extent independent chapters and goes into the the meaning of mathematics, art, music and computing. Though the book looks voluminous it gives a possibility for selected reading.
2)What is the name of this book?: The riddle of Dracula and other logical puzzles, By Raymond M Smullyan ("The most original, most profound, and most humorous collection of recreational logic and math problems ever written." — Martin Gardner.)
3)Letter to young mathematician: the art of mentoring, By Ian Stewart : Subjects ranging from the philosophical to the practical--what mathematics is and why it's worth doing, the relationship between logic and proof, the role of beauty in mathematical thinking, the future of mathematics, how to deal with the peculiarities of the mathematical community are all discussed here.
4)An Imaginary Tale: The Story of i [the square root of minus one], By Paul J. Nahin : Discusses imaginary numbers from a historical perspective.
(added later)The Creative Process: Reflections on the Invention in the Arts and Sciences : Collection of essays written by some of the best minds like Poincare, Einstein, Mozart on how they got original ideas. Especially Poincare's essay is very insightful.
(added later)I Want to Be a Mathematician. Springer-Verlag. By Paul Halmos : It is good to introduce students to Halmos writing. I am surprised nobody mentioned this.