Currently rereading (for the umptenth time):
Godel, Escher, Bach: An Eternal Golden Braid by Douglas R. Hofstadter (ISBN: 0465026567)
God, i wish i had Mr. Hofstadter as Math-Teacher back in my study days at my U. If his lessons are only a tenth that fascinating like the way he writes... it would have been sooooo cool. But well, unfortunately for me he is prof at Indiana University in Bloomington - and not in Germany.
About the book, a short quote from the authors description of topics covered:
J.S. Bach, M.C. Escher, Kurt Gödel: biographical information and work, artificial intelligence (AI) history and theories, strange loops and tangled hierarchies, formal and informal systems, number theory, form in mathematics, figure and ground, consistency, completeness, Euclidean and non-Euclidean geometry, recursive structures, theories of meaning, propositional calculus, typographical number theory, Zen and mathematics, levels of description and computers; theory of mind: neurons, minds and thoughts; undecidability; self-reference and self-representation; Turing test for machine intelligence.
Why i like the book?
Well... hard to describe. The book is a travel through your mind, through music, philosophy, painted art and math. Fictional characters (A turtle, the greek hero Achilles, a Crab and an Anteater) discussing mathematical problems of Kurt Goedel, Zen, the art of painting of M.C.Escher and the musical works of Bach and what they have in common. And no wonder that Hofstadter won the Pulitzer Price for it - i mean: Writing complete dialogues exactly the way like works of BACH, with even the first letters on the beginning of each sentence fitting to the musical notes of the discussed canons and fugues... totally insane.
It is wonderful to read, sometimes the author talks to the reader directly and asks him to do certain things - its like he sits just over the table discussing his work with you.
By the way: It is something even really good accessible for TOTAL Math-Dummies. The first chapters are cool for that - discussing math and why it works, what the rules behind math are and how you can play with axiomatic systems and how that changes the way in setting up a calculation. What is Logic?... And from there, he takes you down the rabbit hole - from why 2+2=4 to what the works of Escher, Bach and Goedel have to do with theories behind A.I programming and neuronal computing. A 900 pages thick page-turner.