What are we measuring when we put two yardsticks together? Are a fortune teller's predictions about numbers mathematical propositions? What does the kWhat are we measuring when we put two yardsticks together? Are a fortune teller's predictions about numbers mathematical propositions? What does the knowledge that an infinity of different proofs could prove the same proposition do to our understanding of any particular proof and what exactly it proves? Wittgenstein dares to ask sublimely inane questions about basic mathematical concepts like, um, counting--the results are wonderful. My favorite crazy little question comes in section V: "The class of cats is not a cat." --How do you know?...more

a wonderful and concise look at the prehistory, main concepts and implications of goedel's 1931 proof, detailing how he used a complex mapping systema wonderful and concise look at the prehistory, main concepts and implications of goedel's 1931 proof, detailing how he used a complex mapping system (almost like talking with integers, still mind-boggling to me in a time when computers are commonplace) to blur the line between mathematics and meta-mathematics and reveal irreparable holes in systems of deduction. i like it on page 103 when nagel and newman, apparently bored with writing the words "proof" and "theorem", refer to goedel's "recipe". not included, however, in this edition, is goedel's recipe for formally undecidable bundt cakes....more

keith recommended i read this months ago and i'm glad that i finally did. a quite unusual take on the problem of selfhood/identity that uses godel's tkeith recommended i read this months ago and i'm glad that i finally did. a quite unusual take on the problem of selfhood/identity that uses godel's theorem and video feedback loops as its central analogies. by far my favorite chapters of the book were the chapters devoted to catching the uninitiated reader (myself) up with a simplified (but not terribly over-simplified) understanding of the basics of number theory, set theory, principia mathematica, and godel. it had been so long since i'd read about mathematics that i forgot how much i enjoy it. i was actually giddy during hofstadter's little math lessons. i found it interesting that hofstadter makes attempts here and there to distance his thinking from freud's. he can easily do this when he brings to mind the well-known models of ego, superego, and id and how these differ from his "strange loop" model of consciousness, but when you compare the strange loop to the "mystical writing pad" and some of freud's contemporaneous writings on "pcpt.cs.", they seem (to me at least) to be running on much the same loop. i was often annoyed by the clunky analogies that hofstadter employs throughout this rather prolix book, but in the end these analogies helped me think about some difficult concepts that might have otherwise lost me. so i didn't particularly like the "simmballs" but what would i have done without them?...more