Computers are providing solutions to math problems that we can't check - "A computer has solved the longstanding Erdős discrepancy problem! Trouble is, we have no idea what it's talking about — because the solution, which is as long as all of Wikipedia's pages combined, is far too voluminous for us puny humans to confirm." (via; previously ;)
The Adventures of Fallacy Man, from Existential Comics.
Using computer systems for doing mathematical proofs - "With the proliferation of computer-assisted proofs that are all but impossible to check by hand, Hales thinks computers must become the judge." [more inside]
"Perhaps twenty or thirty people in England may be expected to read this book." G.H. Hardy's review of Whitehead and Russell's Principia Mathematica, published in the Times Literary Supplement 100 years ago last week. "The time has passed when a philosopher can afford to be ignorant of mathematics, and a little perseverance will be well rewarded. It will be something to learn how many of the spectres that have haunted philosophers modern mathematics has finally laid to rest."
92 years young, the delightful Raymond Smullyan is a mathematician, logician, magician, concert pianist, and Taoist philosopher - who also pioneered retrograde chess problems.
Douglas Hofstadter's Gödel, Escher, Bach: An Eternal Golden Braid has been recorded as a series of video lectures for MIT's Open Courseware project.
Consider the Philosopher. The early metaphysical investigations of David Foster Wallace.
Parmenides. "The pre-Socratic philosopher sparked an intellectual revolution that still echoes today. Yet for philosophy and science to continue to progress in the 21st century, we may need to embark on an entirely new cognitive journey ."
According to this guy, you’re not ultimately morally responsible for choosing whether to snark or not to snark in response to this FPP. A discussion of the philosophical problems surrounding freewill from British Analytic philosopher Galen Strawson. (Oh, and while we’re at it, let’s throw in this unrelated review of Strawson’s latest work on consciousness, just for an extra splash of color.)
Douglas Hofstadter says, "What troubles me is the notion that things that touch me at my deepest core -- pieces of music most of all, which I have always taken as direct soul-to-soul messages -- might be effectively produced by mechanisms thousands if not millions of times simpler than the intricate biological machinery that gives rise to a human soul.". That was prompted by his reception to the output of David Cope's project Experiments in Musical Intelligence.
Gödel and the Nature of Mathematical Truth : A Talk with Verena Huber-Dyson
Godel's theorems have been used to extrapolate a great many "truths" about the world. Torkel Franzen sets the record straight in his new book Godel's Theorem: An Incomplete Guide to Its Use and Abuse. Read the introduction (PDF). If you want, check out his explanation of the theorems.