Kurt Gödel proved that math has an incurable flaw that will plague it, and us, forever. Learn what it is, and why it has to do with everything from your computer to your brain!
A fantastic blog post detailing how the incompleteness theorem can be logically deduced from the halting problem, and vice-versa - http://www.scottaaronson.com/blog/?p=710
A lecture by Stephen Hawking about how Gödel’s theorem might contain a central truth of physics, namely its inevitable failure - http://www.hawking.org.uk/godel-and-the-end-of-physics.html
A paper detailing how a human brain isn’t immune to the incompleteness theorem’s effects - http://www.columbia.edu/~hg17/godel-incomp4.pdf
Alfred Whitehead and Bertrand Russell’s “Principia Mathematica,” a triumph in mathematical rigor - https://archive.org/details/PrincipiaMathematicaVolumeI
Russell’s “Introduction to Mathematical Philosophy,” a fantastic work demonstrating just what sort of thinking was turned upside-down by Gödel’s proof - http://people.umass.edu/klement/imp/imp-ebk.pdf
Yes, I know that most non-German/Austrian people pronounce it “Girdle.” That’s not quite right; I’m trying to say it closer to how it’s pronounced in Gödel’s language - http://www.forvo.com/word/kurt_g%C3%B6del/