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Stephen_Cole comments on Rationality Quotes Thread August 2015 - Less Wrong Discussion
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Beyond all doubt sounds fairly dogmatic, no? Godel proved in 1931 that Hilbert's program for a solid mathematical foundation (circa 1900) was impossible.
Not everything can be proven, but those that are are proven beyond (virtually all) doubt.
While I don't quite agree with your claim about what Gödel accomplished, 'beyond all doubt' is an overstatement. The history of mathematics provides many examples of apparent proofs accepted by the profession later being rejected for containing devastating errors. Even a single instance of this occurring would, strictly speaking, rule out a literal 'beyond all doubt' claim.
Well no, Goedel proved that nonstandard models of the natural numbers exist. Chaitin went on to prove that any formal axiomatic system, only containing a finite amount of axioms, will eventually face true theorems it cannot prove, for lack of information in its axioms.
That doesn't mean proven mathematical theorems are actually wrong, and unfortunately, Goedel's Platonism has resulted in most of society thinking about mathematics and proof in the wrong way.
Maybe you should write a post about that.
I've got a bunch of, let's call them, philosophical intuitions and positions that I got from reading this book. I've been meaning to do at least one post explicitly about Hierarchical Bayesian inference as a way to express what it is that keeps banging through my head.
First, though, I want to get Venture up and running to actually implement and train such a model, in progress, to make sure that my thoughts about the way it "should" work accord at all with how it actually does work.