Anon, Bostrom's book is Anthropic Bias: Observation Selection Effects in Science and Philosophy.

Raj, Godel's Theorem does not say what people believe it does (would take a separate post). But in any case, Godel's Theorem surely does not show that natural numbers don't exist. It says you'll have trouble proving certain theorems. The observed universe is like the natural numbers, not like a theorem about them.

James, you would seem to be committing the Mind Projection Fallacy. Whether reality is a perfectly regular mathematical object, and whether we can ever have perfect confidence of this fact even if it is true, are two entirely different questions.