Most math does not attempt to describe real phenomena, and so is not empirically wrong but empirically irrelevant.
Suppose we lived in a universe where the sum of two and two wasn't any number in particular. You couldn't predict in advance how many objects you would have if you had two collections of two objects and added them together, or if you divided or multiplied a collection of objects, etcetera. We have no system for manipulating numbers or abstract symbols in a coherent and concrete way, and the universe doesn't appear to operate on one.
Then, one day, a person declares, "I've developed a set of axioms which allow me to manipulate 'numbers' in a coherent and self consistent way!" They've invented a set of rules and assumptions under which they can perform the same operations on the same numbers and get the same results... in theory. But in practice, they can't predict the real result of adding two peanuts together any better than anyone else can.
In such a universe, you'd have considerable evidence to suspect that they had not stumbled upon some fundamental truths which simply don't happen to apply to your universe, but that the whole idea of "math" was nonsense in the first place.
If maths is supposed to apply to the universe and doesn't then, that is a problem. But most of it doesn't apply to the universe, And it is physics that is supposed to apply to the universe.
One of the most annoying arguments when discussing AI is the perennial "But if the AI is so smart, why won't it figure out the right thing to do anyway?" It's often the ultimate curiosity stopper.
Nick Bostrom has defined the "Orthogonality thesis" as the principle that motivation and intelligence are essentially unrelated: superintelligences can have nearly any type of motivation (at least, nearly any utility function-bases motivation). We're trying to get some rigorous papers out so that when that question comes up, we can point people to standard, and published, arguments. Nick has had a paper accepted that points out the orthogonality thesis is compatible with a lot of philosophical positions that would seem to contradict it.
I'm hoping to complement this with a paper laying out the positive arguments in favour of the thesis. So I'm asking you for your strongest arguments for (or against) the orthogonality thesis. Think of trying to convince a conservative philosopher who's caught a bad case of moral realism - what would you say to them?
Many thanks! Karma and acknowledgements will shower on the best suggestions, and many puppies will be happy.