A strong AI should be better than humans at pretty much every facet of reasoning, essentially as a starting premise. It’s not like humans aren’t computers, we’re just wetware computers built very differently from our own current technology. “As good as the best humans” should be the absolute floor if we’re positing the abilities of an optimally designed computer.
Humans are also bad at numerical derivatives. Derivatives are really messy when we don’t have a closed analytical form for the function f’. Basically the problem is that the derivative formula
-f(x)}{h})
involves subtracting nearly equal numbers and then dividing by almost zero. Both of these things destroy numerical accuracy very very quickly, because it takes very tiny errors and turns them into very large numbers. As long as the solution to the n-body problem is expressed in terms of a differential Taylor series without analytic components, it’s going to be very very difficult to solve accurately.
For practical problems, where we don’t know the initial state of the system to infinite accuracy this is a big problem. It also forces you to use of lots and lots of memory storing all your numbers accurately, because you burn through that accuracy really quickly.
Side note—finite differencing (which, you’re right, typically throws away half of your precision) isn’t the only way to get a computer to take a derivative. Automatic differentiation packages will typically get you the derivative of an explicitly defined function to roughly the accuracy with which you can evaluate the function itself.
I’m not familiar with the n-body problem series solution, though; there’s lots of other ways that could turn out to be impractical to evaluate.
Can you expand on this?
A strong AI should be better than humans at pretty much every facet of reasoning, essentially as a starting premise. It’s not like humans aren’t computers, we’re just wetware computers built very differently from our own current technology. “As good as the best humans” should be the absolute floor if we’re positing the abilities of an optimally designed computer.
Humans are also bad at numerical derivatives. Derivatives are really messy when we don’t have a closed analytical form for the function f’. Basically the problem is that the derivative formula
involves subtracting nearly equal numbers and then dividing by almost zero. Both of these things destroy numerical accuracy very very quickly, because it takes very tiny errors and turns them into very large numbers. As long as the solution to the n-body problem is expressed in terms of a differential Taylor series without analytic components, it’s going to be very very difficult to solve accurately.
For practical problems, where we don’t know the initial state of the system to infinite accuracy this is a big problem. It also forces you to use of lots and lots of memory storing all your numbers accurately, because you burn through that accuracy really quickly.
Side note—finite differencing (which, you’re right, typically throws away half of your precision) isn’t the only way to get a computer to take a derivative. Automatic differentiation packages will typically get you the derivative of an explicitly defined function to roughly the accuracy with which you can evaluate the function itself.
I’m not familiar with the n-body problem series solution, though; there’s lots of other ways that could turn out to be impractical to evaluate.