> The pocket calculator is doing a huge amount of computation to find those results, an amount which would be impractical for the human to do (hence the use of slide rules instead of laborious pen-and-paper arithmetic). It’s just a different flavor of brute force. The calculator is basically going back to the pre-logarithm method, carrying out elementary school arithmetic algorithms very fast.
I think the point is more that all those computations are packaged up into a black box where the user doesn't need to think about its internals. Elegant/short proofs are often like this too: they build on deep/high-power/complicated-to-prove results, using them as black boxes. Of course the actual proofs of those theorems might be ugly (e.g. a proof that uses the four colour theorem), but the statement can still be neat.
I think the point is more that all those computations are packaged up into a black box where the user doesn't need to think about its internals. Elegant/short proofs are often like this too: they build on deep/high-power/complicated-to-prove results, using them as black boxes. Of course the actual proofs of those theorems might be ugly (e.g. a proof that uses the four colour theorem), but the statement can still be neat.