Josh, if you think about a picture like the one Eliezer drew (but in however many dimensions you like) it’s kinda obvious that the leading term in the difference between two n-cubes consists of n (n-1)-cubes, one per dimension. So the leading term in the next difference is n(n-1) (n-2)-cubes, and so on. But that doesn’t really give the n! thing at a glance. I’m not convinced that anything to do with nth differences can really be seen at a glance without some more symbolic reasoning intervening.
James Bach, I suspect that the really good mathematicians can’t be had for cheap because doing mathematics is so important to them, and the quite good mathematicians can’t be had for cheap because they’ve taken high-paying jobs in finance or software or other domains where a mathematical mind is useful. But maybe it depends on what you count as “cheap” and what fraction of the mathematician’s time you want to take up with tutoring...
Isabel, I think perhaps differentiation really is easier in some sense than differencing, not least because the formulae are simpler. Maybe that stops being true if you take as your basic objects not n^k but n(n-1)...(n-k+1) or something, but it’s hard to see the feeling that n^k is simpler than that as mere historical accident.
Josh, if you think about a picture like the one Eliezer drew (but in however many dimensions you like) it’s kinda obvious that the leading term in the difference between two n-cubes consists of n (n-1)-cubes, one per dimension. So the leading term in the next difference is n(n-1) (n-2)-cubes, and so on. But that doesn’t really give the n! thing at a glance. I’m not convinced that anything to do with nth differences can really be seen at a glance without some more symbolic reasoning intervening.
James Bach, I suspect that the really good mathematicians can’t be had for cheap because doing mathematics is so important to them, and the quite good mathematicians can’t be had for cheap because they’ve taken high-paying jobs in finance or software or other domains where a mathematical mind is useful. But maybe it depends on what you count as “cheap” and what fraction of the mathematician’s time you want to take up with tutoring...
Isabel, I think perhaps differentiation really is easier in some sense than differencing, not least because the formulae are simpler. Maybe that stops being true if you take as your basic objects not n^k but n(n-1)...(n-k+1) or something, but it’s hard to see the feeling that n^k is simpler than that as mere historical accident.