IMHO, the figure of speech "n-th order" generally comes from Taylor series approximation, where n-th term is indeed expressed using n-th derivative. Also, if the series converges, the higher order terms will become smaller, thus the practice of approximating the reality up to n-th order term.
Also, it's often practical to ignore 2nd order and higher terms, because then you have a linear function - something really easy to work with. Therefore, "2nd order effects" have effectively become synonymous to "non-linear effects".
I really doubt this is where "nth order" comes from colloquially.
There are many examples in physics, for example: https://physics.stackexchange.com/questions/310875/second-or...
Yes, but physics is not colloquial. Most people are unfamiliar with physics.
This is what I've always assumed as well. The feeling of "things that usually don't matter but may explode (if it turns out the series doesn't converge)" seems to fit the expression pretty well.
Higher order functions have nothing to do with derivatives however.
Re:Second-Order Thinking
by earleybird on Monday August 15, 2022 @01:24PM
(#32472555)
However, regular types do: "The Derivative of a Regular Type is its Type of One-Hole Contexts"
Re:Second-Order Thinking
by cbm-vic-20 on Monday August 15, 2022 @11:21AM
(#32470843)
E = mc^2 is a notable example.
Re:Second-Order Thinking
by shsbdncudx on Monday August 15, 2022 @11:30PM
(#32478892)
I think it’s simply order of events. If you do N, N+1 will happen
Or think chess - 'first order thinking' would make a move that merely escapes mate, or puts you in a position where another move immediately could be a great one; 'second order thinking' considers what the opponent will do from there (probably disallow the great move or threaten it back); etc.