The Wolfram link for K-B completion was surprisingly unhelpful, Wikipedia worked much better for me because it has a detailed example.
Symbolic integration is not actually a complete algorithm because it sometimes needs to check if an expression is equivalent to zero, which is not yet known to be decidable, so in practice people use heuristics for that part. Though of course I agree that teaching math students to find antiderivatives by tricks and guesses is dumb.
I actually cited the Wolfram article because I preferred it, but I went ahead and added a link to the wikipedia article for those whose taste is closer to yours! Thanks.
The Risch algorithm for symbolic integration is what first gave me a hunger to learn “the actually good ways of doing things” in this respect, and a sense that I might have to search for them beyond the classroom. However, I never did learn to use the Risch algorithm itself! I don’t really know whether it turns out to be good for human use.
My impression is that many if not most algorithms for computers are not quite directly usable by humans.
For example, back-tracking is a simple algorithm that works very well for some problems, but there are just too many steps for anything but the smallest problems for a human to follow, even with pen and paper. A human will need fudge parts of it (skip steps, make decisions based on guesses instead of systematically) to be able to finish it quickly.
But knowing about real back-tracking is still useful: one has a better intuition which steps should be fudged and which shouldn’t, a better estimate of how hard the problem is (which helps, e.g., for deciding whether you’re likely to find a solution if you spend a bit more time, or if it’s better to go to a computer), or how to pick solutions systematically when deciding it’s worth to do it “by hand”. This applies to many algorithms.
Nice to see you here again!
The Wolfram link for K-B completion was surprisingly unhelpful, Wikipedia worked much better for me because it has a detailed example.
Symbolic integration is not actually a complete algorithm because it sometimes needs to check if an expression is equivalent to zero, which is not yet known to be decidable, so in practice people use heuristics for that part. Though of course I agree that teaching math students to find antiderivatives by tricks and guesses is dumb.
I actually cited the Wolfram article because I preferred it, but I went ahead and added a link to the wikipedia article for those whose taste is closer to yours! Thanks.
The Risch algorithm for symbolic integration is what first gave me a hunger to learn “the actually good ways of doing things” in this respect, and a sense that I might have to search for them beyond the classroom. However, I never did learn to use the Risch algorithm itself! I don’t really know whether it turns out to be good for human use.
My impression is that many if not most algorithms for computers are not quite directly usable by humans.
For example, back-tracking is a simple algorithm that works very well for some problems, but there are just too many steps for anything but the smallest problems for a human to follow, even with pen and paper. A human will need fudge parts of it (skip steps, make decisions based on guesses instead of systematically) to be able to finish it quickly.
But knowing about real back-tracking is still useful: one has a better intuition which steps should be fudged and which shouldn’t, a better estimate of how hard the problem is (which helps, e.g., for deciding whether you’re likely to find a solution if you spend a bit more time, or if it’s better to go to a computer), or how to pick solutions systematically when deciding it’s worth to do it “by hand”. This applies to many algorithms.
Agreed! My intention is definitely more toward the second approach then the first.