4 ms·

I'm not sure that's really an example of "understanding the steps without understanding the why". I'd say it's an example where there's no more to "understandin

by gjm11 4y ago

I'm not sure that's really an example of "understanding the steps without understanding the why". I'd say it's an example where there's no more to "understanding the why" than "understanding the steps", and the only actual problem is the expectation that something you can describe fairly simply ought to have a proof that doesn't require any calculation.

knorker 4y ago
The comic? That's not how I read it. To me it's exactly that you have all the pieces, all the steps, but still find it completely unreasonable that it works. > the only actual problem is the expectation that something you can describe fairly simply ought to have a proof that doesn't require any calculation. No, to me it's not like that. It's easy to explain that the "angle" of a graph of ln x is 1/x, but after you've explained calculus to someone, and they understand it, they will NOT say "but why?". I'm also not sure why anyone would expect a proof of a math trick to not involve math or calculation.
- gjm11 4y ago
  Hmm. I thought what the comic was aiming at was not "Even though I have seen a proof, I still cannot really believe that it works" but "Even though I have seen a proof, I don't feel that I fully understand why it works" and I think the latter is because (so far as I can currently tell) there isn't any simple way to see at a glance that it works. You just have to do the calculation. (I am not sure whether I am actually disagreeing with you here. It depends on what you mean by "completely unreasonable". I would not, personally, at all say that it feels completely unreasonable that the algorithm works.) Compare, e.g., a different multiplication algorithm: "Russian peasant multiplication", the thing with the doubling and halving. (Look it up if not familiar.) This is another technique where at first glance you might be completely baffled. But with this one, you can, once you understand what's going on, pretty much see at a glance how and why it works. (It's just binary. The halving and checking for remainder delivers the bits of one number, least significant first. The doubling gives you the other number times powers of 2, starting from 1. You add the 2^k.n values corresponding to bits 2^k present in the binary representation of m.) You could write that explanation out in a form that looks like calculation, but I don't think you should. Once you see the idea behind it, the only need for the calculation is to verify that there isn't an off-by-one error or something. In contrast, for the Polish hand thing, there isn't (so far as I know) an explanation that doesn't involve calculation, and (for me) the calculation is a bit too big to fit comfortably in my working memory. So there's the sense of "I can see that this is right but I can't see it at a glance, even though the thing it's proving is rather simple", and that's the nearest thing I have to a "but whyyyyyyy?" reaction. For me (as for you) d/dx log x = 1/x doesn't provoke the same reaction, and I think there are two reasons. 1. The thing being proved is a bit less elementary so there's not so much temptation to expect something "obvious". 2. The explanation isn't very calculation-heavy; to whatever extent you understand it, what you've got is a series of basically-simple steps leading to the conclusion. (For instance, if you're happy to start from d/dx exp x = exp x, then you really can see at a glance that d/dx log x = 1/x, because since log and exp are one another's inverses their graphs are one another's reflections in the line y=x. No calculation at all required.)

by gjm11 4y ago

knorker 4y ago

The comic? That's not how I read it. To me it's exactly that you have all the pieces, all the steps, but still find it completely unreasonable that it works. > the only actual problem is the expectation that something you can describe fairly simply ought to have a proof that doesn't require any calculation. No, to me it's not like that. It's easy to explain that the "angle" of a graph of ln x is 1/x, but after you've explained calculus to someone, and they understand it, they will NOT say "but why?". I'm also not sure why anyone would expect a proof of a math trick to not involve math or calculation.

gjm11 4y ago
Hmm. I thought what the comic was aiming at was not "Even though I have seen a proof, I still cannot really believe that it works" but "Even though I have seen a proof, I don't feel that I fully understand why it works" and I think the latter is because (so far as I can currently tell) there isn't any simple way to see at a glance that it works. You just have to do the calculation. (I am not sure whether I am actually disagreeing with you here. It depends on what you mean by "completely unreasonable". I would not, personally, at all say that it feels completely unreasonable that the algorithm works.) Compare, e.g., a different multiplication algorithm: "Russian peasant multiplication", the thing with the doubling and halving. (Look it up if not familiar.) This is another technique where at first glance you might be completely baffled. But with this one, you can, once you understand what's going on, pretty much see at a glance how and why it works. (It's just binary. The halving and checking for remainder delivers the bits of one number, least significant first. The doubling gives you the other number times powers of 2, starting from 1. You add the 2^k.n values corresponding to bits 2^k present in the binary representation of m.) You could write that explanation out in a form that looks like calculation, but I don't think you should. Once you see the idea behind it, the only need for the calculation is to verify that there isn't an off-by-one error or something. In contrast, for the Polish hand thing, there isn't (so far as I know) an explanation that doesn't involve calculation, and (for me) the calculation is a bit too big to fit comfortably in my working memory. So there's the sense of "I can see that this is right but I can't see it at a glance, even though the thing it's proving is rather simple", and that's the nearest thing I have to a "but whyyyyyyy?" reaction. For me (as for you) d/dx log x = 1/x doesn't provoke the same reaction, and I think there are two reasons. 1. The thing being proved is a bit less elementary so there's not so much temptation to expect something "obvious". 2. The explanation isn't very calculation-heavy; to whatever extent you understand it, what you've got is a series of basically-simple steps leading to the conclusion. (For instance, if you're happy to start from d/dx exp x = exp x, then you really can see at a glance that d/dx log x = 1/x, because since log and exp are one another's inverses their graphs are one another's reflections in the line y=x. No calculation at all required.)