3 ms·

This is fascinating, but I am still curious: would I ever have a good reason to repackage seven binary trees as one (or other way around)?

hither_shores 4y ago
The particular example is not the point. The question you should be asking is > would I ever have a need to understand what happens when you perform algebraic manipulations on data structures and the answer is as usual that it depends on what you're working on. Parsers? Absolutely. CRUD apps? Yes, but only if you look for them. Numerical linear algebra? Probably not.
- feoren 4y ago
  > CRUD apps? Yes, but only if you look for them. I upvoted just for this breath of fresh air right here. CRUD apps are boring and crappy because people make them boring and crappy. I find uses for this type of thing all the time in "boring" CRUD apps, because I look for them.
svat 4y ago
I can't imagine any application. :-) The fascination is more that of explanation: if you treat T = 1 + T^2 as an algebraic equation in the complex numbers, you can get a lot of expressions that are meaningless: it's hard to make sense of T = 1/2 ± i√3/2 or T^3 = -1 or even T^6 = 1 (there is certainly no bijection sending six trees to the empty tree). But then, T^7 = T is suddenly very much meaningful and we can turn it into an explicit description. What's going on? Which equations are "meaningful" and which are not? Blass's paper gives an explanation: an equation is meaningful if you can derive it by working in (something like) N[T]/(T = 1 + T^2) (with some more conditions). Baez's discussion (linked above) puts it in terms of "categorification", and is also interesting.
jerf 4y ago
This is like giving an artist a new shade of color they've never had before, like a new rich blue or something. It isn't that everything is improved by using that new color as much as possible everywhere. It's that it opens up new possibilities you may never have considered before. This insight may be useless, but there are other useful ones that can emerge that you wouldn't have thought of on your own in a long time.
feoren 4y ago
Sharding a huge database index (or other binary tree structure) onto 7 different servers or hard drives? Or 49 (recursively apply it)? I don't know if that's actually useful here but it's the first thing that came to mind.

hither_shores 4y ago

The particular example is not the point. The question you should be asking is > would I ever have a need to understand what happens when you perform algebraic manipulations on data structures and the answer is as usual that it depends on what you're working on. Parsers? Absolutely. CRUD apps? Yes, but only if you look for them. Numerical linear algebra? Probably not.

feoren 4y ago
> CRUD apps? Yes, but only if you look for them. I upvoted just for this breath of fresh air right here. CRUD apps are boring and crappy because people make them boring and crappy. I find uses for this type of thing all the time in "boring" CRUD apps, because I look for them.

svat 4y ago

I can't imagine any application. :-) The fascination is more that of explanation: if you treat T = 1 + T^2 as an algebraic equation in the complex numbers, you can get a lot of expressions that are meaningless: it's hard to make sense of T = 1/2 ± i√3/2 or T^3 = -1 or even T^6 = 1 (there is certainly no bijection sending six trees to the empty tree). But then, T^7 = T is suddenly very much meaningful and we can turn it into an explicit description. What's going on? Which equations are "meaningful" and which are not? Blass's paper gives an explanation: an equation is meaningful if you can derive it by working in (something like) N[T]/(T = 1 + T^2) (with some more conditions). Baez's discussion (linked above) puts it in terms of "categorification", and is also interesting.

jerf 4y ago

This is like giving an artist a new shade of color they've never had before, like a new rich blue or something. It isn't that everything is improved by using that new color as much as possible everywhere. It's that it opens up new possibilities you may never have considered before. This insight may be useless, but there are other useful ones that can emerge that you wouldn't have thought of on your own in a long time.

feoren 4y ago

Sharding a huge database index (or other binary tree structure) onto 7 different servers or hard drives? Or 49 (recursively apply it)? I don't know if that's actually useful here but it's the first thing that came to mind.