5 ms·

This is precisely what made Haskell's monads difficult for me to understand. I kept trying to fit the square block in the round hole. I finally had to divorce

by reeses 13y ago

This is precisely what made Haskell's monads difficult for me to understand. I kept trying to fit the square block in the round hole. I finally had to divorce the terms in my head, much like watching the film version of a book that I have read. They are..."related" but frustration will result from assuming they will be the same.

jfarmer 13y ago
Nice! Yeah. I had two avenues of understanding available to me when it came to monads. The first was my knowledge as a mathematically-inclined software engineer. The second was my knowledge as someone fluent in high-level mathematics. I'll say, here was one key to my understanding once I saw it. In a purely functional language you want everything to be referentially transparent. That is, an expression should be able to be substituted for its value at any given moment in time. At the very least, you'll want the ability to denote which parts of your program are referentially transparent and which aren't. This can be expressed as utopian vision: in a purely functional language we'd like everything -- everything -- to be a value. We now need some way of translating code with side effects into code that returns a value. The logical abstraction is to say that the "value" is not the work itself -- once the work is done it's out of the bag, after all -- but rather the "value" is the machine which can perform that work. To a mathematician this screams "functor." If you want to move between the two spaces repeatedly this screams "adjoint functors." There's some work you want to do in Space A that's more easily expressed in Space B, so you take a thing in A, transform it to a thing in B, do the work in B, and then transform back to A. Think in, e.g., Ruby: sentence.split(' ').map(&:reverse).join(' ') split(' ') and join(' ') aren't quite inverses of each other, but they are adjoint. These pairs take us from the land of space-separated strings to the land of arrays back to the land of space separated strings. You see that all the time in programming and mathematics. I sometimes call it the "Super Mario Brothers" maneuver when I'm trying to teach it to students. Mario wants to get to the end of the level, so he hits a block, goes up the vine to cloud land, runs through cloud land, and then comes back down.
- minikomi 13y ago
  I would love to see you write more in this style - it's very clear and straight forward.
  - omaranto 13y ago
    I disagree, I found it very muddled. For example, how are split(' ') and join(' ') supposed to be adjoint functors when they're not even functors?
    - chas 13y ago
      He's playing fast and loose with the formalisms to try to convey the intuition to people without a strong algebraic background. Ruby isn't very well behaved from a categorical perspective (heck, even Haskell doesn't get all of the way there if you want to be really formal) so it's not perfect, but I think the reasoning is morally correct and insightful.
- omaranto 13y ago
  split(' ') and join(' ') aren't quite inverses of each other, but they are adjoint. Do you mean you think they are adjoint functors? How are they functors, between which categories?
  - tel 13y ago
    I wouldn't say they're the tightest functors ever, but you could see them as endofunctors between the subcategories String and [String]. Some small amount of intuition would translate, although by and large this setting isn't polymorphic enough to have good behavior. After all, there are a lot of functions on strings that would not have a functorial relationship over split(' ').
    - jfarmer 13y ago
      Yes, this is more correct, and I was also imagining the category being something like "space-delimited strings." I realize that'd require redefining things like string concatenation to be "space-delimited concatenation" and other such bits. Anyhow, I was playing loose and I know it. If I were writing in a more serious context I'd cross all the t's and dot all the i's. To me the more important thing was this slow gnawing in my gut, seeing hints of natural transformations and adjoint functors here and there and thinking, "Oh come on, there has to be a way to connect these dots." Non-functional languages like Ruby don't think in this way, so there are hole all over the language.

by reeses 13y ago

jfarmer 13y ago

Nice! Yeah. I had two avenues of understanding available to me when it came to monads. The first was my knowledge as a mathematically-inclined software engineer. The second was my knowledge as someone fluent in high-level mathematics. I'll say, here was one key to my understanding once I saw it. In a purely functional language you want everything to be referentially transparent. That is, an expression should be able to be substituted for its value at any given moment in time. At the very least, you'll want the ability to denote which parts of your program are referentially transparent and which aren't. This can be expressed as utopian vision: in a purely functional language we'd like everything -- everything -- to be a value. We now need some way of translating code with side effects into code that returns a value. The logical abstraction is to say that the "value" is not the work itself -- once the work is done it's out of the bag, after all -- but rather the "value" is the machine which can perform that work. To a mathematician this screams "functor." If you want to move between the two spaces repeatedly this screams "adjoint functors." There's some work you want to do in Space A that's more easily expressed in Space B, so you take a thing in A, transform it to a thing in B, do the work in B, and then transform back to A. Think in, e.g., Ruby: sentence.split(' ').map(&:reverse).join(' ') split(' ') and join(' ') aren't quite inverses of each other, but they are adjoint. These pairs take us from the land of space-separated strings to the land of arrays back to the land of space separated strings. You see that all the time in programming and mathematics. I sometimes call it the "Super Mario Brothers" maneuver when I'm trying to teach it to students. Mario wants to get to the end of the level, so he hits a block, goes up the vine to cloud land, runs through cloud land, and then comes back down.

minikomi 13y ago
I would love to see you write more in this style - it's very clear and straight forward.
- omaranto 13y ago
  I disagree, I found it very muddled. For example, how are split(' ') and join(' ') supposed to be adjoint functors when they're not even functors?
  - chas 13y ago
    He's playing fast and loose with the formalisms to try to convey the intuition to people without a strong algebraic background. Ruby isn't very well behaved from a categorical perspective (heck, even Haskell doesn't get all of the way there if you want to be really formal) so it's not perfect, but I think the reasoning is morally correct and insightful.
omaranto 13y ago
split(' ') and join(' ') aren't quite inverses of each other, but they are adjoint. Do you mean you think they are adjoint functors? How are they functors, between which categories?
- tel 13y ago
  I wouldn't say they're the tightest functors ever, but you could see them as endofunctors between the subcategories String and [String]. Some small amount of intuition would translate, although by and large this setting isn't polymorphic enough to have good behavior. After all, there are a lot of functions on strings that would not have a functorial relationship over split(' ').
  - jfarmer 13y ago
    Yes, this is more correct, and I was also imagining the category being something like "space-delimited strings." I realize that'd require redefining things like string concatenation to be "space-delimited concatenation" and other such bits. Anyhow, I was playing loose and I know it. If I were writing in a more serious context I'd cross all the t's and dot all the i's. To me the more important thing was this slow gnawing in my gut, seeing hints of natural transformations and adjoint functors here and there and thinking, "Oh come on, there has to be a way to connect these dots." Non-functional languages like Ruby don't think in this way, so there are hole all over the language.