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Monads Made Difficult
- jfarmer 13y agoYes. Thank you. Yes yay yes. I studied mathematics. I've been programming for 15 years, 10-or-so professionally. I've always wanted a "real math" explanation of monads. They now make sense to me. Seeing why mu needs to go from T^2 -> T and the role it plays concretely WRT the list and IO examples makes it really clear to me. God bless you for not using some crazy analogy like bacon or whatever.
- nbouscal 13y agoBurritos! Monads are Burritos! (http://byorgey.wordpress.com/2009/01/12/abstraction-intuition-and-the-monad-tutorial-fallacy/ http://byorgey.wordpress.com/2009/01/12/abstraction-intuitio...)
- tome 13y agoHaskell's Monad is a very, very special case of the mathematical monad. The problem with Stephen Diehl's approach is that it treats the topic in full generality making it much harder to understand how the special case is actually much simpler. I prefer http://web.jaguarpaw.co.uk/~tom/blog/posts/2012-09-02-what-is-a-monad-really.html http://web.jaguarpaw.co.uk/~tom/blog/posts/2012-09-02-what-i..., but that's probably because I wrote it.
- effn 13y agoI think this is for people that already know how to use monads effectively in Haskell and wants to learn more about the mathematical background.
- freyrs3 13y agoThis is explicitly not a writeup for beginners ( ergo the title ), it's a side-by-side description of the /very/ general categorical description and how you can model that in Haskell. The three parameter form of morphism composition ( i.e. c y z -> c x y -> c x z ) would be quite unwieldy in day-to-day programming but it does illustrate the underlying definitions better to have categories explicit.
- trobertson 13y agoI don't mean to nitpick, but the three parameter form is exactly what you use in day-to-day Haskell programming. You simply let c be the infix (->), and you get (y -> z) -> (x -> y) -> (x -> z). We just tend to think of it as a two parameter form because a lot of the time, you can think of (->) as syntax, as opposed to the type that it is.
- lelf 13y agoYup. This is straight cut from Control.Category module: instance Category (->) where id = Prelude.id (.) = (Prelude..)
- tome 13y agoThat's pretty much my point: I don't think mathematical monads are a good background for Haskell Monads. It's a bit like saying the ZF axioms are the mathematical background for Data.Set.
- jfarmer 13y agoIt's great for two classes of people: those who understand monads from a Haskell/software engineering perspective and want to learn the mathematical perspective, and those like me who understand them from the mathematical perspective and want to learn them from the Haskell/software engineering perspective. Given that, I think it's great! :D
- rwosync 13y agoThis is a post about categorical monads, I assume this is why it doesn't use the standard symbols that the Haskell monad uses (>>=, return) and instead refers to the natural transformations (η, μ) as mathematicians do. [1] http://ncatlab.org/nlab/show/monad#definition_18 http://ncatlab.org/nlab/show/monad#definition_18
- tel 13y agoUnderstanding the very generalized algebraic monad helps you to understand the nature of encapsulating effects and adding complexity to categories through layering. It's absolutely not beginner material—but it's valuable material. I don't like your take because it refuses to take that deeper dive into the mathematical Monad, despite [alluding] to it. Knowing the relationship between `Monad` and `Category` the type classes is almost never useful in programming Haskell. Knowing the relationship between `Monad`, monads, and their categorical underpinning is a different matter.
- tome 13y ago> Understanding the very generalized algebraic monad helps you to understand the nature of encapsulating effects and adding complexity to categories through layering. I don't see how the general notion of category theoretical monad helps with this. In fact it's debatable whether the notion of monad is really the right treatment for universal algebra at all. Lawvere theories are much simpler and capture almost all of what you need. See: https://www.dpmms.cam.ac.uk/~martin/Research/Publications/2007/hp07.pdf https://www.dpmms.cam.ac.uk/~martin/Research/Publications/20... > Knowing the relationship between `Monad` and `Category` the type classes is almost never useful in programming Haskell. I disagree. I use (<=<) all the time. It's basically (.) with the Kleisli wrapping and unwrapping done for you.
- tel 13y agoI'm not arguing that (>=>) isn't great or that Monads are better than Lawvere Theories, nor that what you presented isn't interesting and useful. Instead, I'm responding from the point of view of someone who read a lot of terse accounts of what typeclass Monads were back when I was first learning them. The category law account for the Monad laws is useful once you've already learned to think categorically, at which point a full definition is great and provides indications of the next places to examine. Defining that a "Monad is really a Category k with an identification between k a b and a -> k () b" seems most useful to people who could already write "Monads Made Difficult", but not those who are trying to struggle to see how all these connections play out. Monads Made Difficult is difficult, but lays the groundwork for a lot of the categorical machinery that forms the environment where Monads were born. That's important.
- nbouscal 13y agoThat's not a problem with Stephen's approach, that's simply what he was writing about. I found it to be a great concise presentation. Your post and his post have different goals, that does not make either one better than the other.
- tome 13y agoTo all those who are replying to me that I have misinterpreted this as beginner material, well the first sentence does say that it's an "introduction to Haskell monads".
- nbouscal 13y agoNo, the first sentence says that it’s an “introduction to Haskell monads derived from a categorical perspective.” Big difference.
- jfarmer 13y agoAs a counterpoint, I'm fluent in mathematics but know virtually no Haskell. Your write up presumes the exact opposite and is therefore totally unsuitable for someone like me. :) I mean that quite literally: I can make neither heads nor tails of your write up. It looks like a bunch of text with funny characters to me. This write up, OTOH, is written in my language using idioms I understand and relates it to Haskell without involving too much of the machinery of the language.
- reeses 13y agoThis 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 agoNice! 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.
- dasil003 13y ago> The problem [...] is that [...] making it much harder [...] But the title of the article is "Monads made difficult", so I don't understand why you think this is a problem.
- cgag 13y ago"This assumes you are familiar with Haskell typeclasses and basic category theory." Where do I learn basic category theory? Anything better than just perusing wikipedia?
- johnsoncarity 13y agoA college CS or pure math course should cover it.
- cgag 13y agoPerhaps should but didn't (CS).
- jfarmer 13y agoHrm, not sure about that. I know of no college CS course that requires category theory, let alone having the expectation that a decent CS degree will give you exposure to it. As an example, the University of Chicago, my alma mater, doesn't expose CS students to Category Theory. Of all the places it might be where you's expect it the most. It has a notoriously "theoretical" computer science department tied closely to the mathematics department and is where Category Theory was invented. Even in a math degree, one wouldn't typically touch on Category Theory in a classroom setting until one studied Algebraic Topology. Before that students might come across it as a neat sideshow, but nothing they'd be interested in using to solve an actual mathematical problem. There are plenty of students who get a BS in mathematics without ever making a serious go at it. It's also not really that useful as a first-order field of mathematics. It's mostly useful as a way of organizing other mathematical things and as I kind of general vernacular. Even mathematicians sometimes lovingly call it "abstract nonsense". I see CT as a kind of mathematical interstate system. If you want to go back and forth between Chicago to LA it's fantastic, but most of your work is being done in Chicago or LA itself. The interstate system itself isn't that interesting most of the time.
- gjm11 13y agoYes, I agree: you can easily do a bachelor's degree in either mathematics or computer science without ever being taught category theory (though the chances are you'd hear it mentioned a few times). More specific evidence to go with jfarmer's: I read mathematics at the University of Cambridge -- like UChicago, an absolutely first-rate institution and not one that gives its students an easy ride -- and there category theory is not part of the undergraduate curriculum. It is one of the courses you can take in "Part III", which is a one-year taught master's degree[1]. [1] Until very recently, the best way to describe it was "a one-year taught master's degree that inexplicably isn't actually a master's degree". But they've fixed that now.
- marshray 13y agoI love it! I just wish I understood it.
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- nilkn 13y agoI thought Hask was supposed to be the category of Haskell types with functions between them. My Haskell is pretty rusty... how exactly does (->) resolve to this meaning? Also, on a more theoretical level, since Haskell types themselves belong to the category Hask, by definition, to what extent can Haskell truly model category theory when the objects of a category are evidently being represented here as Haskell types?
- quchen 13y ago1. Hask has Haskell types as objects, and functions between them as morphisms. "a" is a type, "b" is a type, and "a -> b" is an arrow between them. 2. Haskell cannot model (the entirety of) category theory. Everything in Haskell is in Hask, and the only way to get to an entirely different category (say Set) is on paper. This great talk gives some insight into the issue: https://vimeo.com/67174266 https://vimeo.com/67174266
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- tel 13y ago1. Hask has exponentials, so (a -> b) is the exponential object final in the category of c's such that we have arrows from a * c to b. 2. It's incorrect that the only category in Haskell is Hask, after all it's easy enough to formulate subcategories of Hask and do your algebra there. I don't know how to answer to what degree you can model category theory, but you can do a lot.
- nilkn 13y agoRegarding 2, you're right of course about subcategories of Hask, but I'm more interested in completely different categories. My question really boils down to this: given the Category class provided in this article, which mathematical categories occur as instances of this Category? Are they necessarily all subcategories of Hask?
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- exDM69 13y agoI would like to give a word of encouragement to anyone who, like me, thought this insight into Monads was all greek. Understanding any of the material in this tutorial is not a prerequisite to writing practical computer programs in Haskell. You don't have to completely understand Monads in order to use them. Just as you learn adding two numbers together in pre-school and learn about axioms of associativity and commutativity later, you should start by learning IO first and think about the more general ideas later. Nevertheless, this was quite an interesting article. Maybe I'll try to put some thought into it and see if I can learn something that would help me level up my Haskell skills.