Well, this is certainly blowing up into something bigger than I thought it would. (At one point last night, 12 people had my "Haskell in Plain English" essay/chapter open on Google Docs, a very unintentional side effect.) Me: "I finally got (most of?) what monads really mean for practical programming when none other than Simon Peyton-Jones said F# called a very similar construct "workflow," a word he likes." A response: "Was it literally just a single sentence introducing a new word for a concept that made you "get it"? Could you elaborate? This is really quite remarkable." Richard Eisenberg: "For me, coming from a mostly Java background (but with a healthy dollop of functional programming thrown in the mix -- but no Haskell), the phrase that unlocked monads was "programmable semicolon"." I think one explanation for the Monad Tutorial Explosion (and how it still didn't work for a lot of people) is that there's no single right metaphor that will finally make monads "click". It depends on what combination of conceptions (and perhaps all misconceptions to be dispelled) is floating around in given mind. And minds vary. Workflow is a metaphor at one useful level. "Programmable semicolon" is a metaphor at another useful level. Semicolon as imperative-language separator actually hides a lot that's going on in a real CPU. Even if the imperative language is assembly language, the compiled instructions go to an instruction fetch/decode unit that -- these days especially -- can be engaged in a bewildering number of tasks, including finishing up several previous instructions, translating the decoded instruction to what amounts to microcode that get sent to an instruction cache, and fetching the next few instructions even as it's been decoding the instruction fetched into a little to-do list for the CPU. Perhaps we can think of instruction fetch/decode as "monadal" -- it's wildly (but not chaotically) mapping a previous state of the machine into new one. After all, it's all still Turing machines down there somewhere (but even the Turing machine was mathematically defined as a function that maps one tape-state to the next while it's also mapping its internal state to a new state.) So the semicolon can be thought of as lexically exposing the fetch-decode operator in the CPU's own little program. The "workflow" metaphor is at a higher level. If you think of how incoming data can be sorted and parceled out as if to a number of desks in an office, with forms processed at a desk going from one outbox to another desk's inbox, you're ignoring exact sequencing. It's the view where you're labeling desks by their specialized tasks and drawing outbox-inbox connections. In this view, it matters little that, in the one-instruction-at-a-time CPU model, there's only one "clerk" who (ridiculously, I know) puts a form in an inbox, then takes it out again, puts it in the inbox of another desk, then sits down at that other desk, takes the same form out and does the very specific form-processing task assigned to that desk. You can instead imagine it as some sluggish bureaucracy where everyone else but the single busy clerk is waiting for a form to hit their empty inboxes. The fact is, "one-instruction-at-a-time CPUs" hardly exist anymore except in the cheapest microcontrollers -- there are all multitasking like crazy. And this is because CPU designers work very hard to see how much of the little bureaucracy they can actually keep busy, ideally at all times, but at least at high throughput for average instruction streams, even if 100% utilization is seldom reached. (Lambda, the Ultimate Semicolon?) Bryan: "Listing sequential actions one after the other is so intuitive you see it everywhere. Take baking recipes as one example. Or assembly code. Or poems." Yes, sequencing just grows out of the fact of time. You'd think that poems aren't really sequences of tasks to do. But in a way, they are, and in a sense that matters for my attempts at natural-language understanding. There IS an implicit imperative in the energy required to speak: the subtext of any statement addressed to another person (even when it's dead poet to living reader) is: think about what I'm saying. Each new word in a sentence being spoken maps the listener's mind-state to a new one. The purpose of the sequencing is to build up state in the listener's mind. "When a Haskeller says, "Monads are great! They let you chain effectful actions together!" it can take a very long time to understand they actually mean exactly what they're saying — the usual intuition is that sequencing actions can't possibly be a real problem, so you go round and round in circles trying to understand what "effectful" means, what "action" means, and how these oh-so-important "laws" have anything to do with it." Yes, it's sort of like aliens from outer space with a very alien consciousness, and a different relation to the dimension of time, and they are exulting (to each other, and to themselves, really, more than to you): "Periods at the ends of sentences are great! They enable these strange blobs of protoplasm who call themselves 'human' to more easily map the states of these internal things they call 'neurons' into new states. No wonder we've been having such trouble trying to tell them what to do, using our intuitively obvious gestaltic visual patterns that allow us to grasp, with our quantum-computer brains, a whole meaning at once, through our retinas." (Yes, I am riffing a little here on the SF movie Arrival.) Math is this world where we can feel a little like that. Instead of thinking of a function as a machine that takes inputs, grinds away for a little while, and extrudes outputs, you can think of all inputs and output as already existing, and reason logically from the resulting properties. Haskell's laziness helps you fake what you can't really have: infinite workspace and the ability to do everything in an instant. It's a fake-it-til-you-make-it language, and you don't have to be Buzz Lightyear going to infinity and beyond, perhaps to all the way to the galaxy Aleph-omega. You just have to make a Haskell program do something that people like as much as their Buzz Lightyear doll. The math view is a useful view at one level, but it can only help you so much. I greatly enjoyed the level where I could define a Turing machine as a mapper from one infinite tape to another, then imagine all possible Turing machines as a countably infinite set. I could then prove, with Cantor diagonalization (abstractly, at least, though not very intuitively) that there are things Turing machines can't do. I loved that. But I also listened to grad students mutter about how recursive function theory may be beautiful and all, but it's now classical, and therefore dead for their own practical purposes (getting a PhD, getting onto tenure track.) And by then I knew I wasn't good enough at math to make a go of computing theory in their world, as a career, no matter what. I think Simon Peyton-Jones had a similar experience of academia -- he wasn't a John Hughes -- but he found a way through anyway. Bryan: "I can understand why that [puzzled newbie] viewpoint might be long-forgotten by those who were involved in the effort to solve it. :) And I appreciate that it was solved in such a general way that it can be applied to so many seemingly unrelated things! That's the beauty of mathematics!" Yes, it's there. But in user experience design (and writings about Haskell ARE part of its user experience) there's a saying: the intuitive is the familiar. At some point in gaining sophistication, it's no longer intuitively obvious to you why something wasn't intuitively obvious for you before. You're so absorbed in where you are that you don't remember quite how you got there. You might even assume that everyone who has gotten where you are has gotten there by the same route. And, if you can dimly remember how you got a flash of inspiration, you might end up writing yet another monad tutorial that few people understand. "Workflow" helped me. And now "programmable semicolon" is helping a little, in another way. I presented a strategy at the NSMCon-2021 conference: think of NSM reductive paraphrasing as writing very complete and precise code for natural-language terms and grammar rules, using NSM primes and syntax as a kind of logic programming language -- one that might even be very simply translatable to Prolog. I did some coding experiments and saw a problem: viewed more procedurally -- i.e., with each line of an NSM explication adding to a conceptual schema of a natural-language meaning -- NSM wasn't single-assignment. A Prolog interpreter couldn't just try a match, looking at each statement of an explication in relative isolation, and bail out when a statement failed to be true in the given context of the attempted match. It had to drag some state along just to resolve what the NSM prime "this" referred to. (Think of "it" in GHCI.) OK: effects. But wait, more: scoped effects. You want to strip some schema-stuff back out if a match failed midway through, then try another match somewhere else. OK: I need a context stack to store assertions that succeeded. Uh-oh, Prolog doesn't give me that paper trail. Dang. And I still want the one-for-one translation of sentences in an NSM reductive paraphrase to something like Prolog. OK: I need to grab control of sequencing, so that I can not only use the intrinsic Horn clause resolution without pain. But I also want it to hold onto the assertions that succeeded, and knit them into the growing conceptual schema that's part of the state. AND I want it roll those back out, transactionally, if a match didn't succeed. It turns out that Prolog makes this not too much harder than, say, a Lisp with transactional memory. So: I think I might now return to my Prolog code and see if I was unconsciously reinventing some kind of monad. Because because I felt forced by Prolog's limits to write my own semicolon. It wasn't too hard, though the resulting code required for NSM explications looks clunkier now. (Prolog is very limited in syntax extensibility.) Maybe there's some way in which Haskell makes it easier still, and nicer-looking too. None of which is to say that "programmable semicolon" will immediately help everyone pick the conceptual lock of monads. But for those who just bonk on "programmable semicolon", it might help to point out that the load-operate-store model of CPUs is a lie these days. Say a little about why it's not true. Then again pose semicolon as a kind of operator. People like to think, "OK, this statement will finish, and the one after the semicolon will start." It's a convenient view, even though in fact, while the imperative statement is in progress, the CPU might have already fetched instructions beyond it, and scheduled them for speculative execution. Or, you can tell them, "Just learn lambda calculus, then study monads algebraically, and here's a side dish of category theory while I'm it. Bon appetit." How's that working for you, guys? It doesn't work for me. And I don't think it's because I can't do the math. It's that I often write code to see whether I'm thinking about a problem the right way. And is that so bad? I believe it was in Stewart Brand's "II Cybernetic Frontiers" where a researcher at Xerox PARC, Steve ("Slug") Russell, one of their star hackers, was described by his peers as "thinking with his fingertips." I'm no star hacker, but "workflow" feels like it will make my fingertips smarter. "Programmable semicolon" feels like it will make my fingertips smarter. You think top-down mathematically? Well, bon appetit. It's not to my taste. No single way is likely to work for all. Math is not some fundamental reality. It's not even reality -- in fact, it gains a lot of its power precisely from being imaginary. And it's just a bunch of metaphors too. Regards, Michael Turner Executive Director Project Persephone 1-25-33 Takadanobaba Shinjuku-ku Tokyo 169-0075 Mobile: +81 (90) 5203-8682 turner@projectpersephone.org Understand - http://www.projectpersephone.org/ Join - http://www.facebook.com/groups/ProjectPersephone/ Donate - http://www.patreon.com/ProjectPersephone Volunteer - https://github.com/ProjectPersephone "Love does not consist in gazing at each other, but in looking outward together in the same direction." -- Antoine de Saint-Exupéry