When people talk about "algebraic reasoning", explanations fall flat because we neglect to first figure out kind of reasoning we're contrasting it against. When we write computer code using an operational model, we do think about what we write — so what manner of reasoning are we using? Can we give it a name? And can we explain why it seems so incompatible with the sort of reasoning that Lambda Man is always going on about?
I propose that the culture of programming at present may be divided into two approaches, explained by the following competing conceptions of the act of programming:
- An operational programmer goes into the computer;
- An algebraic programmer remains outside the computer.
We all have some need to shift between the two perspectives, but many of us become more entrenched in one or the other.
Who calls the calls
To start piecing together the orientation of each kind of programmer with respect to the computer, I'd like to look at a curious question of agency regarding function calls.
- When I write the expression "f x", I may describe my act of programming by saying that "we apply the function f to an argument x".
- When the machine executes the program, I expect that "it will evaluate the function f at the argument x".
But this separation between my action as an author and the machine's action as an automaton is only so distinct in the parlance of an algebraic programmer. If I were a Python programmer:
- When I write the expression "f(x)", I am "calling the function f".
- When the machine executes my program, I expect that the Python interpreter will "call the function f".
So who calls the function: me or Python? The answer is both; I am Python, and its actions are my actions, regardless of whether I am present and typing into a REPL or whether I have scripted them out ahead of time in a program that may run in my absence.
One with the machine
In either variety of programming, we sometimes put ourselves in the shoes of the machine to reason about the anticipated outcome of what we write. But how this imagination works depends on a great deal on whether we are outside or in. When we trace an operational program flow, the text of the program forms a space we can move within, and each variable is a statue that comes alive and begins to talk. Algebraic expression evaluation is a much more sterile and dull affair, and we remain seated in our desk chair. Evaluating is rewriting an expression in another form; it does not take us into different headspace from the one in which we wrote the code in the first place.
Removing the lime from the coconut
When an experienced inside-the-computer author begins in a programming language that forces us to approach programs from the outside, we can expect the question: If "I have an
IO String", then "how do I get to the
While others have already addressed this question in detail, what I want to draw attention to here is that the misunderstanding originates from trying to apply the inside conceptual mapping to a programming model that is strongly outside. An
IO String is a process that produces a
String result. So if I were standing inside a Haskell program, holding such a thing in my hands, it stands to reason that I could run the process and get the string. But we do not have such values because we do not go inside to get anything. We remain at the text editor, writing definitions. One such definition might be for a process which consists of the machine 1. first running some
IO String process; and then 2. doing some other action with the resulting string.
What you got in that room
The term "global variable" reveals something interesting about our mental picture. Such a variable does not span the globe, nor even a local network. To what scope does a word so grand as "global" refer? Humbly, the scope of a process. Or perhaps an entire machine, if I am a kernel developer. When I code operationally, I reside in a tiny world — the landmasses on my little blue marble are the memory segments to which I have access.
When I switched from operational to algebraic programming, first I learned that there are no global variables, then that terminology began to fade from consciousness altogether. As a Haskell programmer, I'm not in a little globe on the desk; I live on the Earth and I type definitions. Among those definitions may be a datatype that represents the state of a process, true. But this datatype is not my world, and the vast majority definitions I write in service of the program will not be functions of it.
Private "member variables" in a Java class can only be accessed from within the class. Perhaps the preposition can be taken to refer somewhat literally to the lexical scope of the class definition — that is, the code that is written between the opening and closing braces. But do we employ a deeper container metaphor here? Maybe this one is just me, but I see the instance as a room, and the members as the stuff I have at my disposal when I'm working inside that room.
In Haskell we also have lexical scoping, as well as a notion of modules with definitions that are either exported or not exported, which for many purposes mirrors the public/private field distinction. But I do not have the experience of mentally going inside a module in the same sense as reasoning inside of a Java instance. I believe it is Java's coupling of modules with mutable state that encourages this spacial reasoning.
Getting your steps in
When you use a step debugger, you actually step into the program! This is true regardless of whether you are using the debugging facilities of Python or Haskell. Though the code may be algebraic, when we use a step debugger we are always looking at it from an operational perspective.
Working on documents
Lately I like to refer to my role as "author" more than "programmer" — regardless of whether the file extension is ".md" or ".hs". It's because I don't feel like I work inside a computer anymore. I work sitting at a computer, I write about programs, and — although much of what I write can be executed by a machine — I do not often become lost within, because I remain safely on the outside.