> Even though they may take any number of arguments, they must each have one return value. [...] This is true of all functions.
Mathematical functions take one value and produce one value. This is true of all mathematical functions.
Programming functions can be modelled this way by treating multiple input arguments as a single product, and non-scalar outputs containing multiple values as a single product.
Programming functions don't even have to always return one value:
- Haskell's Void type
- Rust's ! type
- TypeScript's never type
all have zero values, meaning the function can't meaningfully return.
Arguably, programming functions must take at least one value as input, otherwise how can they be called?
In that sense, programming functions are asymmetric: It rarely makes sense to write a function you can't call, but it often makes sense to write a function that never returns.
When does it make sense to write a function that you can't call? When the point of the function is to prove something as a result of being compiled. The value lies in the compilation, not in being called.
A better title for this article: Some Functions Are Asymmetric.
> A better title for this article: Some Functions Are Asymmetric.
If some functions are asymmetric, then functions (as a class of things) are not symmetric.
Just as some rectangles are squares, still leaves rectangle as a class/type as transpose asymmetric. I.e. width and height are not constrained to be equivalent, which is required for transpose symmetry.
Not really arguing for or against your comment. Just noting that it’s easy for multiple valid arguments to pass each other without connecting due to slight differences in what people mean with the same words.
Functions map members of a set A to members of a set B. These can simply be Cartesian products whose members are tuples. In my dream PL syntax a function call would be a function name followed by a tuple, and that tuple would be no different than the tuples you would use in any other part of the program (and so you could use all the tuple manipulation library goodies). If the function preserves any other structure of the type, like an identity element, that could be stated so you could have morphisms. And that identity element or other properties could be declared just as other stuff like 'const' are declared, and since the compiler can't verify all these stated properties, it's on the user to provide correct info, just like it's on the user to write a correct program, so nothing lost here, and anything more, like verification by the compiler, would be a bonus.
Mathematicians have been packing all this stuff nicely for a couple of centuries now, maybe we could use more of their work on mainstream computing, and it could also be a nice opportunity to get more people to appreciate math and structure.
Something that has side effects all over the place should just not be called a function, but something else, maybe "procedure" would be an appropriate, clear term.
Haskell is much like this? A function like `borp :: a -> b` maps from type a to type b. If you want to have side effects like mutable state, you need to encode that in the function signature, like `borpWithState :: a -> State s b`, where s is the type of the mutable state.
In this case it's almost the opposite of most programming languages. In (say) Ruby or Java, any function or method can do anything; write to stdout, throw exceptions, access the network, mutate global state, etc. In haskell, a function can only do calculations and return the result by default. All the other things are still possible, but you do have to encode it in the type of the function.
EDIT: The annotations you mention with regards to identity elements etc do exist, but they live mostly on the data structures rather than on the functions that operate on those data structures.
Languages with dependent types (Agda, Idris, Coq, Lean) seem even closer to the description, with a user supplying proofs of properties. For instance, as in idris2-algebra [1]. One can similarly define isomorphisms, or other kinds of morphisms, by listing their properties, and requiring proofs to construct.
Thanks. Maybe I should have stressed this more, but what I had in mind was a more mainstream language suitable for everyday programming too. All the languages mentioned in this thread are great but they are not getting any more popular and you would not use them to build, say, a game or a linear algebra library, right?
Well, at least Idris aimed to become practical, and there is the CompCert compiler in Coq (called Rocq now, actually), demonstrating viability of large verified projects. Besides, one does not have to verify everything in those, and could program more or less as in Haskell (which may also seem impractical, but it is used for all sorts of things, and is my primary language these days, for both work and hobby). I think the primary issues with those have to do with their unpopularity, and particularly lack of libraries. Though C libraries are usually just an FFI call away, so even such smaller languages are not that painful if you try to simply get things done in those.
I recall there being at least SDL2 bindings for Idris (not to mention those for Haskell, which also has game libraries), and some linear algebra libraries in those languages (complete with verification), but probably not particularly extensive.
They are not the most practical choice if you do not need verification, but if you would like to use languages like that, they are available and usable, but with additional effort/drawbacks. I wish they were more mature and had a better infrastructure, too, but that would take people pushing them to that point.
Math is great and should be well studied by programmers, but in general I oppose this idea. Mathematicians define things the way they do because they have neither flip-flops nor do they have a defined execution method as part of their foundational system. These two things radically change the interaction we have with any given formal system put on top of it.
> Functions map members of a set A to members of a set B.
> Something that has side effects all over the place should just not be called a function
Leibniz defines functions as a quantity that depends on some geometry like a curve. Bernoulli later defined it as a quantity that results from a variable. The latin word "functio" means process, not implying a mapping but an arbitrary sequential performance. Mathematicians are prone to taking words from elsewhere, either twisting their meaning or inventing wholly new meaning out of thin air, all according to their whimsy for their own particular needs. I do not think a reasonable case can be made to assert we have to respect ZFC's narrow conception of a function when we do not live in a ZFC world.
>Mathematicians are prone to taking words from elsewhere, either twisting their meaning or inventing wholly new meaning out of thin air, all according to their whimsy for their own particular needs.
True but one benefit of those guys is that they actually define what they mean in a formal way. "Programmers" generally don't. There is in fact some benefit in having consistent names for things, or if not at least a culture in which concepts have unambiguous definitions which are mandated.
There was one attempt at creating a language splitting both pure function and effectful procedures. Any construct with a procedure call was automatically/effectively typed as a procedure. But I can't recall the name so far..
I'm pretty sure OCaml functions are monadic. They don't really accept tuples (unless you intentionally do something weird).
Rather, if you define (in your mind) a function of three variables, the compiler makes that a function of one variable that returns another function. And that return-value function takes one variable and returns a third function. And that third function takes one variable and returns the result of the triadic function you intended to write.
That's why the type of a notionally-but-not-really triadic function is a -> b -> c -> d and not (a, b, c) -> d. It's a function of one variable (of type a) whose return value is of type b -> c -> d.
OCaML has tuples like all the other MLs. Thus given a function f with a signature (int * int) -> int, invocation is written as something like:
f (1, 2)
So yes, they do accept tuples. This isn't weird, and is the standard way to write a multi-argument function in most of the ML family. In the 5th stage baudrillardian mess that OCaML has become in 2025 it might be weird since at some point they added implicit currying to the language and that became the default idiom. But personally I hate implicit currying and avoid it whenever possible.
It's been a while since I used OCaml or SML, so I went back to check. You're correct that in OCaml functions of multiple parameters are idiomatically represented as curried functions:
It's a bit harder to tell the situation in SML, because it isn't really an actively used language. "ML for the Working Programmer" (lol) appears to idiomatically represent them as functions a single tuple parameter:
> SML [...] appears to idiomatically represent them as functions of a single tuple parameter
Tuple is not special, though. Functions accept a single argument of any type.
To use the argument in the function body you can name it, or you can use any valid pattern to destructure it and bind parts of it to local variables.
Tuple is just one of the valid patterns. Coincidentally, it looks like argument list with positional arguments in many other languages. You can also use a record, which makes it look like "keyword arguments". You can also use patterns of custom types.
All the above is still about the single "argument" case, single value that is passed to the function. The pattern matching is what makes it possible to bind parts of it to multiple local variables in the body of the function.
> In my dream PL syntax a function call would be a function name followed by a tuple, and that tuple would be no different than the tuples you would use in any other part of the program
So PRQL (prql-lang.org) is kind of like that, with the limitation that control flow is limited to the List Monad bind, i.e. the tuples from one step are piped to the function call in the next step one at a time producing 0..* result tuples and the resulting multiset is flatmapped. At the moment it just transpiles to SQL but a couple of months ago I was exploring different Lambda Calculi and how to extend this to a more general PL. Alas, that won't take shape until AI is at the level that it can write that code for me. I guess LINQ and similar Language Integrated Query Languages already provide this functionality.
P.S. Writing the above made me think that it's not quite what you asked for; in the PRQL case each function receives an implicit `this` argument which is the tuple I was thinking of. However the function can also take other arguments, including keyword arguments. Those are arbitrary. I guess they are implicitly ordered and could be represented as a tuple as well. What would you see as the benefit of that?
> (and so you could use all the tuple manipulation library goodies)
Other than indexing into tuples, I can't really think of anything else, at least for single tuples. I initially thought of something like `zip(*args)` but that's only really useful when you have list of tuples or tuple of lists and then you're back in PRQL land. Indexing into tuples is also brittle and does not produce self-documenting code so I prefer the PRQL and SQL namedtuples/structs where fields are referenceable by name.
I have this suspicion that PRQL functions are parameterised natural transformations but my Category Theory at that level is too rusty to check without extra work. If that's the case though then having the explicit function arguments be simple values feels justified to me since they're just indexing families of related transformations and are not the primary data being transformed (if that makes sense?).
> Functions map members of a set A to members of a set B. These can simply be Cartesian products whose members are tuples.
Well, a function can't be a Cartesian product unless set B has cardinality 1. It's perfectly coherent to view a function as a set of tuples, but it's not legal for that set to contain two tuples (a, b) and (a, c) where b ≠ c.
> In my dream PL syntax a function call would be a function name followed by a tuple, and that tuple would be no different than the tuples you would use in any other part of the program (and so you could use all the tuple manipulation library goodies).
This already exists. For example, that's how `apply` works in Common Lisp.
> Maybe that doesn’t seem strange. It’s just how we’re used to functions working. Ah, those steeped in functional programming might say, but maybe this is the wrong way to look at it. Because if we curry functions and use partial application, we can say that they always have one argument and return one value, and then they are symmetric.
Ah, those steeped in functional programming might say, but maybe this is the wrong way to look at it. Because if we represent the represent the functions with explicit continuations, we can say that they have N arguments and pass N arguments to their continuations, and then they are symmetric.
It seems like this has fertile overlap with Scheme and the (concurrent computatation) Actor model.
Of course, I can imagine the Execution control library authors know full well about those, with existing C++ goals and designs making that a bridge too far.
The range can be a product type, as can the domain. Most languages are expressive enough that you can create the product type (struct). You're right on point.
No, bijective functions are symmetric.
Injectivity is enough to invert a function.
> Even though they may take any number of arguments, they must each have one return value. [...] This is true of all functions.
Mathematical functions take one value and produce one value. This is true of all mathematical functions.
Programming functions can be modelled this way by treating multiple input arguments as a single product, and non-scalar outputs containing multiple values as a single product.
Programming functions don't even have to always return one value:
all have zero values, meaning the function can't meaningfully return.Arguably, programming functions must take at least one value as input, otherwise how can they be called?
In that sense, programming functions are asymmetric: It rarely makes sense to write a function you can't call, but it often makes sense to write a function that never returns.
When does it make sense to write a function that you can't call? When the point of the function is to prove something as a result of being compiled. The value lies in the compilation, not in being called.
A better title for this article: Some Functions Are Asymmetric.
Which is less of a profound insight.
If some functions are asymmetric, then functions (as a class of things) are not symmetric.
Just as some rectangles are squares, still leaves rectangle as a class/type as transpose asymmetric. I.e. width and height are not constrained to be equivalent, which is required for transpose symmetry.
Not really arguing for or against your comment. Just noting that it’s easy for multiple valid arguments to pass each other without connecting due to slight differences in what people mean with the same words.
Mathematicians have been packing all this stuff nicely for a couple of centuries now, maybe we could use more of their work on mainstream computing, and it could also be a nice opportunity to get more people to appreciate math and structure.
Something that has side effects all over the place should just not be called a function, but something else, maybe "procedure" would be an appropriate, clear term.
In this case it's almost the opposite of most programming languages. In (say) Ruby or Java, any function or method can do anything; write to stdout, throw exceptions, access the network, mutate global state, etc. In haskell, a function can only do calculations and return the result by default. All the other things are still possible, but you do have to encode it in the type of the function.
EDIT: The annotations you mention with regards to identity elements etc do exist, but they live mostly on the data structures rather than on the functions that operate on those data structures.
[1] https://github.com/stefan-hoeck/idris2-algebra
I recall there being at least SDL2 bindings for Idris (not to mention those for Haskell, which also has game libraries), and some linear algebra libraries in those languages (complete with verification), but probably not particularly extensive.
They are not the most practical choice if you do not need verification, but if you would like to use languages like that, they are available and usable, but with additional effort/drawbacks. I wish they were more mature and had a better infrastructure, too, but that would take people pushing them to that point.
> Functions map members of a set A to members of a set B.
> Something that has side effects all over the place should just not be called a function
Leibniz defines functions as a quantity that depends on some geometry like a curve. Bernoulli later defined it as a quantity that results from a variable. The latin word "functio" means process, not implying a mapping but an arbitrary sequential performance. Mathematicians are prone to taking words from elsewhere, either twisting their meaning or inventing wholly new meaning out of thin air, all according to their whimsy for their own particular needs. I do not think a reasonable case can be made to assert we have to respect ZFC's narrow conception of a function when we do not live in a ZFC world.
True but one benefit of those guys is that they actually define what they mean in a formal way. "Programmers" generally don't. There is in fact some benefit in having consistent names for things, or if not at least a culture in which concepts have unambiguous definitions which are mandated.
C++ sort of has this, with const.
Rather, if you define (in your mind) a function of three variables, the compiler makes that a function of one variable that returns another function. And that return-value function takes one variable and returns a third function. And that third function takes one variable and returns the result of the triadic function you intended to write.
That's why the type of a notionally-but-not-really triadic function is a -> b -> c -> d and not (a, b, c) -> d. It's a function of one variable (of type a) whose return value is of type b -> c -> d.
OCaML has tuples like all the other MLs. Thus given a function f with a signature (int * int) -> int, invocation is written as something like:
f (1, 2)
So yes, they do accept tuples. This isn't weird, and is the standard way to write a multi-argument function in most of the ML family. In the 5th stage baudrillardian mess that OCaML has become in 2025 it might be weird since at some point they added implicit currying to the language and that became the default idiom. But personally I hate implicit currying and avoid it whenever possible.
https://ocaml.org/docs/values-and-functions#defining-functio...
It's a bit harder to tell the situation in SML, because it isn't really an actively used language. "ML for the Working Programmer" (lol) appears to idiomatically represent them as functions a single tuple parameter:
https://www.cl.cam.ac.uk/~lp15/MLbook/PDF/chapter2.pdf
Tuple is not special, though. Functions accept a single argument of any type.
To use the argument in the function body you can name it, or you can use any valid pattern to destructure it and bind parts of it to local variables.
Tuple is just one of the valid patterns. Coincidentally, it looks like argument list with positional arguments in many other languages. You can also use a record, which makes it look like "keyword arguments". You can also use patterns of custom types.
All the above is still about the single "argument" case, single value that is passed to the function. The pattern matching is what makes it possible to bind parts of it to multiple local variables in the body of the function.
So PRQL (prql-lang.org) is kind of like that, with the limitation that control flow is limited to the List Monad bind, i.e. the tuples from one step are piped to the function call in the next step one at a time producing 0..* result tuples and the resulting multiset is flatmapped. At the moment it just transpiles to SQL but a couple of months ago I was exploring different Lambda Calculi and how to extend this to a more general PL. Alas, that won't take shape until AI is at the level that it can write that code for me. I guess LINQ and similar Language Integrated Query Languages already provide this functionality.
P.S. Writing the above made me think that it's not quite what you asked for; in the PRQL case each function receives an implicit `this` argument which is the tuple I was thinking of. However the function can also take other arguments, including keyword arguments. Those are arbitrary. I guess they are implicitly ordered and could be represented as a tuple as well. What would you see as the benefit of that?
> (and so you could use all the tuple manipulation library goodies)
Other than indexing into tuples, I can't really think of anything else, at least for single tuples. I initially thought of something like `zip(*args)` but that's only really useful when you have list of tuples or tuple of lists and then you're back in PRQL land. Indexing into tuples is also brittle and does not produce self-documenting code so I prefer the PRQL and SQL namedtuples/structs where fields are referenceable by name.
I have this suspicion that PRQL functions are parameterised natural transformations but my Category Theory at that level is too rusty to check without extra work. If that's the case though then having the explicit function arguments be simple values feels justified to me since they're just indexing families of related transformations and are not the primary data being transformed (if that makes sense?).
Well, a function can't be a Cartesian product unless set B has cardinality 1. It's perfectly coherent to view a function as a set of tuples, but it's not legal for that set to contain two tuples (a, b) and (a, c) where b ≠ c.
> In my dream PL syntax a function call would be a function name followed by a tuple, and that tuple would be no different than the tuples you would use in any other part of the program (and so you could use all the tuple manipulation library goodies).
This already exists. For example, that's how `apply` works in Common Lisp.
https://www.lispworks.com/documentation/HyperSpec/Body/f_app...
Ah, those steeped in functional programming might say, but maybe this is the wrong way to look at it. Because if we represent the represent the functions with explicit continuations, we can say that they have N arguments and pass N arguments to their continuations, and then they are symmetric.
It seems like this has fertile overlap with Scheme and the (concurrent computatation) Actor model.
Of course, I can imagine the Execution control library authors know full well about those, with existing C++ goals and designs making that a bridge too far.
// In typescript const [a, b, c] = foo(d, e, f)
You could even pass this to itself
foo(…foo(d, e, f))
Also one definition of a function is a map from a domain to a range. There’s nothing that forbids multiple values, or is there?
[1]: https://simon.peytonjones.org/verse-calculus/
Monty Python fan detected :D