3 ms·

I didn’t think you meant that numbers are functions because you said: “multiplication is a function so one [multiplication] is just a specific instance of the o

I didn’t think you meant that numbers are functions because you said: “multiplication is a function so one [multiplication] is just a specific instance of the other [function composition]”. Thus I thought your point was that, since multiplication is a function, it’s a form of function composition. But that wouldn’t follow, for the reasons I said. As for multiplication being “composed” addition, addition being “composed” succession, etc.: The multiplication function (at least of integer arguments) is composed of addition. Function composition is a function that returns a composition of two other functions. The way you’re using the term blurs the distinction between the thing that is composed and the thing that does the composing. Function composition is a specific operation that takes two functions, f and g, and returns the function f∘g defined by the behavior (f∘g)(x) = f(g(x)). The function which does function composition is not itself a composed function, and the act of multiplication is the act of a composed function, but not an act of function composition.

lisper 6y ago
> The way you’re using the term blurs the distinction between the thing that is composed and the thing that does the composing. Yes, that is intentional. Both of these things are functions. Even numbers are functions, they just happen to be functions of zero arguments, or functions that ignore their arguments, or functions whose value is constant regardless of what the argument(s) is(are). It's all the same thing. That is the whole point. Numbers and addition and multiplication happen to be particularly important functions, but they are not structurally different from any other functions. Giving them special notation, especially when you are first introduced to them, obscures this fact. This kind of mental damage is very hard to recover from in later life. I believe it's one of the reasons so many people think they hate math. Math can be beautiful and elegant, but the standard notation used for school-book arithmetic is arbitrary and perverse, a bizarre accident of history with no actual merit. IMHO of course. [UPDATE:] > Function composition is a specific operation that takes two functions, f and g, and returns the function f∘g defined by the behavior (f∘g)(x) = f(g(x)). Function composition is a function, no different from any other function. There is no more reason to use infix notation for it than there is for any other function. In fact, if you drop the infix notation it immediately becomes obvious how ubiquitous and non-special function composition actually is: compose(f,g)(x) = compose(f)(g)(x) = f(g(x)) On that view, the COMPOSE function is actually the identity function!
- deleted 6y ago
  [deleted]
- DavidSJ 6y ago
  You seem to be jumping around in terms of what position you're taking. Is times(a, b) function composition because a and b are functions, or is times(a, b) function composition because it's composition of the addition function, or is times(a, b) function composition because multiplication is a function? Those are very different assertions, with very different implications to the original discussion, and so far as I can tell you’ve made all three of them. Maybe they're all true, maybe one or two of them is true, maybe it's a matter of perspective which are true. But I don't think you're being clear or consistent in which assertion your position is based on. And no matter what, there's still a difference between composing a with b, and composing b with itself a times, which is what I mean by the distinction between composition and iteration (sort of like the distinction between a brick and a brick wall). At this point I feel we're going around in circles, so I'll bow out.
  - lisper 6y ago
    > Those are very different assertions But they are not mutually exclusive. That's the whole point.

lisper 6y ago

> The way you’re using the term blurs the distinction between the thing that is composed and the thing that does the composing. Yes, that is intentional. Both of these things are functions. Even numbers are functions, they just happen to be functions of zero arguments, or functions that ignore their arguments, or functions whose value is constant regardless of what the argument(s) is(are). It's all the same thing. That is the whole point. Numbers and addition and multiplication happen to be particularly important functions, but they are not structurally different from any other functions. Giving them special notation, especially when you are first introduced to them, obscures this fact. This kind of mental damage is very hard to recover from in later life. I believe it's one of the reasons so many people think they hate math. Math can be beautiful and elegant, but the standard notation used for school-book arithmetic is arbitrary and perverse, a bizarre accident of history with no actual merit. IMHO of course. [UPDATE:] > Function composition is a specific operation that takes two functions, f and g, and returns the function f∘g defined by the behavior (f∘g)(x) = f(g(x)). Function composition is a function, no different from any other function. There is no more reason to use infix notation for it than there is for any other function. In fact, if you drop the infix notation it immediately becomes obvious how ubiquitous and non-special function composition actually is: compose(f,g)(x) = compose(f)(g)(x) = f(g(x)) On that view, the COMPOSE function is actually the identity function!

deleted 6y ago
[deleted]
DavidSJ 6y ago
You seem to be jumping around in terms of what position you're taking. Is times(a, b) function composition because a and b are functions, or is times(a, b) function composition because it's composition of the addition function, or is times(a, b) function composition because multiplication is a function? Those are very different assertions, with very different implications to the original discussion, and so far as I can tell you’ve made all three of them. Maybe they're all true, maybe one or two of them is true, maybe it's a matter of perspective which are true. But I don't think you're being clear or consistent in which assertion your position is based on. And no matter what, there's still a difference between composing a with b, and composing b with itself a times, which is what I mean by the distinction between composition and iteration (sort of like the distinction between a brick and a brick wall). At this point I feel we're going around in circles, so I'll bow out.
- lisper 6y ago
  > Those are very different assertions But they are not mutually exclusive. That's the whole point.