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They’re both inverses: one with respect to multiplication, the other with respect to function composition. In abstract algebra, we observe that there are many

They’re both inverses: one with respect to multiplication, the other with respect to function composition. In abstract algebra, we observe that there are many types of “products”: multiplication, addition, function composition, composition of rotations, matrix multiplication, etc. A common, unified “power” notation for repeatedly taking the product of a single element, or of its inverse, has some value. There is some ambiguity, since multiplication of functions can also often make sense. This is usually resolved by context, or by being explicit where the context isn’t clear.

lisper 6y ago
> one with respect to multiplication, the other with respect to function composition And, of course, multiplication is a function so one is just a specific instance of the other. Infix notation is another blight on the mathematical notation landscape. The amount of human effort that has been put into figuring out how to parse a+b*c is staggering. All of this confusion could have been avoided if we'd just started with s-expressions in the first place.
- DavidSJ 6y ago
  Multiplication is a function of numbers, and function composition is a function of functions, so neither is an instance of the other (unless you take the unusual perspective of thinking of numbers themselves as functions, but I don’t think that’s what you meant).
  - lisper 6y ago
    > I don’t think that’s what you meant Why not? Multiplication is a composition of addition, and addition is a composition of the successor function. The successor function is not a composition of anything, but you can define it in terms of set theory if you don't want to simply accept it as a primitive. But sets are functions too.
    - JadeNB 6y ago
      In both of these instances, I think that you are using composition to mean iteration. Iterating is repeated self-composition, so it is composition, but it seems likely that you meant the more specific term. (For example, "the successor function is not a composition of anything" is definitely false in the literal sense of composition. It's also false in the literal sense of iteration, but there it's clear what you meant: there's no natural operation which we iterate some number of times greater than 1 to get the successor function.)
      - deleted 6y ago
        [deleted]
      - lisper 6y ago
        > you meant the more specific term No, I meant what I said. The fact that iteration is a kind of composition is true, but it's a tangent from the point I was trying to make, which is that infix notation is a Really Bad Idea. No one in their right mind would use it if they were not indoctrinated into it. > "the successor function is not a composition of anything" is definitely false That's news to me. I genuinely thought that successor could legitimately be considered a primitive. What is successor a composition of?
        JadeNB 6y ago
        > Multiplication is a composition of addition, and addition is a composition of the successor function. The successor function is not a composition of anything, but you can define it in terms of set theory if you don't want to simply accept it as a primitive. But sets are functions too. > the point I was trying to make, which is that infix notation is a Really Bad Idea Not to be snarky, but, reading these two (from your two thread-successive posts https://news.ycombinator.com/item?id=23312725 https://news.ycombinator.com/item?id=23312725 and https://news.ycombinator.com/item?id=23314776 https://news.ycombinator.com/item?id=23314776) in succession, I still can't see anything about the first one that indicates the point that infix notation is a bad idea. Not that I'm disagreeing with the point, just that I can't find it in the first post. Could you clarify the connection? > What is successor a composition of? It's a composition of, for example, itself and the identity function, like everything else; or you could view it as a composition (-1) . (+2). Those kind of silly solutions are why I thought you meant 'iteration'. Iterative solutions are easy if you don't restrict yourself to natural numbers—for example, (+1) is (+(1/2)) composed with itself—but that's clearly not what you meant. (As soon as you leave the natural numbers, even the idea that addition is iterated successor becomes false.) If you do so restrict yourself, then it becomes true that the successor is not a non-trivial iterate. (I just skated the edge of claiming the opposite in my post, but avoided error by not specifying what domain I meant. That's just luck, though; I meant a particular thing, and I was wrong. To prove it, supposing you start your natural numbers at 0 and that f is a function such that f^{\circ k} is the successor function for some k > 1, then note that f is injective (because a composition power is). If f(0) = 0, then succ(0) = f(f(0)) = f(0) = 0, which is a contradiction. Put n = f(0) and note that f^{\circ n k}(0) = succ^n(0) = n, but n k > 1.) By the way, you quoted (https://news.ycombinator.com/item?id=23314776 https://news.ycombinator.com/item?id=23314776): > > you meant the more specific term Just to be clear, what I said (https://news.ycombinator.com/item?id=23314530 https://news.ycombinator.com/item?id=23314530) was "I think you meant the more specific term". And I was wrong, but I intentionally didn't just assume I knew you what you meant!

lisper 6y ago

> one with respect to multiplication, the other with respect to function composition And, of course, multiplication is a function so one is just a specific instance of the other. Infix notation is another blight on the mathematical notation landscape. The amount of human effort that has been put into figuring out how to parse a+b*c is staggering. All of this confusion could have been avoided if we'd just started with s-expressions in the first place.

DavidSJ 6y ago
Multiplication is a function of numbers, and function composition is a function of functions, so neither is an instance of the other (unless you take the unusual perspective of thinking of numbers themselves as functions, but I don’t think that’s what you meant).
- lisper 6y ago
  > I don’t think that’s what you meant Why not? Multiplication is a composition of addition, and addition is a composition of the successor function. The successor function is not a composition of anything, but you can define it in terms of set theory if you don't want to simply accept it as a primitive. But sets are functions too.
  - JadeNB 6y ago
    In both of these instances, I think that you are using composition to mean iteration. Iterating is repeated self-composition, so it is composition, but it seems likely that you meant the more specific term. (For example, "the successor function is not a composition of anything" is definitely false in the literal sense of composition. It's also false in the literal sense of iteration, but there it's clear what you meant: there's no natural operation which we iterate some number of times greater than 1 to get the successor function.)
    - deleted 6y ago
      [deleted]
    - lisper 6y ago
      > you meant the more specific term No, I meant what I said. The fact that iteration is a kind of composition is true, but it's a tangent from the point I was trying to make, which is that infix notation is a Really Bad Idea. No one in their right mind would use it if they were not indoctrinated into it. > "the successor function is not a composition of anything" is definitely false That's news to me. I genuinely thought that successor could legitimately be considered a primitive. What is successor a composition of?
      - JadeNB 6y ago
        > Multiplication is a composition of addition, and addition is a composition of the successor function. The successor function is not a composition of anything, but you can define it in terms of set theory if you don't want to simply accept it as a primitive. But sets are functions too. > the point I was trying to make, which is that infix notation is a Really Bad Idea Not to be snarky, but, reading these two (from your two thread-successive posts https://news.ycombinator.com/item?id=23312725 https://news.ycombinator.com/item?id=23312725 and https://news.ycombinator.com/item?id=23314776 https://news.ycombinator.com/item?id=23314776) in succession, I still can't see anything about the first one that indicates the point that infix notation is a bad idea. Not that I'm disagreeing with the point, just that I can't find it in the first post. Could you clarify the connection? > What is successor a composition of? It's a composition of, for example, itself and the identity function, like everything else; or you could view it as a composition (-1) . (+2). Those kind of silly solutions are why I thought you meant 'iteration'. Iterative solutions are easy if you don't restrict yourself to natural numbers—for example, (+1) is (+(1/2)) composed with itself—but that's clearly not what you meant. (As soon as you leave the natural numbers, even the idea that addition is iterated successor becomes false.) If you do so restrict yourself, then it becomes true that the successor is not a non-trivial iterate. (I just skated the edge of claiming the opposite in my post, but avoided error by not specifying what domain I meant. That's just luck, though; I meant a particular thing, and I was wrong. To prove it, supposing you start your natural numbers at 0 and that f is a function such that f^{\circ k} is the successor function for some k > 1, then note that f is injective (because a composition power is). If f(0) = 0, then succ(0) = f(f(0)) = f(0) = 0, which is a contradiction. Put n = f(0) and note that f^{\circ n k}(0) = succ^n(0) = n, but n k > 1.) By the way, you quoted (https://news.ycombinator.com/item?id=23314776 https://news.ycombinator.com/item?id=23314776): > > you meant the more specific term Just to be clear, what I said (https://news.ycombinator.com/item?id=23314530 https://news.ycombinator.com/item?id=23314530) was "I think you meant the more specific term". And I was wrong, but I intentionally didn't just assume I knew you what you meant!