6 ms·

> x^{-1} means \frac{1}{x}, but f^{-1}(x) doesn't mean \frac{1}{f(x)}. That notation for inverse functions is truly appalling. I don't know how the first mathe

by twic 6y ago

> x^{-1} means \frac{1}{x}, but f^{-1}(x) doesn't mean \frac{1}{f(x)}. That notation for inverse functions is truly appalling. I don't know how the first mathematician to think of that didn't immediately discard it as nonsensical and misleading.

abnry 6y ago
But the inverse function is the "multiplicative inverse" in the group of functions with composition as "multiplication". In that way, it makes a ton of sense. It's only a problem because you are mixing together two group operations.
StavrosK 6y ago
> I don't know how the first mathematician to think of that didn't immediately discard it as nonsensical and misleading. Maybe they did.
DavidSJ 6y ago
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?
ColinWright 6y ago
There is logic to be had. Think of "x" as being the process/operation of "multiply by x". Now you want to invert that, so you multiply instead by x^{-1}. Raising to the power of "-1" means that we are inverting the operation in question, so inverting the application of function "f" to argument "x" will be to apply f^{-1} to x instead. Now I'm not saying it's good notation, but some of the fields I've worked in, the notation is not only OK, it's genuinely empowering, as a good notation should be. I have a lot to say about notation in math, but this isn't the forum. There are a lot of crimes committed, agreed, but some of the things people pick out are only bad because they don't know the context where they redeem themselves.
- wcarey 6y ago
  > I have a lot to say about notation in math, but this isn't the forum. There are a lot of crimes committed, agreed, but some of the things people pick out are only bad because they don't know the context where they redeem themselves. Indeed - I wish we could conceal those notations until the redemptive context became more apparent, in the same way that Latin teachers can conceal Sallust's inconcinitas until students are ready for it.
rocqua 6y ago
It's not because 'function powers' make sense, yhey are just iterated function application. That's how they work for the natural numbers and when you extend that to the integers, you immediately get f^-1 for the inverse. Notation in higher level maths is almost always very ambiguous. Because many concepts are analogues of each other and to reflect that notation is just taken from the analogue. Within a single domain, (like high-school arithmetic) you will usually not have this ambiguous problem. But once you move past that, it is something to get used to.
- wcarey 6y ago
  The idea that "when you extend that" is something human beings can do is an absolute revelation to young mathematicians. The idea that our notations are, at least in some sense, choices that we make that come with tradeoffs is a huge point of mathematical maturity for them, and usually causes my students to look back with either deep awe or deep suspicion when they realize that even our choice of base 10 is a choice among many and we can make other choices. As von Neumann said, very much something to get used to.
- JadeNB 6y ago
  > It's not because 'function powers' make sense, yhey are just iterated function application. To be clear, I think you aren't saying "It's not because 'function powers' make sense" (which seems to apply that's not the reason it's done, and possibly that the reason isn't correct), but rather "It's not [an appalling notation], because 'function powers' make sense"—to me, that extra comma changes the meaning!
  - rocqua 6y ago
    Indeed!
MaxBarraclough 6y ago
A good example of this can be found in an incredible algorithm with a terrible name: Fast inverse square root. It sounds like an algorithm for computing the inverse of the square root function, so one might think it's for squaring non-negative numbers, or something along those lines. Not so. It's for computing the reciprocal (the 'multiplicative inverse') of the square root of a number. [0] Related to this: the way a superscript '2' next to a function means the function shall be applied twice (that is, composition with itself)... unless it's a trigonometric function, in which case it means the square of the result. [1] [2] [0] https://en.wikipedia.org/wiki/Fast_inverse_square_root https://en.wikipedia.org/wiki/Fast_inverse_square_root [1] https://en.wikipedia.org/wiki/Function_composition#Functional_powers https://en.wikipedia.org/wiki/Function_composition#Functiona... [2] https://en.wikipedia.org/wiki/List_of_trigonometric_identities#Pythagorean_identities https://en.wikipedia.org/wiki/List_of_trigonometric_identiti...
- saagarjha 6y ago
  That one is clear to me, although it might be for a reason you find displeasing: if the thing being computed was an "inverse square root" in the sense of a functional inverse, then you'd be computing a square and it would make much more sense to call it that instead.
simiones 6y ago
To me it always seemed pretty natural. \frac{1}{f(x)} would be (f(x))^{-1}, by analogue to x^{-1}+1 vs (x+1)^{-1}. I would have been very surprised if raising just the 'f' part of the function application expression to some power were to mean raising the whole expression to that power. I also would expect f^{2}(x) to mean f(f(x)), not (f(x)) * f(x)) (which would be (f(x))^2).

by twic 6y ago

abnry 6y ago

But the inverse function is the "multiplicative inverse" in the group of functions with composition as "multiplication". In that way, it makes a ton of sense. It's only a problem because you are mixing together two group operations.

StavrosK 6y ago

> I don't know how the first mathematician to think of that didn't immediately discard it as nonsensical and misleading. Maybe they did.

DavidSJ 6y ago

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?

ColinWright 6y ago

There is logic to be had. Think of "x" as being the process/operation of "multiply by x". Now you want to invert that, so you multiply instead by x^{-1}. Raising to the power of "-1" means that we are inverting the operation in question, so inverting the application of function "f" to argument "x" will be to apply f^{-1} to x instead. Now I'm not saying it's good notation, but some of the fields I've worked in, the notation is not only OK, it's genuinely empowering, as a good notation should be. I have a lot to say about notation in math, but this isn't the forum. There are a lot of crimes committed, agreed, but some of the things people pick out are only bad because they don't know the context where they redeem themselves.

wcarey 6y ago
> I have a lot to say about notation in math, but this isn't the forum. There are a lot of crimes committed, agreed, but some of the things people pick out are only bad because they don't know the context where they redeem themselves. Indeed - I wish we could conceal those notations until the redemptive context became more apparent, in the same way that Latin teachers can conceal Sallust's inconcinitas until students are ready for it.

rocqua 6y ago

It's not because 'function powers' make sense, yhey are just iterated function application. That's how they work for the natural numbers and when you extend that to the integers, you immediately get f^-1 for the inverse. Notation in higher level maths is almost always very ambiguous. Because many concepts are analogues of each other and to reflect that notation is just taken from the analogue. Within a single domain, (like high-school arithmetic) you will usually not have this ambiguous problem. But once you move past that, it is something to get used to.

wcarey 6y ago
The idea that "when you extend that" is something human beings can do is an absolute revelation to young mathematicians. The idea that our notations are, at least in some sense, choices that we make that come with tradeoffs is a huge point of mathematical maturity for them, and usually causes my students to look back with either deep awe or deep suspicion when they realize that even our choice of base 10 is a choice among many and we can make other choices. As von Neumann said, very much something to get used to.
JadeNB 6y ago
> It's not because 'function powers' make sense, yhey are just iterated function application. To be clear, I think you aren't saying "It's not because 'function powers' make sense" (which seems to apply that's not the reason it's done, and possibly that the reason isn't correct), but rather "It's not [an appalling notation], because 'function powers' make sense"—to me, that extra comma changes the meaning!
- rocqua 6y ago
  Indeed!

MaxBarraclough 6y ago

A good example of this can be found in an incredible algorithm with a terrible name: Fast inverse square root. It sounds like an algorithm for computing the inverse of the square root function, so one might think it's for squaring non-negative numbers, or something along those lines. Not so. It's for computing the reciprocal (the 'multiplicative inverse') of the square root of a number. [0] Related to this: the way a superscript '2' next to a function means the function shall be applied twice (that is, composition with itself)... unless it's a trigonometric function, in which case it means the square of the result. [1] [2] [0] https://en.wikipedia.org/wiki/Fast_inverse_square_root https://en.wikipedia.org/wiki/Fast_inverse_square_root [1] https://en.wikipedia.org/wiki/Function_composition#Functional_powers https://en.wikipedia.org/wiki/Function_composition#Functiona... [2] https://en.wikipedia.org/wiki/List_of_trigonometric_identities#Pythagorean_identities https://en.wikipedia.org/wiki/List_of_trigonometric_identiti...

saagarjha 6y ago
That one is clear to me, although it might be for a reason you find displeasing: if the thing being computed was an "inverse square root" in the sense of a functional inverse, then you'd be computing a square and it would make much more sense to call it that instead.

simiones 6y ago

To me it always seemed pretty natural. \frac{1}{f(x)} would be (f(x))^{-1}, by analogue to x^{-1}+1 vs (x+1)^{-1}. I would have been very surprised if raising just the 'f' part of the function application expression to some power were to mean raising the whole expression to that power. I also would expect f^{2}(x) to mean f(f(x)), not (f(x)) * f(x)) (which would be (f(x))^2).