5 ms·

We don’t allow multiple values for a given input of a function. Ever. This is never done. f(x) = x^2 is not invertible as a real valued function of a real va

We don’t allow multiple values for a given input of a function. Ever. This is never done. f(x) = x^2 is not invertible as a real valued function of a real variable. On the restricted domain of [0, infinity) it is invertible and that inverse is sqrt(x). If you want to talk about negative roots you use the function -1*sqrt(x). You can construct a new function as you did to encapsulate this fact but one never has a function with multiple values for a given input. I don’t know of any mathematician that thinks of functions as allowing them to have multiple values for a given input. You will not find this in the literature.

thaumasiotes 3y ago
Don't be ridiculous. It's a common need, it's easily satisfied, and so it is commonly done. Compare the concept of the "inverse element" of a value given by an invertible function with the "inverse image" of a value given by a non-invertible function. Of course the function that gives you inverse images is nothing more than an inverse function that is allowed to give multiple values. Why do we do that? Because our need to invert functions has nothing to do with whether those functions are one-to-one, and because in most contexts it isn't actually important that functions yield a single value. It's a formalism. There is no high principle that says "collecting multiple values and giving them a collective name that implies they are a single value is metaphysically superior to collecting multiple values and admitting that they are multiple values".
- eternityforest 3y ago
  Seems like a useful formalism though, at least coming from a programming background. Maybe some kind of "it's all of them at once" mental model is useful in math, but object oriented programming gives you a hierarchal mindset, everything is in a container of some sort, so multiple return values having a container like a set makes perfect sense and just a bunch of loose values isn't a very familiar concept. But perhaps if you're actually doing math, things are different?
  - thaumasiotes 3y ago
    It's useful in many places and less useful in others. Over in the programming world, Common Lisp allows you to return multiple values without wrapping them in a container. If you want the primary value, you just treat the function as if it returned one value. If you want additional return values, you use multiple-value-bind. (I believe the general idea is along these lines: your function does a certain amount of computation, producing a set of related values. Most of the time, the caller will be interested in just one of those, which you return as the primary value. But some of the time, the caller will be interested in more than just that one value, and you have to compute the secondary values whether the caller wants them or not, so you return those too.) For a different example of formalism in math, it is conventional to say that there are two boolean logical operators, negation and implication. You can still write about conjunction and disjunction, but everyone understands that when you write "p and q are both true", what you really meant to write was "it isn't the case that the truth of p implies the falsity of q". The point of the formalism is that you can do your proofs by considering negation and implication and then ignoring everything else. (It isn't conventional to say that there's just one logical operator NAND. You might think that would be even better, but the effort saved by only considering how one operation works ends up being less than the extra effort involved in doing proofs about NAND.) The situation with functions is more or less the same thing; at many points we want to rely on the assumption that when a = b, f(a) = f(b). So we define functions that way, and functions that give multiple values have to be treated as giving a single composite value instead. But in a context where you have some value a and what you want to know is "what is f(a)?", the fact that the answer may consist of multiple values won't bother you. Now, I have never seen someone take the position that boolean conjunction and disjunction don't exist as concepts just because that is how logic is normally defined, but the analogous position seems to be more popular for multiple-valued functions.
    - AmpsterMan 3y ago
      I can think of two simple, practical answers to your query: The expressive power of single value functions is very powerful and the constraint is not necessarily restrictive but may even drive a stronger analysis. (Where does this function have multiple returns? Is it for the whole domain? Etc) By contrast the expressive power of negation and implication are relatively low given very intuitive and well defined alternatives exist. Second, there already exist good enough paradigms for dealing with multi-valued functions. Splitting the function up into multiple functions, mapping to an ordered pair, etc. Defining a function the traditional way is more than just notational convenience. The single value constraint allows for many simplifying assumptions, enough that is worth to pay the cost when dealing with relations that you want to talk about in functional ways
- adrian_b 3y ago
  In mathematics (and in programming) it is essential to use non-ambiguous names, i.e. names whose meaning is certain when you see them independently of any context, without having to add an explanation of what is meant by them. The concept of a special kind of mapping between two sets, where to any element of the first set corresponds a unique element of the second set is very important and it needs a special name. The choice of the names is arbitrary and one could use for instance the term "univocal function" to mean a mapping like described above and "function" for any mapping between two sets. Nevertheless, by far the most widespread convention in mathematics is to use the term "relation" for any mapping between two sets and the term "function" only for those relations where to any element of the first set corresponds a unique element of the second set. There exists no reason for not following this convention, from which it also results that an invertible function is a function where for any element of the second set corresponds a unique element of the first set, so this convention also provides a simple meaningful name for another important concept that needs a special name. The functions in programming languages that return multiple values, unless they return partially or totally random values (in which case they are not functions of only the input arguments, but also of an internal state or of time), are just functions that return values which belong to the set that is the Cartesian product of the types of the individual values. So the name "function" is usually correct in the mathematical sense even for such functions. If they had not been functions, the programmer would not have known what values to return, when writing them. Moreover, I disagree that in most contexts when you want to invert a function "it isn't actually important that functions yield a single value". In the overwhelming majority of the cases that appear in engineering and science when you want to solve equations a.k.a. to invert functions, you want to obtain a unique solution that can be directly implemented in practice. Whenever you cannot obtain a unique solution, you need to add extra criteria that allow the selection of a unique solution that is usable. Those extra criteria are actually equivalent with transforming the original non-invertible function into a function that can be inverted.
- cykotic 3y ago
  There is no high principle other than the agreed upon definition of what a function is. Of course the function that gives you inverse images is nothing more than an inverse function that is allowed to give multiple values. You can’t use the word function for something that isn’t a function. It makes no sense to say inverse function that is allowed to give multiple values. Hence the need for terms like pre-image of a set. In all contexts it is important that a function yield a single value for a given input because that is the definition of the word.
tsimionescu 3y ago
The point GP was making is that you can also define the function sqrt(x) as a function from reals to pairs of reals, and say that the sqrt(4) is the pair (-2, 2). With a proper definition of squaring a pair of reals, you could even have this be the inverse of the square function (we could define square((x, y)) = x²-xy+y², so square((-2,2))=4-4+4=4). Of course, these are still functions which take exactlt one element in their input set (the reals for sqrt, or pairs of reals for square) and return for each a single element in their output sets (the pairs of reals for sqrt, or the reals for square).
bigbillheck 3y ago
> You will not find this in the literature. It would have cost you nothing except a minute or so of your time to have refuted yourself before posting: https://arxiv.org/abs/1912.08274 https://arxiv.org/abs/1912.08274 https://arxiv.org/abs/1703.01700 https://arxiv.org/abs/1703.01700 (See also: https://en.wikipedia.org/wiki/Multivalued_function https://en.wikipedia.org/wiki/Multivalued_function)
- cykotic 3y ago
  A multivalued function is not a function. Hence the need to call it something else. In complex analysis you have branch cuts and things like that. We don’t call something a function that doesn’t have a unique output for a given input. EDIT: Complex analysis was developed well before the formal definition of functions, relations, set theory. As such there are legacy terms in use. Mathematics has a lot of abuse of notation. For instance x=2 can mean assignment or an equation to solve. It depends on context. In each branch of a multi-function the mapping is a function and, as a convenience, a single term “multi-function” is used to encapsulate all this. It cuts out the verbosity. A multi-function is not a function though. My pedantry in this comes from the blog post. It was poorly written and left out some, from a mathematical point of view, important details. For instance the author uses a Schauder basis and has an uncountable sum at one point whilst ignoring the need distinguish between a Hamel basis and Schauder basis or the need to use a linear ordering in the reals to properly make sense of the uncountable sum.
  - bigbillheck 3y ago
    A set-valued function is a perfectly valid kind of function, and a multivalued function in complex analysis is just a set-valued function with certain continuity and other restrictions.
    - cykotic 3y ago
      If set up that way then yes it meets the definition of a function. I’m not a complex analyst but I imagine they think of a function from C to C on a given branch rather than multi-function being a function from C to the countable product of C (or however one wants to formally define the set value). I think this because they tend to care about analytic properties and saying function from C to countable product of C includes far too many functions. Saying multifunction will immediately make one think of branch cuts. They don’t think of Log(z) as a set of values. At least I don’t. I assume one means the principal branch when writing Log(z) unless specifically stated otherwise. Maybe in their mind they do think of an infinite set when seeing Log(z). I’m skeptical of this possibility though.

thaumasiotes 3y ago

Don't be ridiculous. It's a common need, it's easily satisfied, and so it is commonly done. Compare the concept of the "inverse element" of a value given by an invertible function with the "inverse image" of a value given by a non-invertible function. Of course the function that gives you inverse images is nothing more than an inverse function that is allowed to give multiple values. Why do we do that? Because our need to invert functions has nothing to do with whether those functions are one-to-one, and because in most contexts it isn't actually important that functions yield a single value. It's a formalism. There is no high principle that says "collecting multiple values and giving them a collective name that implies they are a single value is metaphysically superior to collecting multiple values and admitting that they are multiple values".

eternityforest 3y ago
Seems like a useful formalism though, at least coming from a programming background. Maybe some kind of "it's all of them at once" mental model is useful in math, but object oriented programming gives you a hierarchal mindset, everything is in a container of some sort, so multiple return values having a container like a set makes perfect sense and just a bunch of loose values isn't a very familiar concept. But perhaps if you're actually doing math, things are different?
- thaumasiotes 3y ago
  It's useful in many places and less useful in others. Over in the programming world, Common Lisp allows you to return multiple values without wrapping them in a container. If you want the primary value, you just treat the function as if it returned one value. If you want additional return values, you use multiple-value-bind. (I believe the general idea is along these lines: your function does a certain amount of computation, producing a set of related values. Most of the time, the caller will be interested in just one of those, which you return as the primary value. But some of the time, the caller will be interested in more than just that one value, and you have to compute the secondary values whether the caller wants them or not, so you return those too.) For a different example of formalism in math, it is conventional to say that there are two boolean logical operators, negation and implication. You can still write about conjunction and disjunction, but everyone understands that when you write "p and q are both true", what you really meant to write was "it isn't the case that the truth of p implies the falsity of q". The point of the formalism is that you can do your proofs by considering negation and implication and then ignoring everything else. (It isn't conventional to say that there's just one logical operator NAND. You might think that would be even better, but the effort saved by only considering how one operation works ends up being less than the extra effort involved in doing proofs about NAND.) The situation with functions is more or less the same thing; at many points we want to rely on the assumption that when a = b, f(a) = f(b). So we define functions that way, and functions that give multiple values have to be treated as giving a single composite value instead. But in a context where you have some value a and what you want to know is "what is f(a)?", the fact that the answer may consist of multiple values won't bother you. Now, I have never seen someone take the position that boolean conjunction and disjunction don't exist as concepts just because that is how logic is normally defined, but the analogous position seems to be more popular for multiple-valued functions.
  - AmpsterMan 3y ago
    I can think of two simple, practical answers to your query: The expressive power of single value functions is very powerful and the constraint is not necessarily restrictive but may even drive a stronger analysis. (Where does this function have multiple returns? Is it for the whole domain? Etc) By contrast the expressive power of negation and implication are relatively low given very intuitive and well defined alternatives exist. Second, there already exist good enough paradigms for dealing with multi-valued functions. Splitting the function up into multiple functions, mapping to an ordered pair, etc. Defining a function the traditional way is more than just notational convenience. The single value constraint allows for many simplifying assumptions, enough that is worth to pay the cost when dealing with relations that you want to talk about in functional ways
adrian_b 3y ago
In mathematics (and in programming) it is essential to use non-ambiguous names, i.e. names whose meaning is certain when you see them independently of any context, without having to add an explanation of what is meant by them. The concept of a special kind of mapping between two sets, where to any element of the first set corresponds a unique element of the second set is very important and it needs a special name. The choice of the names is arbitrary and one could use for instance the term "univocal function" to mean a mapping like described above and "function" for any mapping between two sets. Nevertheless, by far the most widespread convention in mathematics is to use the term "relation" for any mapping between two sets and the term "function" only for those relations where to any element of the first set corresponds a unique element of the second set. There exists no reason for not following this convention, from which it also results that an invertible function is a function where for any element of the second set corresponds a unique element of the first set, so this convention also provides a simple meaningful name for another important concept that needs a special name. The functions in programming languages that return multiple values, unless they return partially or totally random values (in which case they are not functions of only the input arguments, but also of an internal state or of time), are just functions that return values which belong to the set that is the Cartesian product of the types of the individual values. So the name "function" is usually correct in the mathematical sense even for such functions. If they had not been functions, the programmer would not have known what values to return, when writing them. Moreover, I disagree that in most contexts when you want to invert a function "it isn't actually important that functions yield a single value". In the overwhelming majority of the cases that appear in engineering and science when you want to solve equations a.k.a. to invert functions, you want to obtain a unique solution that can be directly implemented in practice. Whenever you cannot obtain a unique solution, you need to add extra criteria that allow the selection of a unique solution that is usable. Those extra criteria are actually equivalent with transforming the original non-invertible function into a function that can be inverted.
cykotic 3y ago
There is no high principle other than the agreed upon definition of what a function is. Of course the function that gives you inverse images is nothing more than an inverse function that is allowed to give multiple values. You can’t use the word function for something that isn’t a function. It makes no sense to say inverse function that is allowed to give multiple values. Hence the need for terms like pre-image of a set. In all contexts it is important that a function yield a single value for a given input because that is the definition of the word.

tsimionescu 3y ago

The point GP was making is that you can also define the function sqrt(x) as a function from reals to pairs of reals, and say that the sqrt(4) is the pair (-2, 2). With a proper definition of squaring a pair of reals, you could even have this be the inverse of the square function (we could define square((x, y)) = x²-xy+y², so square((-2,2))=4-4+4=4). Of course, these are still functions which take exactlt one element in their input set (the reals for sqrt, or pairs of reals for square) and return for each a single element in their output sets (the pairs of reals for sqrt, or the reals for square).

bigbillheck 3y ago

> You will not find this in the literature. It would have cost you nothing except a minute or so of your time to have refuted yourself before posting: https://arxiv.org/abs/1912.08274 https://arxiv.org/abs/1912.08274 https://arxiv.org/abs/1703.01700 https://arxiv.org/abs/1703.01700 (See also: https://en.wikipedia.org/wiki/Multivalued_function https://en.wikipedia.org/wiki/Multivalued_function)

cykotic 3y ago
A multivalued function is not a function. Hence the need to call it something else. In complex analysis you have branch cuts and things like that. We don’t call something a function that doesn’t have a unique output for a given input. EDIT: Complex analysis was developed well before the formal definition of functions, relations, set theory. As such there are legacy terms in use. Mathematics has a lot of abuse of notation. For instance x=2 can mean assignment or an equation to solve. It depends on context. In each branch of a multi-function the mapping is a function and, as a convenience, a single term “multi-function” is used to encapsulate all this. It cuts out the verbosity. A multi-function is not a function though. My pedantry in this comes from the blog post. It was poorly written and left out some, from a mathematical point of view, important details. For instance the author uses a Schauder basis and has an uncountable sum at one point whilst ignoring the need distinguish between a Hamel basis and Schauder basis or the need to use a linear ordering in the reals to properly make sense of the uncountable sum.
- bigbillheck 3y ago
  A set-valued function is a perfectly valid kind of function, and a multivalued function in complex analysis is just a set-valued function with certain continuity and other restrictions.
  - cykotic 3y ago
    If set up that way then yes it meets the definition of a function. I’m not a complex analyst but I imagine they think of a function from C to C on a given branch rather than multi-function being a function from C to the countable product of C (or however one wants to formally define the set value). I think this because they tend to care about analytic properties and saying function from C to countable product of C includes far too many functions. Saying multifunction will immediately make one think of branch cuts. They don’t think of Log(z) as a set of values. At least I don’t. I assume one means the principal branch when writing Log(z) unless specifically stated otherwise. Maybe in their mind they do think of an infinite set when seeing Log(z). I’m skeptical of this possibility though.