Re:Functions are Vectors
by adrian_b on Sunday July 30, 2023 @04:07AM
(#36928970)
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.
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.