Ambiguity of addition

Question

In the case of addition, we are taught from early to just put a $+$ between the addends, 'this is $2+2$ this means $4$' and hence we define the value of $2+2$, however, I've seen 'addition' defined as a function, and the notation as $+(a,b)$ for functional notation.

This creates an issue, if different functions have different domains, they define different functions, so if we see $+$ as a function, how do we know which function is defined if we are just given a formula like:

$y=x+c$

My previous understanding is that 'addition is addition', but clearly this is not the case if my operator represents a function, I'm aware that 'addition' is defined differently for vectors, numbers etc, but this 'function' way of viewing it means that $+$ could be different functions even if we are just using real numbers.

Is the definition of the $a+b$ notation equivalent to $+(a,b)$? in which case how do we get to the situation that we can say $3+2=5$ is it that for all functions $+$ of different domains, that are defined at $(3,2)$ will have the value of $5$ so say the value of $3+2=5$?

Answer 1

Yes, this is one of the conventions most people follow writing math. However, it is almost always clear from context what is meant.

This is not only the case for binary operators though, any function can be "reused" in different contexts, for example, as you pointed out how vector addition is simply addition done component-wise, take any function $f$ that is defined on a field $\mathbb{F} $ and you implicitly have a function defined on any $\mathbb{F} $-vector space by simply applying $f$ to all the components of a vector (or more generally, transform the linear combination of a vector with regards to a basis by applying $f$ to the coefficients)

Granted, I have only actually seen this done in Machine Learning literature, but the point still stands.

A more common abuse of notation that is more traditionally done in maths is the equivalent application of a function $f$ defined on a set $A$ to the set $A$.

Answer 2

Addition is a function but it's not any function. Unless explicitly said otherwise, $+$ is the usual addition of numbers.

Answer 3

If you see something of the form $A + B,$ generally the objects $A$ and $B$ will both be members of some domain $D$ and the addition function $+: D\times D\to D$ will already have been defined.

If the domain is a commonly-used domain such as pairs of natural numbers, integers, rational numbers, real numbers, complex numbers, quaternions, vectors or matrices of real or complex numbers, etc., in most contexts (other than an introductory textbook to the subject) it may be assumed that the reader is already familiar with the definition of addition usually used on that domain. Therefore the addition operator/function is often used without giving a definition.

The properties of the usual domains of $+$ are such that we are guaranteed that the result will always be in the domain, hence the function is well-defined.

I don't know if anyone ever dares to redefine $+$ differently than the usual definition when the domain of $+$ is one of the commonly-used ones. So there isn't much chance for ambiguity. The larger source of ambiguity is the varied uses of the names of the domains: for example, someone might say the sum of two complex numbers is a real number when what they really mean is that the result is a complex number with zero imaginary part.