What is the correct reading of $A+B+C$?

Question

We have many operations where when we write it in a form that does not make it explicit which expressions are the inputs to which operands, for example $1-1-1$ is a form. Is there a name for this kind of form?
Given this expression, we can use the fact that $-$ is left associative so $1-1-1$ has a specific form of a $-$ operation that takes $1-1$ as the left input and $-1$ as the right input giving us $(1-1)-1$.

If we take $A+B+C$, we don't know as clearly if it is $(A+B)+C$ or $A+(B+C)$. Does $A+B+C$ have a standard form that such as subtraction where we can deduce a clear structure where we know which values are the inputs of which operations? Is there a standard way where $+$ is treated as either left or right associative, or do we just accept that due to the general associativity of $+$ that $A+B+C$ will always have a specific value and it will be the same no matter the choice of left or right associativity?

Because it seems strange to have an expression where we know it's value but not precisely the order of operations, does the expression have a specific well-ordered representation or is it a shorthand for any sum?**

In this way is $A+B+C$ the same as saying: 'Any possible sum of $A$, $B$ or $C$.'

I've been told that addition is both left and right associative, which would imply there's two ways of viewing A+B+C.

Answer 1

If we take $A+B+C$, we don't know as clearly if it is $(A+B)+C$ or $A+(B+C)$.

Then $+$ is not properly defined. In most cases, + is the operation in some (abelian) group, thus the operation is associative by definition, hence the placing of () does not play a role.

And even if it would make a difference, () can be dropped, if the operation is to be defined left- or right-associative. Most prominent example is IEEE floating-point arithmetic, where neither + nor * are associative. The exact definition is in the specification of the arithmetic or programming language — or in the mathematical definition.

Because it seems strange to have an expression where we know it's value but not precisely the order of the inputs.

It's not about the order of the inputs, it's about the priority of operations. If it's about the order of operations, then it's about whether the operation is abelien or commutative or symmetric.

Anyway, it's quite common that representations are not unique. Much more common than unique representations. It's just the case that if a representation happens to be unique, then it's usually much more emphasized or even crucial in proofs, like

• Uniqueness of prime factorization of natural numbers (up to ordering of the primes).
• Uniqueness of power series expansion of an analytic function at some point.

Representations like "represent a number as a sum of three numbers" is usually inherently non-unique, and it can be a field of study in it's own right to find how many of specific representations / decompositions there are. See for example partition function.

for example $1−1−1$ is a form. Is there a name for this kind of form?

It's non-associative (in general. And also not abelian in general). Usually such expressions are evalated left-to-right. But there are notable exceptions, like $$a^{b^c} = a^{(b^c)}\neq (a^b)^c = a^{b\cdot c}$$ This might even be the case in programming, for example the above can be expressed in python as

a**b**c = a**(b**c)

which is how Python implements it.