How addition will be if we don't define it only on pairs of numbers? In calculus by Michael spivak  book
he say in page 4

It might seem reasonable to regard addition as an operation which can be performed on several numbers at once and consider the sum $a_1+...+a_n$ of n number $a_1,...,a_n$ as a basic concept . it is more convenient , however to consider addition of pairs of numbers only and to define other sums in terms of sums of this types

My questions :
1- How addition will be if we don't define it only on pairs of numbers ?
2- why it is more convenient , however to consider addition of pairs of numbers only and to define other sums in terms of sums of this types?
 A: How addition will be if we don't define it only on pairs of numbers ?
The short answer is, it will be summation, as in $\sum_{i = 1}^n a_i$.
The long answer is, Spivak was writing in the 1960s, for people fresh out of high school, who hadn't necessarily ever heard of associative or commutative laws, but were familiar with expressions like "$a + b + c$." He's about to show that that expression can be justified by use of the associative law of addition; he's anticipating objections from readers who don't see why it needs to be justified at all.
why it is more convenient , however to consider addition of pairs of numbers only and to define other sums in terms of sums of this types?
Because then you can define the algebraic properties of the real numbers as consequences of the field laws and order laws, Spivak's properties P1-P12, and these involve only the binary operation.
A: Unless I am misunderstanding the question, addition this way would be exactly the same. We don't only define addition on pairs of numbers - we define it on any number of numbers by using the associative property. Because we have defined
$$a_1+a_2$$
Then we can express a sum of any number of terms
$$a_1+a_2+...+a_n$$
as a bunch of sums in the form that we have defined:
$$(((a_1+a_2)+a_3)+...+a_n)$$
The reason this is more convenient is because if we were going to define it for any number of addends, we would have to address an infinite number of cases. But if we simply define $a_1+a_2$ and then create the commutative property, we can let it do the heavy lifting for us, since all other cases, using the commutative property, can be reduced to the case $a_1+a_2.$
I suppose you could try to define it for many addends. But if you did, here's what would happen:
Currently, addition is defined, at the most basic level, as an iterative operation. If $S(n)$ is the successor function
$$S(n)=n+1$$
then addition can be expressed in terms of the successor function
$$a+b=S^b(a)$$
However, trouble arises when you try to do it for three numbers. You would need a "successor function" with multiple arguments, but what would that even represent? Or you could do something like
$$a+b+c=S^{b+c}(a)=S^{S^c(b)}(a)$$
or
$$a+b+c=S^c(a+b)=S^c(S^b(a))$$
but that is the same as pairing up the addends
$$(a+b)+c$$
That's why it is much more convenient just to define it for two addends.
