Binary Operators

There is something that has bothered me for a long time in mathematics: what is up with the number 2? Why do we have "and" and "or" (conjunction and disjunction) in logic and type theory, meet and join in lattice theory, unions and intersections in topology (and set theory), Cartisian products and disjoint unions as polynomial data types (and set theory), products and sums in abstract algebra (sometimes we even have two "layers" of two operators which take two arguments, like in an algebra), and categorical products and coproducts in category theory. There are two things going on here- why are these binary operators (in the sense that they act on two objects) and why are there always two of them (why are they paired up)?

The reason this interests me is that we can easily define unary operators, or ternary, or what-ever-ary we want, so why is binary so common? Also, certainly not all structures have binary operators, and just having one doesn't always imply having the other, in general. For some, its because of their relationship with another pair of operators, like how lattice theory is historically a model theoretic view of truth-valued logic. However, the point is that there are often the two binary operators, not four ternary operators or one 5-ary operator or anything else. I don't know why this is the case, but I will take a stab at it.

One of the reasons is that its often really just one operator and its dual. This is particularly clear in category theory, where they are dual constructs, but similar duality conditions seem to hold for the others (perhaps not exactly? I don't know). This covers why two operators, but why are they binary?

The answer seems to be that they aren't, really. They can be given as binary operators, but as long as they are associative there is no ambiguity when defining them over finite lists of objects. Some can even be defined over infinite lists/sets. Giving them as binary operators is easy to define, but we can just as well define X V Y (with V the join) as the join of X and Y as we can define V{X, Y} as the join of all elements of a set (in this case X and Y).

So why two binary operators? It seems like we are really just looking at some operation, its dual, and the special case of defining it over two things instead of groups of things, which is just for convenience.