math, are those conditions in the definition of a field, natural?

So, in algebra, you have a field. It's a set and 2 functions f and g, of the form f(a,b) and g(a,b), and X nesting properties of f and g (called commutativity, associativity, distributive, invertible, etc).

So, what happens if we have more than 2 functions, 3 parameters each? So, such study is called universal algebra. (i haven't studied, but i wonder what happens there, in general. The mix of nesting of function, i gather, would create more complex concept similar to associativity and distributive, involving 3 functions, but we don't have a name.)

But, WHAT is the fundamental nature, that real number (a field) is this specific X nesting properties? What is it, that real numbers, which we consider as naturally occurring or developed, form this “field” with such specific nesting properties?

am thinking, there must be some logical answer.

to describe my questions further, for example, real number developed because, first we have counting, 1, 2, 3, then naturally we developed 0, then we have rational, which is ratio, e.g. 1/2 as cutting a pie, then we have negative numbers (from, say, I OWE YOU). And from rational we discovered irrational, as in pythagorean. So there, we have real numbers. And, addition came from simple counting. Multiplication can be considered as a short for repeated additions.

so, am guessing, addition, and multiplication (repeated addition) necessitates the commutativity, associativity, distributive, properties?

now, having written this out, it seems obvious and is the answer to my own question.

... because, by looking at the definition of field, i've always thought, it's somehow arbitrary and complicated. I'd be interested, in a systematic approach of studying structures, e.g. a set, with n operation of m-arity, starting with n=1 and m=1. Then, we develop, all possible ways of nesting n such functions of m arity, so that associativity, commutativity (order of arg), distributivity, are just 3 of the possible properties.

but i gather that, universal algebra may began like this, but actually has become a bit something else.
similar situation is group theory

Group, is much simpler than field. But if you look at definition of group, you see that, it seems also arbitrary and complex.

But then, if you look at symmetry, such as symmetry of polyhedra, you see that all the requirement of group is necessary, and no more, no less.

but the question remain, why is the group definition seems arbitrary?

i mean, is there some point of view, so the associativity (a • b) • c = a • (b • c) requirement, dissolved as if it is natural?

again, i really like to see, a combinatorial exploration of all such possible condition of n functions of m arity. (as in universal algebra)

perhaps after seeing that, then one can judge, comparatively, whether the associativity condition is natural.

now, to be sure, function with 1 or 2 arity is actually perhaps the most natural. And 1 or 2 functions is also pretty bare. As opposed to, a structure with function that has 3 arity, or more than 2 functions.

If you just have 1 function, or just function wit 1 arity, than it may become too simple to have interesting things going on.

but again, would like to see a systematic combinatorial list of the conditions that may arise of n functions and m arity.
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