There is an essential difference between binary functions and unary functions.

With binary functions you can compose them using a very complex composition graph.

With unary functions you can compose them only linearly, so in general it is impossible to make a binary function with unary functions.

You can make binary functions from unary functions only by using at least one other binary function. For instance, you can make multiplication from squaring, but only with the help of binary addition/subtraction.

So the one function that can be used to generate the others by composition must be at least binary, in order to be able to generate functions with an arbitrary number of parameters.

This is why in mathematics there are many domains where the only required primitives are a small number of binary functions, but there is none where strictly unary functions are sufficient. (However, it may be possible to restrict the binary functions to very simple functions, e.g. making a tuple from components, for instance the CONS function of LISP I.)

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thaumasiotes 3 hours ago | parent [-]

What are you responding to?

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adrian_b 3 hours ago | parent [-]

I think that you may have replied before I saved my entire response, so I am not sure how much of it you had read before replying yourself.

I have replied to your last statement:

> "you can use the second parameter of a binary function to identify a unary function just as you can use the fourth parameter of a quaternary function to identify a trinary one."

As I have explained above, what you propose does not work. It works in functions with 3 or more parameters, but it does not work in binary functions, because you cannot make binary functions from unary functions (without using some auxiliary binary functions).

	▲	thaumasiotes 2 hours ago \| parent [-]
		> As I have explained above, what you propose does not work. It works in functions with 3 or more parameters, but it does not work in binary functions, because you cannot make binary functions from unary functions (without using some auxiliary binary functions). I have no idea what you're trying to say. If you can use one parameter to identify a desired function, then obviously you can use a function of arity n+1 to define as many functions of arity n as you want, and it doesn't matter what the value of n is. For example: selector(3, "sin") = sin 3 selector(3, "log2") = log₂ 3 This works going from arity 4 to arity 3, and it also works going from arity 2 to arity 1. Your "response" talks about going from arity 1 to arity 2, a non sequitur.