Why does the author prefer function names after arguments?

Question

As you can see above, the author prefers to write functions after elements, which is contrary to the century old practice of writing arguments after the functions. I wonder why he does that? Is there a technical advantage of doing so?

Answer 1

The paragraph you quote strongly suggests that the paragraph(s) immediately above it describes a notational problem that he claims derives from the tradition of writing the function name before the argument. You will have to read those paragraphs in order to get the knowledge you seek.

Answer 2

As the preceding paragraph says:

If $f,g\in S_A = \operatorname{Aut}_{\text{Set}}(A)$, then the ‘product’ of $f$ and $g$ should be written $g\circ f$ and should act as follows: $$(\forall p\in A): g\circ f(p) = g(f(p))$$ But the prevailing style of notation in group theory would write this element as $fg$, apparently reversing the order in which the operation is performed.

(Algebra, Part 0, Paulo Aluffi, p. 50)

If this isn't clear, perhaps you should ask a question about the particular aspect that you find unclear.

Answer 3

Elaborating on what Omnomnomnom says in his comment to your original post, if you compose multiple functions in a row, say $f$, $g$, and $h$, first applying $f$ to $p$, then applying $g$ to the result of that, then applying $h$ to the result of that, you write $h\circ g\circ f (p)$, so they are written on the opposite order (according to our writing conventions) to how they are applied.

Compare this with how you would write out addition of $a$, $b$, and $c$. It would be $a+b+c.$ Similarly, with multiplication of $a,$ $b,$ and $c$, you write $a\times b\times c.$ The terms are written from left to right in each case. Function composition is the rebellious one. It's an annoying outlier (to some people anyway) in the more general context of groups.

(Things would also fall into line if we started writing everything else in the reverse order and maintained the convention for function composition, but one cannot change centuries old traditions.)