The author of the CT/math book I'm reading (Paolo Aluffi) suggested that if functions defs put output first (

f : B <- A

instead of

f : A -> B

) then function composition like

f . g

would make more sense. Then a couple chapters later he goes on to say that writing

(x)f

instead of

f(x)

would make

(f.g)(x)

be

((x)g)f

(or even maybe

xgf

) Both are nice suggestions, but the mix really does hurt my head

If you read older algebra books, people actually do this. I think herstein's book does it. The context switching I think is why people have mostly settled on one notation

De Bruijn Notation is a variant notation for lambda calculus where you write

(x)f

to call a function. The wikipedia page describes how this simplifies the algebraic manipulation of lambda terms, and makes certain proofs easier. https://en.wikipedia.org/wiki/De_Bruijn_notation

I think another perspective is evaluation order / dominance of expressions. For example, in a lazy language like Haskell, an expression like

f x

- is "dominated" by

f

which may choose to evaluate or not evaluate

x

at all, and in any case

f

and its application are evaluated first. So it makes sense to see things ordered by their importance:

f (g (h x)).

In an eager language, it's nicer to see things read by their evaluation order:

x |> g |> f

(though

f

or

g

as function yielding expressions are actually be evaluated first, their application is evaluated later).

The Forth is strong.