[semolinapudding]

This makes me wonder whether the usual order for function application is an artifact of the S-V-O sentence structure in western European languages. Maybe if Euler had been raised in a language with a S-O-V sentence structure, we would write (x)f instead.

Plus, reverse function application/composition works better with our tendency to write the domain of the function before its range. Given f : A -> B and g : B -> C, the composite function is written as g o f which is mildly annoying.

[schoen]

> Maybe if Euler had been raised in a language with a S-O-V sentence structure, we would write (x)f instead.

This is an interesting question but it's further complicated by how frequently Euler was writing in Latin, which usually prefers (but doesn't require) SOV order.

I just looked up a random text by Euler (https://scholarlycommons.pacific.edu/euler-works/298/) and I see

> Genus autem secundum eiusmodi problemata complectetur, ad quae soluenda coordinatae ad duo curuae puncta pertinentes simul considerari debent; cuiusmodi erat problema de traiectoriis reciprocis, illudque problema catoptricum, quod ante aliquot annos tractavi. Cum enim in his continuo bina curuae puncta inter se conferantur, et coordinatae ad ea pertinentes in computum ingrediantur; per principium continuitatis effici debet, ut bina haec puncta ad eandem lineam curuam referantur, sicque aequatio inter coordinatas unicum punctum spectantes eliciatur.

I put all of the clauses' main verbs in italics. All of them except for "erat" ('was') are at the ends of their respective clauses! So, Euler had quite a lot of experience writing SOV sentences in technical contexts.

[schoen]

I just had another thought about this.

Programmers are pretty accustomed to thinking of functions as verbs. In some programming styles, we're even explicitly encouraged to name our functions after verbs describing what they do.

Sometimes mathematicians think of functions more as expressing a complete (and sort of timeless) relationship between a domain and a range. That relationship, considered in its totality, can be thought of as a noun rather than a verb.

Earlier mathematicians might also have tended to think of functional notation as referring to individual specific instances of the result of the calculation that the function refers to, like "the sine of 1/2" being a specific number (the result of computing the sine of one half, or, alternatively, the output of the function given a specific input). So when writing something like sin(1/2) they might not be thinking "please [person or machine] perform this computation on the number 1/2 right now" so much as "[I am here indicating] the specific [timeless and inherent] number that is the sine of 1/2". That's a different way that a function could be seen as a noun, essentially seeing the function as merely a way of referring to its output.

Calculus and real analysis start to give stronger reasons for thinking of functions as abstract objects in their own right (e.g. because we can say something like "d/dx sin(x) = cos(x)" or even "the derivative of sin is cos" without thinking about specific values. Or questions like "can a function exist that is continuous but not differentiable?". Maybe computer science then starts to give reasons for thinking of functions as descriptions of how to perform them (like source code or even machine code punched on punch cards) as opposed to references to their specific results (like a printed book of function tables, which someone else calculated ahead of time so you wouldn't have to).

This is all to say basically that maybe it's more natural for us as programmers to think of functions as verbs, but maybe Euler didn't conceive of them that way at all!

[susam]

Even in modern mathematics, the postfix notation is used in some places. One example I can recall is permutations.

Consider the following permutation:

  f = (1 2 3 4)
      (1 4 2 3)

This is typically written in cycle-notation like this:

  f = (1)(243)

If we look at the cycle notation, f maps each symbol to the one on the right. For example, it maps 1 to 1, 2 to 4, 4 to 3, and 3 to 2.

Now if we want to find out what happens to the sequence (4 3 2 1) when we apply f to it, we normally write it as

  (3 4 2 1)f

The argument comes first. The function name comes next. It makes it quite convenient to work with permutations, composition of permutations, etc. For example, the result for the above application is:

  (3 4 2 1)f = (3 4 2 1)(1)(243) = (2 3 4 1)

[wging]

You see it in some treatments of abstract algebra too. (Which are not unrelated to permutations in any case.) I think at least one book used in my undergrad used postfixes for function application, but I can't remember which.

[JadeNB]

> This makes me wonder whether the usual order for function application is an artifact of the S-V-O sentence structure in western European languages. Maybe if Euler had been raised in a language with a S-O-V sentence structure, we would write (x)f instead.

Herstein's Topics in algebra uses that notation for function application. It's surely not a coincidence that he was Polish, and that that reminds one of RPN.