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Does anyone know how historically we came to notate functions as f(x) instead of (x)f? For example g(f(x)) instead of ((x)f)g or for matrices B A x instead of x A B.

After all (in the Western world) we write from left to right, but in mathematics we notate operations from right to left. Also if anyone of Arabian origin sees that, does it mean you don't have that particular confusion?

J. M. ain't a mathematician
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Danvil
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  • $f$ is applied to $x$, so $f$ goes first (i.e. on the left, reading from the left). – Dave Nov 13 '17 at 00:28
  • Also, when we read $f(x)$ verbally, we often say "f of x", so we start with $f$ on the left and put $(x)$ to the right of it to be consistent with our reading from left to right of "f of x". – Dave Nov 13 '17 at 00:31
  • If you look at g(f(x)), f is evaluated before g. So the operation which comes first in writing is last in execution. – Danvil Nov 13 '17 at 00:32
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    In this case, $g$ is applied to $f(x)$, and $f(x)$ is the value obtained from applying $f$ to $x$. We would read this as "g of f of x", which is again consistent with our reading left to right. – Dave Nov 13 '17 at 00:36
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    Many mathematicians do write function applications with the function on the right. I suggest you take this question up on https://hsm.stackexchange.com. – Rob Arthan Nov 13 '17 at 00:41
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    @Dave: You argue that notation is consistent with pronunciation. My question is why is notation (and apparently also pronunciation) not consistent with evaluation. The common left-to-right notation is often a hassle. – Danvil Nov 13 '17 at 00:46
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    @Rob Arthan: Do you have examples? I would love to look at their notations and in which domains they are using that. – Danvil Nov 13 '17 at 00:47
  • Because it looks better. – Friedrich Philipp Nov 13 '17 at 00:56
  • @Danvil It's least uncommon in algebra/group theory. I know Isaacs lets functions act on the right. There was a more famous intro to algebra/group theory book author that did the same, but I can't remember who it was (EDIT: Herstein, apparently). – pjs36 Nov 13 '17 at 01:12
  • I am saying that I think one of the reasons the notation is this way is because of the way we would say these things verbally. – Dave Nov 13 '17 at 01:40
  • Just thinking through the arguments, this does suggest that at the time they were inventing these syntaxes, they were not thinking of functions as things to compute (which would lead to the "evaluate f, then evaluate g" argument you put forth). I would note that in at least 90% of the cases where I use functions, the function is vastly more important than the arguments going into it, so it would be a bit inconvenient to hide the important part at the end of the phrase. – Cort Ammon Nov 13 '17 at 05:17
  • This kind of syntax (or reverse - function on the left) is used in some programming languages as it is by far easier for the computer to parse. Have a look at Polish notation: https://en.m.wikipedia.org/wiki/Polish_notation – jaskij Nov 13 '17 at 07:58

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This is consistent with English syntax and with the concept of "function" being a special kind of relation on sets.

Look at the relationship "Husband of": In English this is typically expressed, for example, as Husband of Hillary Clinton is Bill Clinton.

In mathematical notation this gets naturally translated maintaining the syntax order: H(Hillary) = Bill

Successor of Obama is Trump (in US presidency)

$S(n) = n+1$ (Successor function in natural numbers)

P Vanchinathan
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    You also say "Hillary's husband is Bill". Apart from that, instead of seeing it as a relation like "h of g of f of x", you can as well see it as an operation like "f then g then h". – Danvil Nov 13 '17 at 01:06
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    Valid point. How about this? Chief of Staff of the President of the United States of America" – P Vanchinathan Nov 13 '17 at 02:01
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    @PVanchinathan America's States' Union's President's Staff's Chief? – IanF1 Nov 13 '17 at 07:08
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    You’re dwelling on peculiarities of English syntax, whereas other languages have different rules. Since Euler published in Latin, where the genitive case is shown by noun declension (with word order unimportant), I’m not sure this argument would really have applied to his decision when introducing the notation... – eggyal Nov 13 '17 at 10:04
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    @eggyal I've already linked this question elsewhere, but note that Euler didn't introduce the order of function of argument; in this he was following earlier examples from Bernoulli, who was apparently publishing in French. The word order for this in French is the same as described in this answer, so could well have been an influence in why Bernoulli wrote it that way. – Jules Nov 13 '17 at 10:35
  • @Jules: Excellent point. Moreover, in French, there is no alternative way to express possession that's equivalent to the apostrophe form in English—which also addresses Danvil and IanF1's comments above. – eggyal Nov 13 '17 at 10:40
  • Humanity is stuck in local minima – Danvil Nov 16 '17 at 00:44
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There are times when $f(x)$ is written $xf$. It's merely a case of trying to use the accepted convention except when it is inconvenient to do so; in which case you state that you are doing it differently and then proceed to do so.

Steven Alexis Gregory
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To my recollection this notation goes back to Leonard Euler.

David Reed
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    The top answer here gives Bernoulli as having written the oldest known example of a function symbol on the left and argument to the right of it. Euler was an early adopter, and added the use of parentheses to delimit the argument. – Jules Nov 13 '17 at 10:29