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In most of Mathematics in English, the argument $x$ of a function (or partial map) $f$ with value $y$ is written $y=f(x)$ instead of, at least in Semigroup Theory and Formal Languages & Automata, $xf=y$. Why?

Thoughts:

A function $f$ is a subset of the Cartesian product of its domain $D$ and its codomain $C$ (i.e., $f\subseteq D\times C$) such that for each $d$ in $D$, $\lvert\{(d, c)\mid c\in C\}\rvert=1$. We write either $y=f(x)$ or $xf=y$ for $(x, y)\in f$.

The $xf=y$ notation has its strengths: composition of functions is usually taken left-to-right with this notation, as read in English, instead of right-to-left, with the argument $x$ first, as, I suppose, one usually has when evaluating $y$; one can read it as "$x$ through $f$ is $y$" instead of, say, "$y$ is $f$ at $x$"; and the left-to-right order respects that of going from the domain to the codomain.

Let me know if you can think of any strengths & weaknesses of the $xf=y$ notation.

I think history might have gotten the better of convention, that's all.

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    $\begingroup$ It's just a matter of conventions; in module theory it's fairly common to use arguments both on the left and on the right: homomorphisms of right modules take their argument on the right, homomorphisms of left modules take their argument on the left. This avoids juggling with the opposite ring. The usual notation is $f(x)$ because Euler used it. $\endgroup$ – egreg Apr 17 '17 at 16:57
  • $\begingroup$ @egreg I suppose a thorough answer to this question, then, would take into account different conversations for different areas of Mathematics. $\endgroup$ – Shaun Apr 17 '17 at 17:01
  • $\begingroup$ For people writing right-to-left Euler’s notation is handier (no reversal, of course) exactly because composition is more natural. The same happens with numbers: 1230 is written, in Arabic, as ١٢٣٠ (no reversal) $\endgroup$ – egreg Apr 17 '17 at 17:18
  • $\begingroup$ Years ago I. N. Herstein's algebra text was one of the best available. He wrote his functions on the right so that composition worked intuitively. I think it turned off people who might have adopted as a text. (You can find the latest edition on line.) $\endgroup$ – Ethan Bolker Apr 17 '17 at 19:08
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Because before algebra, western mathematicians used prose, and presumably wrote things like "the sine of the angle." In making the transition from words to symbols, things were still read the same way, and it makes little sense to read $x\sin$ as "sine of $x$." The choice $\sin{x}$ is far more natural, and from there, $f(x).$ This example was relevant to Euler, to whom the choice is usually attributed.

Similar observations apply to, for example:

  • the other trig functions,
  • the exponential and logarithm functions,
  • the notations $D$ and $\frac{\mathrm{d}}{\mathrm{d}x}$ for "the derivative (with respect to $x$) of" a function,
  • $\operatorname{sgn}$ for "the sign of" a real number,
  • $\operatorname{area}$ for "the area of" a plane region (or whatever).

Some slightly more modern, but still common, examples:

  • $\operatorname{Pr}$ or $\operatorname{Prob}$ for "the probability of" an event
  • $\mathcal{P}$ for "the power set of"
  • $Z$ for "the number of zeroes of" a real- or complex-valued function
  • $Z$ for the centre of a group (and other such things, like $N$ for the normalizer of a subgroup)
  • $\operatorname{Gal}$ for "the Galois group of"

Doubtless, there are countless others; whenever people come up with a new function name, they're thinking "the such-and-such of...". The function notation is almost certainly also the inspiration for notations like ${GL}_{n}(R)$ and $T_{p}M$ for "the space tangent to $M$ at $p$", though I admit this last is a bit of a stretch unless you think of it slightly more abstractly as "the tangent space of...".

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  • $\begingroup$ True, consistency is an issue. That doesn't explain why the argument is on the right of, say, $\sin$, for example. Having function-operation might read better in left-to-right English too. It's consistent when doing Semigroup Theory. $\endgroup$ – Shaun Apr 17 '17 at 18:39
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    $\begingroup$ @Shaun: Thank you for your feedback. In light of your comment I have changed my answer. $\endgroup$ – Will R Apr 17 '17 at 19:03
  • $\begingroup$ But Euler published in Latin, where the genitive case is indicated by noun declension (with word order unimportant)... $\endgroup$ – eggyal Nov 13 '17 at 10:11
  • $\begingroup$ I am not a linguist, so I can't really argue. But I would guess that although Euler published in Latin, he thought, and moreover did his mathematics, in some other language (German? Russian?). That being said, I don't know about those languages enough to make a concrete counterargument out of this. $\endgroup$ – Will R Nov 13 '17 at 10:14
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    $\begingroup$ Going by this answer, it seems Euler may have been expanding on Bernoulli’s practice (who published in French, where possession is always in the form “object of owner”)—so you may well be right. $\endgroup$ – eggyal Nov 15 '17 at 8:38
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Hint: According to A History of Mathematical Notations by F. Cajori this notation was introduced by Johann Bernoulli in 1718 . See this MSE post for a complete citation of the relevant section in the book.

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