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Wolfram's MathWorld website, at the page on functions, makes the following claim about the notation $f(x)$ for a function:

While this notation is deprecated by professional mathematicians, it is the more familiar one for most nonprofessionals.

From context, it appears that this is referring to the use of $f(x)$ to refer to the actual function, rather than just to a particular value, when $x$ is (in the context) a dummy variable.

Is this true? Do professional mathematicians "deprecate" this notation?

To avoid long and windy discussions as to the values or otherwise of this notation (which would be much more appropriate in a blog), this question should be viewed as a poll. As MO runs on StackExchange 1.0, it doesn't have the feature whereby the actual "up" and "down" votes for an answer can be easily seen. Therefore I shall post two answers, one in favour and one against, the following statement. Please only vote up. A vote for one answer will be taken as a vote against the other. The Law of the Excluded Middle does not hold here. The motion is:

This house believes that the notation $f(x)$ to refer to a function has value in professional mathematics and that there is no need to apologise or feel embarrassed when using it thus.

This poll has now run its course. The final tally can be seen below.

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    $\begingroup$ Take the advice on the Wolfram site with a dose of skepticism. People write $x \longmapsto f(x)$ rather than simply writing $f(x)$ when one needs to make clear what the variable is -- for example try to walk through the proof that a finite-dimensional vector space is isomorphic to its double dual without using this notation and you'll see why people like it. But it really only becomes important to make these distinctions when it's difficult to "identify" the variable in a functional expression. $\endgroup$
    – Ryan Budney
    Commented Oct 1, 2010 at 4:06
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    $\begingroup$ The question asked in the title is not the question asked in the body. Can you make up your mind as to what exactly you would like to know, and edit accordingly? $\endgroup$
    – Gerry Myerson
    Commented Oct 1, 2010 at 4:23
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    $\begingroup$ I don't think that this is a good question for MO. Then again, I don't feel sufficiently strongly to vote to close --- moreover, I don't have a great reason for feeling this way. It's just that I think this would make a better discussion on a blog (or in a bar) than on MathOverflow (or in a mathematics seminar). $\endgroup$
    – Theo Johnson-Freyd
    Commented Oct 1, 2010 at 4:36
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    $\begingroup$ @Theo, it might make sense to keep this question open. Although mathematicians may know when to trust the Wolfram site and when not to, I think the site has credibility in the general population as Wolfram is the producer of a popular mathematics software package. So this thread would have something of an "outreach" role, at least as a rebuttal to the statement "depreciated by professional mathematicians". This is definately a borderline question though so I have no strong feelings either way. $\endgroup$
    – Ryan Budney
    Commented Oct 1, 2010 at 4:47
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    $\begingroup$ meta thread: tea.mathoverflow.net/discussion/692/poll-question-about-wolfram (please vote for this comment so that it appears "above the fold") $\endgroup$
    – Andrew Stacey
    Commented Oct 1, 2010 at 8:38

2 Answers 2

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Vote for this answer if you agree with the statement:

This house believes that the notation $f(x)$ to refer to a function has value in professional mathematics and that there is no need to apologise or feel embarrassed when using it thus.

(Note: the answer is CW so that this is a genuine poll)

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    $\begingroup$ It's just an abbreviation for λx.f(x) $\endgroup$
    – Richard Borcherds
    Commented Oct 1, 2010 at 19:38
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Vote for this answer if you disagree with the statement:

This house believes that the notation $f(x)$ to refer to a function has value in professional mathematics and that there is no need to apologise or feel embarrassed when using it thus.

(Note: the answer is CW so that this is a genuine poll)

Improve this answer
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    $\begingroup$ Andrew, at present this isn't CW. $\endgroup$
    – Robin Chapman
    Commented Oct 1, 2010 at 8:26
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    $\begingroup$ Whoops! Fixed. As you can probably tell, I just cut-and-pasted from one to the other, but the CW bit doesn't get included. I'll donate all the rep that I got to charity. Oh, wait, no-one's voted for this yet so that's nothing. (Seriously, I hope this hasn't skewed the results) $\endgroup$
    – Andrew Stacey
    Commented Oct 1, 2010 at 8:36

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