Mathematica allows this syntax, but I cannot find any documentation (or books) that discuss the various uses/advantages of it

g[1] := Plus;
g[2] := Times;
g[1][3, 5]
(* 8 *)

g[2][3, 5]
(* 15 *)

What keywords can I use to find this in the documentation?

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2 Answers 2


Your g[1] and g[2] are simply acting as Head:

g[1] := Plus;

So there is no mystery in this syntax:

 {g[1][a, b], Plus[a, b]}

{a + b, a + b}

 Head /@ {g[1][a, b], Plus[a, b]}

{Plus, Plus}

So you need to read:

But maybe there is a bit more to it than meets the eye. You actually almost wondered into programming concept called Currying which according to Wikipedia "is the technique of transforming a function that takes multiple arguments in such a way that it can be called as a chain of functions each with a single argument (partial application)."

So you can do things like:

f[x_][y_] := Sin[x y]

f[x] /@ Range[5]

{Sin[x], Sin[2 x], Sin[3 x], Sin[4 x], Sin[5 x]}

Documentation mentions it here. For deeper insight see discussion in @SalMangano "Mathematica Cookbook".

BTW, @acl nice addition (to see how Mathematica thinks) can be visualized as

TreeForm[Trace[g[1][a, b]]]

enter image description here

So the thinking goes from top to bottom and from left to right.

  • $\begingroup$ If you're mentioning Currying, then you might add in a word or two and some links to SubValues (I remember there were a few posts here too) $\endgroup$
    – rm -rf
    Aug 22, 2012 at 22:44
  • $\begingroup$ @R.M SubValues (the function) isn't in the documentation, but here's a nice question which covers them. $\endgroup$
    – rcollyer
    Aug 22, 2012 at 22:51
  • $\begingroup$ @rcollyer Right, that and this one are the ones that I remembered $\endgroup$
    – rm -rf
    Aug 22, 2012 at 22:52
  • $\begingroup$ @R.M I had forgotten about that one. $\endgroup$
    – rcollyer
    Aug 22, 2012 at 22:53
  • $\begingroup$ the discussion on currying I think deserves an upvote, but I can't upvote twice $\endgroup$
    – acl
    Aug 22, 2012 at 23:15

Just to add to Vitaliy's answer: You can see what happens with

g[1] := Plus;
FullForm /@ (g[1][3, 4] // Trace)

Mathematica graphics

So, on evaluating g[1][3,4], Mathematica looks up g[1] and sees it evaluates to Plus; it's then left with Plus[3,4] which evaluates to 7.

  • $\begingroup$ +1 Cool idea to use Trace. I referred to your answer too ;-) $\endgroup$ Aug 22, 2012 at 22:41
  • $\begingroup$ @VitaliyKaurov does that mean the entire fabric of reality is going to rip apart? Or, just stackexchange? $\endgroup$
    – rcollyer
    Aug 22, 2012 at 22:52
  • $\begingroup$ @rcollyer This just means that you need to answer too and do a three way cross-referencing $\endgroup$
    – rm -rf
    Aug 22, 2012 at 23:00
  • 1
    $\begingroup$ @R.M there's a French term for that, but I don't think it is appropriate for polite company. $\endgroup$
    – rcollyer
    Aug 23, 2012 at 0:09
  • $\begingroup$ @rcollyer it just means household of three so I don't see the problem :) $\endgroup$
    – acl
    Aug 23, 2012 at 0:12

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