# Keywords to find documentation for this syntax

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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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]]]


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

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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) –  rm -rf Aug 22 '12 at 22:44
@R.M SubValues (the function) isn't in the documentation, but here's a nice question which covers them. –  rcollyer Aug 22 '12 at 22:51
@rcollyer Right, that and this one are the ones that I remembered –  rm -rf Aug 22 '12 at 22:52
@R.M I had forgotten about that one. –  rcollyer Aug 22 '12 at 22:53
the discussion on currying I think deserves an upvote, but I can't upvote twice –  acl Aug 22 '12 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)


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`.

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