Basic misunderstanding of Begin

Question

In a notebook with a new kernel, execute the following in an Input cell:

$Context
Begin["Test`"];
$Context
f = x \[Function] x*x;
f[2]
End[];
$Context
f[2]  (* why is this recognized 2nd time?? *)
Test`f[2]

The result is as expected (i.e., f[2] is unevaluated in the Global context). But now evaluate the same Inpute cell, without any changes. The contexts are reported as before, but this time f[2] evaluates to 4 in the Global context. Why?

Answer 1

There are two things that influence symbol lookup: $ContextPath and $Context. This is described in:

• https://mathematica.stackexchange.com/a/43629/12

In short: $ContextPath controls where the system looks for existing symbols. This search is done first. $Context controls where new symbols are created if a name was not found in $ContextPath any contexts.

Begin changes only $Context. BeginPackage changes both $ContextPath and $Context.

You are only using Begin, but not BeginPackage. This means that $Context will be set to "Test`" but $ContextPath will stay what it was before.

When you mention a symbol name such as f, the first thing Mathematica does is that it looks for it in the contexts contained in $ContextPath. If it is found there, it uses that instance. This is what happens during the second evaluation: f is found in Global`, so f now refers to Global`f, not Test`f.

Why does f exist in Global`? Because you mentioned it when evaluating f[2]. Note the distinction between the fact that a symbol exists and that a symbol has associated definitions. Just evaluating sym alone will create that symbol.