The address could be a symbol, too.
The Python statement
xyz = 3
places a number in the address "xyz".
It doesn't matter whether addresses are numeric or symbolic, Python does
not use that model for variables. Consider this example. There are a few
steps, so let's go through them. First we prove that so-called address
"abc" and "xyz" are distinct. If they were the same address, then they
would logically have to contain the same value, but that is not the case:
abc = 23
xyz = 42
assert abc != xyz
This proves that the two addresses are different.
Now we prove that they are the same address:
abc = xyz = []
xyz.append(42)
assert abc == [42]
If they were different, then modifying the object at xyz cannot modify
the object at abc. So we have a contradiction: "addresses" abc and xyz
are both the same address, and different addresses.
There is a way to wiggle out of the contradiction: accept that the
addresses are distinct, but claim that a single object can be in two
different locations at once. But if we make that argument, we are
stretching the metaphor of location quite considerably. Absent time-
travel, objects cannot be in two locations at once. Claiming that they
can be requires stretching the metaphor of location past breaking point.
Self-confession time: some years ago, I used to make exactly that
argument. I used to argue that the right way to visualise Python's
variable model was to consider that objects were stored in variables in
exactly the way you are suggesting. I didn't describe them as symbolic
addresses, but otherwise the model was the same. In order to make the
model work, I argued that objects could be in more than one place at
once, including *inside* itself:
L = []
L.append(L)
and explicitly argued that this didn't matter. After all, if I can watch
Doctor Who and accept the concept of his TARDIS materialising inside
itself, I can understand the idea of a list being inside itself.
That was before I really grasped the difference between the name binding
and fixed location variable models. While I'm still fond of the concept
of a box being inside itself, I've come to understand that having to
stretch the metaphor of location so far simply indicates that the
metaphor does not work with Python's semantics.
Python's actual behaviour is not a good fit for the "variables are
locations" model, not even if you think of locations in the abstract with
symbolic names rather the numeric addresses.