Any symbolic ring in Sage has a dictionary called "symbols". (I'll refer
specifically to "SR" for the following.) "SR.symbols" contains all the
previously defined variables through "SR.var()" and/or "SR.symbol()".
However, this generates a problem: Many symbols remain stored there even
when they have been overwritten for other purposes. For example, when one
starts a new Sage session, Sage executes the line
from sage.calculus.predefined import x
This is done in order to have "x" as a default predefined symbolic
variable. However, the whole file "sage/calculus/predefined.py" is executed
which calls "SR.var()" on every single letter of the alphabet (lowercase
and uppercase). So, when one has a freshly started Sage session (without
any additional execution), "SR.symbols" already has length 48:
SR.symbols = {'a':a, ...'z':z, 'A':A, ...,'Z':Z}
Since "x" is the only predefined variable, this should be simply
"SR.symbols={'x':x}".
An even bigger problem is seen with the following code:
SR.var('foo')
# Do something meaningful with "foo" until you don't need it anymore
foo = 2
# Do something meaningful with the new meaning of "foo"
This keeps "foo" listed as a symbolic variable in "SR.symbols" even though
it is now an Integer. In time, "SR.symbols" gets cluttered with
non-existing symbolic variables.
Do we really need this behavior? Is it "SR.symbols" necessary at all? Could
we remove it?
*Note: *The motivation for this proposal comes from this Ask SageMath
question
<http://ask.sagemath.org/question/56041/differences-between-srvar-and-srsymbol/>,
where @tmonteil proposed a command "SR.symbols()" for defining symbolic
variables. The current "SR.symbols" could be renamed to "SR._symbols_"
(this complies with Python's conventions on private variables), but perhaps
it is unnecessary to have "SR._symbols_" at all.
(Sorry for the long post.)
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