It is an often-used rule in mathematics and statistics that if one representation of an object has $n$ "free" parameters, then another representation of that same object requires at least that number of parameters. E.g. I was just now reading along in this paper and came along this sentence:
A necessary condition is that the number of free parameters be preserved.
as a premise in a proof.
Although this is somewhat intuitive, it seems to me that such a rule is not entirely obvious, especially since the cardinality of $\rm I\!R$ is equal to the cardinality of $\rm I\!R^2$. However, $\rm I\!R$ is not homeomorphic to $\rm I\!R^2$, so perhaps therein lies the meaning of a "number of free parameters".
What is the meta-mathematical basis for this "law of conservation of free parameters"? Is something like a "law of conservation of free parameters" derivable from ordinary axiomatic systems? Where can I learn more about the theory of this rule?
