Consider the set of all infinite arithmetic sequences of real numbers:
$$\forall f,d\in\mathbb R\ (f,f+d,f+2d,\cdots,f+id,\cdots)$$
Most people would say that the set of all of these sequences is parameterized by $f$ and $d$ thus there is a mapping from $\mathbb R^2$ to these sequences. Even further people would say that, since this is the lowest number of parameters possible, the dimension of the space of sequences is $2$.
Now consider everything I said above but for integers. It would still take at minimum $2$ integers to parameterize it.
But what about space-filling curves? A space-filling curve can map the real line to the real plane, meaning only $1$ variable is needed to parameterize the plane. Does that mean the plane is $1$ dimensional? You might say the parametrization has to be continuous (something the space filling curve is not) but then what about the integer sequence case? Can that case not even be parametrized because the integers are not continuous?
And in general, if two sets have the same cardinality then there must only be 1 variable needed two parameterize one to the other, no? The definition of cardinality is that there exists some bijective function from one to the other. Or equivalently we could consider the set of pairs $(f,d)$ to map to the sequences rather than the two individual variables.
I've been trying to pick this apart for a while now since parametrizations are everywhere and, especially in my calculus class, we talk of the dimension of objects as being associated with the minimum number of parameters needed to paramaterize it (i.e a sphere while embedded in 3-space is actually 2-dimensional).
In that case it might be as simple as an implication that the parameters be differentiable since we are doing calculus but that fact that this is never addressed has me confused.
But in general, is parameterize a term that doesn't really mean anything concrete? Does it change meaning depending on the type of math you're doing? What if these two definitions collide like with the sequences?