I have been putting off writing the section on parameters in abstractmath.org because I don’t very well understand the connections between the different ways the word is used.
A parametrized family of functions, say f_a(x) = x^2+a, is easily handled by using the adjunction that makes the category of sets cartesian closed. That is, the three ways of looking at it, function of x parametrized by a, a function of a parametrized by x, and a function of two variables, are all naturally equivalent in a completely satisfactory way.
When you talk about taking a curve defined by an equation, e.g. x^2 + y^2 =1, and parametrizing it, that is a different situation; I have never thought about it from a categorical point of view but it is not hard to explain at the level of abstractmath.org.
What I don’t understand is if there is any real connection between these two meanings. I would welcome comments!
There are other meanings of “parameter”. One is the usage, “An important parameter of a finite group is its order.” I am not sure how often this occurs, and I look at the usage with some disapproval. The order of a finite group is more like an invariant (e.g. under isomorphism) than a parameter. Now do I have to write about invariants, too? Ye gods, do I have to do everything myself?
Another usage is in programming languages, where it can be analyzed simply as a variable. But they have different kinds of parameters that are implemented differently.
What I really need to do is spend a day searching for the word in JStor. I swore when I started doing abstractmath.org I wasn’t going to do any more lexicographical research (after the Handbook.). But maybe looking around for a few hours won’t hurt.
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My understanding of the word comes from physics. Parameters there generally represent constants, but are left as letter representatives for the formulas. Like “g” for gravitational constant represents 9.8 m/s^2 on Earth at sea level, but could be some other value at different elevations or on another planet/moon.
Similarly, “m” is usually meant to represent some constant mass, but could vary from object to object or even in some cases replaced with some other function if mass is variable.
In general, a parameter is any piece of information that helps specify an object in a family. The information may be incomplete (you can talk about a single parameter even if several parameters are needed to specify everything fully) or redundant (you can parametrize a circle as (sin(t),cos(t)) despite the fact that t and t+2pi lead to the same point).
As for finite groups, the usage you mention would be slightly nonstandard since one doesn’t usually conceive of all finite groups as a family in this sense. There’s no logical reason why one shouldn’t, but the terminology of parameters is generally used only in simpler situations. The idea is to make a geometrical analogy, and the set of all finite groups is just too complicated for this to be worthwhile.
It seems to me that to parametrize a curve, like x^2+y^2=1, is to find a path whose image is the curve. Then when you think of f(x)=x^2+a as a family of curves parametrized by a, you’ve got yourself a path _in the space of functions_ whose image is ‘f’.
I haven’t thought too much about allowing more parameters, in this context, but it seems like it should go through.