>Explicitly, it usually isn't; but I believe that abstract algebra and >linear algebra cannot be understood without implicitly seeing variables as >objects. > >For example, if you are going to think about the vector space of >polynomials, you can't be thinking of "x" solely as a placeholder for a >number -- you're sitting there waiting to find out what "x" *really* is, >when, in that sense, it isn't.
Ah, the light begins to break! I must say, I have been thinking a lot about the "what is a variable?" question, and your question of my answer. Math is so cool! <begin _light-hearted_ joke>(Or should I say, "The philosophical games that no American high school teacher has time for but are associated with mathematics are so cool!"?)<end joke>.
I would say that the "x" you refer to here is not appropriately referred to as a variable, but a representation of the function, f(x)=x. (Perhaps this is a bias that comes from having studied classical applied analysis more recently than linear algebra.) In other words, when we state that the set
forms a basis for the space of polynomial functions, we are using the (collections of) symbols "x", "x^2", etc. in their capacity for denoting functions (in shorthand)--functions from which we may "build up" other functions, in this case the polynomial functions. (We won't get into the amazing fact that they can get us arbitrarily close to any continuous function on a finite interval!)
By shorthand, I mean that the "list" above, is "really" the list
Not that this necessarily clarifies what a variable is...
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