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Topic: what is a variable
Replies: 4   Last Post: Dec 8, 1995 3:50 PM

 Messages: [ Previous | Next ]
 Michael j. South Posts: 61 Registered: 12/6/04
what is a variable
Posted: Dec 7, 1995 10:58 AM

Judy Roitman says:

>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
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

{1,x,x^2,x^3,...}

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

{f(x)=1,f(x)=x,f(x)=x^2,...}

Not that this necessarily clarifies what a variable is...

Michael South
Shodor Education Foundation, Inc.
628 Gary St
Durham, NC 27703
(919) 688-2176 (voice)
(919) 688-2697 (fax)

Date Subject Author
12/7/95 Michael j. South
12/7/95 roitman@oberon.math.ukans.edu
12/7/95 Chih-Han sah
12/7/95 Andre TOOM
12/8/95 Bartel, Anne