Talk:Constant

I removed this footnote:

There is some evidence that the speed of light is gradually decreasing, but such speed is constant over the whole universe at any given time.

Most physicists are certain that the speed of light isn't changing with time, and there isn't even agreement among them as to just what such a claim means, if anything. The issue should certainly be discussed in Speed of light, but not here. — Toby 06:16 Sep 18, 2002 (UTC)

This is pretty much moot in light of my recent changes. — Toby 07:40 Sep 18, 2002 (UTC)

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I just noticed that there are also articles titled mathematical constant and physical constant. I think some re-organization is in order. If all articles are to be kept, then constant should be the general concept, and link to the more specific articles. Or more ambitiously, someone might make one, really good article by merging all three. I don't think I'm the person to do this (mathematically stupid)! ike9898 15:23, 21 Jan 2004 (UTC)

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I find the section "Variables vs. constants" very confusing and disturbing. It doesn't match my own understanding of the terms.

Constants vs variables

A number that is constant in one place may be a variable in another.

First of all, all numbers are really constants, not variables. I believe what was intended was something about symbols that may stand for a variable in one place, constant in another. But even here, I don't really agree with the statement.

This whole distinction over "constant vs. variable" is really a matter of distinguishing between free variables and bound variables. (free variables are "constants", bound variables are "variables"). I believe that article does a much better job of explaining the distinction in practice. In general, this distinction is a fairly abstract concept that requires precise definitions, and I admit I'm not familiar with them off the top of my head, I would have to consult a logic book. But the explanation below seems misleading.

Consider the example above, with a function f defined by f(x) = sin x + c.

Now consider a functional F, a function whose argument is itself another function, defined by

F(g) = g(π/2).

Then for the function f given above, we have

F(f) = c + 1.

In the formula for f(x), we are fixing c and varying x, so c is a constant. But in the formula for F(f), we are varying both c and f, so c is a variable.

I believe this is a bit misleading. It is true that c is a variable, but it is a variable that depends on f. What this means is that f is actually a constant (free variable!) Seems ridiculous, but watch. Let's call

f_c = sin x + c

Then what we're really saying is,

F(f_c) = c + 1

To be really precise, we should say that f is a function from the reals R into the domain of F, where

f(c) = f_c

A really good way of expressing this is to say that

(f(c))(x) = sin x + c

So, f isn't a function like sin x now, it's a function that gives out a function, given a real number.

What the first statement is saying is:

(FOR ALL x) f(x) = sin x + c.

Here, clearly, x is bound, and c is free. (In other words, x is a variable, and c is a constant.)

The other statement is really saying:

(FOR ALL c) F(f(c)) = c + 1.

Here, clearly, c is bound, but just as clearly, f is free! The point is that the original "f" given in the original formula was actually f_c, not f. This dependence on c was concealed by the notation. We could have defined f differently, say

f(c) = cos x + c

and then the second statement would have been false. The second statement depends on f, so in that statement, f is a constant.

The only way f could be a bound variable was if we interpreted the statement as saying,

(FOR ALL c)(FOR ALL f) F(f(c)) = c + 1

But this is not what we mean by F, e.g. we don't mean to say that (taking c = 3, f(x) = x²)

F(9) = 4

even if we do interpret "9" to be the constant function 9. In that case, we would want

F(9) = 9

Even this statement might be false in the presence of some larger context that gives yet another point of view.

I think maybe this is supposed to be some very vague way of talking about the scope of quantifiers?

Thus, there is no precise definition of "constant" in mathematics; only phrases such as "constant function" or "constant term of a polynomial" can be defined.

Well, yes there is a precise definition, it's defining "constant" to mean "free variable". The other definitions given are really distinct from this notion, but related intuitively. The point is, they can all be defined without referring specifically to free variables, so in that sense, they are terms which have only "co-opted" the term "constant" as a matter of habit or tradition.

There is a mathematicians' joke to the effect that "variables don't; constants aren't." That is, the term variable is frequently used to mean a value that is fixed in a given equation, albeit unknown;

I never call these "variables" at all, except when I'm just being sloppy in my speech...a more accurate term would be "parameter" (or constant, or free variable, etc.)

while the term constant is used to mean an arbitrary quantity which may assume any value, as in the constant of integration.

This "paradox" arises from the conflation of terminology surrounding the term "constant". When mathematicians use the word "constant of integration", they are not using it the sense of a free variable, they are using it in the sense of a "constant function". The constant of integration doesn't vary at all, at least not with respect to the scope of the function; for any given fixed antiderivative, the constant of integration is a fixed number.

Revolver 04:44, 14 Apr 2004 (UTC)