Thinking back on my math education, part of the difference in viewpoint may be the first time I rigorously met the continuous-domain exponential.
This was in real analysis. Exponentiation is defined first for positive integer exponents, and then for rational exponents. All elementary. Then it's extended to real-valued exponents by taking the limits of rational numbers, and appealing to continuity.
I just looked, this is exercise 6 in chapter 1 of baby Rudin.
So, because the notion of limit and continuity is embedded in this definition of the exponential function, it's natural to "approach" (groan) 0^0 as a special case, because the conditions of this definition (continuity) don't hold.
Mathematician here; we do not.
It seems I was a little too bold with my claim. All the mathematicians I know (and I'm a mathematician as well) agree with 0^0 = 1. It's a folklore specific thing, I guess.
As am I, by training if not by profession. As is lotharbot. You're in a thread full of mathematicians. :)
Which is what I would expect on this site, actually. I'm always timid making technical claims here unless I'm sure I'm correct; it seems to be a place frequented by arbitrarily large fish.
To answer your question, if 0^0 = 1 is _the_ answer in discrete setting, and it's _an_ answer in continuous setting, why don't we just agree that 0^0 = 1 and stop creating confusing situation where sometimes it's defined and sometimes it's not.
I'm not persuaded it is always the answer. I think the fact that it is an indeterminate form in limits is a forceful enough demonstration of that. It all depends on context. If I came across a 0^0 in, say, an engineering context, my first instinct would be to check whether the formula was defined in that case, not to just assume that 1 would work.
I mean, it's like 1/0. If you're working in R, that's simply illegal. If you're working in R*, it's the infinite point. If you're taking a limit, it means "unbounded". If you're working in my favorite field, the hyperreals, it could be any number of flavors of infinity depending on the flavor of zero it was.
It would be foolhardy to try to define the symbol; without a context to supply some sort of sense, it is nonsense. And that is how I feel about 0^0 as well.
From this point of view, the whole notion of "indeterminate form" makes just as little sense as distinguishing some arbitrary class of functions and calling them "elementary". Why are some points of discontinuity of some functions more special than other points of discontinuity of other functions? Why sin is more elementary than gamma? Historical heritage of confusion, I guess.
Similarly, if you're evaluating a limit and you get 0^0 you need to do more work. You can't just stop and say "oh, that's 1". It depends on what function you used to get there -- x^x will give you a different answer from ( e^(-1/x) )^x. Again, it has nothing to do with the continuity of exponentiation. The issue is whether the convention of 0^0=1 is correct in the specific part of mathematics you're working in.
The same argument can be made if you're working in the hyperreals, or if you're working with field axioms -- the convention 0^0 doesn't work in that context.
Please, by all means, use the convention 0^0=1 when it's appropriate. But understand that it's not always appropriate. Not every mathematician works in the particular subschool that you do; not every mathematician is going to find your convention appropriate.
What do you mean by "getting 0/0" in the process of evaluating limits?
The issue is not the continuity of x/x; the issue is whether setting a convention for 0/0 would give you the right value for a limit.
Please, tell me - what is the relation between lim f(a_n) and f(lim a_n) ?
Because sometimes it's better not to. Sometimes it's inconsistent with our definitions.
Just like sometimes we agree that you can't divide by zero, and sometimes we agree that you can. Sometimes infinity is an actual value (say, in the extended reals), and sometimes it's just a symbol for "unbounded". Sometimes we agree that you can't take the square root of a negative number, and sometimes you can. Sometimes we use the axiom of choice, and sometimes we don't (and you can have an awful lot of fun either way!)
Mathematics is contextual. How various operations behave depends on which axioms and conventions are being used.
I'd love to see even one example of 0^0=1 being inconsistent with a definition. The closest I've ever seen is that it bothers people that for reasons of their own had their hearts set on (x,y) -> x^y having no discontinuities...
Perhaps it's more precise to say "Because sometimes it's better not to. Sometimes there is no canonical choice that follows from our definitions, and it doesn't help to assign an arbitrary value that doesn't help solve any related problems."
What is "x" equal to? In general, I mean, not in the context of any equation like "x+1=2". You could say "x=7 in the study of free variables over integers when no other constraints are given", and that is completely consistent with the rest of mathematics, and yet would not be particularly useful and introduces an ugly (philosophical weasel word, yes) asymmetry in the theory (I'd say it introduces a gauge invariance (https://secure.wikimedia.org/wikipedia/en/wiki/Gauge_theory), but I'm really not qualified to discuss that in a rigorous way.)
Someone like Scott Aaronson could put this claim on more solid footing, but I would state that, intuitively, "assigning a value to an indeterminate form leads to a more complex definition of a mathematical system" in some formal complexity-theory sense.
Mathematician here; we do not. See http://math.stackexchange.com/questions/11150/zero-to-zero-p....
More precisely, as Arturo Magidin points out at http://math.stackexchange.com/questions/11150/zero-to-zero-p..., if we view exponentiation in the 'discrete setting', then $0^0$ must be $1$; whereas, if we view it in the continuous setting, there is simply no good answer—unlike $e^{i\pi}$, which also lives in the continuous setting, but has a perfectly good, unambiguous answer. (lotharbot gives a nice explanation below of the ways that this is consistent with existing mathematics; but it can also be derived from the definition of the exponential function, with no further arbitrary conventions needed.)