0^0

No, because there's no such thing. Either we can define 0^0 to have a value, or we can not do so. Either way, there's no separate option "indeterminate" that is different from "undefined".

Now, it's common to teach in calculus classes that 0^0 is one of the "indeterminate forms", along with 0/0 and so forth, which, so the story goes, is a different thing from being undefined, like 1/0. Since, after all, if f(x) approaches 1 and g(x) approaches 0, then the limit of f(x)/g(x) is undefined, whereas if f(x) and g(x) both approach 0, then the limit of f(x)/g(x) cannot be predicted in advance. So 1/0 is undefined, but 0/0 is indeterminate.

Now this certainly is getting at a real distinction! But it's not a distinction between the value of 1/0 and that of 0/0; both are undefined. "Indeterminate" is not some separate actual value. Rather, they are getting at the distinction of the behavior of the division function near the point (1,0) vs. how it behaves near the point (0,0). Not at the points! At both those points, the function is not defined.

And similarly with 0^0. Of course, it's pretty common to define that 0^0=1, and it's a definition I'd agree with -- but this is not inconsistent with the calculus teacher's statement that 0^0 is "indeterminate", because the latter (once made sense of) is not really a statement about the value of 0^0 at all; it's a statement about how the exponentiation function behaves near the point (0,0) (not at it; at it, it's equal to 1, or at least by my definition it is, at any rate).

In short, there's no such value as "indeterminate"; the calculus teacher's "indeterminate forms" (as opposed to "undefined"), while getting at a real destinction, is not actually about the value of the function at the point at all.

I love math, but I have never really enjoyed these types of debates.

I guess I have always been drawn to the application of the concepts in the real world rather than the abstract beauty of it.

It's very important to note here that 0^0=1 is a shorthand and not a truth.

Mathematicians are absolutely not stating that they have proven, or that it is true, that 0^0=1. It is a definition, not a claim of equality. They're not saying "0^0 is 1" in the sense that they say "1+1 is 2" or "0.999... is 1". They're saying "we define 0^0 to be 1". The difference is more than just pedantry, it strikes at the core of why mathematics is the most powerful tool for determining truth that humanity has either discovered or invented (hat tip to anyone familiar with that debate).

One of the cold, austere, beauties of mathematics* is that if you do not accept a definition, you can reject it as false and reason with the result. To give the traditional example, if you accept as true that two parallel lines never intersect, then along with the other 4 of Euclid's Axioms you can prove all of Euclidian Geometry (what you learn in high school). If you do not accept it as true (an explicitly accept it as false), then you can prove all of Hyperbolic Geometry (one form of non-Euclidian Geometry).

In the case here, you're free to reject the convention that defines 0^0 as 1 and reason with the result; you will not break any mathematics. But you should know that mathematicians have never run into any issues with this convention--or can handle it trivially when they arise--so you're only adding a lot of work for yourself.

When you really think about it, it fills you with awe. It's awesome.

*“Mathematics, rightly viewed, possesses not only truth, but supreme beauty—a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show.” -Bertrand Russell

I'm not a mathematician, but this seems the most reasonable/believable answer I've heard. The last sentence especially.

Mathematicians are engaged in a dramatically different kind of project than almost any other human discipline. Mathematical details emerge from definitions, but they appear exactly the same way for everyone else using the same definitions. And what's really surprising is how robustly those purely rational results compare to messy empirical reality. There is no good reason to believe that this should be the case!

In most other human endeavors, when something doesn't work we just work until find something else that does work, and marvel at our ingenuity. Mathematics doesn't quite give you that option. There are right and wrong answers to questions that we create ourselves, but those answers are fixed once we ask the questions, even if we didn't know what they were.

One enduring mystery in this vein was whether or not Euclid's fifth axiom was actually a logical result of the first four: https://en.wikipedia.org/wiki/Parallel_postulate#History