Even for things as "objective" as mathematics, definitions ultimately come down to what makes it convenient to manipulate symbols like an expert would.
This comes up all the time in fields as diverse as software engineering, law, finance, business, even hard sciences like physics - things are "right" because they're convenient, and because the consequences of them being that way make it possible to build on those results with new constructs, while doing it a different, more intuitive way would result in those constructs being impossible.
What the article is saying is that your intuition for it being 0 or undefined is because you've been exposed only to exponentiation as repeated multiplication or as the limit of some series; if you consider other theorems like the binomial theorem, and figure out what is necessary for them to hold without special casing, you'll decide otherwise.
For me, intuition-wise, I'd order it "undefined, 1, 0".
There are 3 cases for 1 and one case for 0 that immediately spring to my mind when considering the problem:
0) Limit of 0^x, as x approaches 0 (from above).
1a) Limit of x^0 as x approaches 0 (from either direction).
1b) Limit of x^x as x approaches 0 (from above).
1c) "What did you multiply by 3 once, to get 3^1? So, multiplying 1 by zero, zero times..."
Limits here, simplified to intuition level, being "what would you need to fill that hole in the graph?" The fact that these disagree would be why I'd assume undefined, but the case for 1 seems stronger (to me).
The limit in 1b is 1 from below as well, right? I'm not sure how limits work with complex numbers, but the imaginary part of x^x approaches zero as x approaches zero from below, so can we say that the limit of x^x as x approaches zero from below is also zero?
Yes, perhaps "intuition" isn't the best word. Formal limits certainly aren't "intuitive" to me, at least by one definition of the word. I suppose I used "intuitive" to mean "according to my mathematical understanding, ignoring the mathematics explicitly dealing with 0^0."
Not really - the explanation from the "mathematician" perspective gives some of the rationale, and it makes perfect sense & in line with intuition.
Defining 0^0 as anything but 1 is weird since x^0 is 1 given the other definitions we have adopted, i.e. x^n = x * x * ... * x (n times) for positive integers, (x^n)^(-1) being defined as the unique number such that x^n * (x^n)^(-1) = 1, (x^n)^(-1) * x^n = 1 for non-zero numbers x, and the simple result that (x^n)^(-1) = (x^(-1))^n from proof by induction & the uniqueness condition of the multiplicative inverse, we then get the natural formula that 1 = x^n * x^(-n) = x^(n - n) = x^0 for non-zero numbers. Defining 0^0 = 0 or undefined doesn't agree with the formula given for all non-zero real numbers, and goes against the limit of x^x as x approaches 0 (as detailed in the article), making x^x a discontinuous function at x = 0 if it is defined as another value.
Even for things as "objective" as mathematics, definitions ultimately come down to what makes it convenient to manipulate symbols like an expert would.
This comes up all the time in fields as diverse as software engineering, law, finance, business, even hard sciences like physics - things are "right" because they're convenient, and because the consequences of them being that way make it possible to build on those results with new constructs, while doing it a different, more intuitive way would result in those constructs being impossible.
What the article is saying is that your intuition for it being 0 or undefined is because you've been exposed only to exponentiation as repeated multiplication or as the limit of some series; if you consider other theorems like the binomial theorem, and figure out what is necessary for them to hold without special casing, you'll decide otherwise.