I understand the "math"...the numbers...the work on paper. But how does that translate to something useful in the real world? That, after all, is what useful math helps us do...solve problems for the real, tangible world. Saying that 0^0 = 1 is a cool math game; but translate 0 into something in the real world (i.e. nothing, none, etc.)...and trying to make something out of it other than 0 or "indeterminate" starts to make less sense.
From the article's example the simplicity of the binomial formula is extremely useful compared to a formula that would have to account for the case where k=0. Another commenter pointed out the useful elegance of 0log0=0 for physicists. These are the real world applications for mathematicians choosing definitions directly. Saying that an idea may be defined in many ways is correct, but choosing a working definition for the system helps to apply the definition to appropriate concepts. This is what mathematicians are doing when choosing a specific definition instead of saying that any definition will work.
One could argue the entire field of Real Analysis was formed because Calculus showed the world that we didn't really have those definitions, but they were needed. There are cases where the integral of the derivative does not equal the derivative of the integral (violation of the fundamental theorem of calculus) without having a specific epsilon-delta definition of limit.
Also, zero is not always the same as nothing or none. Zero is an abstract number that some have decided is useful to represent nothing or an empty set, but really comes from an abstract idea that you can count nothing and have a number. This goes back to the fact that numbers are pretty useful ideas regardless if you may consider one of them a function or not.
> But how does that translate to something useful in the real world?
I think this is a by-product of the way we are taught maths at the very start; that it must somehow relate to real things. We start our understanding of maths by using real world objects like apples and we show how addition works and subtraction. I think perhaps it sticks in our head that everything must somehow relate to real objects and the real world.
We somehow get past that when we are introduced to things like square roots and integrals and higher mathematical concepts, but even with those we often try and relate them back to the real world.
I wonder of there are other ways to start teaching maths that doesn't start by using balls or apples? How would that work?
Perhaps not really great examples - I mean you can relate to the length of a diagonal line in a square room, but you can't really relate to the square root of three apples, can you? I'm not trying to say that those example don't relate to the physical world. My point is we abstract more and more away from the "real world" until you get something like this post.
What about complex numbers - the square root of -1, incredibly, is useful in electronics and other real physical systems, but you cannot really relate it to physical objects - or at least not obviously. Or more incredibly something like Banach Tarski [1].
The strongest value of math is the most consistently efficient approach for solving non-trivial abstract problems, not necessarily the answer to the question solved by the article. This is due to understanding how to probe a problem, experiment with possibilities, and repeat in a logical manner until you strike gold - it is a skill that is applicable in just about every walk of life.
