“0^0 = 1 exclusively” is the right definition in many contexts where the exponent is considered as a discrete variable. But I can’t right now think of any context in which the exponent varies continuously in which there is good reason to say “0^0 = 1 exclusively”.
So this would be like saying “9^(1/2) = 3 exclusively”. Well, there are contexts where this is true (working only with positive values, say), and contexts where this is less true (there are times one wants to consider 9^(1/2) as equally encompassing -3). Or like saying “ln(1) = 0 exclusively”, which again, there are contexts/interpretations where this is true, and also contexts/interpretations where one wants to say ln(1) equally well includes any integer multiple of 2πi. Everything is contextual.
When people say “0^0 is just as well any value”, they are attempting to convey a genuine phenomenon, and this is indeed probably a better way to think of things in contexts where the exponent and base can vary continuously. While in other contexts where the exponent varies only discretely, often the better way to think of things is that “0^0 = 1 exclusively, unambiguously”. So it goes.
What is “God’s proof” of this marvelous little theorem? I think we’ll agree it isn’t the proof via integrating sine functions! :eek:
Both the second and the third proofs at that webpage are very elegant — and they are very different from each other. The clever 2nd proof with its cute device is, arguably, much easier than the 3rd. But the cute device it uses is tangential, so perhaps the 3rd proof is “God’s proof” ?
Everywhere throughout this post, wherever I said “floor(whatever)”, I meant “whatever - floor(whatever)”; i.e., the fractional component of whatever. [I see now no one read this post closely enough to call me out on this; oh well.]
As I said, it is an instance of the more abstract proof which notes that any functions of appropriate period can be used as integrating factors along each dimension; here, the periodic function used is the fractional part function rather than the sine function.
Thinking in terms of “periodicity” is not, in itself, naturally framed to allow the even more general result for arbitrary equivalence relations along each axis (for example, such conditions such as coordinates differing by algebraic numbers; “Theorem #3” at the website).
But as I essentially said above, instead of using any specific periodic functions such as sine or the fractional part coordinate function, one can work instead abstractly in tensor products of free abelian groups on coordinates modulo whatever equivalence relations. (That is, one can say precisely “Make up the most generic subtract-and-multiply-able functions along each axis such that two points’ values are equal under the function iff they are to be considered equivalent along this axis; since no other details of the functions matter, we won’t assume any”). And this gives the general result for arbitrary equivalence relations; it’s the same as an abstract analysis of the Stokes’ theorem type cancellation going on in the more concrete integration proofs. [I suspect what I’m saying here is not particularly clear to most people; I will return in the morning with an example block diagram and illustrate it better]
This is what I would consider “God’s proof”. It’s a little different from, but close to, the graph-theoretic Proof #3 on the website; it abstracts away from specific distances and recognizes that the problem is purely combinatorial; irregular stretchings and squashings of the block diagram make no difference to the underlying phenomenon.
The difference between what I’m calling “God’s proof” and the Proof #3 on the website is like the difference between two classic styles of proof of Hex’s determinacy or, just as well, Brouwer’s fixed point theorem: one by analyzing “winding numbers” and how they add up as regions are combined, and the other by tracing out a path according to a specific rule for which direction to move at any moment till one finds one has connected two corners. I prefer the winding number style proof because it points the way to a host of further important machinery in algebraic topology, but I suppose there must be some perspective from which these two approaches can be unified; I will have to think about that more.
[Ah, I don’t think much of what I’ve been trying to say in this post has been clearly expressed, but I shall settle for clarifying it later.]
My second-year differential equations prof mentioned that when she was teaching engineering majors and made a minor mistake on the board, they would immediately raise their hands and ask/complain about it. But when she was teaching math majors they would just quietly put the correct answer in their notes and not break the flow of the lecture.
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