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0^0

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Re: 0^0

#91

Post by baddox »

dvanduzer wrote: post #90
baddox wrote: post #57

Earlier quoted context omitted.

Correct me if I'm wrong, but I infer that you're using "we made it all up" as a pejorative toward mathematicians. Of course mathematicians invented the terminology, notation, and methodology, but that's not a bad thing. It's a great thing, just like it's great that engineers "make up" bridges, chemists "make up" pharmaceuticals, writers "make up" novels, etc.

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…

I don't think the fifth axiom is considered particularly mysterious anymore. The traditional fifth axiom clearly isn't a logical result of the first four, since it can be replaced with other parallel postulates to yield non-Euclidean geometries which are themselves perfectly workable and consistent. In fact, that section of the Wikipedia article notes that Beltrami proved the independence of the parallel postulate.

Re: 0^0

#92

Post by 6d0debc071 »

Dylan16807 wrote: post #68
6d0debc071 wrote: post #59

Earlier quoted context omitted.

> It is [arbitrary]. We invented the arabic numerals because... Then it's not arbitrary; chosen at random or on a fleeting whim, without reference to a reason or system. It was invented to fill a specific need based on certain limitations.

But it is arbitrary; subject to individual will or judgment without restriction; contingent solely upon one's discretion.

I doubt many people would agree that you were doing maths if you made your rules and symbols up randomly.

It's true enough that mathematicians define certain things certain ways, but they generally have reasons for doing so that tie into other aspects of whatever system they're working within at the time, or with particular areas of investigation: 'If I alter this rule, or make this assumption, what does it do to the system as a whole? Does it let me find some answer more easily than another way? Does it preserve consistency/truth values? Under what conditions?'

That's far from being dependent solely on their individual whim, the decisions they make in that regard, and the answers they will get, are strongly influenced by the form the system has taken and it's uses and limitations.

Of course if you want to maintain that maths as a whole is arbitrary because you could make whatever you liked up and say you were doing maths... well, I won't argue you're not, but it seems to me you've made the objection general enough that it could safely be ignored. Anyone doing something purposeful could simply assert: 'Your's, maybe. We're trying to do our-maths-goal.' And move on.

Re: 0^0

#94

Post by baddox »

dllthomas wrote: post #85
baddox wrote: post #51

Earlier quoted context omitted.

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, but getting there steps out of the realm of "intuition" for me.

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."

Re: 0^0

#96

Post by rntz »

stephencanon wrote: post #78

The real problem here is that x^y is a single shorthand which refers to a few fundamentally different mathematical concepts (which happen to have significant overlap with each other). First, it refers to a function f:C x N --> C, defined in terms of repeated multiplication. f(x,0) is 1 for all x != 0, and so we adopt the convention that f(0,0) is also 1. But it also refers to a function g:C x C --> C, defined as g(x,…

In fact, exponentiation f : C x N -> C has a natural generalization to any monoid C, where f(x,n) is x "multiplied" by itself using the monoid operator n times. In this setting, the only sensible choice is that x^0 = 1 for any x, where 1 is the unit of the monoid in question. In particular, that is the only definition of exponentiation which is parametric in our choice of monoid.

Naive Haskell example code (using Int for Nat):

    f :: Monoid m => (m,Int) -> m
    f (x,0) = mempty
    f (x,n) = x `mappend` f (x,n-1)
    -- or equivalently
    f (x,n) = msum (replicate n x)

Re: 0^0

#97

Post by bhrgunatha »

ryannevius wrote: post #63
bhrgunatha wrote: post #26

Earlier quoted context omitted.

> 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 w…

Square roots and integrals both relate back to the real world...

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].

[1] https://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox

Re: 0^0

#98

Post by my_username_is_ »

blackrabbit wrote: post #93

Can anyone explain why this is a valid operation? http://wp.com/latex.php?latex=\lim_{x%20\to%200^{+}}%20x^{x}...

it's using log to mean log base e, also known as the natural log or ln. you can do that because they're inverse operations (I think that's the right term for it, it's been a while since a formal math class) so they undo each other

Re: 0^0

#99

Post by dhaivatpandya »

SilasX wrote: post #43

- "0^0. Why? Because mathematicians said so. No really, it’s true." - [Detailed explanation of the tradeoffs involved in choosing different definitions of exponentiation.] So, it's not "because mathematicians said so", it's because of a deep review of the tradeoffs of defining how exponentiation generalizes, the kind of thing that mathematicians happen to study more than other identifiable groups.

It is still a "said so" because it is a definition. Even if thought went into the definition, it doesn't make any less of a definition.

Re: 0^0

#100

Post by my_username_is_ »

biot wrote: post #58
analog31 wrote: post #53

Perhaps a related question: How should it be defined in a math library for a programming language? Should it return 1, or throw an exception?

Why is that even a question? It should return the correct answer, of course. Edit: At the very least, that behaviour should be a configurable option for those who desire something other than what most mathematicians accept as being the correct answer.

The whole point of this is that there isn't necessarily a single right answer. As another user pointed out, mathematicians define 0^0 to be 1, but you don't necessarily have to accept that as truth in the way that 1+1=2.
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