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Re: 0^0

#141

Post by rickhanlonii »

rntz wrote: post #105
rickhanlonii wrote: post #86

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…

In a certain sense, "1+1 is 2" is also merely a definition, in the same sense that "0^0 is 1" is a definition. Addition can be formally defined in mathematics; we habitually omit this definition because it is tedious, and because addition is such an intuitive operation that we do not require a definition in order to reason about it. Much as the question "what if the parallel axiom didn't hold?" leads to alternative g…

I agree with the point you're making, but I want to be a little pedantic: It's true that addition is a definition, but 1+1=2 is not--it logically follows from the definition of addition.

Re: 0^0

#142

Post by tel »

This question doesn't have an answer because, as this post suggests, the meanings of the first 0, the second 0, and the (^) are overloaded. The reason mathematicians don't mind the overloading is that they all have "kind of" similar meanings.

So we can talk about a recursively defined function on naturals where it's convenient to "define" 0^0 = 1. We can talk about various kinds of limit techniques in the reals where 0^0 is not actually a value but instead a shorthand for a particular kind of limit.

This probably forms the most interesting notion of what 0^0 means where we define it as the limit of a particular kind of path in the complex plane and then notice that 0^0 can take any value we choose---depending on exactly what path was taken to get there. [0]

[0] http://en.wikipedia.org/wiki/Complex_logarithm

Re: 0^0

#143

Post by chacham15 »

mathgrad wrote: post #137
chacham15 wrote: post #132

Earlier quoted context omitted.

I didnt discuss a function, I discussed a mapping. They are different constructs. Imagine the question like this: you have two groups of people, students and teachers. How many possible ways are there to assign students to teachers? To put it another way, how many arrows would it take pointing from the student to the teacher to illustrate every possible assignment? Now, if there are no students and no teachers, you h…

I am a mathematician I don't need examples. What is your formal definition of a mapping. In mathematic it is normally used as a synonym for a morphism in a given category and in the category of sets this would be a function. So in our context a mapping is a function.

> I am a mathematician I don't need examples.

I gave the example not to teach you, but so that we could get away from terminology and get to meaning. Look at the question I posed: "how many arrows would it take pointing from the student to the teacher to illustrate every possible assignment?" So, if you have two students Alice and Eve, and two teachers Bob and Carl, there are four arrows: Alice->Bob, Alice->Carl, Eve->Bob, and Eve->Carl. How many arrows are there if there are no teachers and no students?

Re: 0^0

#144

Post by einhverfr »

chacham15 wrote: post #121
rickhanlonii wrote: post #86

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…

> If you do not accept it as true (an explicitly accept it as false), then you can prove all of Hyperbolic Geometry What is the equivalent in this analogy if you do not accept that 0^0=1 (i.e. accept that 0^0=0)?

I think you break algebra if you start treating y = x^2/x as distinct from y = x, and the both the former and x/x have the same problem with x=0.

Edit:

As has occurred to me, you might actually have a point, in that x/x may be a continuous function, but so is 0^x, which gives you a different limit. So it may actually determine whether in context it makes sense to treat 0^0 as x/x or whether it makes sense to treat it as a limit of 0^x as x -> 0. This may be application dependent.

Edit2:

Ouch. 0^x can't be defined for negative numbers, so I am back to siding with 1 for reasons that any other answer breaks algebra.

Re: 0^0

#145

Post by placeybordeaux »

rntz wrote: post #105
rickhanlonii wrote: post #86

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…

In a certain sense, "1+1 is 2" is also merely a definition, in the same sense that "0^0 is 1" is a definition. Addition can be formally defined in mathematics; we habitually omit this definition because it is tedious, and because addition is such an intuitive operation that we do not require a definition in order to reason about it. Much as the question "what if the parallel axiom didn't hold?" leads to alternative g…

There are a couple of ways to define this part of mathematics, I believe I have seen some definitions that do in fact define `1 + 1 = 2`

Re: 0^0

#146

Post by chunky1994 »

mason55 wrote: post #135
chunky1994 wrote: post #60

Earlier quoted context omitted.

What exactly do you mean by fundamental truths here? Mathematics is an internally consistent (for the most part) logical framework that is extremely powerful in expressing our knowledge about the world. However, that doesn't mean that there is some intrinsic correctness about it or its concepts.

If I have three objects and you give me two more then I'll always have five objects. You can call it cinco or 五 but there are still five of them. Likewise, you'll always be able to determine the length of the hypotenuse of a right triangle by its two legs. No matter what system you set up, if you're cutting three boards to build a triangle the length of the big one is absolutely defined by the length of the other two…

> No matter what system you set up, if you're cutting three boards to build a triangle the length of the big one is absolutely defined by the length of the other two, assuming Euclidian geometry. The symbols (a, b, =, c, +, superscript 2) are all totally arbitrary but if you're working with three boards there is absolutely an intrinsic correctness to the Pythagorean theorem.

This is only true because we are assuming Euclidean geometry, and as I had said earlier, this implies that we can logically conclude these facts because mathematics is logical and internally consistent given these axioms, so we cannot say that they are intrinsically true; that would require us to know without a doubt that the axioms are correct. Since we can use Gödel's incompleteness theorem to show that our axioms are necessarily assumed, we can say that we do not know if they are correct or not, only that assuming they are, we can make a lot of really good predictions about the world around us.

> The amazingly cool thing is that our system is so good that we can use our abstract symbols to make predictions about physical laws and they actually come out to be true!

This is definitely amazing, however it does not mean that our system is necessarily correct, in fact it is demonstrably lacking in Quantum mechanics for example, where we need to renormalize infinities, which makes almost no mathematical sense whatsoever, but we do it because our experiments tell us that if we do, we can make predictions about how things work.

> I would argue that all of physics is basically "math that's a fundamental truth of the universe".

I agree with the spirit of what you are saying here, but think I would phrase it in the following way: I would argue that physics is basically fundamental truths about the universe that we can describe to the best of our ability using an abstract framework such as math. The fact that we can do that does not mean mathematics consists of these fundamental truths.

EDIT: Modified last paragraph to be slightly clearer.

Re: 0^0

#147

Post by j2kun »

einhverfr wrote: post #139
rickhanlonii wrote: post #86

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…

I see it as somewhat complex but in the end I sort of agree that it is a definition. Let's start with a problem: 0/0 The problem with this is not that the equation itself is meaningless. the limit of n/x as x->0 is infinity for any positive real number, and negative infinity for any negative real number. In essence 0/0 ends up reducing to 0 * infinity, which isn't very helpful. I would argue that discontinuity in a f…

But what if you're in a context where you're not reasoning about continuous functions at all? Why would you have to be subject to reasoning that doesn't apply to your situation?

Re: 0^0

#148

Post by gohrt »

mathgrad wrote: post #128
j2kun wrote: post #119

Earlier quoted context omitted.

You're just pushing the arbitrariness of defining things one step further to the definition of a number. It makes it no less arbitrary that you've defined it and force us to accept the definition to get to your conclusion.

There is a very natural definition of numbers as sets. We define 0 to be the empty set and we define the successor function by S(x) = {x} union x. Then the natural numbers are the smallest set containing 0 and closed under the successor operation. This is the standard way to define the natural numbers within ZFC set theory. This is admittedly very formal and not how the lay person thinks of natural numbers. However,…

> within the system we work.

within a system we work. Which is the point. The system you are working in determines which value you use. You are more an algebraist than analyst.

Re: 0^0

#149

Post by yaks_hairbrush »

anonymoushn wrote: post #71

I did not encounter this convention while working on my math degree. I am surprised that none of the characters in the article said "0^0 is nothing, but limits of the form 0^0 can be any nonnegative real number or infinite".

Yes, you did. It may not have been stated explicitly, but it was implied in a couple places:

1. Binomial theorem. Its statement does not conventionally include any caveats about the x^0 case.

2. D(x^n) = n x^(n-1). When n = 1 and x = 0, you get a 0^0 on the RHS. If 0^0 is taken to be 1, the derivative rule holds. Else, the statement gets messy: D(x^n) = n x^(n-1) if (x,n) != (0,1)

Re: 0^0

#150

Post by gohrt »

scythe wrote: post #102
didgeoridoo wrote: post #11

Students: Let's come up with some crazy proofs based on our individual levels of understanding. Teachers: Let's do it by the book and come up (somehow) with conflicting answers. Mathematicians: Yeah, sorry guys. We made it all up. Pretty much captures most mathematicians I know.

0^0, like any indeterminate form, can be made to equal anything via sufficient cleverness. Consider the limit: y = lim_[x->0] x^[a / log(x)] We have log y = lim_[x->0] (a / log(x)) log(x) = a. So 0^0 equals any number at all! Of course, nobody in their right mind would define exponents this way, but the indeterminacy is inherent in the definition of the symbols.

How does your definition align with x^y in the case x!=0 or y!=0 ?
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