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0^0

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Re: 0^0

#171

Post by mbrutsch »

rickhanlonii wrote: post #141
rntz wrote: post #105

Earlier quoted context omitted.

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.

It took Alfred Whitehead and Bertrand Russell 379 pages to prove that it "logically follows", and that was before they even defined addition! http://quod.lib.umich.edu/u/umhistmath/aat3201.0001.001/401?...

Re: 0^0

#172

Post by xamuel »

gweinberg wrote: post #10

Missing Q and A: But if mathematicians insist it is 1, why do high school teachers act like they know more than the mathematicians do? A: They don't. The statement that mathematicians uniformly say it is 1 is simply false. My high school teacher had a PhD in math, I think it's fair to say she was a mathematician. And yes, she said it was undefined.

>"The statement that mathematicians uniformly say it is 1 is simply false"

Not really. No serious mathematician would dispute the fact that for every real x, e^x=sum(n=0 to infinity)x^n/n!.

But the above fails at x=0 if we don't define 0^0=1. So even mathematicians who claim to not use 0^0=1, can almost always be convinced to admit that they do indeed use 0^0=1, using the Taylor series for e^x.

Re: 0^0

#173

Post by SamReidHughes2 »

rickhanlonii wrote: post #141
rntz wrote: post #105

Earlier quoted context omitted.

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.

It's a definition of what "2" means.

Re: 0^0

#174

Post by xamuel »

my_username_is_ wrote: post #100
biot wrote: post #58

Earlier quoted context omitted.

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.

If the calculator is powerful enough (as in the case of Mathematica) to evaluate Taylor series, for example the Maclaurin series for e^x when x=0, then in doing so it implicitly admits 0^0=1. If it simultaneously says 0^0 is not 1, then the calculator is inconsistent. (Mathematica IS inconsistent in this example)

Re: 0^0

#175

Post by xamuel »

gus_massa wrote: post #81
azinman2 wrote: post #22

Another good reminder on how math itself is arbitrary and made up by humans (often for what's simplest/easiest), and not handed down to us by God. Luckily it's an extremely useful and extendable made up system. I see this all the time with AI/machine learning. Most algorithms are based on assumptions that make the math work out better rather than being aligned with some "fundamental truth." The world is not linear, b…

This is a problem of definitions. The definitions are arbitrary and are chosen to make the life (of the mathematicians) easier. It’s easier to write a lot of results if we define 0^0=1. On the other hands, he proofs express a fundamental truth and are handed down to us by God (or whatever deity you believe in).

The definitions only decorate the truths. By unwrapping the definitions, the truths can be expressed using just the barest predicates and function symbols of the background language, and logical operators. Whether you define 0^0 to be 1 or not doesn't change the unwrapped truth.

Re: 0^0

#176

Post by bumbledraven »

Although standard mathematica notation glosses over the difference, it's important to distinguish between the following two versions of the power operation, which I'll call pow1 and pow2: pow1(x, y) = x^y where y is any integer pow2(x, y) = x^y where y is any real

The value of 0^0 is 1 for (pow-a) and 0 for (pow-b).

Re: 0^0

#177

Post by shurcooL »

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…

Agree, and you've basically described the concluding message in the last minute of this awesome video: https://www.youtube.com/watch?v=TINfzxSnnIE

Re: 0^0

#178

Post by _nedR »

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…

Its funny how quickly the argument devolved into an argument over how (or even whether) 1 + 1 = 2

Re: 0^0

#179

Post by smsm42 »

SamReidHughes2 wrote: post #173
rickhanlonii wrote: post #141

Earlier quoted context omitted.

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.

It's a definition of what "2" means.

It's one of the possible definitions. You could as well define next(x) = min(y such as y>x) and define 2 as next(next(0)) (0 can be defined as either x such as for all y y+x = y, or as x such that for all y x<y, depending on what set you are working with). Proving that 2 is 1+1 would be a theorem then. Of course, that works only for integers :)

Re: 0^0

#180

Post by xyzzyz »

chacham15 wrote: post #132
mathgrad wrote: post #129

Earlier quoted context omitted.

A function is a relation for elements in a to elements in b such that for every element in a there is a unique element in b. This is vacuously true of the empty relation when a is empty.

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…

So mapping is exactly a function, unless domain and codomain are empty, in which case it doesn't exist?

Empty set is as good and useful as nonempty ones, and you seen to have no problems with using them. Why do you find it hard to accept an empty function f: E -> X, where E is empty set, that doesn't associate any element of E with any element of X?

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