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Infinity is not a number

en.wikibooks.org

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Re: Infinity is not a number

#11

Post by T_S_ »

impendia wrote: post #6

The problem is that "a number" is meaningless. Is i (square root of -1) a number? Is x? Is 3/2? Depends on your perspective. You can define the integers, the rationals, the real numbers, the complex numbers, and plenty of more exotic systems. What about C[x], the ring of polynomials in one variable. Are these "numbers"? There is no a priori reason to say no. The integers mod 7? Quaternions? Etc. etc. etc. And, yes, y…

Understanding what mathematical objects are can lead to insanity. Much better to focus on what you can do with them.

Or to quote John von Neumann, "In mathematics you don't understand things, you just get used to them."

Re: Infinity is not a number

#12

Post by thaumasiotes »

The argument "Addition breaks" proves just as well that zero "isn't a number", since it breaks division rather badly. (Yes, division is a badly-behaved version of the more upstanding multiplication. The article uses subtraction in what is nominally a complaint about addition). It doesn't address ordinal numbers at all, since addition works just fine with infinite ordinals, exactly the way the article claims you'd expect (what have I missed?).

The immediately obvious uses of infinity in calculus (the topic of the wikibooks article) are, according to wikipedia, termed the "affinely extended real number system" (I learned to just refer to the "extended reals"), which heavily implies that the points within are considered extended real _numbers_. http://en.wikipedia.org/wiki/Extended_reals

The terms "cardinal number" and "ordinal number" both definitively include infinite quantities -- infinitely many, even.

The IEEE standard for floating point defines two infinite numbers.

Essentially, I'm in full agreement with tokenadult; the only relevant question is "what do you mean by number?". But we can easily observe that varying infinities, including the calculus uses of infinity, are referred to as numbers all up and down the chain, including in the most unimpeachably correct sources, and that it walks and quacks like a duck, even if it may not quack in the precise manner of Anas Platyrhynchos.

Re: Infinity is not a number

#13

Post by harshreality »

T_S_ wrote: post #11
impendia wrote: post #6

The problem is that "a number" is meaningless. Is i (square root of -1) a number? Is x? Is 3/2? Depends on your perspective. You can define the integers, the rationals, the real numbers, the complex numbers, and plenty of more exotic systems. What about C[x], the ring of polynomials in one variable. Are these "numbers"? There is no a priori reason to say no. The integers mod 7? Quaternions? Etc. etc. etc. And, yes, y…

Understanding what mathematical objects are can lead to insanity. Much better to focus on what you can do with them. Or to quote John von Neumann, "In mathematics you don't understand things, you just get used to them."

You have to know whether your definition of "numbers" makes them into a field or not, before you can go applying field operations to them and expecting usual results.

Including infinite ordinals in the set of numbers is legitimate, as parent points out that "number" is not strictly defined, but if you include infinity, numbers are no longer a field, and you have to cut someone off when they try to use field axioms in theorems, the uniqueness of multiplicative and additive inverses, which must be how people end up with nonsense like 1=0.

Re: Infinity is not a number

#14

Post by jpallen »

impendia wrote: post #6

The problem is that "a number" is meaningless. Is i (square root of -1) a number? Is x? Is 3/2? Depends on your perspective. You can define the integers, the rationals, the real numbers, the complex numbers, and plenty of more exotic systems. What about C[x], the ring of polynomials in one variable. Are these "numbers"? There is no a priori reason to say no. The integers mod 7? Quaternions? Etc. etc. etc. And, yes, y…

The problem is that "a number" is meaningless.

No it isn't. It's fairly clear from the article that in this context a number is something which you can have that many of. You can have 3 dollars. It even makes sense to have -2.5 dollars. Ok, an irrational number of dollars is pushing it a bit, but you don't need to go that far to see that infinity doesn't work when trying to count things consistently.

A lot of the comments here are getting hung up on trying to pin down the mathematics of what you can and can't do with infinity. That's fine, it's been keeping mathematicians busy for centuries, but this article is for the layman who doesn't know about rings, groups, algebras or any other mathematical structure which you might call 'numbers'. It's for someone who thinks the obvious when someone says 'number'.

Re: Infinity is not a number

#15

Post by powertower »

Infinity can be relative and can be defined unlike a division by zero which in undefined (as in ... it can't be relative to anything else, and can't be used in a formula).

And hence infinity can be used in a formula and can cancel out with another relative infinity...

Example:

1) There are an infinite amount of real numbers between 1 and 2.

2) The amount of real numbers between 2 and 4 is twice the amount of real numbers between 1 and 2.

I would guess that if numbers are defined in terms of relativity/relationship, then infinity is a number.

But it seems that people wrongly define numbers in absolute terms, as if they exist outside the mind, and are separate from one another. Like the Universe cares about 1.24545434 and 7656.45433477.

But that's just my guess.

Re: Infinity is not a number

#17

Post by mcpherrinm »

jpallen wrote: post #14
impendia wrote: post #6

The problem is that "a number" is meaningless. Is i (square root of -1) a number? Is x? Is 3/2? Depends on your perspective. You can define the integers, the rationals, the real numbers, the complex numbers, and plenty of more exotic systems. What about C[x], the ring of polynomials in one variable. Are these "numbers"? There is no a priori reason to say no. The integers mod 7? Quaternions? Etc. etc. etc. And, yes, y…

The problem is that "a number" is meaningless. No it isn't. It's fairly clear from the article that in this context a number is something which you can have that many of . You can have 3 dollars. It even makes sense to have -2.5 dollars. Ok, an irrational number of dollars is pushing it a bit, but you don't need to go that far to see that infinity doesn't work when trying to count things consistently. A lot of the co…

In this context, infinite limits, I think the distinction is beginning to become important. I agree that colloquially it often doesn't matter, but I think it does here.

"Something which you can have that many of" is a terrible definition: You can have sets of infinite size. Do you mean physical things? What, then, is a "thing"? I can have infinity intervals of different length on my arm. So are intervals not a "thing"? You appear to think irrationals are numbers, so what physical thing can you have an irrational number of?

This is a dangerous path to walk down, but that doesn't make it an unimportant one. And that's why this article is ultimately flawed. The idea is important though. We need to understand how to operate formally on mathematical objects. If there's one thing I learned in my Philosophy of Mathematics class as a math undergrad, it is that we shouldn't struggle over defining what it means to be a number. We can just use them. I'll leave the philosophy to the philosophers.

Re: Infinity is not a number

#18

Post by fferen »

powertower wrote: post #15

Infinity can be relative and can be defined unlike a division by zero which in undefined (as in ... it can't be relative to anything else, and can't be used in a formula). And hence infinity can be used in a formula and can cancel out with another relative infinity... Example: 1) There are an infinite amount of real numbers between 1 and 2. 2) The amount of real numbers between 2 and 4 is twice the amount of real num…

Correct me if I'm wrong, but I believe the cardinality of the two sets you describe are equal, as there exists a bijective mapping between them, meaning there are an equal "number" of real numbers in both.

Re: Infinity is not a number

#19

Post by drt1245 »

fferen wrote: post #18
powertower wrote: post #15

Infinity can be relative and can be defined unlike a division by zero which in undefined (as in ... it can't be relative to anything else, and can't be used in a formula). And hence infinity can be used in a formula and can cancel out with another relative infinity... Example: 1) There are an infinite amount of real numbers between 1 and 2. 2) The amount of real numbers between 2 and 4 is twice the amount of real num…

Correct me if I'm wrong, but I believe the cardinality of the two sets you describe are equal, as there exists a bijective mapping between them, meaning there are an equal "number" of real numbers in both.

Not only does there exist a bijective mapping between [1,2] and [1,4], there exists infinite different bijective mappings between a subset of [1,2] and [1,4].

i.e.: One could map bijectively from [1,1.5] to [1,4] and map bijectively from [1.5,2] to [1,4] (1)

To talk about there being "twice as much" in one uncountable infinity than in another uncountable infinity is nonsense, since you can't apply words like "twice", since the infinities can't be counted.

(1) https://imgur.com/NkKEI

Re: Infinity is not a number

#20

Post by jpallen »

mcpherrinm wrote: post #17
jpallen wrote: post #14

Earlier quoted context omitted.

The problem is that "a number" is meaningless. No it isn't. It's fairly clear from the article that in this context a number is something which you can have that many of . You can have 3 dollars. It even makes sense to have -2.5 dollars. Ok, an irrational number of dollars is pushing it a bit, but you don't need to go that far to see that infinity doesn't work when trying to count things consistently. A lot of the co…

In this context, infinite limits, I think the distinction is beginning to become important. I agree that colloquially it often doesn't matter, but I think it does here. "Something which you can have that many of" is a terrible definition: You can have sets of infinite size. Do you mean physical things? What, then, is a "thing"? I can have infinity intervals of different length on my arm. So are intervals not a "thing…

I pretty much agree with you. I'm just worried that with all the comments about 'what is a number?', and 'sure, we can make infinity a well defined number', it will dilute the message of the article. Which is that infinity is not a 'number' (in the sense of real numbers). It's important to understand why it's not a number and what it means when we write it on one side of an equality.
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