Infinity is not a number • Hacker News Board

Re: Infinity is not a number

#31

Post by yequalsx » Wed, Feb 15, 2012, 12:55 AM UTC

tokenadult wrote: post #26

The discussion that has begun in this thread suggests that the Wikibooks chapter submitted here could use some more work. (Its last revision was a 9 October 2011‎ reversion of I.P. edits to restore a version from 22 May 2011.) Evidently, not every reader of Hacker News is convinced that infinity is not a number, despite several websites by mathematically learned people who say exactly that, http://scienceblogs.com/go…

We have to define what is meant by "infinity". Do you mean the infinity symbol one encounters in calculus with regard to limits? Do you mean infinite cardinal number? Infinite ordinal numbers? Do you mean the symbols added to the reals in order to compactify the set (extended real numbers)? Some other concept?

Once that gets settled then we have to define what is meant by a number. I suggest that the most broad, reasonable definition of this concept would be elements of a ring. There should be some algebraic structure that is broad but not too broad. If we use such a definition then the various concepts of infinity mentioned above are not numbers since they are not elements of a ring.

The limit example in the Wikibooks article is merely a shorthand notation and isn't a number. It's mostly to be suggestive and make things a bit easier to comprehend. It's not intended that a person think of it as a number.

Re: Infinity is not a number

#32

Post by [deleted] » Wed, Feb 15, 2012, 1:05 AM UTC

tokenadult wrote: post #26

The discussion that has begun in this thread suggests that the Wikibooks chapter submitted here could use some more work. (Its last revision was a 9 October 2011‎ reversion of I.P. edits to restore a version from 22 May 2011.) Evidently, not every reader of Hacker News is convinced that infinity is not a number, despite several websites by mathematically learned people who say exactly that, http://scienceblogs.com/go…

[deleted]

Re: Infinity is not a number

#33

Post by nandemo » Wed, Feb 15, 2012, 1:06 AM UTC

tokenadult wrote: post #26

The discussion that has begun in this thread suggests that the Wikibooks chapter submitted here could use some more work. (Its last revision was a 9 October 2011‎ reversion of I.P. edits to restore a version from 22 May 2011.) Evidently, not every reader of Hacker News is convinced that infinity is not a number, despite several websites by mathematically learned people who say exactly that, http://scienceblogs.com/go…

(Check your link to the WolframMathWorld article)

> Evidently, not every reader of Hacker News is convinced that infinity is not a number, despite several websites by mathematically learned people who say exactly that

Like impendia said, it depends on the context.

To give a simpler example: at elementary school I was told that I couldn't write "2 - 3". Essentially my teacher was saying that numbers less than zero did not exist! She wasn't lying to me. She was telling me that we were working with natural numbers only, and that our operation "-" was supposed to be closed on the natural numbers.

In the context of calculus, infinity is not a number. lim(n->c) foo = ∞ is just another way of saying that "foo does not converge as n goes to c".

But in realm of surreal numbers (or combinatorial game theory), infinities are first-class numbers. You can add them and subtract them alright! Besides, in that realm ω + 1 is not the same as ω. It is greater than ω, as expected. I can even show you a children's game -- that is, we can teach kids how to play it without ever uttering the words sets and cardinals -- that has positions with such values.

Re: Infinity is not a number

#34

Post by powertower » Wed, Feb 15, 2012, 1:40 AM UTC

drt1245 wrote: post #30

powertower wrote: post #29

Earlier quoted context omitted.

For every real number between 1 and 2... 1) I can find that number also in the set between 1 and 4. 2) But I can't find that number in the set between 2 and 4. To myself numbers are always relative to one another, don't exist outside the mind, and the set between 2 and 4 is twice the set between 1 and 2. But like I said, it's a personal spin on it.

Where 1 f(x) = (x-1)/9 + 1, maps bijectively from [1,4] to [1,4/3] g(x) = (x-1)/9 + 4/3, maps bijectively from [1,4] to [4/3,5/3] h(x) = (x-1)/9 + 5/3 maps bijectively from [1,4] to [5/3,2] Therefore, [1,2] must contain three times as many numbers as [1,4], right? It doesn't work like that.

I'm not sure what you are doing with the above.

Re: Infinity is not a number

#35

Post by mturmon » Wed, Feb 15, 2012, 1:40 AM UTC

drt1245 wrote: post #28

thaumasiotes wrote: post #12

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 exp…

> The argument "Addition breaks" proves just as well that zero "isn't a number", since it breaks division rather badly. Mathematically, numbers (be it natural, rational, real, or complex) are defined as a field. Fields (or, more accurately, rings, which all fields are) are defined by addition and multiplication, not both. [1] [1] https://en.wikipedia.org/wiki/Ring_(mathematics)

Sometimes you may use a "number", but you just want ordering properties from it.

You may not want addition and multiplication in such a "number".

This may be the case in numbers used for ranking outcomes, or counting, or optimization. Using infinity in this context does not cause problems.