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On 7/20/2013 11:28 AM, Warp wrote:
> Nekar Xenos <nek### [at] gmail com> wrote:
>> I was thinking of adding any particular infinity to itself.
>
> In a sense, adding infinity to itself doesn't make any sense because
> infinity is not a number. It's a concept used to describe an abstraction.
Huh? Certainly infinity isn't an integer/real/rational/etc, but there
are consistent and sensible definitions for "number" which include
infinity(s) (I'm pretty sure you're aware of this, so maybe I'm missing
your point?).
Of course infinity is *also* used as an abstraction (as in the case with
limits you mention), but this doesn't mean that in other cases you can't
use it as a perfectly fine number.
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Kevin Wampler <nob### [at] nowherenet> wrote:
> Huh? Certainly infinity isn't an integer/real/rational/etc, but there
> are consistent and sensible definitions for "number" which include
> infinity(s) (I'm pretty sure you're aware of this, so maybe I'm missing
> your point?).
So it's a number that's not any kind of number? It's not an integer,
it's not rational, it's not irrational, it's not transcendental, and it
doesn't follow any of the rules of any of the other sets (eg. if you
add 1 to any number, you get a new number that's larger than the
original; or if you multiply any number by 0 you get 0.) All mathematical
operators and functions work completely different for it than for any
other numbers (moreover, most of them aren't even well-defined for
infinity.)
Infinity is not a number. It's just a concept that's used to describe
a more abstract notion. You can use it a bit like if it were a number
when dealing with things like limits, but even then it's just a shortcut
notation.
--
- Warp
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Am 21.07.2013 01:27, schrieb Warp:
> Kevin Wampler <nob### [at] nowherenet> wrote:
>> Huh? Certainly infinity isn't an integer/real/rational/etc, but there
>> are consistent and sensible definitions for "number" which include
>> infinity(s) (I'm pretty sure you're aware of this, so maybe I'm missing
>> your point?).
>
> So it's a number that's not any kind of number? It's not an integer,
> it's not rational, it's not irrational, it's not transcendental, and it
> doesn't follow any of the rules of any of the other sets (eg. if you
> add 1 to any number, you get a new number that's larger than the
> original; or if you multiply any number by 0 you get 0.) All mathematical
> operators and functions work completely different for it than for any
> other numbers (moreover, most of them aren't even well-defined for
> infinity.)
>
> Infinity is not a number. It's just a concept that's used to describe
> a more abstract notion. You can use it a bit like if it were a number
> when dealing with things like limits, but even then it's just a shortcut
> notation.
Well, I guess it simply boils down to definitions again: What, actually,
/is/ a "number" (in the system you're currently examining)?
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On 7/20/2013 4:27 PM, Warp wrote:
> So it's a number that's not any kind of number? It's not an integer,
> it's not rational, it's not irrational, it's not transcendental, and it
> doesn't follow any of the rules of any of the other sets (eg. if you
> add 1 to any number, you get a new number that's larger than the
> original; or if you multiply any number by 0 you get 0.) All mathematical
> operators and functions work completely different for it than for any
> other numbers (moreover, most of them aren't even well-defined for
> infinity.)
Before I say anything, it's important to point out that I don't find the
term "number" to be particularly mathematically meaningful in the first
place. It's probably best to just be explicit to say "integer", "real",
"cardinal number" etc., and not bother too much about what counts as a
"true number" and what doesn't. That said, if you're going to take some
sort of definition as to what counts as a "number", then there are some
good reasons that you might want to include some notions of infinity as
numbers.
What you say above is true for the particular type of infinity you seem
to be thinking of, but it is *not* true in general for other definitions
of infinity. I'll address it point by point:
Q) It's not an integer, it's not rational, it's not irrational, it's not
transcendental
A) If this is your criteria then you've basically said from the
beginning that infinity isn't a number by definition -- no matter how
much like a number it behaves. You can certainly take this approach,
but you'll need to accept the existence of "number-like things which are
not numbers". This is completely valid, in which case I don't think we
really disagree in any substantial way but just draw the line between
"number" and "number-like thing" is different places.
Q) it doesn't follow any of the rules of any of the other sets (eg. if
you add 1 to any number, you get a new number that's larger than the
original; or if you multiply any number by 0 you get 0.)
A) This is not true in general. There are definitions of infinity for
which infinity+1 > infinity, and for which infinity*0 = 0. And besides,
there are similar situations with "normal" numbers anyway. For instance
in the reals you can calculate 1/x for every number x except 0, and
something like x+1 > x doesn't makes sense with complex numbers. Heck,
in the integers x/y only makes sense if y if a factor of x. "Normal"
numbers have arithmetic exceptions too, and it doesn't prevent you from
thinking of them as genuine numbers.
Q) All mathematical operators and functions work completely different
for it than for any other numbers (moreover, most of them aren't even
well-defined for infinity.)
A) I actually find the change in how things work by including infinite
numbers to be *less* than the difference between the integers and the
reals. Saying "All mathematical operators and functions work completely
different for it than for any other numbers" simply isn't true in
general. It's certainly true for *some* definitions of infinity, but
for other definitions you get perfectly well defined addition and
multiplication, and for a few you even get commutative addition and
multiplication, as well as division, subtraction, etc. Heck there are
some with these properties where you even maintain that either a <= b or
b <= a, which isn't even true of the complex numbers!
I mean, obviously you need to draw the line somewhere between what you
call a number, and what behaves differently enough that the term
"number" no longer makes sense, but I think it's pretty reasonable to
consider some (but all) definitions of infinity as sufficiently
number-like to be genuine numbers. I also think that it's fine to call
complex numbers, quaternions, elements in cyclic groups etc. as numbers
too, so maybe I just take a pretty broad stance as to what I think of as
a number.
> Infinity is not a number. It's just a concept that's used to describe
> a more abstract notion. You can use it a bit like if it were a number
> when dealing with things like limits, but even then it's just a shortcut
> notation.
This is math we're talking about, everything is abstract in some sense.
I agree with limits that it's a notational shortcut, but there are
many many other ways of defining infinity quantities, and some of them
behave quite a bit like numbers.
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Kevin Wampler <nob### [at] nowherenet> wrote:
> I agree with limits that it's a notational shortcut, but there are
> many many other ways of defining infinity quantities, and some of them
> behave quite a bit like numbers.
So how do you define infinity so that it's a number?
--
- Warp
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On Sun, 21 Jul 2013 03:26:02 +0200, Kevin Wampler <nob### [at] nowherenet>
wrote:
> On 7/20/2013 4:27 PM, Warp wrote:
> Q) All mathematical operators and functions work completely different
> for it than for any other numbers (moreover, most of them aren't even
> well-defined for infinity.)
>
> A) I actually find the change in how things work by including infinite
> numbers to be *less* than the difference between the integers and the
> reals. Saying "All mathematical operators and functions work completely
> different for it than for any other numbers" simply isn't true in
> general. It's certainly true for *some* definitions of infinity, but
> for other definitions you get perfectly well defined addition and
> multiplication, and for a few you even get commutative addition and
> multiplication, as well as division, subtraction, etc. Heck there are
> some with these properties where you even maintain that either a <= b or
> b <= a, which isn't even true of the complex numbers!
>
It almost seems that any mathematical operation using infinity has the
same answer: Infinity
:)
--
-Nekar Xenos-
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On 21/07/2013 8:37 AM, Nekar Xenos wrote:
> It almost seems that any mathematical operation using infinity has the
> same answer: Infinity
> :)
And some answer only to themselves. O.o
--
Regards
Stephen
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On 7/21/2013 12:37 AM, Nekar Xenos wrote:
> On Sun, 21 Jul 2013 03:26:02 +0200, Kevin Wampler <nob### [at] nowherenet>
>
> It almost seems that any mathematical operation using infinity has the
> same answer: Infinity
> :)
>
Well, for some of these you get a whole slew of different "infinite
numbers", so the result of many mathematical operations using infinity
just give you different infinities as their answers (which is pretty
much what you'd expect if you count these infinite quantities as actual
numbers).
In situations where such things are allowed, dividing by infinity can be
useful, in which case it gives you either zero or an infinitesimal,
depending on which system you're using.
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On 7/21/2013 12:34 AM, Warp wrote:
> Kevin Wampler <nob### [at] nowherenet> wrote:
>> I agree with limits that it's a notational shortcut, but there are
>> many many other ways of defining infinity quantities, and some of them
>> behave quite a bit like numbers.
>
> So how do you define infinity so that it's a number?
>
I would count any of the following, but you may find the last two to be
better examples than the first two:
1) cardinal numbers: http://en.wikipedia.org/wiki/Cardinal_number
2) ordinal numbers: http://en.wikipedia.org/wiki/Ordinal_number
3) extended complex plane: http://en.wikipedia.org/wiki/Riemann_sphere
4) surreal numbers: http://en.wikipedia.org/wiki/Surreal_number
There are probably plenty of other ways too, but these are the ones that
come to mind for me.
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I wanted to post about this earlier but couldn't remember the name.
Anyway stumbled across it again today, possibly relevant to this thread:
http://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox
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