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if you do, then it's not infinity. how then are we supposed to understand it? or are we not entitled to know it? or is infinity just a human thought and there is no such thing? or you can firmly believe in some definition of infinity and stick to it and it will be your infinity in your own world.

Nope. Anybody who tells me they inderstand infinity doesn't understand it. Can we understand it? Probably, but not in our current, limited biological forms.

Just as there must be more dimensions, infinity will be better understood as we have a greater vocabulary from future discoveries.

Infinity is merely a direction taken to an infinite amount of time.  Whether or not it actually exists depends on whether or not the universe is finite or not.  This currently is not known.

To be honest, the concept of infinity is rather useless outside of certain mathematical uses, such as Calculus.

The concept of infinity can be proven in thought experiment.  For example, you could walk an infinite distance around the earth, provided that you lived forever, because you would just be walking in circles.  But when you try to put infinity to practical use, you encounter problems.  Mostly, it's the fact that nobody has an infinite amount of time to test something, unless someone has drunken an immortality potion that I am unaware of.

Quote from: lady amarant on February 24, 2008, 07:50:10 AM
Nope. Anybody who tells me they inderstand infinity doesn't understand it. Can we understand it? Probably, but not in our current, limited biological forms.

<nerd voicestyle="nasal">
I beg to differ.

Infinity is simply the property of a set (http://mathworld.wolfram.com/Set.html) that it has a mapping (http://mathworld.wolfram.com/Map.html) with a proper subset (http://mathworld.wolfram.com/ProperSubset.html) that is one-to-one (http://mathworld.wolfram.com/One-to-One.html) but not onto (http://mathworld.wolfram.com/Onto.html).

For example, each natural number (0,1,2,...) has a number that is its successor (0->1, 1->2, 23346->23347, etc.); not all numbers natural numbers are successors of another (0 is not the successor of any natural number). There are many such relationships: each natural number has a double, a square, a cube, etc., that is another natural number, but not the other way around. Any set with such a relationship is infinite. That's all there is too it.

(and then there's the epsilons-and-deltas calculus/real analysis formulation, but it's basically the same idea)
</nerd>

Here's someone who understood infinity pretty well, I think:

(https://www.susans.org/proxy.php?request=http%3A%2F%2Fupload.wikimedia.org%2Fwikipedia%2Fen%2F1%2F17%2FGeorg_Cantor.jpg&hash=cb4025b8c566733fd14f96a870bc01afd32a9c42)

That would be good ol' Georg Ferdinand Ludwig Philipp Cantor (http://en.wikipedia.org/wiki/Georg_Cantor). He had some of the coolest ideas of any mathematician ever IMO, and surprisingly accessible ones too.

universe is finite

That the universe is finite does not preclude infinity. 

But hell no, I don't understand it.  I have a hard time keeping my finances in line, much less the infinity of spacetime.

Quote from: Alyssa M. on February 27, 2008, 08:40:48 PM

Quote from: lady amarant on February 24, 2008, 07:50:10 AM
Nope. Anybody who tells me they inderstand infinity doesn't understand it. Can we understand it? Probably, but not in our current, limited biological forms.

<nerd voicestyle="nasal">
I beg to differ.

Infinity is simply the property of a set (http://mathworld.wolfram.com/Set.html) that it has a mapping (http://mathworld.wolfram.com/Map.html) with a proper subset (http://mathworld.wolfram.com/ProperSubset.html) that is one-to-one (http://mathworld.wolfram.com/One-to-One.html) but not onto (http://mathworld.wolfram.com/Onto.html).

For example, each natural number (0,1,2,...) has a number that is its successor (0->1, 1->2, 23346->23347, etc.); not all numbers natural numbers are successors of another (0 is not the successor of any natural number). There are many such relationships: each natural number has a double, a square, a cube, etc., that is another natural number, but not the other way around. Any set with such a relationship is infinite. That's all there is too it.

(and then there's the epsilons-and-deltas calculus/real analysis formulation, but it's basically the same idea)
</nerd>

Here's someone who understood infinity pretty well, I think:

(https://www.susans.org/proxy.php?request=http%3A%2F%2Fupload.wikimedia.org%2Fwikipedia%2Fen%2F1%2F17%2FGeorg_Cantor.jpg&hash=cb4025b8c566733fd14f96a870bc01afd32a9c42)

That would be good ol' Georg Ferdinand Ludwig Philipp Cantor (http://en.wikipedia.org/wiki/Georg_Cantor). He had some of the coolest ideas of any mathematician ever IMO, and surprisingly accessible ones too.

Mathematically we also understand and use the concept of the square root of -1, but that doesn't mean we can point to such a ridiculous thing in nature or really get our minds around WHAT it actually is - yet it must exist at some level, 'cuase it gets used in electrical engineering on a daily basis, or so my engineering friends kept telling me.

Infinity as a mathematical concept is quite simple, I agree, but as a real, actual thing, I feel our minds are just too limited. I mean, we're still bound to this idea of 3 dimensional space bound to the fourth one, time, when it might actually all just be a holographic projection (Bohm's holographic principle) or a collection of 'maybe's' (classical quantum physics) or existing in many slightly different configurations at once or sliced into moments of time all existing at the same ... 'time'?! Even our language is inadequate, and since our thoughts are bounded by our language, we've still got a long way to go, I feel.

In mathematics and physics it is a fairly well defined term.  It is simple to use.

In cosmology, it is also useful in defining things.

I suppose that it isn't so much understanding infinity as accepting it.

Cindi

Quote from: lady amarant on February 28, 2008, 12:45:07 AM
Mathematically we also understand and use the concept of the square root of -1, but that doesn't mean we can point to such a ridiculous thing in nature or really get our minds around WHAT it actually is - yet it must exist at some level, 'cuase it gets used in electrical engineering on a daily basis, or so my engineering friends kept telling me.

Infinity as a mathematical concept is quite simple, I agree, but as a real, actual thing, I feel our minds are just too limited. I mean, we're still bound to this idea of 3 dimensional space bound to the fourth one, time, when it might actually all just be a holographic projection (Bohm's holographic principle) or a collection of 'maybe's' (classical quantum physics) or existing in many slightly different configurations at once or sliced into moments of time all existing at the same ... 'time'?! Even our language is inadequate, and since our thoughts are bounded by our language, we've still got a long way to go, I feel.

Hmm.... Well, human language -- I mean, spoken language -- is pretty inadequate to the job. But the mathematical language developed by Cantor, Peano, Hilbert, Gödel, etc. removes the barriers pretty darn well.

I guess I'm not sure what one would mean by "infinite" outside of the mathematical concept -- I'm not really sure what "finite" means outside of math either! I certainly don't understand what the square root of -1 would mean -- or -1 itself!

pls pls pls don't get me started on interpretations of QM -- Copenhagen, Many World, etc.!!!  :P :-X

Quote from: Alyssa M. on February 28, 2008, 01:57:07 AM
Hmm.... Well, human language -- I mean, spoken language -- is pretty inadequate to the job. But the mathematical language developed by Cantor, Peano, Hilbert, Gödel, etc. removes the barriers pretty darn well.

Only as functions of n-dimensional space, usually with a separate time element, but (and this is purely from an amateur interpretation) when you start to try and understand ideas like higher-dimensional branes and stuff, I get lost with the math real quickly - which admittedly might have more to do with my lack of mathematical skill than a problem with the math.

Quote
I guess I'm not sure what one would mean by "infinite" outside of the mathematical concept -- I'm not really sure what "finite" means outside of math either! I certainly don't understand what the square root of -1 would mean -- or -1 itself!

Practically I just don't think we can translate the math into a real concept that can be held in the mind and examined. All my physics and math professors always said that that was the point when I started bugging them on what the concepts and equations actually represented. Which always seemed a cop-out to me - Science is just a subset of Magic, in my mind - I wanna SEE and UNDERSTAND the cool stuff, not just be able to talk about and describe it.

That's why I hold to the point that we cannot understand infinity, just do an inadequate job of describing it according to the limits of our imagination.

Quote
pls pls pls don't get me started on interpretations of QM -- Copenhagen, Many World, etc.!!!  :P :-X

>:D   Might be fun ...

Quote from: Natasha on February 24, 2008, 06:25:20 AM
if you do, then it's not infinity. how then are we supposed to understand it? or are we not entitled to know it? or is infinity just a human thought and there is no such thing? or you can firmly believe in some definition of infinity and stick to it and it will be your infinity in your own world.

I would say that you can come up with your own working definitions of infinity, in various contexts.

If those working definitions borrow from the works of others, that is fine.  It puts you on the same page they are so you have a common language.

Now you can get a feel for infinity through various exercises.  Doing tree diagrams, where each set of branches represents additional options can get you to VERY large numbers very fast.  "Infinity" could
be the largest tree diagram you can comprehend, plus one.

Quote from: lady amarant on February 28, 2008, 04:47:44 AM
Practically I just don't think we can translate the math into a real concept that can be held in the mind and examined. All my physics and math professors always said that that was the point when I started bugging them on what the concepts and equations actually represented. Which always seemed a cop-out to me - Science is just a subset of Magic, in my mind - I wanna SEE and UNDERSTAND the cool stuff, not just be able to talk about and describe it.

That's why I hold to the point that we cannot understand infinity, just do an inadequate job of describing it according to the limits of our imagination.

Ah, that's just the thing: I look around, and I see the world in mathematical terms: I see integrals and derivatives and divergences and gradients and curls. It's sort of like the Matrix in reverse: On the surface I see the girl in the red dress, but as she walks I see the dynamics of continuous media in the way the dress sways, the minimization of the time integral of the Lagrangian; Newton's Laws and fluid dynamics made manifest. I see the flux of energy and momentum as she uses friction to alter her trajectory, harmonizing her motion with the gravitational potention. And of course many other levels of understanding -- biology, chemistry, quantum and statistical mechanics, electrodynamics of the scattering of light. There are so many of levels of beauty beneath the surface image. But they expressible only in the language of mathematics; this poetry doesn't translate well.

If I see a poem written in Chinese, I can only see the beauty of the typography; a Chinese person doesn't notice the typography, only the poem itself. If you are fluent in the language of math, you can't help but see how these notions translate to the world; if you aren't, it is magic and incantations.

Quote

Quote
pls pls pls don't get me started on interpretations of QM -- Copenhagen, Many World, etc.!!!  :P :-X

>:D   Might be fun ...

Perhaps another time.... ;) (the trout circles, brushes against the lure, and then darts under a rock.)

Quote from: Alyssa M. on February 29, 2008, 01:09:57 PM
Perhaps another time.... ;) (the trout circles, brushes against the lure, and then darts under a rock.)

Ah I look forward to this most intense meeting of minds, this exchange of ideas, this ... geez I do go on, don't I...

Quote from: lady amarant on February 29, 2008, 01:32:50 PM

Quote from: Alyssa M. on February 29, 2008, 01:09:57 PM
Perhaps another time.... ;) (the trout circles, brushes against the lure, and then darts under a rock.)

Ah I look forward to this most intense meeting of minds, this exchange of ideas, this ... geez I do go on, don't I...

Surely as momentous an occasion as the meetings of Bohr and Heisenberg themselves!

Quote from: Alyssa M. on March 01, 2008, 05:21:35 PM
Surely as momentous an occasion as the meetings of Bohr and Heisenberg themselves!

hehe.

Well infinity from the perspective of looking at infinity on its own, yes, it's quite difficult to comprehend.

But take 1 divided by infinity, or pretty much any natural number divided by infinity, and that's the closest number you can get to zero without being zero.  It's still not zero, but the difference is so incredibly small it mind as well be zero, except that it's not.

Infinity is the opposite of that.  Can it be proven or understood empirically? Yes by using the above it's much easier to understand infinity by thinking of infinitely approaching zero than infinity itself. 

It's just a general restatement of Zeno's Paradox but you don't comprehend infinity by looking up, you comprehend it by looking down.

Zeno's paradox shortly explained:

"Zeno's Paradox may be rephrased as follows. Suppose I wish to cross the room. First, of course, I must cover half the distance. Then, I must cover half the remaining distance. Then, I must cover half the remaining distance. Then I must cover half the remaining distance . . . and so on forever. The consequence is that I can never get to the other side of the room." (from http://www.mathacademy.com/pr/prime/articles/zeno_tort/ )

-Kit

Quote from: DarthKitty on March 01, 2008, 07:01:01 PM
"Zeno's Paradox may be rephrased as follows. Suppose I wish to cross the room. First, of course, I must cover half the distance. Then, I must cover half the remaining distance. Then, I must cover half the remaining distance. Then I must cover half the remaining distance . . . and so on forever. The consequence is that I can never get to the other side of the room." (from http://www.mathacademy.com/pr/prime/articles/zeno_tort/ )

Thank goodness Leibniz invented limits eh? (I think Newton was the one who cheated ... Never trust an Englishman ... ;) )

I used to play with Zeno's paradox.

If you take have of what you have left of your life and divide it by two.....

If you follow, you will really never die.  ;)

I suppose that's a form of immortality isn't it?

Cindi

Quote from: Cindi Jones on March 02, 2008, 02:01:18 AM
I used to play with Zeno's paradox.

If you take have of what you have left of your life and divide it by two.....

If you follow, you will really never die.  ;)

I suppose that's a form of immortality isn't it?

Cindi

Damn you Leibniz!!!

In a mathematically rigorous sense, you can't add or subtract or multiply or divide infinity. Infinity isn't really a number; it's more of a concept.  That said, if you want to try to assign some meaningful values to infinity, I'd use these:

Infinity plus infinity would be infinity. If you have two things that are without end, and you put them together, you're left with something without end.

Infinity times infinity should also be infinity, for a similar reason. (If you have so many objects in one bag that you'll never run out, and you also have so many bags that you'll never run out, you certainly have infinitely many objects in total.)

Infinity minus infinity is really tough to define! This is usually called an "indeterminate form" in mathematics. Let's see why.

Suppose that I have all the positive integers, {1, 2, 3, 4, ...}. If you take them all away, I'm left with zero. So maybe infinity minus infinity is zero.  But suppose I have all the positive integers, {1, 2, 3, 4, ...}, and you just take away the ones that are bigger than 10; that is, you take {11, 12, 13, 14, ...}. Then you've taken infinitely many things from my original infinity, and I'm left with ten numbers. So maybe infinity minus infinity is 10.

Now, finally, suppose I have all the positive integers, {1, 2, 3, 4, ...}, and you take away just the odd ones, {1, 3, 5, 7, ...}. Then I'm left with all the even ones, {2, 4, 6, 8, ...}. You've taken infinitely many from me, but I STILL have infinitely many left!

So, for this reason, infinity minus infinity is usually called "indeterminate"--it could be anything, depending on the circumstances.  Infinity divided by infinity is also indeterminate, for a similar reason. Ask yourself "if I have infinitely many objects, and I divide them into infinitely many equal-sized piles, how many are in each pile?"

The answer is "It depends how you do it." (This is never true for a normal number!) If I have all the positive integers {1, 2, 3, 4, ...}, I can divide them into infinitely many piles of one:

{1}, {2}, {3}, {4}, ...

So maybe infinity divided by infinity is one. But I can also divide them into infinitely many piles of three:

{1, 2, 3}, {4, 5, 6}, {7, 8, 9}, {10, 11, 12}, ...

So maybe infinity divided by infinity is three. If I get really clever, I can also divide infinity into infinitely many piles of infinity, as follows:
1. First put one object in the first pile.
2. Now put one object in the first pile, and one in the second pile.
3. Now put one object in the first pile, one in the second pile, and one in the third pile.
4. Now put one object in the first pile, one in the second pile, one in the third pile, and one in the fourth pile.

If I keep going (I'll have to keep going forever, because I have infinitely many objects), then when I'm done, all my piles will have infinitely many objects (I put one object in the first pile on each step; one object in the second pile on each step from the second step onward; one object in the third pile on each step from the third step onward; and one object in the millionth pile each step from the millionth step onward.) So maybe infinity divided by infinity is infinity!

So we say infinity divided by infinity is "indeterminate" as well. Good question though Natasha.

Quote from: Leslie on March 02, 2008, 03:02:06 AM
In a mathematically rigorous sense, you can't add or subtract or multiply or divide infinity. Infinity isn't really a number; it's more of a concept.  That said, if you want to try to assign some meaningful values to infinity, I'd use these:

Infinity plus infinity would be infinity. If you have two things that are without end, and you put them together, you're left with something without end.

...

Infinity minus infinity is really tough to define! This is usually called an "indeterminate form" in mathematics.

Of course, it's not really so fluid as that: Real analysis yields definitive answers as to convergence, boundedness, and divergence of sequences and series. For example, the limit of the ratio of two divergent (or vanishing) functions can be defined using l'Hopital's rule. Certainly you're correct that in analysis, it's more of a concept that a number in its own right: it's sort of an "extra" number that behaves a bit differently, and basically represents a way that sequences or series can fail to converge -- "For every positive integer n, there exists and M..."

But infinity in the set-theoretical sense certainly is a number -- or more accurately, in infinite class of numbers! (see Cantor's Diagonal argument (http://en.wikipedia.org/wiki/Cantor's_diagonal_argument), which proved the existence of uncountable infinities -- really nifty!)

What kind of infinity? Mathematical infinity, as a religious concept, as a philosophical idea or just infinity per se?

I think that you can use infinity as a concept (as it was said above) by defining a few ideas of how you think it should work, that would be from outside. Regardless, that's a long ways off from actually imagining infinity and truly understanding how it swirls, that's probably impossible.

Imagine the universe as mathematical. And it is in many ways. Imagine prime numbers then as part of mathematics.  If you ask "Is the universe infinit ?", then you could also ask the question "Is there a last prime number ? And maybe there is.

So .......... maybe the universe is not infinit, it is just awfully big making it look like infinit.



Posted on: April 10, 2008, 01:04:27 AM

---

Quote from: Alyssa M. on March 02, 2008, 04:15:27 AM

Quote from: Leslie on March 02, 2008, 03:02:06 AM
In a mathematically rigorous sense, you can't add or subtract or multiply or divide infinity. Infinity isn't really a number; it's more of a concept.  That said, if you want to try to assign some meaningful values to infinity, I'd use these:

Infinity plus infinity would be infinity. If you have two things that are without end, and you put them together, you're left with something without end.

...

Infinity minus infinity is really tough to define! This is usually called an "indeterminate form" in mathematics.

Of course, it's not really so fluid as that: Real analysis yields definitive answers as to convergence, boundedness, and divergence of sequences and series. For example, the limit of the ratio of two divergent (or vanishing) functions can be defined using l'Hopital's rule. Certainly you're correct that in analysis, it's more of a concept that a number in its own right: it's sort of an "extra" number that behaves a bit differently, and basically represents a way that sequences or series can fail to converge -- "For every positive integer n, there exists and M..."

But infinity in the set-theoretical sense certainly is a number -- or more accurately, in infinite class of numbers! (see Cantor's Diagonal argument (http://en.wikipedia.org/wiki/Cantor's_diagonal_argument), which proved the existence of uncountable infinities -- really nifty!)

Yes you can divide infinity ! You really can.  00 divided by 00 can be any number.

lim sin(x) /n  = 6, when n -->oo (infinity)

Proof: Cancel the n in the numerator and denominator and remove the parenthesis from x.