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Author Topic: Can we move from infinite to finite?  (Read 11158 times)

Offline jeffreyH

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Can we move from infinite to finite?
« on: 30/04/2015 18:26:14 »
If we have two objects that are infinitely far apart and one object moves one metre nearer the other are they still infinitely far apart?


 

Offline alancalverd

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Re: Can we move from infinite to finite?
« Reply #1 on: 30/04/2015 19:56:29 »
Yes, but a different infinity.
 

Offline PmbPhy

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Re: Can we move from infinite to finite?
« Reply #2 on: 30/04/2015 20:38:41 »
If we have two objects that are infinitely far apart and one object moves one metre nearer the other are they still infinitely far apart?
There's no such thing as "infinitely far apart." You've made it sound as if infinite is a distance whereas it's really a concept. It means "that which increases without bound". So while you can meaningfully say that lim 1/x = 0 as x approaches infinity has meaning you can't say that A and B are infinitely far apart with it having any meaning.
 

Offline Bill S

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Re: Can we move from infinite to finite?
« Reply #3 on: 01/05/2015 00:53:28 »
Quote from: Alan
Yes, but a different infinity.

So, infinity B = infinity A – 1m?

As a mathematical concept that may well make sense to a mathematician, I would be the last to argue with that, but as a physical reality it has a lot in common with unicorns.   

I agree with Pete, “There's no such thing as "infinitely far apart."”  I admit that it is a term I have used, but only in an effort to establish that any attempt to measure (non-mathematical) infinity is meaningless.
 

Offline PmbPhy

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Re: Can we move from infinite to finite?
« Reply #4 on: 01/05/2015 06:12:57 »
Quote from: Bill S
So, infinity B = infinity A – 1m?
No. That expression is nonsense.
Quote from: Bill S
As a mathematical concept that may well make sense to a mathematician, ..
Nope, it doesn't.
Quote from: Bill S
I would be the last to argue with that, but as a physical reality it has a lot in common with unicorns.
My sentiments exactly!
Quote from: Bill S
I agree with Pete, “There's no such thing as "infinitely far apart."”  I admit that it is a term I have used, but only in an effort to establish that any attempt to measure (non-mathematical) infinity is meaningless.
I quite agree. It should be remembered that infinity is not a number, it's a concept.
 

Offline alancalverd

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Re: Can we move from infinite to finite?
« Reply #5 on: 01/05/2015 07:04:04 »
By defintion you cannot measure infinity. You can invent as many infinities as you like, so you could indeed define A = 1/x  and B = ((1/x) - 1). Both definitions are valid and for all nonzero values of x, they are distinct and physically realisable, so you have defined two different infinities if you let x → 0.
« Last Edit: 01/05/2015 07:31:35 by alancalverd »
 

Offline PmbPhy

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Re: Can we move from infinite to finite?
« Reply #6 on: 01/05/2015 07:26:00 »
Quote from: alancalverd
By defintion you cannot measure infinity. You can invent as many infinities as you like, so you could indeed define A = 1/x  and B = ((1/x) - 1). Both definitions are valid and for all nonzero values of x, they are distinct and physically realisable, so you have defined two different infinities.
That is incorrect. As I said above infinity is a concept and not a number. I'm certain that you know that it's a number but I'm doubtful if you fully understand precisely what the concept is. It only has to do when discussing limits. Thus

lim (x -> inf) 1/x = 0

lim (x -> inf) (1/x - 1) = 0

Here the "inf" means "increases with out limit. So "increases without limit" means identically the same thing in each example. You should know that my friend!
 

Offline alancalverd

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Re: Can we move from infinite to finite?
« Reply #7 on: 01/05/2015 07:30:30 »
Absolutely, but that wasn't what I wrote or meant! My x's were tending to zero.

Aha! It seems I missed out the crucial bit, now re-edited!
 

Offline PmbPhy

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Re: Can we move from infinite to finite?
« Reply #8 on: 01/05/2015 07:36:03 »
Absolutely, but that wasn't what I wrote or meant! My x's were tending to zero.

Aha! It seems I missed out the crucial bit, now re-edited!
Thanks, pal. I appreciate you catching my bonehead moves and pointing them out to me. Very much appreciated. :)
 

Offline jeffreyH

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Re: Can we move from infinite to finite?
« Reply #9 on: 01/05/2015 08:45:29 »
The point of my question was to establish that infinity can never be reached and so the question is moot. If we say the universe is infinite and no two objects can be infinitely far apart how do we resolve this? No where in the universe can any two objects be infinitely far apart even though it resides in an infinite space. There appears to a circularity in the concept of infinity.
 

Offline alancalverd

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Re: Can we move from infinite to finite?
« Reply #10 on: 01/05/2015 08:48:30 »
Easy. The universe is bigger than the distance between any two objects in it. As it must be, if it contains them.
 

Offline Bill S

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Re: Can we move from infinite to finite?
« Reply #11 on: 01/05/2015 14:03:28 »
Quote from: Pete
No. That expression is nonsense.

My point precisely.

Quote
Nope, it doesn't.

For once I’m on the same page as the mathematicians – well some of them. :)
 

Offline Bill S

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Re: Can we move from infinite to finite?
« Reply #12 on: 01/05/2015 14:09:36 »
Quote from: Jeffrey
The point of my question was to establish that infinity can never be reached and so the question is moot. If we say the universe is infinite and no two objects can be infinitely far apart how do we resolve this? No where in the universe can any two objects be infinitely far apart even though it resides in an infinite space. There appears to a circularity in the concept of infinity.

You will always have this problem as long as you think of infinity as a space.  It's not a number, it's not a space and it's not a time - it's infinity.
 

Offline Bill S

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Re: Can we move from infinite to finite?
« Reply #13 on: 01/05/2015 14:13:40 »
Quote from: Alan
Yes, but a different infinity.

Quote
Easy. The universe is bigger than the distance between any two objects in it. As it must be, if it contains them.

That doesn't even come close to answering Jeffrey's question if you can have different infinities.

 

Offline alancalverd

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Re: Can we move from infinite to finite?
« Reply #14 on: 01/05/2015 16:11:27 »
By definition, any infinity will be larger than any finite number or quantity.

If you can identify two objects, let alone actually measure the distance between them (a) they can't be infintely far apart and (b) any universe, domain or manifold that contains them must have at least one dimension greater than the distance between them.
 

Offline Colin2B

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Re: Can we move from infinite to finite?
« Reply #15 on: 01/05/2015 18:03:56 »
That doesn't even come close to answering Jeffrey's question if you can have different infinities.
Different infinities is something I can have difficulty getting my head round when they get complex. However, at the simple level eg the set of all whole numbers is infinite, but it is contained within the set of all real numbers which is also infinite. I saw a debate between mathematicians who argue the set of real numbers must be larger so is a bigger infinity vs others who say infinite is infinite.
As you say, it doesn't help Jeffery!
 

Offline Bill S

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Re: Can we move from infinite to finite?
« Reply #16 on: 01/05/2015 22:03:09 »
Quote from: Alan
By definition, any infinity will be larger than any finite number or quantity.

Of course, but the statement that "any infinity will be larger than any finite number", while it is perfectly true, says nothing about the relationship between different sizes of infinity; nor does it establish that a value that is smaller than another value is necessarily finite, if you accept that there can be different sizes of infinity.

Quote
If you can identify two objects, let alone actually measure the distance between them (a) they can't be infintely far apart and (b) any universe, domain or manifold that contains them must have at least one dimension greater than the distance between them.

Once again, we agree completely, but it still does not address the OP, because if you accept that there are different sizes of infinity, none of which can be measured, you have to use some other means of establishing sizes.  Comparison seems to be the standard method, and undoubtedly this is meaningful in mathematics.  Outside maths, though, it has no meaning, as you are just comparing one unknown non-quantity with another.
« Last Edit: 01/05/2015 22:16:34 by Bill S »
 

Offline jeffreyH

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Re: Can we move from infinite to finite?
« Reply #17 on: 02/05/2015 05:24:43 »
What we can possibly say is that infinity cannot be based on a particular geometry as that is by definition bounded by surfaces. So a spherical infinity should not exist, nor a cubic or triangular infinity. This then leads on to the geometry of spacetime. As spacetime has geometry then it should not be infinite as infinity is not geometrical.
 

Offline alancalverd

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Re: Can we move from infinite to finite?
« Reply #18 on: 02/05/2015 12:16:07 »
The number of rational vectors within a sphere is infinite, even if the radius of the sphere is finite. The number of irrational vectors is also infinite. It is left a an exercise to the reader to determine which is the larger, but either could be called a spherical infinity because the symmetry of vector distributions is spherical.

That is why balloons inflate the way they do (at least in physics textbooks). 
 

Offline jeffreyH

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Re: Can we move from infinite to finite?
« Reply #19 on: 02/05/2015 15:05:07 »
The number of rational vectors within a sphere is infinite, even if the radius of the sphere is finite. The number of irrational vectors is also infinite. It is left a an exercise to the reader to determine which is the larger, but either could be called a spherical infinity because the symmetry of vector distributions is spherical.

That is why balloons inflate the way they do (at least in physics textbooks).

Which highlights that infinity can only consist of 1 dimensional and not 3 dimensional geometry. The 3 dimensional space is then an expansion of this 1 dimensional space.
 

Offline Bill S

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Re: Can we move from infinite to finite?
« Reply #20 on: 03/05/2015 03:10:55 »
Quote from: Jeffrey
What we can possibly say is that infinity cannot be based on a particular geometry as that is by definition bounded by surfaces. So a spherical infinity should not exist, nor a cubic or triangular infinity. This then leads on to the geometry of spacetime. As spacetime has geometry then it should not be infinite as infinity is not geometrical.

I like that line of thinking; it took me quite a long time to reach that point (lots of dog walks), and even longer to get beyond it to something resembling Barbour's "Platonia".



Quote
Which highlights that infinity can only consist of 1 dimensional and not 3 dimensional geometry. The 3 dimensional space is then an expansion of this 1 dimensional space.

I tend to think in terms of one dimension for infinity, but why do you opt for one, rather than infinite dimensions?
 

Offline jeffreyH

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Re: Can we move from infinite to finite?
« Reply #21 on: 03/05/2015 09:35:15 »
Quote from: Jeffrey
What we can possibly say is that infinity cannot be based on a particular geometry as that is by definition bounded by surfaces. So a spherical infinity should not exist, nor a cubic or triangular infinity. This then leads on to the geometry of spacetime. As spacetime has geometry then it should not be infinite as infinity is not geometrical.

I like that line of thinking; it took me quite a long time to reach that point (lots of dog walks), and even longer to get beyond it to something resembling Barbour's "Platonia".



Quote
Which highlights that infinity can only consist of 1 dimensional and not 3 dimensional geometry. The 3 dimensional space is then an expansion of this 1 dimensional space.

I tend to think in terms of one dimension for infinity, but why do you opt for one, rather than infinite dimensions?

It isn't quite that straightforward. A straight line is unbounded so therefore a directional vector is also unbounded. A closed geometry such as a sphere isn't unbounded. Therefore Alan is correct. One dimensional is probably a bad term for what I mean. I don't think we have a term for what I mean.
 

Offline jeffreyH

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Re: Can we move from infinite to finite?
« Reply #22 on: 03/05/2015 09:40:11 »
It is also interesting to note the fact that as celestial bodies tend towards a more or less spherical shape then all the vectors away from them tend to be straight line vectors. Once angular momentum is introduced this curves the spacetime so may make it finite by definition.
 

Offline Bill S

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Re: Can we move from infinite to finite?
« Reply #23 on: 04/05/2015 00:31:07 »
Quote from: Jeffrey
It isn't quite that straightforward.

It is never likely to be straightforward when trying to compare finite with infinite, unless you remember that the rules that apply to one may not apply to the other.

Quote
A straight line is unbounded so therefore a directional vector is also unbounded.

Absolutely.  The trouble comes if you assume that “unbounded” = “infinite”.  Pete, for one, has argued that etymologically they are the same, because infinite simply means without end. However, that ignores the fact that beginning and end are directional terms and can be interchangeable.   

Quote
I don't think we have a term for what I mean.

Once you get away from mathematical infinities there is a sad, but understandable lack of terms.   
 

Offline alancalverd

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Re: Can we move from infinite to finite?
« Reply #24 on: 04/05/2015 00:44:18 »
Angular momentum is a vector. It has nothing to do with the curvature of anything. 

All vectors in any space can be represented by the shortest path from an origin, e.g. 40.7127° N, 74.0059° W is the vector for New York, as represented by a sector of a great circle that passes through the equator south of London.  There will be an equivalent formulation for any vector in any curved space.

Quote
A straight line is unbounded so therefore a directional vector is also unbounded.

Beware of imprecise terminology. "Directional vectors", conventionally i,j,k in Euclidean 3-space, are indeed bounded  - they have unit length by definition, and other vectors are specified as multiples of those elements. But a direction, as defined by the ratio of elements i,j,k, is of course unbounded.
 

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Re: Can we move from infinite to finite?
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