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Author Topic: Is infinity a misconception?  (Read 58587 times)

Offline jeffreyH

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Is infinity a misconception?
« on: 17/09/2014 22:19:51 »
If we have a right angled triangle and one side adjacent to the right angle is finite while the other adjacent side is infinite then the hypotenuse must be greater than infinity. What does this say about our view of infinity?


 

Offline alancalverd

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Re: Is infinity a misconception?
« Reply #1 on: 17/09/2014 22:41:42 »
It says nothing about "our" view of infininty, but speaks volumes about your understanding of the principal axioms of Euclidean geometry.  The key construction for analysing this triangle is of course to draw a circular square centered on the bisector of the infinite side, using the usual methods.
 

Offline jeffreyH

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Re: Is infinity a misconception?
« Reply #2 on: 17/09/2014 22:59:07 »
Well then find the radius of the circle whose circumference is infinite.
 

Offline jeffreyH

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Re: Is infinity a misconception?
« Reply #3 on: 17/09/2014 23:04:19 »
Conversely if the radius is infinite find the circumference.
 

Offline jeffreyH

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Re: Is infinity a misconception?
« Reply #4 on: 17/09/2014 23:34:17 »
The point here is that any system that can normally be considered as bounded cannot include an infinite component. So if the mathematics of a formerly finite system go infinite somethings is terribly wrong. If we consider the event horizon as our boundary then the black hole can be thought of like a superconductor.

http://arxiv.org/abs/1403.0938
 

Offline PmbPhy

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Re: Is infinity a misconception?
« Reply #5 on: 18/09/2014 01:54:17 »
Quote from: jeffreyH
If we have a right angled triangle and one side adjacent to the right angle is finite while the other adjacent side is infinite then the hypotenuse must be greater than infinity. What does this say about our view of infinity?
Nothing. I think that you're confusing infinity with a number. It means increases without bound. Let me make this very clear first; 7ed9abff4dafd78d08e616c899412e92.gif is not a number.

It's defined as follows;
Quote
The notation

c2fc74c007e280c7de4a3682a1d7dafa.gif

means that the values of f(x) can be made arbitrarily large (as large as we please) by taking x sufficiently close to a (on either side) but not equal to a.

Most of the time 7ed9abff4dafd78d08e616c899412e92.gif is used without this detail and in a more at ease manner (i.e. sloppy). When you start studying calculus and you get into limits you'll have to evaluate expressions like f(x)/g(x) where both f(x) and g(x) increase without bound. Sometimes the ratio approaches a limit and sometimes it doesn't. If you choose to study calculus you'll learn L'Hôpital's rule which is used to handle these cases.

http://en.wikipedia.org/wiki/L'H%C3%B4pital's_rule
 

Offline alancalverd

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Re: Is infinity a misconception?
« Reply #6 on: 18/09/2014 06:36:43 »
The point here is that any system that can normally be considered as bounded cannot include an infinite component.

Precisely. A triangle is by definition bounded by three intersecting sides, so your "triangle with one infinite side" is as meaningless as my "circular square centered on the bisector of the infinte side".

Just because words hold their conventional order in a phrase doesn't imply that the phrase means anything.   

« Last Edit: 18/09/2014 06:42:43 by alancalverd »
 

Offline JohnDuffield

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Re: Is infinity a misconception?
« Reply #7 on: 18/09/2014 12:55:40 »
If we have a right angled triangle and one side adjacent to the right angle is finite while the other adjacent side is infinite then the hypotenuse must be greater than infinity. What does this say about our view of infinity?
I'm not sure that's a great example, but I agree with the thrust of what you're saying. I think infinity is a misconception. IMHO if ever you bump into an infinity in physics, then something is wrong somewhere. As a rule of thumb, there are no infinities in nature. For example, gravitational time dilation is said to be infinite at the black hole event horizon. But did you see the expression pmb referred to?

c2fc74c007e280c7de4a3682a1d7dafa.gif

IMHO it's better to say that the coordinate speed of light is zero at the event horizon.
« Last Edit: 18/09/2014 12:57:22 by JohnDuffield »
 

Online Bill S

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Re: Is infinity a misconception?
« Reply #8 on: 18/09/2014 17:32:30 »
Quote from: Pete
I think that you're confusing infinity with a number. It means increases without bound. Let me make this very clear first;  is not a number.

“Infinity is not a number”!  It’s good to hear someone else make that assertion.

“It means increases without bound”.       Would it not be better to use “boundless”?   

Infinite and boundless are not synonymous.
 

Offline alancalverd

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Re: Is infinity a misconception?
« Reply #9 on: 18/09/2014 18:22:20 »
Here's an old but very strong example (Cantor?)

There is an infinite number of integers 1,2,3,.... because we can always add one more

There are rational numbers between the integers 1, 3/2, 7/4, 2, 9/4, 19/8, 3....

Indeed there is an infinite number of rational numbers between any two integers

So the number of rational numbers must be greater than the number of integers

So there are at least two classes of denumerable infinity, even in one dimension

And we can fill the spaces between rational numbers with nonrational numbers.....
 

Offline PmbPhy

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Re: Is infinity a misconception?
« Reply #10 on: 18/09/2014 18:37:54 »
Quote from: Bill S
“It means increases without bound”.       Would it not be better to use “boundless”?   
Why do you think it'd be better?

Note: The infinity sign is missing when you quoted me. Instead of reading
Quote
Let me make this very clear first;7ed9abff4dafd78d08e616c899412e92.gifis not a number.
Your quote instead reads
Quote
Let me make this very clear first;  is not a number.
« Last Edit: 18/09/2014 18:41:18 by PmbPhy »
 

Offline PmbPhy

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Re: Is infinity a misconception?
« Reply #11 on: 20/09/2014 03:26:58 »
Quote from: JohnDuffield
IMHO if ever you bump into an infinity in physics, then something is wrong somewhere.
Nonsense. It's beginning to become clear that the universe is flat and boundless and as such goes on forever, never ending. That's what it means to be infinite. It also appears to have approximate uniform mass density which means that there's an infinite amount of hadrons in the universe too.
 

Offline jeffreyH

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Re: Is infinity a misconception?
« Reply #12 on: 20/09/2014 11:34:32 »
Quote from: JohnDuffield
IMHO if ever you bump into an infinity in physics, then something is wrong somewhere.
Nonsense. It's beginning to become clear that the universe is flat and boundless and as such goes on forever, never ending. That's what it means to be infinite. It also appears to have approximate uniform mass density which means that there's an infinite amount of hadrons in the universe too.

In which case there can never be an infinite amount of distance between any two particles as that would place a boundary on infinity. So to all intents and purposes the contents of the universe is finite even though the universe itself may not be. Given an infinite amount of time there is the possibility of every particle in the universe interacting with every other particle, every field interacting with every other field. The same exact combination of particles also have an infinite time in order to interact in exactly the same way more than once. I am not disagreeing with you here Pete I think this is a fascinating subject.
 

Offline JohnDuffield

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Re: Is infinity a misconception?
« Reply #13 on: 20/09/2014 15:09:50 »
Nonsense. It's beginning to become clear that the universe is flat and boundless and as such goes on forever, never ending. That's what it means to be infinite. It also appears to have approximate uniform mass density which means that there's an infinite amount of hadrons in the universe too.
It's beginning to become clear that the universe is flat, but it absolutely isn't clear that it goes on forever. IMHO this is a non-sequitur promoted by cosmologists who have an inadequate understanding of general relativity.
 

Online Bill S

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Re: Is infinity a misconception?
« Reply #14 on: 20/09/2014 19:21:10 »
Pete, sometimes you surprise me.

“Without bound” and “boundless” are synonymous.  I doubt that anyone would argue with that.

“Without bound” and “infinite" may mean the same in certain cases, but there would be at least one person on this forum who would argue with any claim that they are synonymous.

Why introduce doubt unnecessarily?

You are absolutely right about the missing infinity sign, which is odd, as I cut and pasted the quote.  I apologise if this caused you disquiet; but I was agreeing with you. 
 

Online Bill S

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Re: Is infinity a misconception?
« Reply #15 on: 20/09/2014 19:37:02 »
Quote from: alancalvard
Here's an old but very strong example (Cantor?)

There is an infinite number of integers 1,2,3,.... because we can always add one more

The sequence of integers is boundless…. “because we can always add one more”.
It is quite reasonable to refer to this as “infinite”, as long as everyone recognises that this is a mathematical infinity – not some sort of physical infinity.
 

Online Bill S

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Re: Is infinity a misconception?
« Reply #16 on: 20/09/2014 19:44:41 »
Quote from: JD
It's beginning to become clear that the universe is flat, but it absolutely isn't clear that it goes on forever. IMHO this is a non-sequitur promoted by cosmologists who have an inadequate understanding of general relativity.

I believe it also arises from unclear thinking about the way in which a physical infinity (if it exists) would differ from a mathematical infinity; which, as Cantor demonstrates, does “exist”.   
 

Online Bill S

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Re: Is infinity a misconception?
« Reply #17 on: 20/09/2014 19:50:35 »
Quote from: alancalvard
Just because words hold their conventional order in a phrase doesn't imply that the phrase means anything.

That I like!  I hope you will not mind if I borrow it at some time.
 

Offline alancalverd

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Re: Is infinity a misconception?
« Reply #18 on: 20/09/2014 23:41:02 »
It is quite reasonable to refer to this as “infinite”, as long as everyone recognises that this is a mathematical infinity – not some sort of physical infinity.


Here's a physical infinity. The gravitational force F exerted by an object of mass m decreases as m/r2, so F→0 as r→∞ . Physical reality? Well we can measure any  F > 0, so it's real 

Now the force exerted by a mass 2m decreases as 2m/r2, so F→0 as r→∞' and clearly ∞' > ∞
« Last Edit: 20/09/2014 23:47:08 by alancalverd »
 

Offline PmbPhy

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Re: Is infinity a misconception?
« Reply #19 on: 21/09/2014 01:35:20 »
Quote from: jeffreyH
In which case there can never be an infinite amount of distance between any two particles as that would place a boundary on infinity. So to all intents and purposes the contents of the universe is finite even though the universe itself may not be.
You're reasoning is wrong. An boundless universe with uniform mass density has an infinite number of galaxies, particles, stars and planets in it and thus an infinite amount of matter.

Think of the universe like you would a 3D Cartesian coordinate system where a particle is located at the intersection of every grid point where a grid point is the point is of the forum (x, y, z) where x, y, z are all integers. Then the distance between all particles is finite yet the number of particles is infinite.
 

Offline PmbPhy

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Re: Is infinity a misconception?
« Reply #20 on: 21/09/2014 01:36:48 »
Quote from: Bill
Pete, sometimes you surprise me.
Is that good or bad? In this post is it good or bad?  :-\
 

Offline jeffreyH

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Re: Is infinity a misconception?
« Reply #21 on: 21/09/2014 13:09:10 »
Quote from: jeffreyH
In which case there can never be an infinite amount of distance between any two particles as that would place a boundary on infinity. So to all intents and purposes the contents of the universe is finite even though the universe itself may not be.
You're reasoning is wrong. An boundless universe with uniform mass density has an infinite number of galaxies, particles, stars and planets in it and thus an infinite amount of matter.

Think of the universe like you would a 3D Cartesian coordinate system where a particle is located at the intersection of every grid point where a grid point is the point is of the forum (x, y, z) where x, y, z are all integers. Then the distance between all particles is finite yet the number of particles is infinite.

Anything bounded cannot be infinite. The particles are bounded by an infinite extent.
 

Offline PmbPhy

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Re: Is infinity a misconception?
« Reply #22 on: 21/09/2014 16:47:38 »
Quote from: jeffreyH
In which case there can never be an infinite amount of distance between any two particles as that would place a boundary on infinity. So to all intents and purposes the contents of the universe is finite even though the universe itself may not be.
You're reasoning is wrong. An boundless universe with uniform mass density has an infinite number of galaxies, particles, stars and planets in it and thus an infinite amount of matter.

Think of the universe like you would a 3D Cartesian coordinate system where a particle is located at the intersection of every grid point where a grid point is the point is of the forum (x, y, z) where x, y, z are all integers. Then the distance between all particles is finite yet the number of particles is infinite.

Anything bounded cannot be infinite. The particles are bounded by an infinite extent.
Who said it was bounded? I don't know where on earth you're getting these ideas from but they sure aren't from me. You should know that I know that already. In fact  I'm the one who made that fact clear in the start of this thread!

I said think of the universe as you would a  3D Cartesian coordinate system. A  3D Cartesian coordinate system is unbounded, i.e. it's infinitely large, i.e. unbounded!
 

Online Bill S

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Re: Is infinity a misconception?
« Reply #23 on: 21/09/2014 19:54:21 »
Quote from: Pete
Is that good or bad?

Assigning “good” or “bad” involves the sort of subjective value judgement I prefer to avoid in a science thread. 

Quote
In this post is it good or bad?   

That probably depends on whether you are asking this question to avoid answering the ones I asked.
 

Online Bill S

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Re: Is infinity a misconception?
« Reply #24 on: 21/09/2014 20:11:07 »
Quote from: alancalverd
The gravitational force F exerted by an object of mass m decreases as m/r2, so F→0 as r→∞ . Physical reality? Well we can measure any  F > 0, so it's real 

Now the force exerted by a mass 2m decreases as 2m/r2, so F→0 as r→∞' and clearly ∞' > ∞
You are aware that my maths is shaky, so let’s be sure I understand what you are saying.

“The gravitational force F exerted by an object of mass m decreases as m/r2,”

Gravitational force decreases as a square of the distance over which it is measured.
 
“so F→0 as r→∞.”

Gravitational force would be measured as 0 only at an infinite distance from the source.

“Well we can measure any  F > 0, so it's real”

We can measure any force greater than 0, so the gravitational force is real.

Is my understanding OK so far?

 

The Naked Scientists Forum

Re: Is infinity a misconception?
« Reply #24 on: 21/09/2014 20:11:07 »

 

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