Can "Infinity" ever be more than a mathematical abstraction?

[Speakpigeon]

Naaah, if you don't get the gist of my reasoning, you never will. I'm not going to waste any more time on this.

???
What reasoning?! You're never explaining yourself! That's very nearly comical.
EB

 

[Speakpigeon]

Don't you see the inherent contradiction in; 1 meter = infinity, and; infinity = 1 meter?
Get Real!

Go on, here's your chance to explain yourself. Tell us why there should be this "inherent contradiction"!
It's not "infinity = 1 metre" as you say, you really don't understand any of this. It's an infinite number of points = 1 metre. That's very, very different and much more interesting. Points are infinitely small but there's an infinity of them. We find the result is a finite distance (doesn't matter if it's 1 metre or 1000 metres). That's much, much more interesting than your gobbledegook.
See?
EB

 

[Write4U]

Go on, here's your chance to explain yourself. Tell us why there should be this "inherent contradiction"!
It's not "infinity = 1 metre" as you say, you really don't understand any of this. It's an infinite number of points = 1 metre. That's very, very different and much more interesting. Points are infinitely small but there's an infinity of them. We find the result is a finite distance (doesn't matter if it's 1 metre or 1000 metres). That's much, much more interesting than your gobbledegook.
See?
EB

Oh indeed, you can find all kinds of interesting aspects to the various interpretations and perspectives of the noun "infinity". Problem is, the OP question does not ask about infinite amount of numbers or points contained in a meter. It asks about Infinity (noun). I guess you have lost sight of that small detail and have gone off on a totally abstract theoretical tangent, which as interesting as may be, is off the mark regarding the OP question, or at least confirms the posed assumption.

 

[Speakpigeon]

Oh indeed, you can find all kinds of interesting aspects to the various interpretations and perspectives of the noun "infinity". Problem is, the OP question does not ask about infinite amount of numbers or points contained in a meter. It asks about Infinity (noun). I guess you have lost sight of that small detail and have gone off on a totally abstract theoretical tangent, which as interesting as may be, is off the mark regarding the OP question, or at least confirms the posed assumption.

Oh, Dear, Oh Dear. Now you're making a fool of yourself. Stop it, please. This is becoming seriously ridiculous.
An infinite amount of points is an infinity of points. An infinity of points is an infinite amount of points.
And the elephant in the room you seem to be overlooking in your desperation is that the OP is asking whether the mathematical concept of infinity might have a physical reality. Everything in the OP is about the possibility of infinity as a physical thing, in the physical world, seen from a scientific perspective. An infinity that would be something like an infinite amount of points or something.
And, of course, you carefully abstained from explaining yourself, as ever.
This is really impressive.
EB

 

[Write4U]

An infinite amount of points is an infinity of points. An infinity of points is an infinite amount of points.

First, you keep qualifying your example of Infinity.
Second, 1 meter is a finite measurement, therefore there are a finite number of points in 1 meter.

You are not talking about the Hilbert Hotel here, where you can add rooms ad infinitum. This is your misconception! You cannot keep adding an infinite number of points in an actual finite space!!!

By contrast, the concept of ACTUAL INFINITY treats the infinite as timeless and complete. Not 1 meter, 1 ocean, 1 universe. Just 1 Infinity.

When are you going to acknowledge that truth?

 

[Speakpigeon]

First, you keep qualifying your example of Infinity.
Second, 1 meter is a finite measurement, therefore there are a finite number of points in 1 meter.

Precisely not. There are an infinity of points and they make up a line of just one metre in length. That is, the number of them is infinite and the length of them side by side is one metre. Not really difficult to understand, that. And certainly interesting. Gobsmacked I am.

You are not talking about the Hilbert Hotel here, where you can add rooms ad infinitum. This is your misconception! You cannot keep adding an infinite number of points in an actual finite space!!!

Oh but look at the OP, we are indeed talking about something like the Hilbert hotel, but even that you misunderstand because, no, you don't need to add rooms, it's each guest who has to move to the next room when a new guest arrives.
And no one needs to add points to a continuum. It already has more than enough.

By contrast, the concept of ACTUAL INFINITY treats the infinite as timeless and complete. Not 1 meter, 1 ocean, 1 universe. Just 1 Infinity.

The concept of actual infinity is precisely what the OP is about. Yes, complete and timeless if you like, but actual infinities can well get to measure 1 metre. Or is that too undignified, do you think?
And possibly not just one infinity. Possibly many infinities, or even an infinity of infinities, why ever not? Think of it.

When are you going to acknowledge that truth?

I see that the Old Preacher can no longer stay hidden inside!
EB

 

[Write4U]

It already has more than enough.

But not enought to qualify for infinity. By your own words , just enough (not more) to make 1 meter.
An infinity of points will always exceed our ability to measure its size. Measuring an infinity of anything as a finite measurement is impossible and contradictory by definition.

1 meter = infinity of points
Infinity of points = 1 meter
Infinity of points = 1 cm ?
Infinity of points = 1 mm?
Infinity of points = 46 billion LY ?

No one knows exactly how large the universe is. It could be infinite or it could have an edge, meaning that traveling for long enough in one direction will bring you back to where you started, like traveling on the surface of a sphere.

https://www.wired.com/2011/12/universe-size/

You've clearly do not understood the implications of the terms "infinite" and "infinity"...

 

[Speakpigeon]

But not enought to qualify for infinity. By your own words , just enough (not more) to make 1 meter.

Oh, well, I guess it's just that you don't really speak English well enough to understand what people say here. And by a long shot.

An infinity of points will always exceed our ability to measure its size.

You're exactly like another guy who also gets mad about infinity and who typically repeats himself a lot, never justify his claims, never get to argue anything, and crucially never addresses the point made by other people. There's no possible debate. You're no even interested. You're just a litany of unsupported claims.

Measuring an infinity of anything as a finite measurement is impossible and contradictory by definition.

One more.

1 meter = infinity of points
Infinity of points = 1 meter
Infinity of points = 1 cm ?
Infinity of points = 1 mm?
Infinity of points = 46 billion LY ?

Ah, interesting point. We don't measure things in the abstract. We measure concrete things, like roads and individual human beings.
There's not just one infinity. There are many different infinities. Each infinity that has a length will have its own specific length just like each road, or each human being. I know you won't understand but the point needed to be explained.

You've clearly do not understood the implications of the terms "infinite" and "infinity"...

Well, well, well, no wonder, you haven't explained anything.
Obviously, the world absolutely needs you here. We're lost whenever you don't explain, which is like, all the time!
EB

 

[Write4U]

We don't measure things in the abstract. We measure concrete things, like.....

1 meter = Infinity? From your post it is clear that you do. All you do is speak in the abstract. Why do you think I object to your examples?

You're just a litany of unsupported claims.

Of course you would say that. You don't bother to read the supporting links I have so copiously provided, unless the author is available to debate his point with you. Talk about hubris.....

And please spare me the gratuitous ad hominems. It really cheapens your arguments.

Here, add this to my litany of unsupported claims.

Infinity (symbol: ∞) is a concept describing something without any bound or larger than any natural number

https://en.wikipedia.org/wiki/Infinity

Go argue with Wiki, they are available 24/7 for editing if you feel qualified.

 

[Write4U]

Oh but look at the OP, we are indeed talking about something like the Hilbert hotel, but even that you misunderstand because, no, you don't need to add rooms, it's each guest who has to move to the next room when a new guest arrives.

You are wrong. The Hilbert Hotel proposes a hotel with infinite number of rooms which is completely occupied.

Hilbert's paradox of the Grand Hotel (colloquial: Infinite Hotel Paradox or Hilbert's Hotel) is a thought experiment which illustrates a counterintuitive property of infinite sets. It is demonstrated that a fully occupied hotel with infinitely many rooms may still accommodate additional guests, even infinitely many of them, and this process may be repeated infinitely often.

https://en.wikipedia.org/wiki/Hilbert's_paradox_of_the_Grand_Hotel

This like saying that an infinite length is able to accommodate an additional infinity of points.
Your example is 1 meter long, hardly infinite.

 

[Speakpigeon]

1 meter = Infinity? From your post it is clear that you do. All you do is speak in the abstract.

No. I really don't, although speaking inevitably involves some abstraction. But measuring something that is one metre long is not abstract. Different things have different lengths. They're all concrete things of course. And they all have an infinity of points between their extremities, which is why infinity can get to measure 1 metre or 1000 kilometres depending on what concrete thing you are measuring.
In a continuum, you can pick any two points and you will still have an infinity of points in between. You can repeat the operations by picking pairs of points that will be closer and closer and yet you will still have an infinity of points in between. And obviously, the distance between the two points you pick will vary. You may start with points a kilometre apart and then move on to pick points only a millimetre apart, and yet in all cases you'll have an infinity of points in between, so an infinity of points within one kilometre and then an infinity of point with one millimetre, so, you have to say that an infinity of points may measure one kilometre or one millimetre, depending of the pair of points you pick initially.
See, that's what explaining means. Something you never do because you don't understand what you're talking about. All you can do is point a childish finger to some grown-up and pretend that this person has already all explained what you mean! Just pathetic.

Of course you would say that. You don't bother to read the supporting links I have so copiously provided, unless the author is available to debate his point with you. Talk about hubris.....

All you can do is point a childish finger to some grown-up and pretend that this person has already all explained what you mean! Just pathetic.

And please spare me the gratuitous ad hominems. It really cheapens your arguments.
Here, add this to my litany of unsupported claims. https://en.wikipedia.org/wiki/Infinity

https://en.wikipedia.org/wiki/Infinity
All you can do is point a childish finger to some grown-up and pretend that this person has already all explained what you mean! Just pathetic.
QUOTE="Write4U, post: 3530738, member: 261885"]Go argue with Wiki, they are available 24/7 for editing if you feel qualified.[/QUOTE]
I don't have a quarrel with the Wiki page on infinity. What Wiki says on infinity is fine with me. You're the one I have a debate with.
What's funny is that you link the Wiki article and yet what you say about infinity is in blatant contradiction with it. The problem is, you see, you just don't understand what people say, like when you talk about the Hilbert Hotel claiming adding more guests required adding rooms! You just don't understand the basic principle! It is precisely the point of the Hilbert Hotel paradox that you can add guests without adding room!
Hey, wake up, buddy!
EB

 

[Speakpigeon]

You are wrong. The Hilbert Hotel proposes a hotel with infinite number of rooms which is completely occupied.
https://en.wikipedia.org/wiki/Hilbert's_paradox_of_the_Grand_Hotel

This like saying that an infinite length is able to accommodate an additional infinity of points.
Your example is 1 meter long, hardly infinite.

Take a careful look at the page you linked and tell me where it says that adding more guests requires adding rooms to the hotel!? You link the Wiki article and yet you just don't understand what it says. Claiming as you do that adding more guests requires adding rooms is just the opposite of what is the point of the Hilbert Hotel paradox! You just don't understand the basic principle! It is precisely the point of the Hilbert Hotel paradox that you can add guests without adding room!
Time to wake up, Baby!
EB

 

[Write4U]

And they all have an infinity of points between their extremities

No that is just plain wrong. A point is not a dimensionless coordinate which you can superimpose ad infinitum. A point has dimensions which is why one can build a measurable dimension consisting of points. But an infinity of points can only result in an unmeasurable infinity of size.

 

[Write4U]

Take a careful look at the page you linked and tell me where it says that adding more guests requires adding rooms to the hotel!?

I did not say that. Jeez, why is it that you constantly read things in my posts that aren't there?
But there happens to be an equation that states Infinity = infinity + 1

The Grand Hilbert Hotel does not start as 1 room, it starts as an infinity of room, (which are all occupied)
You are starting with 1 meter, not a Grand infinity of points (to which you can add as many points as you like).
But 1 meter does NOT have an infinity of points, to which you can add an infinity of points, just like you cannot accommodate an infinite number guests into 1 room.
That's just ludicrous.

However, the pioneering work of Max Planck (1858–1947) in the field of quantum physics suggests that there is, in fact, a minimum distance (now called the Planck length, 1.616229(38)×10−35 metres) and therefore a minimum time interval (the amount of time which light takes to traverse that distance in a vacuum, 5.39116(13) × 10−44 seconds, known as the Planck time) smaller than which meaningful measurement is impossible.

https://en.wikipedia.org/wiki/Infinite_divisibility

 

[Speakpigeon]

I did not say that. Jeez, why is it that you constantly read things in my posts that aren't there?

Yes, you said that. Look here:

You are not talking about the Hilbert Hotel here, where you can add rooms ad infinitum.

Sorry for misinterpreting you verbatim words.
EB

 

[QuarkHead]

But 1 meter does NOT have an infinity of points,

it does you know, just as the Real interval $$[0,1]$$ is in one-to-one correspondence with the Real Line $$R^1 $$

which you can add an infinity of points

try sub-dividing 1 metre and come back when you are done!!

P.S. Planck has NOTHING whatever to do with the subject at hand - do not attempt to learn mathematics from the internet

 

[Write4U]

Write4U said:
But 1 meter does NOT have an infinity of points

it does you know, just as the Real interval $$[0,1]$$ is in one-to-one correspondence with the Real Line $$R^1 $$

try sub-dividing 1 metre and come back when you are done!!

You can only subdivide until you reach the size of a point coordinate. No further subdivision is possible. After that you start dealing with "superposition", which is no longer linear.
Is there such a thing as 1/2 point coordinate? 1/4 point? infinitely divisible point?

P.S. Planck has NOTHING whatever to do with the subject at hand - do not attempt to learn mathematics from the internet

Is Planck length not the accepted minimum measurable length? Anything smaller has no longer any physical attributes and/or relationship to a defined length of 1 meter at all.
It just becomes a mathematical abstraction as suggested by the OP.

 

[someguy1]

it does you know, just as the Real interval $$[0,1]$$ is in one-to-one correspondence with the Real Line $$R^1 $$

QuarkHead, Can you please provide a reference to a peer-reviewed experiment or observation that shows that there are mathematical points in physical space that are in order-preserving bijective correspondence to the mathematical unit interval or real line?

And secondly, let me ask you this. If what you say is true, wouldn't the axiom of choice and the continuum hypothesis then be subject to physical experiment? Not to mention simpler axioms such as Powerset or Replacement. Have any such experiments been done? Or even speculatively proposed? Please provide references.

Where I'm coming from is this:

* Math has this wild crazy symbolic formalization of infinity and infinite sets. If you studied it, and then saw step-by-step how the real numbers are constructed out of the crazy axioms of infinitary set theory, you would cease to believe in the real numbers as physically meaningful.

* Rather, you would see them as a highly idealized, abstract model; which we can use to draw inference, and then apply productively to the real world. But nobody really understands why we can do that!

This conversation goes back to Euclid. He said a point is without breadth or width, and a line is without breadth, and two lines intersect in a point, and based on this abstract, idealized word game, we put men on the moon.

And yet ... there are no points, there are no lines, there are no circles, there is no perfection to be found in the real world.

So I do believe you are conflating the physical world, on the one hand; with the abstract, idealized mathematical world on the other.

It's as if a Greek say, "See, the real world contains points and lines." No, Euclid only said, "Let us play a game with words." And the game is USEFUL. But it is not necessarily ontologically meaningful.

That's my viewpoint. Do you feel that physicists believe differently? That their models are reality? Some do, I'm sure. Others, Newton among them, recognized the distinction.

 

[Write4U]

Yes, you said that. Look here:

Write4U said:
You are not talking about the Hilbert Hotel here, where you can add rooms ad infinitum.

Sorry for misinterpreting you verbatim words.
EB

Which I clarified here and you obviously chose to ignore

Write4U said:
You are wrong. The Hilbert Hotel proposes a hotel with infinite number of rooms, which is completely occupied.

https://en.wikipedia.org/wiki/Hilbert's_paradox_of_the_Grand_Hotel[/QUOTE]

 

[Speakpigeon]

Which I clarified here and you obviously chose to ignore

Only because you "clarification" didn't clarify.
Sorry to take the verbatim of your bad English at face value.
Also, if you think that infinity is not measurable, please read this very, very carefully:
"Under the usual (Euclidean) distance function d(x, y) = |x − y|, the real numbers are a metric space and hence also a topological space. Restricting the Euclidean distance function gives the irrationals the structure of a metric space. "
https://en.wikipedia.org/wiki/Irrational_number
If only you could understand.
We can only hope.
EB