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

[arfa brane]

If it's true that infinitesimals are logically unsound ...

I never said that. Nor did the quotes you posted.

That was something from my textbook. It's in the quote in post 619, and not attributed to you.

 

[someguy1]

That was something from my textbook. It's in the quote in post 619, and not attributed to you.

No worries. I don't think you're right that the hyperreals give Zeno much help one way or the other. That's one substantive thing we might kick around. I agree with you that infinitesimals have a rich history and are interesting. But talking about the hyperreals tend to unfocus conversations because the hyperreals don't do a lot of things people think they do. Nobody's ever proved anything with infinitesimals that couldn't be proved with the standard reals. Of course that might change tomorrow. But till it does, the point stands.

 

[curvature]

Can "Infinity" ever be more than a mathematical abstraction? Without any real world evidence, for me the answer would be "no".

Yet, our Universe is thought to be flat and without a special topology that would seem to imply infinity. Many concede that the Universe likely is infinite. How can science so readily concede that infinity is a real possibility when there is no evidence of infinity outside of math?

Here I'm speaking of infinity of space and time and not some situation where the infinity is just do to the framework imposed such as walking around the North Pole and calling space there a Singularity just because the time zones all converge or some such example.

In physics, a singularity is just another word for some aspect of infinity. We don't like them, and popular attention to singularities where romanticised by Hawing and Penrose... even though later Penrose and the late Hawking had changed their minds trying to find non-singular solutions. Infinity is of course nothing but an abstraction but in physics it means something pertinent, it refers to an non-physical situation; this would explain why there are no infinities observed in nature. Even the universe as large as it is, still has a finite amount of mass in it. Infinities simply do not exist in nature but they are good mathematical abstractions for calculus.

 

[Forceman]

In powers of 10 the repetition of this over and over until the exponent is greater than the base number makes more simple calculations expand into powers and groups of the associative property for a number series and the greater than property of triangles and pi values with circles: pi can then reach into an infinite amount of decimals, and then infinity can spread over this a circle stretches out over the value of the Circumference or even pi without a base number or quad-core more like force or decimals with infinity like a circle, a shape not a decimal or vortex. Vortexes found within singularity can then get the value of pi or i: and then force can then attend to the value of Newtons and more like not the infinity of i. Then this max: an abstraction like a line not seen on a graph can still be numbered on a level of abstraction but Newtonian mechanics may seem behind this like a max not possible with particle spin and particle push by excitation less considering the least of all quantum mechanics and quantum entanglement. The solution seems more like not just a "God" particle, but also the lesser used electron with more quantum states and more flavors of quark assigned to the possibility of the Higgs Boson: not just a particle but facts relayed on information from other orbits to construct more of a universal frame and/or frame reference.

 

[NotEinstein]

Infinity is of course nothing but an abstraction but in physics it means something pertinent, it refers to an non-physical situation; this would explain why there are no infinities observed in nature. Even the universe as large as it is, still has a finite amount of mass in it. Infinities simply do not exist in nature but they are good mathematical abstractions for calculus.

What's your evidence that the universe isn't infinite in (spatial) size?

 

[curvature]

Well for one simple reason, the universe HAS a finite past and the only way a universe in that sense, would have to expand and expand and expand... and it will never reach that infinity because, as infinity is an abstraction, it's not a number that increases like $$(1 + 2 +3 \rightarrow \infty)$$ - under Cantor, he thinks an infinity can be countable but twist is, there is no such thing. Also, we do have an observable horizon and we can trace that horizon back to a hot dense state.... I mean, based on that alone, it doesn't look promising that a universe is infinitely large. Or ever will.

Here's a thought experiment, how many times do you need to count to reach infinity? The answer is you would have to count for all eternity and the end of time itself. So instead we create a computer that will be constructed to count from 1, to 2, to 3 and so on, until it reaches infinity.

Now in that last passage, some things need to be clear, like parts of it were wrong. First of all, you cannot count to infinity (because infinity is not a number) once again, its an abstraction. Second of all, the part in which I state, can you count to infinity and my answer being no I hold to, I also hold the universe is finite in principle because even that expansion has only been going on for about 15 billion years - I'd hardly call that infnite... would you? And even though the number of particles in the observer horizon is very large of order $$2 \times 10^{80}$$ .... this too is no were near infinity, because infinity isn't a thing. It's not a number, at best its a concept and one that has no support in nature.

 

[NotEinstein]

Well for one simple reason, the universe HAS a finite past and the only way a universe in that sense, would have to expand and expand and expand...

It could've been created infinite in size, and then start expanding; as far as I know, there's nothing in the big bang model excluding that possibility.

and it will never reach that infinity because, as infinity is an abstraction, it's not a number that increases like $$(1 + 2 +3 \rightarrow \infty)$$ - under Cantor, he thinks an infnity can be countable but twist is, there is no such thing. Also, we do have an observable horizon and we can trace that horizon back to a hot dense state.... I mean, based on that alone, it doesn't look promising that a universe is infinitely large. Or ever will.

What does the size of the observable universe have to do with the size of the full universe? I can't follow your jump in logic there.

Here's a thought experiment, how many times do you need to count to reach infinity? The answer is you would have to count for all eternity and the end of time itself. So instead we create a computer that will be constructed to count from 1, to 2, to 3 and so on, until it reaches infinity.

Now in that last passage, some things need to be clear, like parts of it were wrong. First of all, you cannot count to infinity (because infinity is not a number) once again, its an abstraction. Second of all, the part in which I state, can you count to infinity and my answer being no I hold to, I also hold the universe is finite in principle because even that expansion has only been going on for about 15 billion years - I'd hardly call that infnite... would you?

Totally irrelevant: I'm talking about infinite size in space, not infinite size in time.

And even though the number of particles in the observer horizon is very large of order $$2 \times 10^{80}$$ .... this too is no were near infinity, because infinity isn't a thing. It's not a number, at best its a concept and one that has no support in nature.

Again you claim the universe isn't infinite in size; please provide actual evidence for that.

 

[curvature]

Big Bang was emergence of time and space, you cannot have one and not the other - you can't talk about an infinite space alone because time is actually part of the Pythagorean triangle. My point is everything I said is relevant, you just don't understand what I have said. Plus try and remember if you talk about an infinity of space, you have to deal with an infinite time, sorry, but your whole arguments crumble because of this.

 

[curvature]

Again you claim the universe isn't infinite in size; please provide actual evidence for that.

I already told you, the fact it has a finite past ensures that we had a moment in both space and time which exploded and became the big bang we know today. Considering this means the universe has to be roughly 15 billion years, I'd call that a penny towards the journey to infinity.

 

[NotEinstein]

Big Bang was emergence of time and space, you cannot have one and not the other

And I never claim otherwise.

- you can't talk about an infinite space alone because time is actually part of the Pythagorean triangle.

What Pythagorean triangle are you talking about? And why is it impossible to have an infinite space but only a finite time? And why would time be finite; it can still be infinite, even if it has a starting point.

My point is everything I said is relevant, you just don't understand what I have said.

No, the problem is that you aren't answering the question: I asked you for evidence that the universe is finite, and you aren't giving any.

Plus try and remember if you talk about an infinity of space, you have to deal with an infinite time,

Why?

sorry, but your whole arguments crumble because of us.

Your strawman indeed does crumble because of that.

I already told you, the fact it has a finite past ensures that we had a moment in both space and time which exploded and became the big bang we know today.

Sure, but that doesn't mean that at the moment the big bang happened, only a finite-size space could have emerged. Why, according to you, couldn't the big bang have produced an infinitely sized space?

Considering this means the universe has to be roughly 15 billion years, I'd call that a penny towards the journey to infinity.

I don't disagree with that, but once again, I don't see how an infinite time must lead to an infinite space. I don't follow that jump in logic in your argument. Can you please elaborate?

 

[curvature]

Oh common, I have a 1o minutes at best you're hitting out with what I believe are genuine questions, but you're way too manic for me if you think I can explain any of that in a night.

 

[NotEinstein]

Oh common, I have a 1o minutes at best you're hitting out with what I believe are genuine questions, but you're way too manic for me if you think I can explain any of that in a night.

I'm sorry, what? "In a night"? Me manic? I have no idea what you are getting at?

 

[iceaura]

Sorry I don't remember what the context was and didn't go back to check. But nothing can "agree perfectly with experiment" unless it's a discrete measurement like counting the eggs in a dozen. The world's best experiment gets 12 decimal places of accuracy.

Oh c'mon.
The world's best calculations also have precision limits. So?
There are physical matters - last I checked - in which the experiments can yield greater precision than the calculations. Nevertheless the calculations agree - perfectly - with experiment.
How about if I use "completely"?
Ok: Nevertheless, our theories of motion and so forth yield calculations that agree completely with our experiments. These calculations make no use of non-computable numbers, and they account for every instance of the modeled physical phenomenon so far.

Some guy wrote a book on computable physics but he's the only guy. It is a VERY obscure pastime to be trying to redo physics on computable math.

Irrelevant. I'm observing that computable numbers already model the known physical world, sufficient, that no smaller set of the reals does, necessary, and so we have evidence that the infinities thereby included match real infinities - that they model "something else", that these infinities are "more than" abstraction.

What model is that?

All of them.

* The noncomputables plug all the holes in the computable real line, so that you can have a continuum!

There are no holes in the computable real line, and it appears to model the continuum of the physical world just fine.

Oh but there are uses for noncomputables. Let me tell you two of them.

I appear to have misled you, accidentally. On this thread, I am talking about the uses of math for modeling some presumed physical world, implying the existence of a correspondence between features of the math and features of the physical world - from which we can discover where to look for infinities that are "more than abstractions".

 

[Write4U]

Sure, but that doesn't mean that at the moment the big bang happened, only a finite-size space could have emerged. Why, according to you, couldn't the big bang have produced an infinitely sized space?

I guess it tried to but failed during the "inflationary epoch".......

p.s. I like to call that infinite "space" by the term; infinite "permittive condition", allowing for space to form.

 

[someguy1]

Oh c'mon..

I've been so greatly enjoying watching a new crew of people arguing on this thread that I realized not posting is more fun than posting. Anything I could possibly have to say on this subject I've certainly said many times over already.

Ok just one. "There are no holes in the computable real line, ..."

Come on, man. You contradicted established mathematical fact. You can't expect me to take you seriously. References: Cauchy sequences, completeness in a metric space, the Intermediate Value theorem. Go do your homework. You're entitled to your opinion but not your own facts. All the best.

 

[curvature]

The additional problem is that you cannot have spacetime without matter or energy. Its not just a matter being good questions, they just won't hold.

What's your evidence that the universe isn't infinite in (spatial) size?

Maybe you would like re-read my post and tell me what you don't understand?

 

[curvature]

And I never claim otherwise.


What Pythagorean triangle are you talking about? And why is it impossible to have an infinite space but only a finite time? And why would time be finite; it can still be infinite, even if it has a starting point.

I have given examples of why it is finite, and I wont repeat myself in this post. I made this post to say, do you not realize your statement is a contradiction... space and time are one object, you bend you twist the other. You stretch space, time becomes stretched with it - this is why it impossible to have space expanding without a notion of time. You're question says you can have infinite space and finite time... that not's how it works and a scientist would call it an oxymoron

 

[someguy1]

Irrelevant. I'm observing that computable numbers already model the known physical world, sufficient, that no smaller set of the reals does, necessary, and so we have evidence that the infinities thereby included match real infinities - that they model "something else", that these infinities are "more than" abstraction.

ps -- I do understand your viewpoint. But you keep saying that because you THINK the universe is a certain way, that you can change the standard definitions of things in math. It doesn't work that way.

I noted earlier that the rationals model the real world too, to any desired degree of precision. You (or someone) disagreed and said that we need all the computables for some reason or other. That's not so, you can arbitrarily approximate any computable number with rationals. Any real number whatsoever. That's what the rationals do. They approximate all real numbers as closely as you want. So your fixation on computable numbers, which comes out of left field, is simply not relevant. All the more so because you have not troubled yourself to explain why you think the world is made of computable numbers. You have just assumed it and won't deign to justify this assumption in the least. Or even recognize that it's nothing more than a metaphysical assumption.

Anyway really I don't want to argue about this. But you are just assuming things and making up your own math. I don't see the point.

Are you arguing from a constructive mathematical foundations of physics perspective? You just keep repeating that the world is quantized by computable numbers. You get the Nobel prize if you can prove that.

But if you're only saying that we can arbitrarily approximate the world using computable numbers, and that that is sufficient for the discussion, then you are abandoning any concept of the real world. In which case you're ignoring the entire point of the thread: to talk about whether there's an infinity in the real world. This discussion is ABOUT the real world, not just mathematical models of it.

 

[arfa brane]

I don't think you're right that the hyperreals give Zeno much help one way or the other. That's one substantive thing we might kick around.

Well, I'm reasonably sure there are lots of people who think the hyperreals, Robinson's "extensions" to the reals with infinitesimals, and whateverelse, do give Zeno some help.

Today's ideas weren't available to the Greeks, who had a lot of what we think today were naive ideas about numbers, what they are and their relations to 'structure'. What we're discussing here is how mathematical ideas of structure are related to our observations of physical structure. We're talking about the structure and topology (what does that word mean exactly?) of certain classes of numbers, but more than that, about those two ideas explaining the physical world, in terms of symmetries and abstract patterns (signals, say). I think our notions of information and what that is are not just a kind of abstraction, but deeply related to what physics is.

Exploring what say, the concept of an abstract point and how measurement means these are really only defined as boundaries of an interval or a subinterval. Measurement can get close to a point but you get an interval: Δt or Δs, points remain abstract, immeasurable. But calculus says they exist. Information theory and physical measurement says they don't.

 

[someguy1]

But calculus says they exist.

Calculus doesn't claim anything exists. It provides a toolbox for reasoning precisely about continuity and smoothness. You'll never open a calculus text and have it claim that the real world is one way or another way or any way at all. Calculus is about the real numbers, a mathematical abstraction. If physicists find it useful, so be it. Math makes no metaphysical claims. It provides toolsets.

Information theory and physical measurement says they don't.

Those disciplines ALSO do not make metaphysical claims. When I measure a piece of wood to be 20 inches I'm not making metaphysical claims about the nature of the world.

You are confusing science with metaphysics.