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

[Write4U]

Please explain what's the logical problem if there's no beginning.
EB

That's not what I said. The logical problem exists with a beginning, something like a BB and the beginning of time.

But then I'm sure you did not watch the Prof Tong lecture at Cambridge U.

 

[Speakpigeon]

So it's a quality of the human mind and not our external reality. You just veered off in yet another direction. An interesting direction to be sure, but quite at odds with the rest of your post. Now you say that math is in our minds and perhaps not in the world! Now this is something that I happen to believe may well be the case. But it completely blows Tegmark out of the water and it makes physics irrelevant. If our math is only some quality of our mind, what is it that physics is studying when it uses math to understand the world? Are we just ancients looking up at the sky and making up stories about hunters and animals?

All we know is our mind. Science is just the systematic and logical study of our mind understood as reflecting the physical world. We all believe there's a physical world and this only on the basis of what we know of our own mind. We even take the mental visual picture we have of the world around us as the world itself and we can't dismiss this impression even once we understand it's untrue. Still, you're free to believe otherwise. At least you can try. Meanwhile, scientists take their subjective experience to be evidence of the physical world and work on this basis, even if most of them would perhaps just sneer at this suggestion. We just have to make the distinction between what we know, i.e. our own, individual, conscious mind, and what we believe, and can't stop believing, i.e. that there's a physical world consistent with what we know in our minds.
Accordingly, we don't know of any way to ascertain the existence of the physical things we think or may think exist, including actual infinities. All we can do is look at the evidence and do our best with that. And in particular, we need to make sure we don't go into logical contradictions. Short of that, that's all we can do that we know of, and that's just at least for now. So, lost of people should tune down the rhetoric and just accept we don't know enough to tell either way and that therefore assumption are legitimate as long as they don't involve logical contradictions.
EB

 

[Speakpigeon]

That's not what I said. The logical problem exists with a beginning, something like a BB and the beginning of time.

There's no logical problem between the idea of a time with no beginning and the scientific notion of the Big Bang.
The Big Bang is usually construed as the beginning of our universe, not necessarily the beginning of time.
So, we could have an infinite past and the Big Bang as the beginning of our universe. No logical problem.
EB

 

[Speakpigeon]

How can my computer screen display text when everything is fundamentally just a binary set of zeros and ones?

First, computers screen do not literally "display text". We see them as displaying text. At least that's the usual, rational, assumption.
Second, we don't normally, rationally, think that "everything is fundamentally just a binary set of zeros and ones".
Third, nothing specific to computer screen. How could the whole of reality be made of physical things given that what we know is what we subjectively experience and that this isn't physical.
So, the same problem arise with the claim that reality is entirely physical as with the claim that reality is entirely mathematical.
There's a difference though. We all believe in the physical world and we all understand the idea of a physical world. However, that's not the case with this notion of a "mathematical universe". We just don't know what it means and we have no reason to believe it.
EB

 

[Write4U]

The Big Bang is usually construed as the beginning of our universe, not necessarily the beginning of time.

Yes it is. t = 0

So, we could have an infinite past and the Big Bang as the beginning of our universe. No logical problem.

A timeless infinite past of what?
The fields which generate particles did not form until directly after the BB, at the very beginning of the inflationary epoch.

 

[Write4U]

How could the whole of reality be made of physical things given that what we know is what we subjectively experience and that this isn't physical.

We subjectively experience differential values of energy quanta.
All of reality is made up from 3 fundamental quantum particles, electrons and atomic nuclei consisting of protons and neutrons which are configurations of up-quarks and down-quarks.

Protons are made of two Up and one Down quark. The neutron is made of two Down and one Up quark.

The Up quarks have a 2/3 positive charge and the Down has a 1/3 negative charge. Fractional charges are a pretty funny concept, but remember we (humans) made up the unit of charge that a proton has, so its very possible that there could be a smaller division of charge.

If you add those charges you will see that sum is positive one for the proton and 0 for the neutron.

https://education.jlab.org/qa/quark_04.html

And there is your binary quantum coding with the electron providing the energy.....

I admit this conclusion is purely speculative on my part. But it sounds good....

 

[someguy1]

All we know is our mind.

Hi, You replied with a lot of stuff for me and I wanted to let you know I'll try to systematically get through it all in the next few days. But really my overall points are humble, and I want to try not to get too far afield. I happen to know about the particular Hilbert space used in QM and from what I can see there is no way in hell there is any such thing in the physical world. It's a highly abstract and idealized model that happens to work. If I could communicate that to the world the world would be better off. Too many pop science articles are giving people the wrong impression about the nature of science. Even many of the high-toned superstars (rhymes with Wegmark as someone typo'd the other day -- I LOL'd) have done their part to confuse the public. For all I know they may be confused themselves.

Now regarding your remark that "all we know is our mind." George Berkeley (pronounced Barkley like the former NBA basketball player) pointed out that everything we experience of the outside world is mediated by our senses. And that, in principle, we have no way of being sure there's anything "out there" at all. There could possibly be nothing BUT our sense impressions, but without anything out there to cause the impressions. And once you realize that the external world is irrelevant to our experience; it follows that the existence of an external world is a superfluous assumption. We might as well just take as given that there's nothing at all "out there" and reason from that.

A slippery slope once you admit that all we know is our mind. If that's true, there's no need to even assume there's an external world at all, much less a mathematical one.

George Berkeley (/ˈbɑːrkli/;[1][2] 12 March 1685 – 14 January 1753) — known as Bishop Berkeley(Bishop of Cloyne) — was an Irishphilosopher whose primary achievement was the advancement of a theory he called "immaterialism" (later referred to as "subjective idealism" by others). This theory denies the existence of material substance and instead contends that familiar objects like tables and chairs are only ideas in the minds of perceivers and, as a result, cannot exist without being perceived.

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

By the way the other cool thing Berkeley did was to supply sharp and logically correct criticism of Newton's method of fluxions, which we now know as the differential calculus. Newton took the quotient of two quantities and let both numerator and denominator get very very close to zero; and when they "became" zero, which gives the nonsensical and undefinable expression 0/0, the mathematically nonsensical result somehow turned out to explain gravity perfectly. Newton could not justify his procedure, he only knew that it worked. Physicists go where mathematicians fear to tread.

Berkeley called the vanishing difference quotient, what today we refer to as the derivative, as the "ghost of departed quantities." It's funny, even back in the day these guys could flame with the best.

So, lost of people should tune down the rhetoric

Is that for me? I had no idea you perceive my posts in that way. I'll await clarification before posting further.

 

[Speakpigeon]

Hi, You replied with a lot of stuff for me and I wanted to let you know I'll try to systematically get through it all in the next few days. But really my overall points are humble, and I want to try not to get too far afield. I happen to know about the particular Hilbert space used in QM and from what I can see there is no way in hell there is any such thing in the physical world. It's a highly abstract and idealized model that happens to work. If I could communicate that to the world the world would be better off. Too many pop science articles are giving people the wrong impression about the nature of science. Even many of the high-toned superstars (rhymes with Wegmark as someone typo'd the other day -- I LOL'd) have done their part to confuse the public. For all I know they may be confused themselves.

I would tend to agree with you as to the little likelihood that QM would be an accurate representation of the real physical world. But I would say the same for General Relativity and Special Relativity. My reason seems broadly similar to the one you have for QM. I feel these theories are just too weird to be true. They look like the theories made up by somebody who is looking in from the outside and only seeing part of the picture.
Still, that's obviously just an impression and I don't give up on the idea that science can improve. And, crucially for our argument here, I think it's probably better to let scientists feel their way.
And compared to that, I consider the question of infinity as really not important at all. We will just have to see in the future if it's a problem to use infinities in physics.

Now regarding your remark that "all we know is our mind." George Berkeley (pronounced Barkley like the former NBA basketball player) pointed out that everything we experience of the outside world is mediated by our senses. And that, in principle, we have no way of being sure there's anything "out there" at all. There could possibly be nothing BUT our sense impressions, but without anything out there to cause the impressions. And once you realize that the external world is irrelevant to our experience; it follows that the existence of an external world is a superfluous assumption. We might as well just take as given that there's nothing at all "out there" and reason from that.

I agree as to the logic of that. However, I would still disagree as to the general idea. I don't think Berkeley forgot to behave as if there was a physical world. Because, as I already said, we just can't stop believing there's a physical world. So, yes, I agree with Berkeley, we don't know there's a physical world, but I disagree with him that we could take the notion of a physical world as superfluous. Or rather, I think we just can't exclude that there's something, even if it's not exactly a physical world, that does what's necessary to result in us having our subjective experience.
In effect, we've already moved from the notion of a material world as very well supported by all our senses to the notion of a physical world made of quarks that we can't even perceive as such. The String theory is in this respect another step away from our sense of a material world. So, we could move even further as long as our new concept seems consistent with our observations.

A slippery slope once you admit that all we know is our mind. If that's true, there's no need to even assume there's an external world at all, much less a mathematical one.

From a logical point of view, all solutions consistent with our observations are equivalent. The notion of Ockham's Razor has no logical foundation. So, I would say we do what we like as long as it's consistent with observations...
Further, physics works remarkably well in terms of predicting future events. Any theory would have to do even better somehow. And I don't believe it would be possible to have a predictive theory that would assume no external world.

By the way the other cool thing Berkeley did was to supply sharp and logically correct criticism of Newton's method of fluxions, which we now know as the differential calculus. Newton took the quotient of two quantities and let both numerator and denominator get very very close to zero; and when they "became" zero, which gives the nonsensical and undefinable expression 0/0, the mathematically nonsensical result somehow turned out to explain gravity perfectly. Newton could not justify his procedure, he only knew that it worked. Physicists go where mathematicians fear to tread.
Berkeley called the vanishing difference quotient, what today we refer to as the derivative, as the "ghost of departed quantities." It's funny, even back in the day these guys could flame with the best.

Berkeley's ideas clearly haven't all gone away and nobody I think could possibly fault his reasoning. And he was clearly a rationalist, not some obscurantist intent on bringing confusion into science.

Is that for me? I had no idea you perceive my posts in that way. I'll await clarification before posting further

You should have noticed that when I want to criticise someone in particular I don't refrain from being very explicit and straightforward, even belabouring the point perhaps a bit too much.
So, no, it wasn't for you.
This question of infinities is apparently a sore point for many people who can become extremely vocal in expressing their bafflement.
And I can admit that you stick to a very reasonable aspect of the question, which is that science doesn't have any evidence of infinities but relies on them in its theories.
EB

 

[someguy1]

So, where would be the problem with scientists assuming infinities, including uncountable ones?

What would be the problem with scientists assuming magical fairies? Nothing, except that there's no evidence or proof.

 

[Speakpigeon]

What would be the problem with scientists assuming magical fairies? Nothing, except that there's no evidence or proof.

Current scientific theories that include infinity have amazing predictive power. The only alternative theory we could have to explain this would be for us ordinary folks to believe scientists are the magical fairies.
Still, it's true there's no absolute criterion. It all comes down to what you believe, precisely because, as I said, we don't know. Yet, there are clearly different kinds of beliefs and we can make the distinction for example between theories that are not falsifiable and those that are. Science is, fairies are not. So, you can choose to believe in things like fairies but it's your choice and nothing anybody could stop you from.
Infinities are not like fairies. I'm not sure how we go about falsifying infinities but the tricky point here is that if infinities don't exist they are falsifiable and if they exist they are not. So, as long as we don't falsify them, we can only remain uncertain.
EB

 

[Write4U]

Infinities are not like fairies. I'm not sure how we go about falsifying infinities but the tricky point here is that if infinities don't exist they are falsifiable and if they exist they are not. So, as long as we don't falsify them, we can only remain uncertain.
EB

I like that analysis.
But we might be able to make some conclusions from known facts.
For instance, we know that particles are constructed in quantum fields. And we have pictures of the early universe and its plasma of quantum fields and it is finite.
Which would lead to the conclusion that even if there is a greater infinite condition, it is unable to create any matter, because it is void of other quantum fields. At least that seemed to be the case at the moment our universe was created.
And maybe that fact creates the apparent contradiction.

 

[Seattle]

I like that analysis.
But we might be able to make some conclusions from known facts.
For instance, we know that particles are constructed in quantum fields. And we have pictures of the early universe and its plasma of quantum fields and it is finite.
Which would lead to the conclusion that even if there is a greater infinite condition, it is unable to create any matter, because it is void of other quantum fields. At least that seemed to be the case at the moment our universe was created.
And maybe that fact creates the apparent contradiction.

We don't know that it is finite. The observable Universe is finite and that's all we know.

 

[Write4U]

We don't know that it is finite. The observable Universe is finite and that's all we know.

But does the picture not show the entire universe, before it became too large to see all of it? As I understand it the picture is the actual size of the background glow, very shortly after the BB, clearly showing it was finite in size, which in itself would indicate that the universe was even smaller shortly before then.
IMO, our universe is clearly finite. If it had a beginning then logically it cannot be infinite in size. There may be an infinite void beyond that, but that is the speculative part, not our universe.

Part of a series on
Physical cosmology

• Big Bang · Age of the universe
• Chronology of the universe

Early universe[show]
Expansion · Future[show]
Components · Structure[show]
Experiments[show]

 

[someguy1]

Current scientific theories that include infinity have amazing predictive power.

Which means nothing in terms of ontology. Newtonian gravity has amazing predictive power but it turns out to be incorrect at high velocities and for large masses.

All scientific theories are approximations. If you don't understand that you don't understand the first thing about science.

Are you making the claim that current scientific theories are the absolute and final truth, and will never be refined or extended or modified in any way? I hope you're not saying that but if you are, admit it now.

The only alternative theory we could have to explain this would be for us ordinary folks to believe scientists are the magical fairies.

Newton famously understood that natural science is DESCRIPTIVE and not EXPLANATORY. Do you understand this point? Science builds mathematical models that match known observations to some satisfying number of decimal places. We do not know anything about ultimate causes. Ultimate causes are metaphysics and not physics.

Still, it's true there's no absolute criterion.

Oh. Ok. You do understand this point. So why are you typing in arguments in support of a thesis you already know is wrong? And aiming them at me?

It all comes down to what you believe,

Oh. So there is no objective reality. It's only about what I believe. I believe in magical fairies and someone else believes in modern physics. What of it? All points of view are equally valid. "It all comes down to what you believe."

Don't you see that you keep contradicting yourself? If it's "all in the mind" and "it all comes down to what you believe" then you aren't in much of a position to be defending the thesis of objective science dispassionately studying an external reality.

precisely because, as I said, we don't know.

Well then what is your point and why are you aiming it at me?

Yet, there are clearly different kinds of beliefs and we can make the distinction for example between theories that are not falsifiable and those that are.

Uh-oh you're about to fall into a trap of your own making.

Science is, fairies are not.

Really? Multiverse theory is falsifiable? True pointlike particles are falsifiable? Tell me about this. Make sure to supply peer-reviewed references. This is going to be interesting.

You see you just wrecked your own thesis. Much of modern physics is not falsifiable. This is a serious problem that a lot of people are noticing. "Landscape" theories are not falsifiable.

So, you can choose to believe in things like fairies but it's your choice and nothing anybody could stop you from.

Well, objective evidence to contrary would convince me. But nobody has provided any objective evidence for the multiverse and for actual physical infinities.

Infinities are not like fairies.

Having studied a bit of infinitary set theory, infinities are very much like fairies. They're imaginary things that we make up stories about and then reason logically from the stories.

Please understand once and for all that infinity exists in mathematics by ARBITRARY FIAT.

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

I'm not sure how we go about falsifying infinities

Turns out you can. The negation of the axiom of infinity is consistent with the other axioms of set theory. When you negate the axiom of infinity you get Peano arithmetic. There's 1, and there's 2, and there's 3, ... but there is not a completed set containing all of them. You can do math in this system just fine, a lot of number theory still works.

Physics, not so much. But this is just the same problem we've already noted. Abstract crazy math is often useful to physicists even though it's not "true" in any meaningful sense. Wigner again.

but the tricky point here is that if infinities don't exist they are falsifiable

Nobody has ever provided evidence for the existence of physical infinities. If anyone ever did they'd get the Nobel prize immediately. I really hope you understand this. There is no evidence for physical infinities. All we have is the fact that mathematical infinities, which are highly abstract and which only exist by arbitrary fiat, turn out to be useful. Wigner again.

and if they exist they are not. So, as long as we don't falsify them, we can only remain uncertain.

Isn't the burden of proof on those who claim something exists that there's no evidence for? There's no evidence for physical infinities.

I have to say I have no idea what you're talking about or why you aimed this post at me. You're not arguing with anything I've said or anything I believe.

QuarkHead dropped in to mention that the Hilbert space formulation of QM uses infinities. I pointed out why this is a mathematical idealization and not to be taken as having ontological value. QuarkHead has not seen fit to come back and explain himself or reply to my points. I'm disappointed in that, I wish I knew why he posted as he did and whether I've addressed his point or missed it.

 

[Seattle]

But does the picture not show the entire universe, before it became too large to see all of it? As I understand it the picture is the actual size of the background glow, very shortly after the BB, clearly showing it was finite in size, which in itself would indicate that the universe was even smaller shortly before then.
IMO, our universe is clearly finite. If it had a beginning then logically it cannot be infinite in size. There may be an infinite void beyond that, but that is the speculative part, not our universe.

No, we are seeing the light/radiation that has had time to reach us. That's the only reason that it is the shape that it is. You aren't looking at a "picture" of a small Universe that falls off into nothingness.

Our Universe could be infinite if it started out that way. All that has happened is that the space between any two points has expanded.

There is no void beyond the Universe. That would still be the Universe.

 

[QuarkHead]

[QUOTE="someguy1, post: 3527474, member: 27037]QuarkHead dropped in to mention that the Hilbert space formulation of QM uses infinities. I pointed out why this is a mathematical idealization and not to be taken as having ontological value. QuarkHead has not seen fit to come back and explain himself or reply to my points. I'm disappointed in that, I wish I knew why he posted as he did and whether I've addressed his point or missed it [/quote]
The reason is I do not fully understand the term "ontology" and what is has to with science.

Plus the Reals are complete (just as an Hilbert space of functions is) if every sequence has a limit in the same space. But the Reals admit of uniform (pointwise) convergence whereas Hilbert spaces require only convergence in the mean to be complete.

I am not sure, but I suspect the theorem that states that a complete infinite space must be uncountable does not apply where convergence is not uniform.

As an aside - any description whatever about the so-called real world is a representation, whether it be a string of words or mathematical symbols

Let me ask you all...... is there ANY description of the the "real world" that does NOT use a representation of this sort?

 

[someguy1]

As an aside - any description whatever about the so-called real world is a representation, whether it be a string of words or mathematical symbols

Let me ask you all...... is there ANY description of the the "real world" that does NOT use a representation of this sort?

Thanks for coming back, my original reply was for you and I'm still curious about what you meant.

Do you agree or disagree that a representation of a thing is not necessarily the thing itself? That the map is not the territory? That although we use the real numbers to model time, that doesn't in an of itself prove that time is identical to the (extremely abstract and strange and UN-real) real numbers?

If you agree that the map is not (necessarily) the territory, then my remarks stand. Hilbert space is a representation or model. It's handy for expressing the theory. But it's not necessarily ontologically correct.

On the other hand if you claim that (at least in this one case) that the map IS the territory, then we can simply have a conversation about the nature of science and Newton's famous point that "I frame no hypotheses." Newton understood that science is DESCRIPTIVE but not EXPLANATORY. Way too many modern "scientism-ists" misunderstand this point.

Building models is science. Believing that those models are identical with ultimate reality (if there even is such a thing) is scientism. It's metaphysics, not physics.

Plus the Reals are complete (just as an Hilbert space of functions is) if every sequence has a limit in the same space. But the Reals admit of uniform (pointwise) convergence whereas Hilbert spaces require only convergence in the mean to be complete.

The real numbers are complete in exactly the same sense as Hilbert space is complete. Every Cauchy sequence converges. There is no uniformity involved.

I do agree that the space of continuous functions on the reals isn't complete unless you require uniform convergence. That's different than the point I'm making but it's perhaps worth exploring more later, once I understand whether you believe map = territory in the case of QM.

 

[arfa brane]

Whether or not map = territory, how many maps are there? Is the territory a singleton set?

It occurs to me that, for instance, because the earth is rotating about an axis, we can't choose a map that has poles on the equator.

 

[Write4U]

No, we are seeing the light/radiation that has had time to reach us.

Yes, but that was what was then. The fact that it shows a regular spherical configuration would indicate that it was the expanding totality.

Another seldom mentioned fact is that even in spite of the inevitable red-shift, this background glow is missing the longest wavelengths, which would indicate a small wave source. Can't make a wave length larger than the length of the vibrating object.

Moreover, the fact that it appears oval would indiate that it is expanding horizontally and not the limiting circumference of some lens.

Moreover, if it was infinite then it would have filled the entire picture can't have a bounded infinity. OTOH it is possible we are looking at an infinitely expanding bounded space.

These are my intuitive reactions to the evidence. I could be wrong.

 

[Seattle]

Yes, but that was what was then. The fact that it shows a regular spherical configuration would indicate that it was the expanding totality.

Another seldom mentioned fact is that even in spite of the inevitable red-shift, this background glow is missing the longest wavelengths, which would indicate a small wave source. Can't make a wave length larger than the length of the vibrating object.

Moreover, the fact that it appears oval would indiate that it is expanding horizontally and not the limiting circumference of some lens.

Moreover, if it was infinite then it would have filled the entire picture can't have a bounded infinity. OTOH it is possible we are looking at an infinitely expanding bounded space.

These are my intuitive reactions to the evidence. I could be wrong.

The Universe isn't expanding "horizontally". That's a Mercator projection. The data is fitted to it just as the Earth is fitted to a Mercator projection. The Earth isn't really distorted like that.

The radial nature of this "picture" is due to the way that light radiates. It's not "dark" outside of the "picture". That light just hasn't had time to reach us yet. The Observerable Universe is larger by one light year every year.