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

Discussion in 'Physics & Math' started by Seattle, Jun 24, 2018.

1. ### someguy1Registered Senior Member

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595
LOL Same point as I just explained to QuarkHead. The claim was made that all real-valued frequencies in an interval of real numbers can physically exist. The fact that some real numbers happen to be integers is irrelevant. No, that's not the word. It's disingenuous. Not intellectually serious. Someone says that "all real values may occur," and I point out that's impossible because most real numbers are noncomputable. which would violate many known physical laws; and two people yammer that, "Oh yeah well 3 is a real number."

Please see my detailed reply to QuarkHead regarding the posts and comments of Q-reeus that I am arguing against. The claim was made -- explicitly, at least three times -- that frequency distributions can be literally (ie physically, in the real world) be continuous. That simply cannot be, on information-theoretic grounds.

Certainly. Most arbitrary real numbers are noncomputable. That means that their decimal (or binary if you like) digits can not be cranked out by an algorithm. Natural numbers, integers, rationals, and even many irrationals like sqrt(2) or pi are computable. So even though pi's decimal expression is an infinitely long non-repeating string of digits, pi only encodes a finite amount of information. We could write a program, using only finitely many characters, that would crank out as many digits of pi as you like, subject only to constraints of computational resources. (Physical constraints on computation are not part of this definition. We assume our program has as much energy and storage and time as it needs to crank out the n-th digit for arbitrary n).

But most real numbers are noncomputable. "Most" in this context has a technical definition. If you throw all the real numbers into a hat and pick one out at random, you will pick a noncomputable real number with probability 1.

A noncomputable number contains a truly infinite amount of information. You can't compress it to a finite-length computer program or algorithm or definition or description or unique characterization. The only way to specify it is it give all its digits, infinitely many of them.

This would violate not only every known law of physics, but many laws that are suspected. It would falsify the Computational universe hypothesis. It would show, once and for all, that the universe is not a computer, that we are not computational simulations. That's not to say someone won't discover a physical noncomputable number tomorrow morning. Only that if it happens, it will overthrow everything we know about computation and physics. The discoverer will get an instant Nobel prize and we'll have our next major revolution after Newton and Einstein.

Now this is all perfectly obvious, and I think you and I are in perfect agreement that electromagnetic frequencies aren't REALLY continuous in the real world. But Q-reeus claims otherwise. He's confusing the mathematical model with reality. In the phrase I've been using, he's confusing the map with the territory.

Last edited: Jul 11, 2018 at 2:43 PM

3. ### Confused2Registered Senior Member

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496
Just to clarify the point..
Imagine a bit of ice gliding across an ice rink (assume zero friction). The claim is that there are places (measured from an arbitray start point in an arbitrary set of units) that the ice block can't occupy because that would require a non-computable number to represent it. Have I understood the claim correctly?

5. ### someguy1Registered Senior Member

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595
Correct. The argument's even simpler than that.

Suppose a physicist with an unlimited budget wants to do a lab experiment to prove that an arbitrary position can be hit. Say the ice surface is one meter, and a physicist claims that the ice can be placed -- it does't even need to slide, I'll let you or the physicist pick the "bit" of ice -- nice pun, since this all comes down to bits -- I'll let you place the "bit" of ice by hand, anywhere you want it.

I claim you can't even prove to me that you can place a "bit" of ice -- what is a bit of ice, by the way? A molecule? Ok, a molecule. I claim you can't place a molecule of ice EXACTLY at the 1/2 meter point.

Why not? Because all laboratory experiments and all measurements are approximate. At the very best, with an unlimited budget to design and build an experimental apparatus, all the physicist can do is claim that the molecule of ice is pretty near 1/2, to 12 decimal points, with an error range of plus or minus a nudge.

In other words forget exotic real numbers like noncomputables. You can't even prove that a bit of ice is at 1/2. You can't even prove it's at some integer distance. All you can do is get a good approximation that's roughly proportional to how much money you spend on your apparatus, and how many Feynman-level brains you have on hand.

I haven't even pressed you on what a "bit" of ice is. Molecules are pretty big. What about hydrogen atoms? They're pretty big too, spread out all over space as a probability wave. What about quarks? Even worse. In QM we can't guarantee that a tiny particle is ANYWHERE in particular with certainty.

I hope you agree with this point. It's crucial to the nature of physics. I was reading the other day that quantum electrodynamics, the theory for which Feynman got his Nobel prize, makes a prediction that's been experimentally verified to 12 decimal places. And that this is famously the most accurate prediction that physics has ever made. Twelve places. Every real number expresses precision to infinitely many places! There's a lot of unknown physics in that tail of infinitely many digits.

All measurement is approximate.

After that, we're into metaphysics. Is the world discrete or continuous? Nobody knows. To be sure, physics MODELS the world as continuous, because that makes the math easy. For suitable definitions of "easy," of course. But the actual, real, physical world "out there?" We don't even know for sure that there is a world "out there," let alone whether it's discrete or continuous.

That's metaphysics, and not physics.

But now suppose we make some assumptions, just to play the game.

* We assume that there is an external, objective reality. This by the way is Tegmark's first assumption. He calls it the ERH, the external reality hypothesis. It's the start of his argument. And Tegmark's smart enough to realize that it is only an assumption. So let's assume it.

* Let's assume that the world is continuous. Let's assume that an interval of spacetime is PERFECTLY represented by an interval of the real numbers. Not modeled, not approximated, not conceptualized. The world IS the mathematical real numbers.

With these two assumptions, there is a hell of a problem. If a bit of ice or even a point-particle (if there really is such a thing) can go from point A to point B and hit every point in between, then at some point -- almost all points, in fact -- its address in spacetime is a noncomputable number, which requires an actual infinity of information. This violates information physics as we currently understand it.

But all this is philosophy and metaphysics. Nobody knows for sure that there's a world "out there," or whether it's discrete or continuous, or whether the question is even meaningful.

But we do know that if the information capacity of a bounded region of spacetime is finite -- that's the Bekenstein bound -- then no noncomputable real number could physically exist.

I hope this is sufficiently clear. The tl;dr is that the question of whether the world is discrete or continuous is metaphysics and not physics. Physics deals only in mathematical models that approximately describe and predict the results of physical experiments.

We MODEL the world with continuous real numbers. That does not mean the world IS actually continuous. The model of the world is not the world. The map is not the territory.

Last edited: Jul 11, 2018 at 4:18 PM
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7. ### Q-reeusValued Senior Member

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I had hoped your #313 was it as far as discussion over realizable photon frequencies was concerned. Let me add a bit extra. Natural linewidth of discrete atomic spectral lines doesn't simply mean there is a continuous possible range of individual photons each with a perfectly sharp frequency. It rather implies each photon contains within itself a continuous spectrum of frequencies. That's because as a necessarily finite length wavepacket, it has to be represented by a Fourier integral and not a discrete series, even an infinite one. A Fourier integral ranges over a continuum of frequencies. The longer the wavepacket, the more concentrated the integral is about some mean value - the narrower the spectral linewidth becomes. And vice versa. But discreteness is completely absent in that representation. Hence to talk of finitely vs infinitely allowable photon frequencies over some arbitrary range is imo really moot.
One article you might like to plough through: https://arxiv.org/pdf/0708.0831

A couple of more links just for 'fun':
The various kinds of continuum radiation - with explanations and sometimes formulas: www.astro.yale.edu/vdbosch/astro320_summary27.pdf
The various contributions to line broadening for atomic spectral lines: http://www-star.st-and.ac.uk/~kw25/teaching/nebulae/lecture08_linewidths.pdf
According to this article (and others say likewise), in principle a single photon could carry not just one bit but an infinite amount of information:
https://arxiv.org/abs/1610.02524

8. ### someguy1Registered Senior Member

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595
I thought I was done too, until not one but TWO people confused "All frequencies are real numbers, " with "All real numbers may be realized as physical frequencies." Logic 101, quantifier confusion. And QuarkHead accused ME of bad logic. Ye Gods. Yes I thought I was done, I really hadn't planned on four more lengthy posts this morning to re-state what I've already clearly stated.

I really wish you'd address the specific points I've made. Even your own exposition supports my point: "That's because as a necessarily finite length wavepacket, it has to be represented by ..." My emphasis.

Nobody's disagreeing that physics MODELS or REPRESENTS things as continuous, from freshman calculus on up. That has no proof value regarding the actual nature of the world. The Schrödinger equation has a real-valued parameter t representing time. The assumption of continuity of the mathematical model is baked in. And there are good fundamental reasons why this presents problems. You can't have an infinite amount of information in a bounded region of spacetime, but every nontrivial interval of the real numbers contains uncountably many noncomputable numbers, each of which requires infinitely many bits to specify it.

I know I'm repeating myself, but could you please address some or all of the points I've already made. Throwing more links and jargon at me about MODELS doesn't support your thesis in the least. Nobody knows if the world is continuous or discrete. You are confusing physics with metaphysics.

You name-checked Fourier integrals. Have you ever seen the rigorous development of Lebesgue integration, which is essential to QM? It's based on the axiom of choice. No axiom of choice, no integration theory (*), no functional analysis, no QM. But the axiom of choice gives you the Banach-Tarski paradox. Physicists just don't appreciate the idealized and un-worldly mathematical abstractions their models are based on.

(*) In the absence of choice, there's a model of the real numbers in which the reals are a countable union of countable sets. Measure theory blows up, it can't even get off the ground.

Last edited: Jul 11, 2018 at 4:59 PM
9. ### Q-reeusValued Senior Member

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Sorry I hopped in to this thread. Seemed a simple enough conundrum to solve at first. someguy1, you have an intensely philosophical/mathematical bent which I don't share.
You have probably come across it already, but as for issue of real-valued t in time-dependent Schrodinger equation, do a search for 'discretized Schrodinger equation'. Maybe that will lead you down a path to happiness. Cheers.

10. ### someguy1Registered Senior Member

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I'm already perfectly happy. I have no objection to basing physics on the real numbers. I just don't confuse that with an ontological position. You do. You're factually wrong.

Do you believe physical models are actual reality? How could they be?

You hopped into the thread to claim that arbitrary real numbers could be physically instantiated. That's inconsistent with known physical law. Is this a claim you stand by or retract?

If you think I object to real numbers being used to MODEL reality, you haven't understood a single word I've written. And if you think real numbers truly ARE reality, you haven't understood a single word of all the physics you think you've learned.

Edit -- By the way I haven't got "an intensely mathematical/philosophical bent. I just understand the difference between models and reality. A point Newton famously made in 1713.

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

pps -- Forget philosophy. Forget math. The claim that any physical quantity could actually, in the physical world, take on all possible real number values in an interval violates physical law.

Do you understand that? Do you agree? Disagree? Do you know what the Bekenstein bound is?

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

Last edited: Jul 11, 2018 at 5:49 PM
11. ### SpeakpigeonRegistered Senior Member

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265
Sure, but I made clear what the problem was in your post. I was addressing only one bit in your post and I quoted it.
So, I guess I have now to quote it again here for clarity's sake:
So, you should have said instead that it is impossible for non-computable real-valued frequencies to exist.
I don't understand why it should be so difficult to get understood on such a simple matter of fact.

OK, I'm going to try to have a debate with you on a specific point. I hope you take the time to understand the issue.
I'm not a specialist but I would assume that the product of a non-computable number by any non-zero rational number is itself a non-computable number. Sounds fairly obvious to me. Do you agree?
Let's assume now that all existing EM frequencies as we mesure them today are in fact non-computable numbers. Suppose further that all frequencies are in fact the product of the same unique non-computable number by some rational number. In other words, this one non-computable number is in fact a common factor to all frequencies. I would assume that such a situation would be compatible with the actual measures done of EM frequencies given that there's always a margin of error in measures.
Suppose now we change our unit system so that the non-computable number which is assumed as common factor to all frequencies is now taken as the unit for EM frequencies. I seems to me that expressed in this new unit system and given my assumptions, all frequencies would be now rational numbers.
So, given those assumptions, would you agree that all frequencies would be rational numbers?
EB

12. ### QuarkHeadRemedial Math StudentValued Senior Member

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1,550
someguy1 I quoted your statement that "it is impossible for real-valued frequencies to exist". You may have mis-spoken, but that does not give you the right to abuse me, merely because I assumed that, like me, you mean what you say.

13. ### someguy1Registered Senior Member

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595
I didn't abuse you. I responded in kind to you claim that my logic was bad. Your logic was awful. You confused "all frequencies are real" with "all reals can be frequencies." And I'd stand by my remark, even the phrasing. Because in context, it's obvious what I'm saying.

Abuse you? Jeez man. I responded exactly in kind.

In any event, you initially mentioned that Hilbert space involves infinities. To which I responded that Hilbert space is an idealized model and not reality itself.

I gather it must be the case that people can get a fine education in physics, without ever being told that idealized mathematical models that work to a given degree of approximation, are not to be taken as reality. I wonder if it's only the undergrads and discussion forum participants who are confused on this point, or if some professional physicists are equally confused. Newton was not confused on this point at all, nor is Allan Adams, the instructor for the MIT Open Courseware course on QM. He was very explicit on this point: "I do physics, not metaphysics." So at least SOME professionals understand that their models are not necessarily reality. I do confess to being surprised that so many physics experts are unaware of that.

Sorry if something I said offended you. I was definitely offended by your post about integers being real numbers. It struck me, and strikes me, as disingenuous, because it's such an obvious mischaracterization of my position. And like I say, in context I stand by my wording. It only looks wrong if taken completely out of context.

14. ### someguy1Registered Senior Member

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595
Hi. You raised a really good point in this post, which I will get to later since I have to think about it. I just wanted to respond to this particular point now, since it's the same point QuarkHead raised and he felt I insulted him. I didn't mean to insult anyone.

I agree that my saying "it's impossible for a frequency to take real values," or whatever I said, taken OUT OF CONTEXT by itself, was imprecise and clearly two of you took it that way and called me on it.

I believe that in the context of my conversation with Q-reeus, who was claiming that frequencies can take ALL real values in an interval, it would not be possible to misunderstand or misconstrue my phrasing. My meaning would be perfectly clear.

But I will concede that nobody reads every word of these lengthy posts, and my phrasing taken OUT OF CONTEXT was subject to the objection you and QuarkHead raised. I still regard it as disingenuous because my meaning was perfectly clear in context. But if I were re-writing my original phrasing I'd be more clear.

My apologies to any and all who were offended by my reaction to what I thought was disingenuous out-of-context quoting and inverted quantifiers.

Let me get to the rest of your interesting post later, I think you raised a good point.

But by the way, the approximation of measurement is still an issue. You can't even prove that an INTEGER frequency is exactly an integer! How could you? You'd have to build an apparatus and measure exactly, which you cannot do, even in theory. Now that doesn't mean a perfect integer frequency couldn't exist ... but you couldn't prove that it does! When we measure 5 centimeters in the lab, that's really 5.0 +/- a confidence interval. That's not an integer!

So there are two levels of this problem: (1) You can't prove that ANY particular frequency or temperature or velocity or any other "continuous" quantity actually takes any particular value; and (2) even if you allow for experimentally unobservable values to exist, none of them can be noncomputable and still be consistent with known physics.

And Q-reeus, that is not a point of philosophy or math. It's a point of physics. You can never have an infinite amount of information in a bounded region of spacetime. In recent years information theory has combined with physics to put hard limits on what types of bitstrings can physically exist.

Last edited: Jul 11, 2018 at 6:34 PM
15. ### SpeakpigeonRegistered Senior Member

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Sorry, I have to quote myself again for clarity's sake:
So, you can see here that I made clear that what you actually said, your wording, was ambiguous and took the time to explain that the second interpretation was probably the best. So, I really don't understand how you could still insist that my comment was "disingenuous". Beats me.
EB

16. ### Confused2Registered Senior Member

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496
No you don't.

17. ### arfa branecall me arfValued Senior Member

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We need to explain why it's possible for engineers to build devices that communicate over a space of frequencies; why is it possible for Fourier analysis to 'yield' a working set of formulas, why do engineers have a continuous time domain and a frequency domain to work with?

If say, I build an oscillating electronic circuit, what happens to its output if I accelerate the circuit to some constant velocity? What should I expect during the acceleration?

18. ### iceauraValued Senior Member

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26,306
I do not see how skipping the non-computable reals destroys continuity in the real world.
And I do not see how, if it does, that prevents the physical manifestation of infinities in the real world.

All limits of the functions describing the entity "frequency" exist, for example - to the extent frequencies exist as defined, they can take on any defined value and change at any defined rate. There appears to be no discontinuity in the real world of frequencies as described by the relevant mathematics - in their values, rates of change, etc.

And so the original question - do infinities really exist - returns as a question about the real existence of a mathematically defined frequency. In provided example, the frequency assigned the number "2" in some description is easily expressed as an infinite series in which every one of the cumulative terms likewise describes a real frequency obtainable on demand in a laboratory from a real physical system. In theory.

19. ### someguy1Registered Senior Member

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I'm going to quit while I'm behind on that particular issue. I still owe you a reply from the rest of your question this morning.

In other news I think perhaps I know what Q-reeus might be talking about. The Schrödinger equation gives a continuous range of probabilities of any particular measurement showing up. So in theory all real number outcomes are present "potentially." Then when we make a measurement and the wave function collapses, the result must be computable. Else we'd violate the CUH, the Bekenstein bound, and the Church-Turing-Deutsch thesis. This is not a point of math or philosophy. It's a point of known physics.

So, how does the wave function know to land on a computable number? Does anyone understand how this works?

Also, suppose we take the "many worlds" interpretation of QM. Then ALL possibilities happen in some universe ... so do SOME universes get those noncomputable collapse events? Or do only the computable universes come into existence? Again, if anyone understands this and can explain it clearly, I'd love to know.

20. ### someguy1Registered Senior Member

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Not entirely sure I'm understanding this para. But if you take the real line and delete all the non-computable reals, you get the constructive real line. It's full of holes. It's not continuous. There are only countably many computable reals. The Intermediate value theorem (IVT) from calculus becomes false.

Constructive mathematicians patch the IVT problem by declaring that only computable functions exist. Then you get the constructive IVT. I've read that there are attempts to base physics on constructive math, but the efforts haven't gotten very far as of yet.

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

"-- as described by the relevant mathematics ..." Agreed. But in reality? Assuming there even is such a thing as external reality? Who knows. What we DO know is that even in classical physics, all measurement is approximate. So suppose the answer to some physics experiment is 47. We never know if that's the answer. Only that our lab apparatus was able to give us 47.000001 +/- some error bars. That's the problem with this entire discussion. All physical measurement is approximate. Nobody's ever measured ANYTHING exactly.

I guess I don't understand how anyone measures '2' of anything from a continuous distribution. We measure 1.9999 +/- an error tolerance. The question of whether there really "is" some actual value there is mysterious, even in classical physics.

In QM it's even worse. A particle has a nonzero probability of being anywhere in the entire universe. It has the highest probability where it "should" be, but it might be somewhere else.

So all questions of what a frequency actually "is" are hopelessly complicated.

"... a real frequency obtainable on demand in a laboratory from a real physical system."

No such thing as I understand it. Only approximations. Nobody has ever measured an exact frequency. Is this basic fact about classical measurement forgotten or no longer taught?

(Edit) -- Possibly so. A Google search for a clear statement that all physical measurement is approximate turned up barely anything at all. Perhaps this fundamental aspect of physical science has somehow slipped out of common knowledge. I did find this:

The most common versions of philosophy of science accept that empirical measurements are always approximations—they do not perfectly represent what is being measured.

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

But to be fair, I'm quite disappointed that this was the best I could find.

Last edited: Jul 12, 2018 at 2:42 AM
21. ### Q-reeusValued Senior Member

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3,003
As you see it, obviously.
No, but they can be good enough models to cover e.g. your objection re continuum of frequencies. Once you get the distinction between 'infinitely many existing all at once', and 'an infinity of possible instantiations'. I have consistently only claimed the latter. While a single photon as wavepacket must be expressed as a Fourier integral, the continuum of frequencies implied does not mean an infinite amount of accessible information is encapsulated. It means that discrete entity is somewhat fuzzy. It means it will trigger a hit by any real-world detector necessarily itself having a finite detection bandwidth. It becomes meaningless to then talk of infinite vs finite in that context. George Ellis again - you can never prove physical infinities exist.
I stand by it. And it seems to me you continue to confuse a need for infinitely many over some span, with complete arbitrariness of any instantiation(s) over that span.
Barking up the wrong tree. You keep misunderstanding my position.
You can't avoid philosophy and maths in making that claim. Try substituting the word any for all in above. See a difference?
Regardless of one's stand on that matter, I have no issue with there being only finite accessible information within a finite region.

22. ### SpeakpigeonRegistered Senior Member

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I'm confused, too.
Well, I'd have to think about that.
Oh, wait, no, maybe I don't.
No I don't.
EB

Last edited: Jul 12, 2018 at 8:44 AM
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23. ### SpeakpigeonRegistered Senior Member

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I thought myself there was no illusion on this point:
Isn't that clear enough?
EB