NotEinstein
Valued Senior Member
No problem, glad I could help.
Can "Infinity" ever be more than a mathematical abstraction?
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Seattle
Valued Senior Member
It depends on whether you are talking about additive or subtractive colors.
Physicists like to talk about the Hilbert space of colors (colours), as a cone over a simplex. (wtf is that?)
You can choose three colors to be like orthogonal basis vectors, then map all visible colors to that basis. Alternatively choose a pair of colors and define a continuous one-dimensional map between them: choose say, red and blue with purple halfway between, for instance. Repeat with another pair, such that each pair lies on a continuous spectrum of wavelengths (i.e. the real line).
Mostly we want color maps we can use in the engineering of display and lighting tech (makes sense).
But, does the real line exist? I remember this question being asked more or less, in a physics course I took, the question was why are intervals of time real numbers? The answer was: because they are, don't worry about it. In fact just about any quantity in physics is a real number, and we don't know if real numbers are just abstract or actually "real", or even if it makes sense to ask about it . . .
You can choose three colors to be like orthogonal basis vectors, then map all visible colors to that basis. Alternatively choose a pair of colors and define a continuous one-dimensional map between them: choose say, red and blue with purple halfway between, for instance. Repeat with another pair, such that each pair lies on a continuous spectrum of wavelengths (i.e. the real line).
Mostly we want color maps we can use in the engineering of display and lighting tech (makes sense).
But, does the real line exist? I remember this question being asked more or less, in a physics course I took, the question was why are intervals of time real numbers? The answer was: because they are, don't worry about it. In fact just about any quantity in physics is a real number, and we don't know if real numbers are just abstract or actually "real", or even if it makes sense to ask about it . . .
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If the universe is continuous, ie that it's accurately modeled to perfection, not to approximation, by the mathematical real numbers; then it immediately follows that there are uncountable sets physically realized; and our original question about actual physical infinity is answered. The thread is done because we've achieved a definitive answer!
Therefore it makes sense to consider various ways in which that thesis might be false.
Perhaps the universe is NOT continuous. At the moment nobody knows but a slam-dunk Nobel awaits whoever figures it out.
Another possibility is that the world is continuous, but it is NOT CORRECTLY MODELED by the standard mathematical real numbers. It might, for example, be the intuitionistic real line; or some flavor of the constructivist real line. Those models are smaller than the standard reals. Or, the true continuum might be LARGER than the standard reals, for example by containing infinitesimals like the hyperreals or the surreals.
We need to be very careful when we reason from the standard 2018 version of the mathematical real numbers and try to draw conclusions about the physical world. Our conception of the continuum is historically contingent and we know of many perfectly sensible alternatives.
By "Platonic" I mean that an abstract entity has existence "out there" somewhere. That things like sets, vector spaces, topological vector spaces, and Hilbert spaces, have an "actual" existence and not just an abstract one. Whatever that might possibly mean.
The question of whether color is an intrinsic property of things or is a true subjective experience; is a subject with an extensive literature in philosophy. I personally don't have much if any familiarity with the literature, so I won't comment.
Also I think it's a little far afield of our topic of the existence of actual physical infinity, but that's just my opinion and clearly people are talking about color theory, so what do I know.
I will mention that the example of physical color, ie electromagnetic radiation in the visible spectrum, does hit our topic of actual physical infinity.
How many colors are there? My understanding is that quantum theory says that there must be only finitely many colors, because wavelengths are not infinitely divisible.
So in this case the situation is NOT well modeled by the mathematical real numbers. I claim no particular expertise in physics so if I'm wrong just let me know.
Quantized emission spectra according to E = hf is a consequence of quantized orbital energy levels in atoms & molecules. That restriction doesn't apply to free charges, and e.g. synchrotron spectra is truly continuous: https://en.wikipedia.org/wiki/Continuous_spectrum
I will go figure out what that means. Are you saying that a physical color can truly take on a continuous range of frequencies indexed by real numbers? Frankly I would be very surprised to find out this is true. Is it possible for you to explain it simply?
ps -- That's a brief article which doesn't follow up on or even acknowledge the profound physical and philosophical consequences of what it claims.
I did find this:
In particular, the position and momentum of a free particle has a continuous spectrum, but when the particle is confined to a limited space its spectrum becomes discrete.
What does that mean? When would a particle not be confined to a limited space? Isn't the universe finite in extent as far as we know? Why doesn't that count? Is this saying that a true continuous spectrum must require the assumption of an unbounded universe?
I find your claim extremely interesting and important for me to understand; but I did not find that Wiki page satisfactory. If ANYTHING in the world takes on all real number values in an interval, then that interval contains noncomputable points. There must therefore be infinite information encoded in that frequency and (as usual) the computable universe hypothesis is immediately falsified, as most of the points in the interval, hence most of the claimed allowable frequencies, could not be specified by any computer program operating on the currently known theory of computation.
Can you put this in perspective for me? Do you claim there can be frequencies described by the uncountable set of mathematical real numbers in an interval? Frequencies that can be observed in a laboratory? Tell me, how would you measure it to infinite precision on lab equipment?
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I had formed an initial answer but then you edited extensively, so, starting again....
Whether charge is bound or unbound, any emission is still of EM quanta i.e. photons. The distinction is that photon emission in latter case could be at any arbitrary frequency within the allowed band appropriate to the physical system. Continuous doesn't mean over an infinite energy range, just that within a physically limited range, there is no restriction as to particular emission frequencies.
See e.g. section 6.4.2 here: https://en.wikipedia.org/wiki/Synchrotron_radiation
As far as the universe providing some ultimate quantized energy bound i.e. 'the ultimate atomic nucleus', no. In the idealized FLRW metric, locally (everywhere) there is no central gravitational field to act as a confining potential. A charged particle orbiting an airless planet is gravitationally confined, so in such a case there is notionally at least quantization of orbital energy levels. In practice even minute fluctuations in gravitational potential from myriads of contributions e.g tectonic motions, convective flows etc., emission spectra as the charge inspiralled would be indistinguishable from perfectly continuous.
It seems to me there are fundamental reasons why arbitrary real number frequencies can not exist. I don't care about infinity ranges, I'm perfectly happy to pick any bounded range, even a tiny one. These are my concerns. I'm continuing to Google around, some references agree with you and others are less clear.
* First, no experimental apparatus could ever detect such a thing as an arbitrary real valued frequency. Any physical apparatus has an experimental error. The best an experiment could ever do is say, "We observed a photon with wavelength of between x and y meters (or nm or whatever). So you could never publish a paper and say, "We experimentally verified a photon whose wavelength is exactly pi or exactly e. You can't experimentally prove ANYTHING exactly, that's basic to science.
In fact you could not experimentally verify an exact rational frequency either, not even an integer one. All physical observations are approximate, don't they teach that to people anymore? So IF there are arbitrary real number frequencies/wavelengths physically instantiated in the universe, they are by definition undetectable by any measurement apparatus. They are invisible to us in principle. So now we're reduced to discussing whether they might exist undetected by any sentient being running lab experiments, "out there" in some kind of invisible world. Like where the theologians and radio preachers say the Baby Jesus exists. It's not an unfair comparison. It's not possible to detect or observe these frequencies you claim exist.
* Secondly, if all real number frequencies (or wavelengths, same argument) between two given finite values are ever attained anywhere in the universe, even for an instant, it blows the CUH completely out of the water. Some of those real numbers (all but a set of measure zero, in fact) are noncomputable. If the universe is required to instantiate some particular noncomputable real number frequency or wavelength, the great computer in the sky and/or our simulation programmers have no way to do it. There is no algorithm that can produce a noncomputable number to an arbitrary degree of approximation. That's the definition of a noncomputable algorithm. Turing was all over this subject in ins 1936 paper, "On Noncomputable Numbers etc."
* Thirdly, well really more a branch of #2, a noncomputable frequency or wavelength falsifies the Church-Turing-Deutsch thesis, which says that "a universal computing device can simulate every physical process." Clearly this is falsified by a noncomputable frequency or wavelength.
https://en.wikipedia.org/wiki/Church–Turing–Deutsch_principle
* Fourth, a noncomputable number contains an infinite amount of information. Contrast this with familiar irrationals like pi or e, which are transcendental yet computable. A newbie programmer can crank out the digits of pi or e by implementing any of the many known algorithms for the digits of those numbers. By contrast, there is no algorithm or finite-length description of a noncomputable number. So if a sender at point A beams a noncomputable photon at a receiver at point B, and the photon travels at the finite speed of light, we have transmitted an infinite amount of information in finite time. I do not believe this is possible by the laws of information theory but I'm not sure of the specific law involved.
I do not believe arbitrary real number frequencies or wavelengths, even ones that are "out there" but cannot even be detected by experiments, can exist basically for information-theoretic reasons.
What say you? I'm way over my head on the physics so even though your erudition and vocabulary are are impressive, they're lost on me. Is it possible for you to simplify your exposition in such a way that you might convince me? Your thesis must be false. David Deutsch surely knows about the alleged continuous spectrum yet he claims all physical phenomena are computable.
ps -- It's the Bekenstein bound that constrains the amount of information in a bounded region of space. If a noncomputable photon exists, it violates the Bekenstein bound because it would instantiate an infinite amount of information in a bounded region of space having finite energy.
https://en.wikipedia.org/wiki/Bekenstein_bound
Ok I think I'm done editing!
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I was going to post a supplementary note to the effect that even for discrete atomic emission spectra, Heisenberg's uncertainty principle limits the sharpness of emission lines to a peak about a finite bandwidth. Owing to the finite time an excited state exists prior to dropping back down to a lower level. Natural broadening:
https://en.wikipedia.org/wiki/Spectral_line#Natural_broadening
Being intrinsically QM, there's nothing one can do to eliminate it. It works afaic against your appeal to finite uncertainty in measurement (an additional uncertainty). The intrinsic QM broadening has no discreteness within that uncertainty relation. To understand where the uncertainty comes from in the first place, the place to go in QFT and 'second quantization'. Where interaction with the vacuum fields in invoked. Beyond my pay grade to discuss that in detail.
Anyway, the same basic mechanism operates in the free charge case too, but only in the sense of imposing, on top of the classical continuum spectra, a fundamental uncertainty as to specific instances of photon emission and frequency spread. The overlap is itself continuous.
I think there is a confusing of complete arbitrariness of frequency (remember - inherently having a non-discrete 'natural linewidth') with information content. Even if spacetime is truly fundamentally quantized, having only finite possible states may not be true. Given the universe is continually expanding at an accelerated rate, there is no way imo one could even in principle define such a complete 'state-space-of-the-universe'. Because it will continually change in fine-grained detail in an unknowable way.
I won't pretend to be up there with those listed gents. But will guarantee there are other equally qualified ones with quite different opinions. Lubos Motl for instance insists QM inherently requires complete continuity i.e. perfectly smooth spacetime. Take your pick. Luckily the local sewerage works doesn't depend on such musings!
Write4U
Valued Senior Member
I have a secondary question; If we are talking about a particle, then are we not by definition talking about a discrete value?
I can understand a wave function to be continuous such as found in fields,........ but a particle?
I am totally not the person to be asking physics questions. But actually I do happen to know the answer to that. QM is based on the mathematical real numbers as a model for time. Continuity is baked into the model. But not into reality itself!! That's the error some people make.
But ok, continuity of time is an assumption. The Schrödinger equation takes t as an argument, where t is a mathematical real number. So all of these continuity assumptions are baked into the model from the start. They are not facts about the universe as revealed by the math. Rather, they're the artifacts of our initial assumption: that time can be modeled by the mathematical real numbers. Of course this gets GREAT RESULTS. That's why the physicists do it. But they don't pretend it's ontology. I watched a QM video from MIT and the professor made that exact point. He said "I do physics and not metaphysics." He totally got it.
So the answer to your question is that these are not discrete points. These are real numbers ... points on a real number line that is an uncountable dense, complete linear order. The real number line you learned in high school or the more sophisticated version you got in real analysis class. Same real line. Infinite decimals and all that. In QM they model time on the real numbers. That's what they mean by points. They're points, but they're not discrete at all. More like maple syrup than bowling balls.
ps -- Yes you guys have me watching QM videos!
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I won't pretend to be up there with those listed gents. But will guarantee there are other equally qualified ones with quite different opinions. Lubos Motl for instance insists QM inherently requires complete continuity i.e. perfectly smooth spacetime. Take your pick. Luckily the local sewerage works doesn't depend on such musings!
As I was researching this fascinating question of real-valued frequencies, I came across a physics.stackexchange thread asking the same question. The checked answer, if I recall, expressed absolute certainty that there were real live actual real-valued frequencies in the universe.
I did note that the author was Luboš Motl. I know his name and I know that he is a bit of a character and contrarian to say the least. I did not take his post as authoritative, for the simple fact that it is impossible for real-valued frequencies to exist. People just don't get how weird the real numbers are.
But anyway I haven't any more to add at my end. I found comments and articles that support my view and arguments to the contrary. The more I learn about the subject the stronger my belief in my own viewpoint gets. So I will leave this subthread and try to catch up on some others.
But I did watch a QM video and it's clear that they start by modeling time as the real numbers. So all those assumptions of continuity are baked in. There's no evidence that they're actually true. They're simplifying assumptions to get their model off the ground that then gets at best 12 decimal points of precision in actual experiments. That's great for all practical purposes, but it doesn't scratch the surface of the actual real numbers. The physicists simply have no idea how weird the real numbers are, otherwise they wouldn't think they're "real." Which in fact they aren't.
tl;dr: Luboš Motl? That's all you got? Anyway I'll stipulate to your far greater knowledge of the physics. And my own opinion hasn't budged, it's been confirmed by my research. I'll continue learning about this subject. Thanks for all the great pointers and areas for me to look into.
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I fully agree.
Speakpigeon
Valued Senior Member
I really don't see how you could justify your claim here of the impossibility of "real-valued frequencies".
First, as formulated, it's rather ambiguous. Real? All Integers are also Reals. So? You think there are no frequencies of 1/s or something?!
So, I suppose we have to be charitable and interpret what you are saying as meaning that at least some Reals can't possibly exist physically, i.e. that there can't be quantities in the physical world that have these numbers as values.
Yet, I'm still puzzled as to what the problem could be.
First, I'm sure you would agree that a number is just a number and that most reasonable people accept that a number doesn't exist as such in the physical world. So, what could it possibly mean exactly for something physically real, such as a frequency or whatever, to have the property you call "real-valued"?
In particular, the question here is whether there is anything at all which would be the property of being "real-valued"?
Take the number 2. Is there a property of being 2 in the physical world? Personally, I don't think we know that. I'm not sure for example that the fact that we observe two electrons logically emplies that there are effectively two physical things at the origin of our observation. Maybe there's just one, i.e. reality for example. How would we possibly know, since all we have access to are our observations?
So, can you specify what it would be for a frequency to have a "number" as property and in such a way that it would be possible to have for example Integer-valued or Rational-valued frequencies, but not to have Real-valued frequencies?
EB
Write4U
Valued Senior Member
Because it makes no difference what symbolic language you use. As long as it follows the same logical mathematical equations, the question is moot. You can write it in morse code if you want.
Write4U
Valued Senior Member
Don't know that there is any debate, but there are different types of primary colors - additive and subtractive. And not everyone's receptors center on the exact same frequencies. Not to mention color blind people - many who only have two primary colors.
someguy's logic deserts him. The integers are Real Numbers, the Natural numbers are Real Numbers. The assertion that a particular frequency is a Real Number does NOT imply that the spectrum of EM frequencies is continuous in the sense that the topologist's Real line is, which would be equivalent to saying that there is non distinction between the integers and the Reals
Oh my God are you trolling or being silly? The claim was made that a frequency may take on arbitrary real values in an interval of real numbers. That would include noncomputable numbers, which totally destroys that claim.
The fact that a frequency could be an integer and an integer is a real number is completely irrelevant, unless you are saying that frequencies could ONLY be integers or perhaps rational or at least computable. That claim was not made. The claim initially made was that all real frequencies in an interval of real numbers can physically exist.
The claim was made that frequencies can take on all real values in an interval. Please go back to the beginning of this part of the thread so you'll understand why I'm pointing out that this couldn't possibly be the case.
You wrote:
"... does NOT imply that the spectrum of EM frequencies is continuous in the sense that the topologist's Real line is ..."
That's the exact claim that was made; and much high-end physics jargon was used to claim it. Please reread the relevant posts and you'll see.
Here is ONE of the relevant quotes:
I did ask Q-reeus (twice) to dial back the jargon and speak plainly but that didn't happen. I assume Q-reeus is claiming that electromagnetic frequencies can take on values in a continuous interval of real numbers. If something else was meant, I'd appreciate if someone would say it plainly.
And again:
My bolded emphasis.
And the post that started this:
So QuarkHead if I misconstrued these three posts of Q-reeus, please do set me straight. Pointing out that integers are real numbers is not helpful. Q-reeus did not say that frequencies are integers and integers are real numbers. He said that the distribution of synchrotron spectra is continuous: not just in the mathematical model, but in nature itself. If that is so, then nature instantiates noncomputable real numbers, each of which carries an infinite amount of information. That is not in dispute. All that's in dispute is whether Q-reeus may be making the classical mistake of taking mathematical modeling for ontology.
What do you say?
ps -- From the Wiki link posted by Q-reeus:
In physics, a continuous spectrum usually means a set of attainable values for some physical quantity(such as energy or wavelength) that is best described as an interval of real numbers, as opposed to a discrete spectrum, a set of attainable values that is discrete in the mathematical sense, where there is a positive gap between each value and the next one.
Now it's clear (if you have in mind the distinction between models and reality) that the article says that some physical quantity is BEST DESCRIBED as continuous; not that nature itself is. This is the error Q-reeus is making, and this is the error I'm trying to correct.
So Wiki did in fact get the philosophy right. A lot of things in physics are conveniently modeled by mathematical real numbers. But it's problematic to claim that nature instantiates all the real numbers in a given interval, because an interval contains uncountably many noncomputable numbers. A physical noncomputable number would contradicts everything we know about the intersection of information theory and physics, for example the Bekenstein bound and the Church-Turing-Deutsch thesis.
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