Thus the signal photon's x-basis state will be directly observable as a detection event at D0a or D0b and predict which of detectors D1 or D2 the partner idler photon will later strike. — Andrew M

Very interesting. I had assumed that beam splitters act entirely randomly, but from your description it seems that they do not. Is that correct? I think I'm slowly getting your point. At the act of entanglement the photons 'decide' how they're going to act, not just in measuring spin, but also in how they will behave at beam splitters and the phase of their interference pattern.

Let me ask you this then: you've mentioned the z-basis and the x-basis. Are there a finite number of bases or is the number infinite? I ask because if there are infinite, that seems like a lot of 'decisions' to make up front.

For an accessible introduction, here's an excellent analysis and video of the experiment by physicist Sabine Hossenfelder. — Andrew M

I actually find your latest example with D0a and D0b more convincing than Sabine's video.

Thanks!

No, the signal pattern never shows interference regardless of what happens to the idler photons. — Andrew M

Thank you for identifying the flaw in my example. I should have known that it wasn't so simple.

Interference is only revealed when the idler photons are detected at D1 and D2 and that information is later used to post-filter the signal pattern. — Andrew M

I'm still confused about how the interference pattern is entirely a postprocessing effect. It seems to me that the signal photons must "know" what will happen to the idler photons so that it can produce the correct signal during postprocessing. If the idler photons don't affect the signal photons, how is it that the "D1" signal photons produce a different signal than the "D2" signal photons?

No, most physicists accept locality. — Andrew M

Fair enough. I'll need to read those threads before bothering you with questions on locality.

See the Local dynamics column in the quantum interpretations comparisons table. — Andrew M

That's a great summary table. Thanks for sharing!

...and thanks for your continued comments. I feel like you're explaining things perfectly clear to me and I'm just not understanding!

If the second sentence were true, I would expect all 4 subsets to produce the same signal pattern.
— keystone

Why would that be? — Andrew M

I'm wondering if my position could be made more clear if we focused on a simpler experiment. Let's assume that I've set up an experiment that starts similar to the DCQE. The entangled signal photons hit d0 and the idler photons have not yet hit PS. At this moment, does the signal pattern show interference? I don't think you can answer the question because you need to know what happens after this moment - you have to know what happens in the future. If after this moment, the idler photons travel through a lens and then hit a detector (akin to what happens to the signal photons), their path is unknown and I would expect to see an interference pattern for both the signal and idler photons. Or what happens to the signal pattern if the idler photons reflect off a mirror forever?

I'm assuming locality — Andrew M

My impression is that you may be holding a minority view here. Is that true? I think there is a subtlety related to quantum nonlocality in that it allows some information to be nonlocal but does not allow for faster-than-light communication. As for "The simple and obvious fact is that information has to be carried by material objects"...it's not that simple or obvious to me...even though it's obvious that Weinberg was a great man.

When the idler photon (qubit) is measured in the z-basis or the x-basis, then we can infer the path or the phase of the signal photon respectively. Making a local measurement of the idler photon has no impact on the signal photon. — Andrew M

I'm finding it hard to see how these two sentences are compatible. The second sentence suggests that measuring the idler photon does not impact the signal pattern at all. The first sentence suggests that measuring the idler photon does give information on subsets of the signal pattern. If the second sentence were true, I would expect all 4 subsets to produce the same signal pattern.

IMHO, it seems like your mathematical explanation only supports your first sentence. How does it support your second sentence? Or even better for a layman like me, how does your dining room table metaphor support your second sentence?

Note that the beam splitter only operates on the idler photon, but it transforms the way the signal photon is represented (i.e., as in a superposition of the red and blue paths). — Andrew M

Thank you for the detailed response. I assume you are being fully clear and it is just me not fully comprehending your answer, but let me ask this: how can the idler photon have any impact on the signal photon representation if the signal photon hits D0 before the idler photon hits PS?

(path information known) shows that photons can strike anywhere on the back screen. — Andrew M

Thank you. I now understand that nothing about the idler photons can be deduced by looking at the signal pattern. However, I'm still reluctant to say that the signal pattern is unaffected by the idler photons. You say:

The choice of measurement instead allows information to be decoded from the observed pattern, i.e., information revealing interference, or not. — Andrew M

When is this information encoded?

In every case, we would simply be left with the puzzle of how that extremely unlikely signal pattern occurred. Like the particles in a box all randomly moving to one side of the box, or an egg unscrambling, it's not something we would expect to observe. — Andrew M

For the double-slit experiment, are there any areas on the back screen which have exactly 0% chance of being hit by the photon when path information is known but >0% when path information is unknown. Or do all areas have >0% chance for both scenarios? I'm trying to understand if we're talking about an unlikely scenario or an impossible scenario.

By the way, I really appreciate your comments so far!!

My conclusion is that "reality" is both Potential and Actual. But we only know the actual stuff by means of our physical senses. The Potential stuff is only known by Reasoning backward from Actual Effects to Potential Causes. — Gnomon

I'll leave it to the more informed to comment on your other thoughts, but I like this conclusion. Classically there is a desire for everything to be actualized but with QM there is more room for the potential. And if I understand the quantum zeno effect correctly, without the existence of potential, change is impossible.

It doesn't matter what happens to the idler photons, the signal pattern will remain the same. — Andrew M

What if all idler photons strike D2? This requires an element of pure luck as the photons pass through BS3, but it also requires an element of choice in deciding what happens after the photons pass through BSa and BSb. In this case, doesn't the element of choice have an impact on the signal pattern?

The important (non-spooky) point with the delayed-choice quantum eraser experiment is that the original observed pattern for the signal photon (upper red and blue paths in Figure 2) is the same, regardless of what happens later to the entangled idler photon (lower red and blue paths) with beam splitters and what not. — Andrew M

What would you say happens to the signal pattern in the unlikely case that all idler photons are (by a stroke of luck) erased?

So it's different to the double-slit experiment where the choice to detect the photons, or not, at the slits does affect what is later observed on the back screen. — Andrew M

Is it fair to say that the difference is that double-slit experiments with and without measurement are performed sequentially while the delayed-choice quantum eraser experiment with and without measurement are performed in parallel. Because it's performed in parallel, it has one extra step at the end to separate the measured from the unmeasured data sets. With this view, it seems that the essence of the experiments are the same. Might you be discrediting the notion of true delayed-choice by attacking an insignificant aspect of the experiment (that it was conducted in parallel)?

I'm not sure anything can be deduced about the history of the photon in either experiment since one's assumptions will be interpretation-dependent. — Andrew M

If one sees an interference pattern can we deduce that there must be some element of waviness about the photon's flight through the double slits?

That would be supernatural interference with the universe. The Wigner interpretation suggests something like that. — noAxioms

True. But if a wave function needs an actual observer to collapse it, doesn't it need an external observer to make the first observation (in my simulation example)? Until then the simulated universe is entirely a superposition of potential states.

Maybe try RQM instead. — noAxioms

From my amateur position, the idea of there being no objective collapse doesn't sit well with me. I'll keep this in mind but right now I'm still grappling with vanilla QM.

The choice of measurement instead allows information to be decoded from the observed pattern, i.e., information revealing interference, or not. — Andrew M

I'm trying to unpack this statement. Could this be related to the Wikipedia entry where it says that "a photon in flight is interpreted as...something that has the potentiality to manifest as a particle or wave, but during its time in flight is neither." So upon flight the photon has a potential state but upon 'decoding' we deduce a history to say that it held an actual state (of a particle or a wave)? If so, the deduction of history follows the measurement, which is what I'm positing.

I don't think my current understanding assumes retrocausality since I'm not claiming that any information is being sent back in time.

@Trestone

My impression remains that you're adding on layers as a way to justify disallowing problematic statements/sets from existing within Layer Logic. My take is that if we wanted to avoid the problem by disallowing problematic statements we might as well do that within the simple framework of Classical Logic...but that's just my view. I hope you can find others to bounce ideas off of. If you can convince TonesInDeepFreeze that will be a good sign!

@Trestone
I haven't had the time to read through your messages thoroughly so I don't expect a response if you've already addressed my comment in your earlier posts.

You disallowed "This proposition is not true in all layers" in Layer Logic. Is there something analogous in Layer Set Theory, perhaps something like "x is an element of R if it is not an element of R in all layers"?

But in a way you are sweeping "this proposition is not true in all layers" under the rug. I wonder if there is an analog to the Russell set which you are also sweeping under the rug...

@Trestone

So my understanding is that your resolution to the Liar's Paradox is not Layer Logic, but disallowing problematic statements.

As for Russell's Paradox, isn't that what is basically done with ZF, where we 'disallow' the set of all sets that are not members of themselves? With this approach Russell's Paradox has been resolved and there's no need for Layer Logic.

Personally I think this sort of disallowing problematic objects sweeps the problems under the rug.

And what is with liars that speak about all layers?
„LA:= This proposition is not true in all layers“
In layer logic meta propositions about layers and truth values have to be nearly classic:
They can only be true or wrong and have to have the same truth value in all layers >=1.
So if LA is true in layer 1 it has to be true in all layers, so LA is false – in all layers, that would be a contradiction.
If LA is not true in layer 1 it has to be not true in all layers, so LA is true – in all layers, that would be a contradiction.
Therefore LA is not an allowed meta proposition in layer logic. — Trestone

If you are fine with disallowing statements in Layer Logic for which a truth value cannot be assigned, why not skip this added layer of complexity and disallow the original liar's statement in Classical Logic?

I see. With layer logic there is no self reference since layers cannot talk about themselves. The problem is that for all statements you've abandoned simple truth values and replaced them with infinite arrays of truth values. Does 1+1=2? Your answer would be [U,T,T,T,T,...]. Aside from being a bloated framework, your view rests upon the existence of actual infinities, 'numbers' which I have hesitations about.

I haven't read your work, but how would you handle the following pair of statements?
Statement A: Statement B is true.
Statement B: Statement A is false.

I don't see that either of us has said anything new in a long time. — fishfry

I agree with this. I need to study more to either accept that it's nonsense or find a way to better communicate it. Until then, we're just wasting our time. Let's not waste any more time. I really appreciate your patience sticking this out with me on this up until now. Thanks!

Depends on the space. In math there are metric spaces, topological spaces, measure spaces, probability spaces, Sobolov spaces, function spaces, and many many other kinds of things called spaces. So the answer is no, without further qualification or clarification. — fishfry

Okay, that makes sense.

What is a continuum? You ask if the real numbers can model a continuum and I don't know what the question means. The real numbers are commonly identified with "the continuum" but one can challenge that on philosophical grounds, hence the history of intuitionism etc. — fishfry

I mean continuum in the context of the geometrical objects of extension studied in elementary calculus, the objects that we typically describe using the cartesian coordinate system.

If you use the word computation it's a Turing machine by default unless you explicitly say otherwise. — fishfry

Okay, my mistake.

In the physical world? I have no idea and neither does anyone else. In math? There's a function f(t) = t that's 0 at time 0, 1 at time 1, and that passes through every intervening point. Or that passes through every intervening location where there could potentially be a point as aletheist noted. — fishfry

I'm talking about the mathematical world. The two sentences in this quote are quite different. The first sentence essentially states that it passes through infinite intervening points. The second sentence states that it passes through all intervening locations where there could be points. I actually agree with the second sentence.

f(t) = t. Or any of infinitely many other functions that have f(0) = a and f(1) = b. I don't follow why you're making a mountain of a mathematical molehill. Or what God has to do with any of this. — fishfry

What I'm trying to convey is that no matter where Atalanta's mathematical universe lives (whether in an infinite computer or the mind of God) it is impossible to construct Atalanta's journey from points because that would amount to listing the real numbers. The only way to build her universe is to deconstruct it from a continuum, working your way down from the big picture to specific instants.

When an engineer tries to solve Zeno's Paradox (of Achilles and the Tortoise) they ask questions about the system as a whole, specifically 'What are the speed functions of Achilles and the Tortoise from the beginning to the end of time?' With that information we don't have to advance forward in time, instant by instant. We just find where their two position functions intersect and conclude that Achilles passes the tortoise at that instant. And if this mathematical universe lives in that engineer's mind, that's the only actual instant that exists. Sure, the engineer could calculate their positions at other instants in time, but the engineer isn't going to calculate their positions at all times. That would be unnecessary...and impossible.

I'm sure you agree with the above paragraph (and perhaps are a little offended that I'm positioning it as the engineer's solution...hehe) but my point is that knowing a function doesn't imply that we can describe it completely using points. Any attempt to do so would be akin to listing the real numbers.
 

On the standard mathematical real line? Yes that's true. You think otherwise? But I don't need to use the philosophically loaded word points. I can say that between any two real numbers all that exists are other real numbers. You disagree in some sense? Be specific. — fishfry

I like when you earlier said 'every intervening location where there could potentially be a point'. It is worth creating a distinction between actual points and potential points. If we make that distinction, then I agree with you that there are only (actual and potential) points between a and b. What I would disagree with is the claim that there are only actual points between a and b. Actual points are discrete while potential points form a continuum. So instead of saying that there are finite actual points and infinite potential points between a and b, I think it is much better to say that there are finite actual points and finite continua between a and b. For example, in the image below, there are 3 actual points and 4 continua between 0 and 1.

You're confusing labeling with existence....I find your claim silly and not at all a serious argument or position. — fishfry

If we start with continua, the actual points only exist when we make a measurement. It seems like you agreed with aletheist on this. With a continuum-based view, when we make a measurement, we are not labeling points that existed all along, we are bringing them into existence (i.e. actualizing them). Until then they are potential points and can only be described as a part of a collection (i.e. a continuum), which I described using an interval. I am totally serious about this argument. My view is only silly when seen from a point-based view because you assume that all we can talk about are actual objects...an infinite number of them.

The zeros (points) is approached by infinite space that is also finite. — Gregory

I don't understand this statement.

Do you expect to be in the news as the guy who solved this ancient problem? As the mathematician said in video linked earlier in this discussion, greater minds than yours have wrestled with this problem and failed. — Gregory

I am probably wrong, but your statement isn't proof of that. Like I said to fryfish, my view has neither been formalized or proven correct. If I provided a formal theory and proved it correct then that would certainly be newsworthy, but as it stands I'm just looking for flaws in my ideas.

The rational numbers are infinitely divisible, but they are not complete. In addition to infinite divisibility, we need to require that every nonempty set of real numbers has a least upper bound. Otherwise the resulting system fails to be an adequate model of continuity. Just noting this for accuracy. — fishfry

I have concerns with infinite sets (including the set of all rational numbers) and real numbers, but those concerns are not essential to my argument. As such, I accept these two conditions.

No, I'd be very surprised if this turns out to be true. The mathematical real numbers are far too strange to be real in the sense of physical reality. — fishfry

Your comment was in response to me asking "Is it fair to assume that you believe that mathematical space can be modelled with the real numbers?" By mathematical space, I just mean continua and given that above you mention that real numbers can adequately model continuity, I will assume that you misread my question.

There are no computers involved in this. Computers are far too limited...So I reject this idea totally. — fishfry

I wasn't granting you a Turing machine, I was granting you an infinite computer with no restrictions. But fine, let's take algorithms out of the picture, and I'll grant you God and the Axiom of Choice. My only requirement is that everything God does must be consistent. Now, going back to my original question, how God move Atalanta from x=0 to x=1? As I mentioned in my last response, he can't advance her point by point because that would be equivalent to listing the real numbers, which is impossible. So how would he do it?

A function that passes through the point a at one time and b at a later time must necessarily pass through every intervening point. — fishfry

Your statement/position implies that all that exist between a and b are points. The way I see it is that the function must pass through the intervening spaces. So in the image below, to get from 0 to 1, the function must pass through the 4 continua represented by the following open intervals: $(0,\frac{1}{3}), (\frac{1}{3},\frac{1}{2}), (\frac{1}{2},\frac{3}{4}), \text{and} (\frac{3}{4},1)$. We both believe that the function cannot skip the intervening objects, I just believe that there are finite intervening objects and you believe that there are infinite intervening objects. I think this difference is what makes Zeno's Paradoxes a problem for the point-based view. Also, my view is not restricted to computable functions.

Just pointing out that you want to refer back to Aristotle as authoritative in some things but "in need of updating" in others. Cherry-picking Aristotle as it were. — fishfry

Oh come on, all I'm saying is that the idea didn't originate from me. I'm not claiming to be right on anyone's authority. You talk as if I must either agree with everything he taught or disagree with everything he taught. Sometimes things come back into fashion...like the mullet, right? That's due for a resurgence soon.

The parts are always there! — Gregory

If we take this reasoning to its limit then we end up with the whole being an assembly of a bunch of 0-Dimensional objects (i.e. nothing). Is that what you believe? Everything is made up of nothing?

That is what I mean when I say that the whole is real and the parts are entia rationis, creations of thought. — aletheist

No, I think that anyone who interprets the Planck length as a discrete constituent part of space is wrong. I interpret it instead as a limitation on the precision of measurement — aletheist

Yay!

As I have asked you several times, is this a mathematical or a physical scenario? — fishfry

I've mentioned QM and physics in an attempt to support my view, but the focus of my argument has always been on the mathematical scenario. From now on, unless explicitly stated otherwise, I'm talking about the mathematical scenario. I'm assuming continuity. The space that I'm talking about is infinitely divisible. There's no planck length/time to suggest any possible notion of discreteness.

In other words space is not described by the mathematical real line. As I've written in this thread at least ten times now. — fishfry

Are you referring to mathematical space or physical space? I assume in this context you're referring to physical space, right? Is it fair to assume that you believe that mathematical space can be modelled with the real numbers?

Mathematically there is no question that she passes through every point indexed by a real number....If this is a mathematical thought experiment, she does pass through every point in the closed unit interval [0,1]. — fishfry

A thought experiment is exactly what I'd like us to do. To perform an experiment, we cannot just say 'she passes through every point', we must actually conduct the experiment. The best way to do this is to envision that Atalanta lives in a simulation and we must understand how that simulation works. I will grant you a computer with no restrictions (e.g. infinite memory and speed). The only constraint is that the simulated universe must be consistent. How does the simulation allow her to pass through every point in the closed interval [0,1]?

One might say that this simulation can be easily performed in 1 second (e.g. the time interval [0,1]). There's a 1-to-1 correspondence between positions and instants in time, so for each instant the simulation outputs the matching coordinate (e.g. at 0.4 seconds, Atalanta is at x=0.4). But hold on. That would mean that the computer is effectively outputting a complete list of the real numbers between 0 and 1 and Cantor showed that such a list is impossible. For her to move, the simulation must be a lot more clever than that. Any ideas?

Aren't you the one saying you agree with Aristotle? He believed that the reason bowling balls fall down is that they're made of "stuff" and so is the earth, and like attracts like. Aristotle needs an update too. — fishfry

Aristotle was wrong on many things. We should not resuscitate his bad ideas without good reason...I just think that his ideas on potential infinity deserve another look now that our intuitions have been altered due to modern physics. (note: I'm talking about intuitions from physics providing further insight into longstanding paradoxes in the philosophy of mathematics, such as the mathematical Zeno's paradox.)

Sorry, I forgot to mention that this is Ryan O'Connor with a new account.

I've been arguing that to be infinitely divisible means that it has infinite parts. These seem identical to me — Gregory

These are not identical. A 'mathematical' string is infinitely divisible because I can cut it to no end. I end up with infinite parts only when done cutting it, but am I ever done cutting it?

My point is much of what QM says is false because they are clothing their findings in the language of an unprovable philosophy. They can predict things, but that is all QM can do. — Gregory

No doubt, QM is in dire need of philosophical progress. But we can't downplay the value of physics. Experiments and corresponding models ground our reasoning.

Is all space infinitely divisible? We can't conceive it as discrete, so our natural lights say yes. So your computer is infinitely packed. — Gregory

Infinitely divisible does not imply infinitely packed. We can cut a 'mathematical' string indefinitely but never will the string be in infinite pieces.

This is Ryan O'Connor under a new account. Hopefully, you'll be able to @mention me now.

Why not? Explain to me exactly why someone can't put one foot in front of the other and take a step. How does the mathematical theory of the real numbers preclude anyone from doing that? — fishfry

Are you suggesting that with each step someone sweeps over infinite points? In other words, are you suggesting that motion involves the completion of a supertask?

Why do you think the modern theory of the real numbers prevents anyone from walking? — fishfry

My problem is that the theory of real numbers implies that continua are made up solely of points. To me, this means that to move from x=0 to x=1, Atalanta must sweep over infinite points (she must complete a supertask). Instead of saying that there are infinite points, I'm saying that she only needs to sweep over a finite number of things. For example, in the image below, Atalanta needs to sweep over only 4 line segments. It’s a task that does not require any supernatural powers.

And -- a question that I keep asking you and that you never answer -- why does any mathematical theory have anything to do with physics? — fishfry

Math and physics are not necessarily related, but as you point out, we move all the time. I take that to mean that nature has found a way to avoid the traps of Zeno's Paradoxes in the physical universe. As such, nature may provide some clues on how to solve Zeno's Paradoxes in the mathematical universe. The cheap solution is to say that continua are discrete. I think nature has found a far more elegant solution.

I genuinely can not understand your point. How does the mathematical theory of the real numbers prevent anyone from taking a step? — fishfry

Let me try from a different angle. Let us say Atalanta lives in a perfect resolution simulation. As such, as she traverses from x=0 to x=1, she must at some instant in time be at each of the intermediate points. If we take out all the fancy graphics of the simulation, the computer is essentially just outputting a list of coordinates from 0 to 1. But we know that the real numbers are not listable (countable). There must be something wrong with the story. Do you agree?

I would say that when you draw the real number line, that's a visual depiction of the mathematical real line, which itself is an abstract object that cannot be depicted. — fishfry

I'm not trying to trick you by using pixels on your screen or atoms on my paper. Take my drawing and 'idealize it'. I know that a truly 1D object cannot be depicted (it takes up no area after all). We should move to 2D to make it depictable but you didn't like my earlier graphs so to keep things simple, let's just assume that we're looking at it with our mind's eye. This is a representation of how I see the idealized image:

I see 4 line segments with the numbers corresponding to the gaps in between the line segments. Points vanish upon idealization.

Fine. What of it. How would anyone's interpretation of that picture prevent them from walking or allow them to walk?......This is your thesis. Don't you see how silly it is?.......Before Dedekind had his clever idea, were people able to walk? And then the day he published, they couldn't? Isn't what you are saying patent nonsense? — fishfry

One can hold incorrect views on the laws of nature, but they will nevertheless follow the true laws. Similarly, one can hold incorrect views on the nature of continua which imply that motion is impossible, but they will still be able to walk. What I'm saying is that your views imply that motion is impossible. I'm not saying anything about your ability to walk.

According to the Copenhagen interpretation, we have no idea what happens when we're not looking. According to Many Worlds, everything happens. How is this relevant to the conversation? — fishfry

Many worlds provides a different interpretation on wavefunction collapse, but it still holds that in between quantum measurements what exists is a superposition, not a definite state. QM is only relevant if it informs our solution to Zeno's Paradox. I'm bringing it up because without QM my view sounds absurd. But you're right, I don't know the correct interpretation of QM. I also don't know what exactly entails a quantum measurement.

Has it occurred to you that perhaps you are not personally possessed of the ultimate truth about how the universe works? — fishfry

Of course. Truth comes out with formalized theories. I'm merely sharing an atypical intuition, which has neither been proven right or wrong. Ideas are cheap, including mine.

Now you're just being silly, since if you claim 1/3 is the smallest positive real number I'll just divide it by 2 (using the field axioms) and note that 0 < 1/6 < 1/3. Your claim stands refuted. — fishfry

No, I'm not being silly. 1/3 is the smallest non-zero number in this system.



In this system 1/6 is the smallest non-zero number.



Once we place another point on that line, we have a distinctly different system. This system is composed of 6 points and 5 line segments. In this system 1/6 is the smallest non-zero number. You have refuted nothing.

As I mentioned to Ryan, Zeno is solved mathematically by virtue of the theory of infinite convergent series — fishfry

As I mentioned to jgill, James Grime in the Numberphile video suggests that the solution kind of involves the completion of infinite tasks. Is this the view you hold?