It's not that it's complicated, but that scientific analysis generally takes place on a different level - that of the scientific analysis of objects, forces and energy. The question of the role of the observer is not complicated in that sense, but it's also not an objective question. That's why it evades scientific analysis - not that it's complicated or remote, but that it's 'too near for us to grasp'. — Wayfarer

The role of the observer in the quantum mechanical sense is complicated. What you are doing is simplifying the question by making some assumptions about what an observer can be. To use your previous reply to me, it is begging the question, as those assumptions are not necessary.

When you interact with others on the forum, you are not interacting with physical objects, but with subjects and their ideas. It is vastly different to how you interact with physical objects. — Wayfarer

First of all I am interacting with light from my computer screen, I am also interacting with the plastic keyboard. Ideas are not flowing directly from your mind to mine. I cannot directly tap into your consciousness. The last bit is not a realist position on my part - I cannot inhabit your consciousness the way I do mine regardless.

However I am assuming that there the ideas in the post by wayfarer come from a conscious being. But there is no good reason for me to assume this if I question whether the screen before me is real, and the light coming from it is real. It seems to me that wayfarer is also not real, just a projection of my consciousness - my consciousness is the only one that I experience specially.

That sounds mostly reasonable, but the branching part based on something making observations still bothers me a bit. What is the branching mechanism? Perhaps I should have started with that question instead. — Marchesk

Per MWI, branching is the process of a system becoming entangled with the environment (of which the observer or measuring apparatus is a part) such that interference between the different parts of the wave function no longer occurs, i.e., decoherence. That process is, for all practical purposes, irreversible. Whereas the entanglement between two microscopic systems is reversible.

To give a classical analogy, suppose you knocked over an empty glass that you quickly stood up again. The action was reversible. But suppose the glass falls on the floor and shatters into a thousand pieces. For all practical purposes, that's an irreversible action.

Scott Aaronson has a good lecture on this:

To see an interference pattern, you'd have to perform a joint measurement on the two qubits together. But what if the second qubit was a stray photon, which happened to pass through your experiment on its way to the Andromeda galaxy? Indeed, when you consider all the junk that might be entangling itself with your delicate experiment -- air molecules, cosmic rays, geothermal radiation ... well, whatever, I'm not an experimentalist -- it's as if the entire rest of the universe is constantly trying to "measure" your quantum state, and thereby force it to become classical! Sure, even if your quantum state does collapse (i.e. become entangled with the rest of the world), in principle you can still get the state back -- by gathering together all the particles in the universe that your state has become entangled with, and then reversing everything that's happened since the moment of collapse. That would be sort of like Pamela Anderson trying to regain her privacy, by tracking down every computer on Earth that might contain photos of her! — Decoherence and Hidden Variables - Scott Aaronson

"The computable numbers are countable since they be put in a one-to-one correspondence with the natural numbers."
— Andrew M
Not to disagree, but an assertion like that requires a demonstration that they’re countable. — noAxioms

Here's one such demonstration, concluding with:

The computable numbers are an infinite set. We have provided an injective function g that maps every computable number to a single natural number: a Godel number. Any set with such a function is countable, and therefore computable numbers are countable. — Alan Turing and the Countability of Computable Numbers - Adam A. Smith

"However the real numbers are not countable per Cantor's diagonalization proof. Thus there are some real numbers that are not computable."
— Andrew M
Interestingly, the real number generated by Cantor's diagonalization proof is a computable number, so I’m not sure if this counts as evidence that there are some real numbers not computable. Once again, not disagreeing with the conclusion, only with how it was reached. — noAxioms

It isn't a computable number (though it is a real number) - see the section entitled "A counter proof?" at the above link.

OK, they managed to test something whose outcome (the CHSH inequality violation) was already predicted by quantum theory. It’s a new test, but not one that changed the theory or any of its interpretations in any way. — noAxioms

Yes, it would be big news if standard quantum theory were ever falsified.

Thanks for the larger context Bell statement. I agree with it fully. What is ‘jumping’ in that quote? “Do we not have jumping then all the time?”. — noAxioms

He's referring to the collapse of the wave function (i.e., a discontinuous change in the otherwise continuous time evolution of the Schrodinger equation).

Meanwhile, I still don’t see what the AI in the box will do. Bell’s statement is pretty clear that a real human in there wouldn’t serve any special role or purpose, so why would an AI be any different? — noAxioms

Presumably it wouldn't. But an AI (unlike a human) could be run on a quantum computer as part of a carefully controlled experiment, thus testing physical collapse theories that differ from standard quantum theory.

"Let’s begin with a thought-experiment: Imagine that all life has vanished from the universe, but everything else is undisturbed. Matter is scattered about in space in the same way as it is now, there is sunlight, there are stars, planets and galaxies—but all of it is unseen. There is no human or animal eye to cast a glance at objects, hence nothing is discerned, recognized or even noticed."
— Charles Pinter, Mind and the Cosmic Order — Wayfarer

So far, so good.

"Objects in the unobserved universe have no shape, color or individual appearance, because shape and appearance are created by minds. Nor do they have features, because features correspond to categories of animal sensation. This is the way the early universe was before the emergence of life—and the way the present universe is outside the view of any observer."
— Charles Pinter, Mind and the Cosmic Order — Wayfarer

Pinter's asserted view of "the way the present universe is outside the view of any observer" is a performative contradiction. That's the problem with the so-called view from nowhere in a nutshell.

My model is Aristotle's hylomorphism. We interact with nature in our capacity as natural creatures. That's the relational perspective. So the moon is round, orbits the Earth and pre-existed life on Earth from a human point-of-view.

Pinter's asserted view of "the way the present universe is outside the view of any observer" is a performative contradiction. — Andrew M

I don't see that. It's a statement based on his knowledge of neural modelling, and supported by an important vein in philosophy, not least Kant's.

. So the moon is round, orbits the Earth and pre-existed life on Earth from a human point-of-view. — Andrew M

I quite agree, but again I don't think you're coming to grips with the point at issue. I think the problem that is highlighted by these debates IS the pretence of science to arrive at a form of perfectly objective knowledge independent of the role of the observer. That is what Nagel is criticizing in his book of that name, he's not advocating it as any kind of ideal.

Pinter's asserted view of "the way the present universe is outside the view of any observer" is a performative contradiction. That's the problem with the so-called view from nowhere in a nutshell. — Andrew M

The real problem here is with the notion of "the present universe". What Einstein reveals with the relativity of simultaneity is that "the present" is frame dependent. So the whole idea that there is such a thing as "the present universe" is an unsound premise because "the present" is something created by the observational perspective.

When we realize that "the present" is purely subjective, and we try to imagine an objective universe, independent from any observer, we have no place to insert "the present", because this would be an artificial insertion, therefore the creation of an observational perspective. Then we cannot possibly imagine such a universe, without a designated temporal perspective, (a point in time of now), because all things would exist everywhere, without some way of determining a specific point in time in their motions.

Interestingly, the real number generated by Cantor's diagonalization proof is a computable number, so I’m not sure if this counts as evidence that there are some real numbers not computable. Once again, not disagreeing with the conclusion, only with how it was reached.
— noAxioms
It isn't a computable number (though it is a real number) - see the section entitled "A counter proof?" at the above link. — Andrew M

I pondered over this for several days trying to understand the arguments. I still hold to what I said. The section you mention nicely shows that the x generated from the list of computable numbers is not itself a computable number, but I was speaking of the x generated from Cantor’s original proof of some real not being a rational number. That x is computable, but not rational, and thus cannot be used as evidence that there are some real numbers not computable.
The page you linked does show other ways to demonstrate exactly this, but the diagonalization method is not one of them.

Back to the quantum business:

What is ‘jumping’ in that quote? “Do we not have jumping then all the time?”.
— noAxioms
He's referring to the collapse of the wave function (i.e., a discontinuous change in the otherwise continuous time evolution of the Schrodinger equation).

Collapse seems to be a choice of classical description of a quantum state, in other words, an interpretation-dependent thing. In interpretations with ‘jumping’, yes, it happens all the time, everywhere. In interpretations without it (such as Everett’s relative state formulation, pre DeWitt’s MWI), it’s just a classical effect, not anything physical that happens.

Presumably [the AI in the box] wouldn't. But an AI (unlike a human) could be run on a quantum computer as part of a carefully controlled experiment, thus testing physical collapse theories that differ from standard quantum theory.

I have serious doubts about that. It is a suggestion that there is an empirical difference between the interpretations, and yet I see not explicit prediction from any pair of interpretations that differ.

Try this:[ tau.ac.il/education/muse/museum/galileo/principle_relativity.html ] — Metaphysician Undercover

You had to reach to Tel Aviv university to find a page closer to your definition?

Galileo formulated the principle of relativity in order to show that one cannot determine whether the earth revolves about the sun or the sun revolves about the earth. — tau

This is blatantly wrong. For one, the appearance of the sun revolving around the Earth once a day is not explained by the Earth revolving around the sun once a day any more than we’re revolving around all those other objects (moon, stars, etc) once a day. Secondly, the sun revolving around the Earth (once a year) violates basic Newtonian physics (lacking a reaction for the action of the sun). Newton’s laws were not in place back then so Galileo wouldn’t have known that.
Anyway, his pitch of principle of relativity used a boat’s relation to the water as the example, not celestial mechanics. The idea was that one could not tell from inside the boat whether the boat was moving relative to the water or not.
[quote-tau]The principle of relativity states that there is no physical way to differentiate between a body moving at a constant speed and an immobile body.[/quote]This definition is still reasonable, despite lack of reference to something relative to which the motion is defined. In short, there’s no way to differentiate an observer’s motion relative to some X if the observer cannot measure X. X is typically implied to be the ground (or water in Galileo’s case) beneath the observer, but it can be anything.
I accept the definition since it is more or less possible to derive either definition from the other, presuming at least that the X is explicit. It is the lack of that explicit reference that makes the definition ambiguous.

It is of course possible to determine that one body is moving relative to the other, but it is impossible to determine which of them is moving and which is immobile. — tau

This statement is especially ambiguous. Which of them is moving/immobile relative to what exactly? Humans tend to imply the ground since that’s their lifelong reference, but the implication is begging in this context.

I spend all my time just having to show you that you don't know what you're talking about.

Maybe you should try to actually understand what I’m talking about instead. I’m referencing far more reputable sources than are you. I’m pointing out explicitly what’s wrong with the pages you choose to quote. I don’t see you doing that with my references.

I do agree that perhaps your inability to understand an alternative to this absolutist view prevents communication between you and the rest of the scientific community. One doesn't have to accept the view with which you don't agree, but to not even understand it just leaves you behind.

This Article is intriguing. At first I thought they had found a way to reverse time in the quantum world, but rather they rejuvenated a photon, taking it back to a previous state.

The mathematics involved is probably linear (much is in the quantum world), since most non-linear systems are not reversible. Who knows? :chin:

You had to reach to Tel Aviv university to find a page closer to your definition? — noAxioms

That's a hell of a lot better than Wikipedia.

This is blatantly wrong. For one, the appearance of the sun revolving around the Earth once a day is not explained by the Earth revolving around the sun once a day any more than we’re revolving around all those other objects (moon, stars, etc) once a day. Secondly, the sun revolving around the Earth (once a year) violates basic Newtonian physics (lacking a reaction for the action of the sun). Newton’s laws were not in place back then so Galileo wouldn’t have known that.
Anyway, his pitch of principle of relativity used a boat’s relation to the water as the example, not celestial mechanics. The idea was that one could not tell from inside the boat whether the boat was moving relative to the water or not. — noAxioms

You clearly have not read any of Galileo's material, and continue to demonstrate that you do not know what you are talking about.

This statement is especially ambiguous. Which of them is moving/immobile relative to what exactly? — noAxioms

One body relative to the other body, is what is being discussed. Obviously, each is moving and neither is immobile. That it is impossible to determine that either one is immobile, yet possible to say that each is moving, implies that neither is immobile. And of course this becomes more obvious when the number of objects considered relative to each other is increased.

Humans tend to imply the ground since that’s their lifelong reference, but the implication is begging in this context. — noAxioms

Why do you incessantly resist and deny the point of the principle of relativity? The basic principle is that nothing is immobile (nothing is at rest). The principle of relativity renders the concept of "at rest" as obsolete. That is what allowed Newton to apply his first law of motion. The traditional concept of "at rest" which implied that a body had to be acted upon to be moved, is replaced with "uniform motion" by Newton, because by the principle of relativity "at rest" is not a valid concept. Then through extension of Newton's first law, a rest frame, or inertial frame, can be derived from any body displaying uniform motion because "uniform motion" is the concept which has take the place of "at rest".

I’m referencing far more reputable sources than are you. — noAxioms

Yeah Wikipedia, great source.

I pondered over this for several days trying to understand the arguments. I still hold to what I said. The section you mention nicely shows that the x generated from the list of computable numbers is not itself a computable number, but I was speaking of the x generated from Cantor’s original proof of some real not being a rational number. That x is computable, but not rational, and thus cannot be used as evidence that there are some real numbers not computable.
The page you linked does show other ways to demonstrate exactly this, but the diagonalization method is not one of them. — noAxioms

Cantor's proof (by contradiction) shows that the set of real numbers is uncountable and thus can't be enumerated. Since the set of real numbers can't be enumerated, the diagonalized number therefore can't be computed. A similar point is made by Carl Mummert (a professor of computing and mathematics) on Mathematics Stack Exchange.

Collapse seems to be a choice of classical description of a quantum state, in other words, an interpretation-dependent thing. In interpretations with ‘jumping’, yes, it happens all the time, everywhere. In interpretations without it (such as Everett’s relative state formulation, pre DeWitt’s MWI), it’s just a classical effect, not anything physical that happens. — noAxioms

Yes. The interpretation provides an account for how a measurement operationally returns a definite state when the formalism describes an indefinite state.

Copenhagen-style interpretations also generally deny a physical collapse. So, in that sense, Copenhagen and Everett/MWI agree (and disagree with physical collapse theories such as GRW).

"Presumably [the AI in the box] wouldn't. But an AI (unlike a human) could be run on a quantum computer as part of a carefully controlled experiment, thus testing physical collapse theories that differ from standard quantum theory."

I have serious doubts about that. It is a suggestion that there is an empirical difference between the interpretations, and yet I see not explicit prediction from any pair of interpretations that differ. — noAxioms

The empirical difference is between physical collapse theories such as GRW, and non-physical collapse interpretations (such as MWI and Copenhagen). From Wikipedia:

The fundamental idea is that the unitary evolution of the wave function describing the state of a quantum system is approximate. It works well for microscopic systems, but progressively loses its validity when the mass / complexity of the system increases.
...
Such deviations can potentially be detected in dedicated experiments, and efforts are increasing worldwide towards testing them. — Objective-collapse theory - Wikipedia

This Article is intriguing. At first I thought they had found a way to reverse time in the quantum world, but rather they rejuvenated a photon, taking it back to a previous state.

The mathematics involved is probably linear (much is in the quantum world), since most non-linear systems are not reversible. — jgill

Yes, that's right. Here's the paper for anyone else interested.

Cantor's proof (by contradiction) shows that the set of real numbers is uncountable and thus can't be enumerated. Since the set of real numbers can't be enumerated, the diagonalized number therefore can't be computed. A similar point is made by Carl Mummert (a professor of computing and mathematics) on Mathematics Stack Exchange. — Andrew M

What do you think this means, to assume numbers which cannot be counted nor computed? To me it's a form of unintelligibility, to say that there are numbers which cannot be accessed by us, or used in any way. Supposing that such a conclusion would be a logical consequence of the axioms assumed, then why would we assume axioms which produce the conclusion that there are numbers which cannot be accessed by us? As real numbers, being implied by our mathematical axioms, this would indicate that there are aspects of reality which we cannot access, grasp. apprehend, or understand in any way, which correspond with these numbers which cannot be accessed.

Since axioms are produced by mathematicians who practise pure mathematics, and those people who apply mathematics have a choice as to which axioms are used, it would appear like we ought not use axioms like these, which necessitate that aspects of reality will be unintelligible to us. Instead, we ought to look for axioms which would render all of reality as intelligible.

Yes, that's right. Here's the paper for anyone else interested. — Andrew M

There was a member here, active a couple years ago, I can't remember the name, but a self-proclaimed physicist who was big on this time reversal stuff. I think the fact that processes at the quantum level might be understood as reversible is simply a reflection of a fundamental misrepresentation of mass.

Since axioms are produced by mathematicians who practise pure mathematics, and those people who apply mathematics have a choice as to which axioms are used, it would appear like we ought not use axioms like these, which necessitate that aspects of reality will be unintelligible to us. Instead, we ought to look for axioms which would render all of reality as intelligible. — Metaphysician Undercover

Those who apply mathematics normally are not deeply conversant with its axiomatic structure. Most mathematicians - in the pure realm - are not either. It's dreadful stuff, but some treasure it. :cool:

There was a member here, active a couple years ago, I can't remember the name, but a self-proclaimed physicist who was big on this time reversal stuff. — Metaphysician Undercover

@Kenosha Kid. He was a Q-physicist who left the profession to play his guitar, as he explained to me. He liked Transactional physics.

Brian May is an example.

What do you think this means, to assume numbers which cannot be counted nor computed? — Metaphysician Undercover

Cantor's diagonal argument assumes that the set of real numbers are countable and then shows that that assumption leads to a contradiction.

Since axioms are produced by mathematicians who practise pure mathematics, and those people who apply mathematics have a choice as to which axioms are used, it would appear like we ought not use axioms like these, which necessitate that aspects of reality will be unintelligible to us. Instead, we ought to look for axioms which would render all of reality as intelligible. — Metaphysician Undercover

You may find that with computable numbers (which are countable):

The computable numbers include the specific real numbers which appear in practice, including all real algebraic numbers, as well as e, π, and many other transcendental numbers. Though the computable reals exhaust those reals we can calculate or approximate, the assumption that all reals are computable leads to substantially different conclusions about the real numbers. The question naturally arises of whether it is possible to dispose of the full set of reals and use computable numbers for all of mathematics. This idea is appealing from a constructivist point of view, and has been pursued by what Errett Bishop and Fred Richman call the Russian school of constructive mathematics. — Computable numbers - Use in place of the reals - Wikipedia

Also from physicist Max Tegmark:

I was seduced by infinity at an early age. Georg Cantor’s diagonality proof that some infinities are bigger than others mesmerized me, and his infinite hierarchy of infinities blew my mind. The assumption that something truly infinite exists in nature underlies every physics course I’ve ever taught at MIT — and, indeed, all of modern physics. But it’s an untested assumption, which begs the question: Is it actually true?

...

Yet real numbers, with their infinitely many decimals, have infested almost every nook and cranny of physics, from the strengths of electromagnetic fields to the wave functions of quantum mechanics. We describe even a single bit of quantum information (qubit) using two real numbers involving infinitely many decimals.

Not only do we lack evidence for the infinite but we don’t need the infinite to do physics. Our best computer simulations, accurately describing everything from the formation of galaxies to tomorrow’s weather to the masses of elementary particles, use only finite computer resources by treating everything as finite. So if we can do without infinity to figure out what happens next, surely nature can, too — in a way that’s more deep and elegant than the hacks we use for our computer simulations.

Our challenge as physicists is to discover this elegant way and the infinity-free equations describing it—the true laws of physics. To start this search in earnest, we need to question infinity. I’m betting that we also need to let go of it. — Infinity Is a Beautiful Concept – And It’s Ruining Physics - Max Tegmark

Thanks Andrew, I especially like this part:

The assumption that something truly infinite exists in nature underlies every physics course I’ve ever taught at MIT — Infinity Is a Beautiful Concept – And It’s Ruining Physics - Max Tegmark

This is completely opposed to what Aristotle presumed himself to have demonstrated, that it is impossible that anything actual is infinite. The closest he gets to an actual infinite, is "eternal", and he even proposes principles which demonstrate that anything eternal must be actual. This is interesting because it places "eternal" (actual) into a different category from "infinite" (potential). The result is that the conception of "eternal" which is derived from Aristotle's principles means roughly "outside of time", because the eternal thing cannot be infinite, being actual rather than potential. However, a commonly used meaning of "eternal" is "infinite time". This is incoherent by Aristotle's principles.

He was a Q-physicist who left the profession to play his guitar, as he explained to me. — jgill

So, why not do both? And participate here in his spare time.

His last participation was about a year ago. He had made some comment about pursuing a "Pot of Gold" with his guitar. :cool:

Our challenge as physicists is to discover this elegant way and the infinity-free equations describing it — Infinity Is a Beautiful Concept – And It’s Ruining Physics - Max Tegmark

Lots of mathematicians don't use the infinity symbol except in a limit sense. The idea of some entity called infinity existing doesn't normally crop up in classical analysis, for example. Tegmark cites the expression $\frac{\infty }{\infty }$, but elementary calculus shows ways around that (in very elementary settings), stating the expression is meaningless. If I were to say $\underset{n\to \infty }{\mathop{\lim }}\,f(n)=\infty$, I only mean that as n increases so does the function, with no upper bound. I don't mean it ultimately ends up at a magical point at infinity.

All this is very simple. I've read that Hilbert spaces used in QM must be infinite in order to preserve continuity and completeness in some circumstances. But there are ways of dealing with these concepts in constructive mathematics. I suspect the noise made by math foundationalists regarding cardinalities and "squeezing in" numbers in R by forcing and such things might make physicists uneasy - if they even care. I don't. :cool:

Enough time spent with the guitar could lead to a resolution of the Fourier uncertainty.

I only mean that as n increases so does the function, with no upper bound. I don't mean it ultimately ends up at a magical point at infinity. — jgill

The problem as I understand it is if you posit the lower limit of zero, to a duration of time, this model negates as impossible, the possibility of a real quantum of time. Furthermore, in analyzing any change (force), relative to what is uniform (inertial), as we narrow down the duration of time, toward any supposed point in time, where the change begins, the magnitude of force required to produce the change approaches infinite. Time is inversely proportional to energy, so as the temporal duration assumed for the force becomes close to zero, the quantity of energy becomes close to infinite. Therefore placing a limit to time at the speed of light, inserts the infinite into calculations which relate mass to energy.

This problem was first expressed by Zeno, and though some mathematicians might claim that calculus resolved the problem with the use of limits, it just provided a practical work-around which was suitable for the practises of the time. In modern times engineers employ extremely short periods of time, so the problem naturally reappears despite the efforts of calculus. Another proposal by philosophers and mathematicians, has been to use infinitesimals. Infinitesimals allow that the apparent infinite change occurs within a non-zero duration of time (infinitesimal), but the actual change within that infinitesimal duration cannot be represented. In other words, each infinitesimal quantum of time may contain an unrepresentable change.

The problem though, is pretty much how Zeno represented it. The ideal (theoretical) representation of time and space is as continuous. However, our descriptions of how things move in real practise represent discrete units of space and time. So there is a fundamental incompatibility between how we describe our observations of things in practise, and how we explain or interpret our observations through mathematical theory. There has been some attempts to make the mathematical theory of continuity consistent with the observations of discrete units (quanta), but the proposals employed for mathematizing quanta are completely ideal, not based in any empirical principles of real discrete quanta of space or time. Until we take notice of the reality of how space and time are actually quantized in real discrete units, these attempts, such as limits and infinitesimals, will remain ideals of theory which do not adequately represent the quanta of reality.

For an analogy, consider how the ancient people modeled the orbits of the planets as eternal circular motions (perfect circles which by that perfection are eternal) which are described by Aristotle. This was the ideal which was employed, but it did not adequately represent reality. Within the practise of modeling the orbits, exceptions (retrogrades etc.) were incorporated to account for the fact that the reality was not as the ideal represented it. Until it is fully acknowledged, and recognized, that reality is not as the ideal (as indicated by the need for exceptions), people just do not get motivated to determine the real representation.

Intuitively, Smooth infinitesimal analysis can be interpreted as describing a world in which lines are made out of infinitesimally small segments, not out of points. These segments can be thought of as being long enough to have a definite direction, but not long enough to be curved.

Until we take notice of the reality of how space and time are actually quantized in real discrete units, these attempts, such as limits and infinitesimals, will remain ideals of theory which do not adequately represent the quanta of reality. — Metaphysician Undercover

Perhaps space and time are not "actually quantized in real discrete units".

There was a member here, active a couple years ago, I can't remember the name, but a self-proclaimed physicist who was big on this time reversal stuff. — Metaphysician Undercover

Maybe he’ll come back in the past. ;-)

Maybe he’ll come back in the past. ;-) — Wayfarer

I think I should revisit some old posts, and see if anything has mysteriously changed. If so, I'd be seriously spooked.

Perhaps space and time are not "actually quantized in real discrete units". — jgill

This is a respectable proposition, but the problems involved in applying mathematics (Zeno's to begin with) demonstrates otherwise. Let me explain, starting with some fundamental principles:

First principle, divisibility is very real, we can physically divide things, therefore division of space or time must be real.
Second principle, if space or time is divisible, and there is nothing real which restricts or limits the divisibility, then it is infinitely divisible.
Third principle, if there is something real which restricts or limits the divisibility of space or time, then these restrictions provide the basic premise for real quantized discrete units.

What has been evident ever since the time of Zeno (and probably even before this formalization, from Pythagoras), is that the second principle (stated above) is problematic. Representing our capacity to divide things with mathematics that provide for infinite divisibility creates various irrational ratios (Pythagoras), and paradoxes (Zeno). The mathematical axioms assume a continuity which is infinitely divisible. However, it can be demonstrated in theory (Pythagoras and Zeno), that these axioms will inevitably lead to problems in application. The conclusion we can draw, or which I would say we ought to draw, is that this idea, of infinite divisibility, is just an ideal which does not truly represent the nature of reality. And this is very intuitive too, because space and time are our fundamental representations for how the world really is, and if you think about it, it doesn't make sense to think that we could keep on dividing the real world, in the very same way, into smaller and smaller bits, forever. It's just not realistic, because this would imply no real substance to the world.

This is why the ancient Greeks proposed a fundamental indivisible base, "atoms", or "matter", to provide substance for our principles of divisibility. So this brings us to the third principle, "substance" is the real thing which limits and restricts our capacity for making divisions in the real world. The conclusion therefore is that we need to understand how "substance" limits and restricts our capacity for division, and formulate our axioms of mathematics to properly represent these restrictions. Then the numbering system which we employ will properly represent the world's real capacity for division.

For example, look at the principles for dividing the octave in music, the basic principles of "scale". Distinct tones, as specific frequencies, are supposed as the substance around which the divisions are made, and the scale is produced. But no tone is absolutely clean, crisp and clear, so there are complexities, overtones, timbre, etc.. Also, the scale of tones must be created so as to be useful in combination for the production of harmony, so "the scale" is intended to relate a multitude of octaves.

Now in dividing the octave into specific tones, to produce a scale, we must also consider what happens when we multiply, because this is how harmony is created, and that is just as important to the scale as the division into tones. Therefore proper, or true division, is not a straight forward act of dividing in any way that the physical world will allow, good proper division practises must be conditioned by the reality of what may be produced through the re-combination of the parts which are divided for. We could say that this is the role of "the good" in the practise of division, or analysis, that the things produced in division are a true representation of the actual parts which went into the production of the whole, rather than a random division. Like for example, cutting with a knife slices through the cells, providing a glimpse inside, but not providing a good principle of division.

So we divide the octave into parts, and produce a scale of individual tones or frequencies. But there is a possibility for random division here, and this would not produce a good scale. So we must keep in mind what will be created from those parts, harmonies, and we must create the scale accordingly. Basing "the scale" in good principles of harmony is what produces consistency in the production of a conceptual multi-leveled spatiotemporal reality. That is the need to represent how the various parts create a harmonic whole, as the reality of substance. In other words, "substance" in its real existence, is a multi-layered harmony through the micro/macro range, so that any production of a base scale must be consistent with intermediate scales, and upper scales, in the way of harmony, so that "substance" is properly formulated.

The mathematical axioms assume a continuity which is infinitely divisible. However, it can be demonstrated in theory (Pythagoras and Zeno), that these axioms will inevitably lead to problems in application. The conclusion we can draw, or which I would say we ought to draw, is that this idea, of infinite divisibility, is just an ideal which does not truly represent the nature of reality. — Metaphysician Undercover

I'm not sure what "problems in application" you refer to since in most applications - if not all - one works with rational or computable numbers. I can pretty much guarantee that Feynman's path integral is a computational problem.

However, you have written a compelling philosophical case for quanta time and space. I don't know about the set theory axioms that should be replaced. Beyond my sphere of interests. Wikipedia says this:

In mathematical physics, the concept of quantum spacetime is a generalization of the usual concept of spacetime in which some variables that ordinarily commute are assumed not to commute and form a different Lie algebra. The choice of that algebra still varies from theory to theory. As a result of this change some variables that are usually continuous may become discrete. Often only such discrete variables are called "quantized"; usage varies.

and

Physical spacetime is a quantum spacetime when in quantum mechanics position and momentum variables x , p x,p are already noncommutative, obey the Heisenberg uncertainty principle, and are continuous. Because of the Heisenberg uncertainty relations, greater energy is needed to probe smaller distances. Ultimately, according to gravity theory, the probing particles form black holes that destroy what was to be measured. The process cannot be repeated, so it cannot be counted as a measurement. This limited measurability led many to expect that our usual picture of continuous commutative spacetime breaks down at Planck scale distances, if not sooner.

It's a very technical subject that bears a slight resemblance to "not turtles all the way down".

Cantor's proof (by contradiction) shows that the set of real numbers is uncountable and thus can't be enumerated. Since the set of real numbers can't be enumerated, the diagonalized number therefore can't be computed. — Andrew M

But that number (from Cantor’s proof) is generated from a countable list of rationals, not an uncountable list of reals. So it can be computed. It doesn’t require the ordering of the reals. That was my point,.

Copenhagen-style interpretations also generally deny a physical collapse. So, in that sense, Copenhagen and Everett/MWI agree (and disagree with physical collapse theories such as GRW).

I am not really clear on what a formal statement of metaphysical Copenhagen interpretation would say. I’m more familiar of its roots as an epistemological interpretation where collapse (of what is known) very much does occur, but it is just a change in what is known about a system, not a physical change. They’ve since created a not-particularly well defined metaphysical interpretation under the same name, and if it doesn’t suggest physical collapse, I’d accept that statement.

The empirical difference is between physical collapse theories such as GRW, and non-physical collapse interpretations (such as MWI and Copenhagen). From Wikipedia:

Such deviations can potentially be detected in dedicated experiments, and efforts are increasing worldwide towards testing them. — wiki

Cool. Interesting to watch then. Thanks.

One body relative to the other body, is what is being discussed. Obviously, each is moving and neither is immobile. — Metaphysician Undercover

Totally agree with this, but it renders meaningless a statement about a single body in the absence of something relative to which motion can be defined.

The principle of relativity states that there is no physical way to differentiate between a body moving at a constant speed and an immobile body.

If there are two bodies with relative motion, then per the definition of motion, both are moving relative to the other, and the differentiation can easily be made by measurement. If there is but the one body, then motion isn’t defined at all.

Why do you incessantly resist and deny the point of the principle of relativity?

I don’t. I said in my prior post that I could accept it, despite the non-relative wording of it.

The basic principle is that nothing is immobile (nothing is at rest).

If what is being discussed is one body relative to the other body, your choice of wording leaves the second entity unspecified, merely implied, like there’s some embarrassment about it. So say it. Relative to what is nothing immobile?

The principle of relativity renders the concept of "at rest" as obsolete.

And here I thought it was the definition of motion that did that. The principle of relativity seems to still hold even if you discard the relative definition of motion, and Einstein’s theories along with it.

… because by the principle of relativity "at rest" is not a valid concept.

If the PoR makes the concept of ‘at rest’ invalid, why does it (or at least the version of PoR that you prefer) reference it?
My apologies for hanging on this point so much, but you seem to contradict yourself regularly, saying that the concept is invalid, but regularly referencing the invalid concept nonetheless. I personally don’t find the concept invalid at all. It’s just a totally different set of definitions with totally different physics than what Einstein proposes. I don’t prefer these alternate definitions, but I cannot prove them wrong.

Then through extension of Newton's first law, a rest frame, or inertial frame, can be derived from any body displaying uniform motion because "uniform motion" is the concept which has take the place of "at rest".

Anyway, I will accept this as well. You don’t seem to be pushing the alternate definitions. In Minkowskian spacetime, a rest frame can be any arbitrarily selected frame and the is the implied second object relative to which motion is defined. The selected frame is an inertial one if Newton’s laws of motion hold in it, but other frames (with different laws) are also allowed.
That said, I don’t know why nothing can be at rest in it. I do realize that I cannot name a single object which isn’t accelerating at least in the coordinate sense, so nothing seems to be stationary for any extended period of time. I say coordinate sense since an object in free fall (Earth say), except for spinning in place, is not particularly acted on by significant forces and thus is effectively stationary in its local inertial frame (it traces a geodesic through spacetime). This is a departure from Newton’s physics where gravity is a force and causes paths to curve.

What do you think this means, to assume numbers which cannot be counted nor computed? To me it's a form of unintelligibility, to say that there are numbers which cannot be accessed by us, or used in any way. — Metaphysician Undercover

I don’t find it unintelligible, but I do find fascinating the complete inaccessibility of such numbers to us. The vast majority of real values are these inexpressible values. I gave a few examples of them, especially ones that don’t require units which themselves are only approximately defined.
It seems that they can be measured to some precision, but those approximations are never the number itself. Hubble’s ‘constant’ (which unfortunately does have units) is one such number, and one that is known to not even an entire significant digit. I put ‘constant’ in quotes because the value changes over time.

Totally agree with this, but it renders meaningless a statement about a single body in the absence of something relative to which motion can be defined. — noAxioms

A statement about a single body is not completely "meaningless", because we can still state properties of the body itself, and this is meaningful. Predication is actually very meaningful. Now, you should consider the distinction between internal and external properties, a distinction which has been quite meaningful in classical philosophy. It seems like the science of physics has no real principles to distinguish these two, the difference between internal change (difference of properties) and external change (locomotion or change of place), and physics wants to reduce, and treat all change as change of place. Internal change would be change of place of internal parts.

The problem with this perspective arises when we get down to the fundamental parts. As I described above in a prior post, there must be base indivisible parts, or else the universe is fundamentally without substance. That's the principle employed by the ancient Greek atomists. If internal change is described exclusively through change of place to internal parts, then internal change to fundamental indivisible parts is ruled out as impossible. Yet if fundamental parts demonstrate internal change, such change is rendered as unintelligible, unless we allow for further division. Then fundamental parts cannot be found unless they are observed to be unchangeable.

If what is being discussed is one body relative to the other body, your choice of wording leaves the second entity unspecified, merely implied, like there’s some embarrassment about it. So say it. Relative to what is nothing immobile? — noAxioms

Relative to the principle of relativity, nothing is immobile. That is, if we take the principle of relativity as our primary premise, and add the observational premise that there are numerous bodies observed to be moving in different ways, we can conclude that noting is immobile. That is the fundamental conclusion, (assumption), which the principle of relativity give us, there is no such thing as "rest" in any truthful, grounded sense.

And here I thought it was the definition of motion that did that. The principle of relativity seems to still hold even if you discard the relative definition of motion, and Einstein’s theories along with it. — noAxioms

Yes, "motion" as defined by the principle of relativity is what renders the concept of "at rest" as obsolete, but only as defined by the principle of relativity. Prior to Galileo's development of the principle of relativity, "at rest" was a valid concept referring to one's position on earth, and motion was defined relative to any location on earth, the earth being "at rest". And, the locations on earth were supposed to be at rest relative to each other. The principle of relativity makes rest a property derived only from locations which are at rest relative to one another (lack of internal change), but these locations are not at rest in the wider context (external change, or change of place). So when motion is defined as change of place relative to what is external to a designated location (the principle of relativity), there is no such thing as "at rest".

If the PoR makes the concept of ‘at rest’ invalid, why does it (or at least the version of PoR that you prefer) reference it? — noAxioms

The principle of relativity creates an artificial, and arbitrary definition of "rest", as "the rest frame". It is not a real rest, because it is moving relative to its environment (external change). So we can designate a number of points, as locations, and say that they are not moving relative to each other (no internal change), and claim to have a rest frame. But this is not a true or real rest, as they move relative to external things. And, as the principles of Einstein's special relativity (time dilation, length contraction) demonstrate to us, there is not even any true internal rest here. We must overlook certain discrepancies to say that they are at rest relative to each other. The locations actually change relative to each other due to the effects of external change, acceleration. Therefore even the internal "rest", or lack of internal change, which is associated with the rest frame, is not a valid "rest".

My apologies for hanging on this point so much, but you seem to contradict yourself regularly, saying that the concept is invalid, but regularly referencing the invalid concept nonetheless. I personally don’t find the concept invalid at all. It’s just a totally different set of definitions with totally different physics than what Einstein proposes. I don’t prefer these alternate definitions, but I cannot prove them wrong. — noAxioms

If you can show me a valid concept of rest, lack of change, which can be maintained consistently along with the principle of relativity as well, I would appreciate the demonstration. Even if we accept locations to be "at rest" relative to each other in an internal way, and deny Einsteinian relativity, these locations are still not at rest in the wider (external) context. And when we start mapping the points which are supposed to be at rest relative to each other, in comparison with external things, we inevitably find minor inconsistencies which cannot be resolved, as demonstrated by Einstein's train example. So we do not have the evidence to adequately support any claim that any multitude of points of location in the material world are actually at rest relative to each other. The need for "spatial expansion" helps to demonstrate the reality of this point.

Cantor's proof (by contradiction) shows that the set of real numbers is uncountable and thus can't be enumerated. Since the set of real numbers can't be enumerated, the diagonalized number therefore can't be computed.
— Andrew M

But that number (from Cantor’s proof) is generated from a countable list of rationals, not an uncountable list of reals. So it can be computed. It doesn’t require the ordering of the reals. That was my point,. — noAxioms

Cantor's proof assumes an enumeration of the set of real numbers (any enumeration, not just an ordered one), and then proves that there can't be one. So it's therefore not possible to compute the diagonal for the set of real numbers. The specific diagonal that is computed as part of the construction is necessarily for a different set of numbers (say, the countable list of rationals).

If it were the diagonal for the set of real numbers, then it would appear somewhere on the list, say, the i^th index. But then to calculate the number at that index to the i^th digit would result in an infinite loop.

Copenhagen-style interpretations also generally deny a physical collapse. So, in that sense, Copenhagen and Everett/MWI agree (and disagree with physical collapse theories such as GRW).
— Andrew M

I am not really clear on what a formal statement of metaphysical Copenhagen interpretation would say. I’m more familiar of its roots as an epistemological interpretation where collapse (of what is known) very much does occur, but it is just a change in what is known about a system, not a physical change. They’ve since created a not-particularly well defined metaphysical interpretation under the same name, and if it doesn’t suggest physical collapse, I’d accept that statement. — noAxioms

:up:

Some quotes to that effect here.

I'd be interested if you could see what I was driving at in this OP on Physics forum particularly #11 and #14. I was trying to argue that the probability wave is outside of space and time, although (probably predictably) the physicist who responded thought this 'gobbledegook' and preferred the (I think inane) dogma that 'particles interfere with themselves'. I don't know enough physics to really articulate the intuition I have about it.

I was trying to argue that the probability wave is outside of space and time... — Wayfarer

Space and time are conceptual. So "outside of space and time", simply means unable to be apprehended through our current conceptions of space and time. This issue is very much analogous to the discussion above, of numbers which our conception of numbers reveals to us as needed to be accounted for, but they lie outside of our capacity to apprehend them. What I argued above, is that this indicates a fundamental flaw in our conception of numbers, that this conception produces numbers which cannot be apprehended by us.

Yes I think I see what you're getting at.

So whether they're discharged one electron at a time, or at a faster (or is that 'higher'?) rate, then you still get the same pattern.

Yes, the pattern will be the same in both cases.

The fact that the effect can't be replicated by a physical (water) wave is, I think, due to the interference pattern not actually being 'waves' as such, but something for which the interference patterns of waves is just an analogy.

I think it's deeper than just an analogy. The wave dynamics are central to the math.

The argument that started this was about whether this means that time (being 'rate') is not a factor; which also that means that space (i.e. proximity of particles) is not a factor (as proximity is an aspect of space-time.) So, what is causing the interference pattern is outside, or not a function of, space-time.

Deux ex machina?

Thanks for taking the time to look. I had the idea that the wave function literally is ‘the degrees of probability’. The reason that seems daft to many people is that there is no concept of ‘degrees of reality’ - usually it is assumed that nothing can be more or less real. But that is the point of this article.

This new ontological picture requires that we expand our concept of ‘what is real’ to include an extraspatiotemporal domain of quantum possibility,” write Ruth Kastner, Stuart Kauffman and Michael Epperson.

Considering potential things to be real is not exactly a new idea, as it was a central aspect of the philosophy of Aristotle, 24 centuries ago. An acorn has the potential to become a tree; a tree has the potential to become a wooden table. Even applying this idea to quantum physics isn’t new. Werner Heisenberg, the quantum pioneer famous for his uncertainty principle, considered his quantum math to describe potential outcomes of measurements of which one would become the actual result. The quantum concept of a “probability wave,” describing the likelihood of different possible outcomes of a measurement, was a quantitative version of Aristotle’s potential, Heisenberg wrote in his well-known 1958 book Physics and Philosophy. “It introduced something standing in the middle between the idea of an event and the actual event, a strange kind of physical reality just in the middle between possibility and reality.”