And I generally agree with Aaronson’s descriptive categories. I am very impressed with Christian Fuchs’ philosophy of QBism which I guess puts me in the second category. Deutsch et al seem to want to preserve the principle of objectivity above all else. — Wayfarer

From the MWI side, the claim is that QBism is solipsist and empty of content:

A good friend of mine, Jacques Pienaar, has recently converted to QBism, as often happens to people that spend too much time around Chris Fuchs. Saddened by these news, I’ve decided to write a blog post explaining why QBism doesn’t contribute anything to understanding Nature. On the contrary, it is a step backwards, as it doesn’t accept the basic premise that there is a world out there and that our job is to understand it. QBism insists that all the puzzles and paradoxes of quantum mechanics don’t actually matter because it’s all subjective anyway. — Why QBism is completely empty - Mateus Araújo

The post is of particular interest because Araújo and Pienaar (both physicists active in Quantum Foundations) have a great exchange in the comments, with Pienaar providing a useful summary of QBism. As a bonus:

“Get out of agents’ heads?” So you think that `experience’ happens inside agents’ heads: you’re stuck in a Cartesian worldview. For you, subjective means `of the mind’, and objective means `of the mind-independent world’. To be fair, this is what most physicists would understand by those words, having been brought up in the Cartesian way of thinking. And on these terms, I think it would be fair to call QBism solipsist. But that would [be] missing the point, because QBism explicitly says that is not how we should think about experience, objectivity, and subjectivity. — Comment by Jacques Pienaar

Pienaar highlights one of the most important issues affecting philosophy, which is Cartesian Dualism. So it's great to see the issue getting some airtime. The issue is manifest in the varied uses of terms such as "observer", "observer effect" and "measurement", including in this thread.

I believe that the notion of a 'passive' observer is actually inconsistent with relativistic physics. Passive observation must be from a "rest frame', or else it cannot be said to be "passive". — Metaphysician Undercover

I understand your point, however the specific contrast being made was between an observer qua reference frame and an observer qua rational agent. Only the latter can be understood as a user of quantum theory, and thus active in that sense.

Unfortunately, I was not able to easily follow the rest of your post. Perhaps you could concisely state your claim and quote specifically from SEP what you're arguing for (or against).

Bohr, Heisenberg, Schrödinger, Pauli, and others of that generation were deeply cultured individuals with deep knowledge of philosophy both Eastern and Western. The ‘shut up and calculate’ generation were mainly Americans after WW2. Totally different mindset. — Wayfarer

Yes, they were actually shutting up. :-)

As for Copenhagen, I’ve described it as “shut-up and calculate except without ever shutting up about it”! I regard Bohr’s writings on the subject as barely comprehensible, and Copenhagen as less of an interpretation than a self-conscious anti-interpretation: a studied refusal to offer any account of the actual constituents of the world, and—most of all—an insistence that if you insist on such an account, then that just proves that you cling naïvely to a classical worldview, and haven’t grasped the enormity of the quantum revolution.
...
But the basic split between Many-Worlds and Copenhagen (or better: between Many-Worlds and “shut-up-and-calculate” / “QM needs no interpretation” / etc.), I regard as coming from two fundamentally different conceptions of what a scientific theory is supposed to do for you. Is it supposed to posit an objective state for the universe, or be only a tool that you use to organize your experiences?
...
This is not like picking a race horse, in the sense that there might be no future discovery or event that will tell us who was closer to the truth. I regard it as conceivable that superintelligent AIs will still argue about the interpretation of QM … or maybe that God and the angels argue about it now. — Interpretive cards (MWI, Bohm, Copenhagen: collect ’em all) - Scott Aaronson

Deutsch is a very imaginative individual, and this whole many-worlds idea appeals on that level. But the Copenhagen attitude is much more modest, in my view. May not be as exciting. — Wayfarer

Yes, Copenhagen can be understood as the operational interpretation - shut-up-and-calculate. But without the shutting up, as Scott Aaronson likes to say!

Apropos:

Imagination will often carry us to worlds that never were. But without it we go nowhere. — Carl Sagan (Cosmos) [via Nurgalieva and Renner]

Analysis of this 'observer effect' can guide us in judging one observational perspective as better than the other (Copernican is better than Ptolemaic). — Metaphysician Undercover

Yes, and excellent post MU.

To take a bit of the woo out of QM, instead one should speak of the "measurement effect", which alludes to the machinery doing that task, not the scientist, Gandolf. — jgill

It's a can of worms - "measurement" is also a contested term. I think Nurgalieva and Renner have some useful comments on this:

2 Observers in quantum mechanics

The notion of an observation is crucial for linking the theoretical formalism of quantum theory to experiment, and in this sense to physical reality. In an experiment, an observation is the outcome of a measurement that is carried out by an observer, which may be a device or a human. How this observer should be treated in theory is however debated. Is the observer just an ordinary system that can itself be described by quantum theory, or should it be regarded as something external to it, or is it even both? The answer to this question exhibits some of the key differences between the different interpretations of quantum theory, and we thus discuss it in more detail in this section.

The notion of an observer gained importance in physics already before the development of quantum theory. One of its most prominent appearances is in Einstein’s original article on special relativity [27]. Here the “Beobachter” plays a key role, for quantities such as time and spatial locations are only defined relative to them. Special and general relativity still portray the observer in a passive light — the observer is understood as a reference frame, thus giving rise to a coordinate system in spacetime [28, 29]. Other theories view observers as users of the theory. For example, Bayesian statistics [30, 31] consists of a set of reasoning rules that a rational agent should follow to make predictions and decide on future actions based on past observations. A Bayesian observer thus takes a more active role. In quantum theory, both of these roles may be relevant, depending on the interpretation. We will therefore use the terms agent and observer interchangeably.

...

While most physicists today don’t resort to the (vague) notion of consciousness,4 they still haven’t reached an agreement on how to resolve the paradox around Wigner’s friend. The ambiguity is not within the formalism of quantum theory, but lies in how the formalism should be applied to such experiments — a question that the formalism itself cannot answer. In their search for answers, quantum physicists came up with a variety of ideas and proposals, which are nowadays known as different interpretations of quantum theory (see Table 1 for a summary and Figure 3 for an illustration). — Testing quantum theory with thought experiments, p4-5 - Nurgalieva and Renner

if we had a quantum computer on which an artificial-intelligence program was running, say, with human level artificial-intelligence then this entity would be able to experience interference in its own consciousness.
— Are There Many Worlds? David Deutsch in conversation with Markus Arndt

You think that is remotely close to what Popper would consider 'falsifiable by empirical evidence'?

‘When we create an artificial human using technology that doesn’t exist yet, all will be revealed!’ — Wayfarer

I put the question to ChatGPT:

Me: Would Popper consider the Wigner-Deutsch quantum experiment to be falsifiable by empirical evidence?

ChatGPT: According to Popper's criterion of falsifiability, a scientific theory must be able to make testable predictions that could, in principle, be shown to be false. The Wigner-Deutsch quantum experiment is a proposed thought experiment in quantum mechanics, which is based on the idea of creating a quantum system that can perform a universal set of quantum gates, a set of operations that can be used to perform any quantum computation. The experiment proposes that if such a system could be created, it would be possible to demonstrate the existence of quantum parallelism and quantum entanglement in a more direct way than is currently possible.

The Wigner-Deutsch quantum experiment is a theoretical concept, and it has not been physically implemented yet. While the experiment is purely theoretical, it's still an interesting concept and it could be considered as scientific by Popper's standard. Popper would argue that the thought experiment is falsifiable, because it makes predictions that can be in principle tested, if the technology and resources become available. If the experiment is performed and it fails to demonstrate the existence of quantum parallelism and quantum entanglement, it would falsify the thought experiment.

In summary, Popper would consider the Wigner-Deutsch quantum experiment as a scientific theory, because it makes testable predictions that could in principle be shown to be false, even though it's not been physically implemented yet.

It would be better to simply recognise there are things science is unable to ascertain and leave it at that. As a general rule, knowing you don’t know something is preferable to thinking you know something that you don’t. — Wayfarer

We don't know that science is unable to ascertain it. Note that Deutsch follows Popper. MWI is a conjecture, and the quantum AI experiment that he describes is a test of that conjecture.

Totally agree here, but I think the effect with which MWI has trouble explaining is the Born rule. It's been a strong piece of criticism. — noAxioms

Yes, though it's not a problem unique to MWI. It's a bit like asking why the Pythagorean Theorem holds (and why Fermat's equation doesn't, for powers greater than 2). The answer is of general interest independent of one's interpretation.

If you one of the types that actually think that MWI implies the creation/generation of ontologically separate worlds, then one also has to deal with how some of them are more probable than others, that meaning must be given to "X exists more than Y". That issue goes hand in hand with the Born rule thing. — noAxioms

You might be interested in Sean Carroll's post on Why Probability in Quantum Mechanics is Given by the Wave Function Squared.

Per your comment, "one also has to deal with how some of them are more probable than others", the basic idea (from Zurek - see the above post) is that paths that are not equally probable can be mathematically reduced to paths that are. For example, a beam splitter with a 2:1 transmission/reflection ratio is equivalent to a beam splitter with a 1:1:1 ratio once a 1:1 beam splitter is added to the transmission path.

There's an 'observer effect' in Einsteinian relativity which nobody objects to. That's not the problem.
— Andrew M
There is? There are dependencies on frames (what velocity has object X?, a completely frame dependent question since Galileo), but I've not heard that observers have any effect at all. That seems to be confined to QM effects. — noAxioms

Yes, I'm referring to frame dependency. The observer on the train platform describes events differently to a passenger in the moving train. Similarly Wigner describes the friend's result differently to how the friend does. While the laws of physics are the same for all observers, they may describe things differently from their respective reference frames.

For a nice visualization of this, see How does a quantum particle see the world?

That sounds cool. In my experience, new evidence just moves the goal posts. An interpretation like the consciousness one will just adjust its story if the linearity of QM can be demonstrated. Others may actually fall out of contention. — noAxioms

Yes, one issue there would be whether the human-level quantum AI should be considered conscious.

I am actually very unfamiliar with how they do such tests. I mean, the double slit thing is pretty obvious, but how do they test for superposition of spin? Far worse, they've succeeded in putting something large enough to see with the eye, in superposition of vibrating or not. My question is, how was that demonstrated? — noAxioms

For spin, via a Stern–Gerlach experiment. For large objects, as I understand it, if you can couple them to a qubit then you can test for interference.

How might one actually attempt to do the sort of test your're talking about with the computer? — noAxioms

Suppose a qubit is prepared in the zero state (i.e., |0>). The friend AI applies a transformation to put the qubit into the plus superposition state (i.e., |0> + |1>, or |+>). Then the friend AI measures the qubit (which, in standard quantum theory, is a unitary transformation resulting in the lab being in superposition).

Wigner then runs the entire transformation in reverse, thus undoing the friend AI's measurement and restoring the qubit to the initial zero state. Then Wigner measures the qubit. If the qubit is measured to be in the one state (i.e., |1>) then that falsifies standard quantum theory. That's because the only way that outcome could occur is if the friend AI's measurement reduced the qubit state to either the zero or the one state, via a physical collapse. In that case, Wigner's inverse transformation would have transformed the qubit to either the plus or minus superposition states, which then gives the possibility of measuring the one state.

Here's the experiment in further detail:

4 Wigner-Deutsch thought experiment

Two central questions that came up repeatedly in our discussions so far are (i) “Does Alice see a definitive measurement outcome?” and (ii) “Is Alice’s lab after the measurement indeed in a superposition state?” Deutsch [23] proposed an extension of the thought experiment described in Section 3, which turns these questions into (in principle) experimentally testable statements. — Testing quantum theory with thought experiments, p17 - Nurgalieva, Renner

The mathematics is linear - a specific kind of function or operation - and this gets confused with "linear interpretation", I think. And then there are "collapses" and so on. How is it possible to even discuss these things without a deep knowledge of the machinery of experiments? That's why its called the measurement problem, ascertaining a quality of an infinitesimal system without altering that quality or another quality. — jgill

It's possible to abstract over the machinery, using quantum logic gates. For example, to place a qubit in superposition, a Hadamard operation is applied.

The crux of Deutsch's experiment is to test whether measurement results in an irreversible physical collapse, or not.

It really is a philosophical problem, about not being able to deal with the fact that there is something fundamental that we can't know. — Wayfarer

It's a different philosophical temperament. Deutsch's argument (from the earlier quote) is that, "that attitude involves saying that there are certain questions about reality that you're not allowed to ask".

This seems to suggest that it is quantum theory that would be falsified given, well, apparently some sort of confirmation of 'consciousness causes collapse', except that in itself is another quantum interpretation (the Wigner interpretation) which was abandoned by Wigner himself due to it leading to solipsism, not because it in any way refuted quantum theory.

In my opinion, Deutsch is answering what seems to be a naive comment, not really describing a test result that differ empirically from one interpretation to the next. Consciousness causing collapse is not an empirical thing. — noAxioms

Deutsch's experiment provides a way of distinguishing between linear interpretations such as RQM/ QBism/Many Worlds and non-linear interpretations such as consciousness-causes-collapse/objective collapse theories like GRW. So it would enable us to rule out an entire class of interpretations.

OK. Do any of the interpretations predict a different outcome of this experiment than the others? It's pretty straight-forward. The friend comes out and only remembers classical stuff. The experience of being in superposition relative to the box exterior is in no way different than the same thing without the box. You can no more get interference of the friend than you can get the dead and live cat to interfere with each other. Perhaps this is not the case with the quantum AI, in which case is kind of isn't the Wigner friend thing exactly. — noAxioms

In the Wigner's friend thought experiment, the friend's lab is a closed/isolated system. A quantum computer provides a way to realize that isolation for a large, complex and artificially-intelligent entity (the friend AI). Then we can test for interference.

That's the crux. To avoid the 'observer effect'. — Wayfarer

There's an 'observer effect' in Einsteinian relativity which nobody objects to. That's not the problem.

As Deutsch puts it, what we want to understand is, "How were the results brought about by the preparation?" Why is it that we observe interference effects?

Deutsch's proposed experiment would implement the Wigner's Friend thought experiment.
— Andrew M

I had read that this had been done in 2019. — Wayfarer

It's similar, and in that experiment Wigner observes interference as predicted by quantum theory (which the paper describes as observer-dependence, since the friend observes a definite outcome). The important difference is that the agents are each represented by single photons. Whereas Deutsch envisages Wigner and the friend as human-level AI's. It would therefore be a test of whether very large, complex and (artificially) intelligent entities exhibit interference.

On face value, the many-worlds interpretation of quantum physics seems the opposite of parsimomious. It seems to say that the world or universe splits or divides at the point of measurement or observation of a sub-atomic particle. — Wayfarer

It depends on how parsimony is understood. Many Worlds has the biggest universe but also the fewest postulates.

So the question is, what problem does the interpretation of quantum physics try to solve? What would its proponents such as Wallace and Deutsch be obliged to acknowledge (apart from the obvious fact that they were mistaken), if by some means it was shown to be untenable? — Wayfarer

The problem that the interpretation should solve is to explain the interference phenomena that we observe. Not merely to predict observations - that's what the formalism does. If Many Worlds were shown to be untenable, Wallace and Deutsch would say that we have no viable explanation (that we know of).

Deutsch comments on just this issue in the video:

0:54 Arndt: The familiar problem that we all try to solve is why is there this unitary evolution of quantum mechanics which seems to explain everything very naturally and, all of a sudden, during a measurement this evolution has to be reduced, collapsed in the Copenhagen interpretation and that's I think something that David doesn't like. He wants to have everything on the same mathematical formalism. But if you follow it through it leads to realities which seem to multiply and then my question to you is what is really the meaning of reality to you because I experience only my single reality here.

1:30 Deutsch: Yes, you put it in terms of how do we make sense of the unitary evolution compared with what we see at a measurement and so on. I would want to start before that. I think we want to understand the world. We want to understand how the world is and that that is not necessarily what we perceive. Our perceptions are at the end of a long chain of physical processes of which themselves we only have scientific knowledge or indirect knowledge.

So I would start with the question, how do we explain quantum phenomena like interference? Not how do we make sense of quantum theory which gives the right prediction but, first, before that, how do we explain quantum phenomena?

So there's an interference process and we have an interference pattern which we can see without any quantum mechanics that the result of the experiment cannot be explained by the events that we see. Now this is not very unusual. This happens a lot, you know, in physics and ultimately every observation is made very indirectly so we have to infer things that are not there. Although infer is the wrong word. We have to conjecture explanations. So that's where I would start.

...

7:44 Arndt: Yeah, but there's something in the formulation of phrasing of our sentences where we're getting also doubtful. When I talk about these things about super positions I always make these quote-unquote when I say a particle is at the same time here and there because there are two words that I don't understand, three words I don't understand.

First the word is 'reality', the second word is what does 'time' really mean, and the third one what does 'space' really mean and we don't have any experimental evidence that the particle is at the same time here and there we just have a physical description the quantum mechanical description that the wave function behaves 'as if' and how can we make the step to the Many-Worlds."

8:23 Deutsch: I think we have something slightly more than that. Again you come from the theory. But, I think, prior to the theory we have the experience that this thing cannot be explained by single trajectories.

8:38 Arndt: Definitely.

8:39 Deutsch: We don't have to believe quantum mechanics to see that. So we rule out single trajectory explanations and that we have before we have quantum mechanics. If we didn't have quantum mechanics, it would be a mystery. We would say that there simply is no explanation.

...

12:40 Arndt: Although other scientists would say shut up and measure - calculate actually. Just don't talk about things that you cannot see and the other part of the other branches you don't see. So why do you dare not to shut up?

12:56 Deutsch: Yes, well, first of all I think that that attitude involves saying that there are certain questions about reality that you're not allowed to ask. You're allowed to ask how the experiment was prepared. You're allowed to ask what will the results be. You're not allowed to ask how were the results brought about by the preparation. So therefore it's not an explanation in my terms.

But as for shut up, that's really another way of trying to evade the consequences in terms of reality. Like my favorite example is of dinosaurs in the past. So there are people who say nobody ever saw a dinosaur and nobody ever will and therefore it's just a frivolity to say that they really exist. At most we can say fossils behave as though dinosaurs existed.

But no paleontologist would accept talking that way even though there is no experimental way of disproving that manner of speaking. And that's because paleontologists are only interested in paleontology because they want to know what really happened. If they were interested in fossils they would be geologists. — Are There Many Worlds? David Deutsch in conversation with Markus Arndt

↪180 Proof He has a very interesting idea on how to put MWI and wave-function collapse interpretations to the test. Assuming we can build a conscious AGI quantum computer. — Marchesk

I didn't get where in the 2nd vid that Deutsch suggested some kind of empirical test that should yield different results from one interpretation to the next. I'm very skeptical of that. — noAxioms

Here's the relevant part of the conversation between physicists David Deutsch and Markus Arndt:

6:12 Deutsch: When we have quantum computers, we will be able to have very large, very complex entities existing in super positions. So, in principle, I suggested long ago before this was remotely on the cards experimentally, that if we had a quantum computer on which an artificial-intelligence program was running, say, with human level artificial-intelligence then this entity would be able to experience interference in its own consciousness.

6:46 Arndt: Well, some people would say that your consciousness would collapse your reality.

6:50 Deutsch: Yes, so if that happened that would refute the Everettian interpretation or, as I would say, it would refute quantum theory. That would be a very interesting problem and that's one of the reasons why scaling up both the size and the complexity and the mass of phenomena that I experimentally observe, but that can only be explained by quantum theory, is very important.

7:17 Arndt: I fully agree. We need to do that.

7:21 Deutsch: We just need to close the gap between that and the AI because the AI would not be having this conversation. Or, at least, the AI would not be able to make the argument that you just made. It would have to say I've only got evidence of many worlds on the scale of my mind, but not bigger so - and I guess that will always be true. — Are There Many Worlds? David Deutsch in conversation with Markus Arndt

Essentially, Deutsch's proposed experiment would implement the Wigner's Friend thought experiment.

In Deutsch's proposal, Wigner would be a human-level AI running on a quantum computer, with the friend (also a human-level AI) and the measured qubit being internal and isolated subsystems of Wigner.

By conducting an interference experiment on the joint friend/qubit subsystem, the Wigner AI would be able to determine whether physical collapse happened or not. If physical collapse were detected, then that would falsify standard quantum theory (and MWI along with it).

That being said, my understanding is that the probabilities we use to calculate the likelihood of what to expect when a measurement is made still needs to be derived within the Schrodinger equation in a self-consistent manner without adding it in post hoc, since the wave function is supposed to describe the universe we live in, if MWI is true. So deriving the Born rule within MWI is an ongoing project. — Marchesk

You might be interested in Sean Carroll's post on Why Probability in Quantum Mechanics is Given by the Wave Function Squared.

A potential issue arises here. What of all the entanglements that don't support observers? Those aren't considered worlds since there's no observers for things to appear classical. Which means observers are fundamental for saying what counts as a world. — Marchesk

Per MWI, worlds are persistent structures that emerge as a consequence of decoherence. In that sense, an MWI world doesn't depend on observers.

David Wallace on worlds:

“Worlds” are mutually dynamically isolated structures instantiated within the quantum state, which are structurally and dynamically “quasiclassical”. The existence of these “worlds” is established by decoherence theory. — Decoherence and Ontology - David Wallace

David Wallace on observers:

However, the structural approach is committed to an approach to the mind which ... denies observers some uniquely special status, but describes them as emergent as structures and patterns in lower-level physics (specifically, in lower-level classical physics, itself to emerge from unitary quantum physics via decoherence); — Everett and Structure - David Wallace

Final paragraph of the article is:
Neither Pan’s nor Fan’s group has yet closed the loopholes in their experiments. — Art48

The loopholes have been closed in a more recent experiment. The remaining aim is to close all the loopholes simultaneously.

By ensuring events in the experiment happened quickly and far enough apart, the researchers say in the paper that, given reasonable assumptions, they have closed three loopholes: locality, independent source, and measurement independence.
...
The new experiment left open the detection loophole, which was closed in Lu and his colleague’s previous experiment. Lu says the team hopes to develop new techniques to enable an experiment that simultaneously closes the loopholes. — New Experiment Suggests Imaginary Numbers Must Be Part of Real Quantum Physics - APS

A Bit More Readable Version . . .

It's a much more complicated subject than I had imagined. — jgill

Yes, me too.

Complex numbers and their properties facilitate this.

Again, this is an almost trivial argument for convenience rather than necessity. — jgill

Yes, so Chen and Li's experiments are an argument for necessity. The gist is that an entanglement swapping protocol is followed such that two observers (Alice and Charlie) end up with one each of a pair of entangled qubits. The subsequent measurements on those qubits (as confirmed by Chen and Li's experiments) match the statistical predictions of standard quantum mechanics.

However the entanglement isn't fully swapped under real-valued QM, and so it makes different predictions to standard QM (and fails to be confirmed by experiment). As the Physics Today article says:

Bob then makes a joint measurement on his two qubits, with four possible outcomes. In complex-valued quantum mechanics, that measurement halves the number of dimensions of the system and cuts the number of entangled pairs from two to one. That is, it transfers the entanglement to qubits A and C. But in a real-valued formulation, Bob’s four-outcome measurement doesn’t cut the dimensionality by enough to fully swap the entanglement—he’d need an eight-outcome measurement to do that—so qubits A and C don’t end up fully entangled. — Does quantum mechanics need imaginary numbers? - Physics Today

At the end, the author adds:

"We must be careful, however, in assessing the implications of these results. One might be tempted to conclude that complex numbers are indispensable to describe the physical reality of the Universe. However, this conclusion is true only if we accept the standard framework of quantum mechanics, which is based on several postulates. As Renou and his co-workers point out, these results would not be applicable to alternative formulations of quantum mechanics, such as Bohmian mechanics, which are based on different postulates. Therefore, these results could stimulate attempts to go beyond the standard formalism of quantum mechanics, which, despite great successes in predicting experimental results, is often considered inadequate from an interpretative point of view" — jgill

Yes, though those alternative formulations are non-local (i.e., require faster-than-light communication for measurements) and therefore don't integrate nicely with relativity. What these experiments demonstrate is that a local-relativistic universe must be based on complex-valued amplitudes.

The Avella paper is a great resource. Thanks everyone. — Art48

:up:

Well, maybe this is not a Philosophy forum after all ... — Alkis Piskas

It's philosophy of physics - specifically the interpretation of quantum mechanics, the nature of space and time, and the relation of mathematics to the universe.

Philosophy of physics is the study of the fundamental, philosophical questions underlying modern physics, the study of matter and energy and how they interact. The main questions concern the nature of space and time, atoms and atomism. Also included are the predictions of cosmology, the interpretation of quantum mechanics, the foundations of statistical mechanics, causality, determinism, and the nature of physical laws.[80] Classically, several of these questions were studied as part of metaphysics (for example, those about causality, determinism, and space and time). — Philosophy of physics - Wikipedia

Question: are complex numbers required for QM, or could Schrodinger's equation be reformulated exclusively in terms of sin and cos? — Art48

Recent experiments seem to show that complex numbers are required for QM.

Two independent studies demonstrate that a formulation of quantum mechanics involving complex rather than real numbers is necessary to reproduce experimental results.
...
[Renou and colleagues] considered two theories that are both based on the postulates of quantum mechanics, but one uses a complex Hilbert space, as in the traditional formulation, while the other uses a real space. They then devised Bell-like experiments that could prove the inadequacy of the real theory. In their theorized experiments, two independent sources distribute entangled qubits in a quantum network configuration, while causally independent measurements on the nodes can reveal quantum correlations that do not admit any real quantum representation.

Chen and colleagues and Li and colleagues now provide the experimental demonstration of Renou and co-workers’ proposal in two different physical platforms.
...
Despite the difficulties inherent in each implementation, both experiments deliver compelling results. Impressively, they beat the score of real theory by many standard deviations (by 43 sigma and 4.5 sigma for Chen’s and Li’s experiments, respectively), providing convincing proof that complex numbers are needed to describe the experiments. — Quantum Mechanics Must Be Complex - Alessio Avella, INRIM (January 24, 2022)

I understand that in this sense, the two gloves are set at their creation. Is SPDC simply asserting the same for these two entangled photons? — universeness

If locality is the case (per QFT) then the correlation is a consequence of the SPDC. However, unlike with the gloves, the final measurement values are not predefined (see counterfactual-definiteness).

So, when you locally open one of the boxes, that act, does not affect the state of the gloves.
But a qubit can have more that two states due to superposition states. A qubit does not just resolve to 1 or 0, it can be in a superposition of 1 and 0. — universeness

Yes. Or in the case of the singlet state, the qubit pair is in a superposition of 01 and 10.

The nature of the gloves as left or right handed is there from the beginning, just like in QFT, a coordinate in spacetime can manifest any of the known particle states/field excitations (almost like an interdimensional vibrating superstring). Entanglement may 'correlated' (as a field excitation 'travels') the states of two spacetime coordinates, regardless of the distance between them.
What's wrong with the imagery I am invoking, if it's incorrect. — universeness

I'm not sure I follow the imagery. There isn't an image or picture that explains the correlation, unless one goes with an interpretation (such as Many Worlds or Superdeterminism).

If locality is the case, then the common cause of the entangled particles' correlation is their initial preparation (See spontaneous parametric down-conversion).
— Andrew M

Phew! That link was to a physics level that is a bit high for me. I clicked on sub-links such as 'non-linear crystal,' 'etc to gain a better insight. But I found I had to click on further and further sub-links eg 'Schwinger limit' and then 'Birefringence,' to gain any clarity. I will go back to it, but you have moved past my current width and depth of physics understanding. — universeness

SPDC converts a photon into two entangled photons, each with half the energy of the original photon. For Type II SPDC, this prepares the singlet state (represented below by qubits):

$\ \ \ \ \ |\psi\rangle = \frac{|01\rangle - |10\rangle}{\sqrt{2}}$

So that anti-correlation is the consequence of a local physical process (i.e., the SPDC process).

Subsequently, the two photons (qubits) can be separated by a large distance. QFT, by construction, says that spacelike-separated events, including measurements, cannot be causally connected. So the measurement of a photon is a single localized event in spacetime which has no effect on the measurement probabilities for the other photon until a signal arrives there (not exceeding the speed of light).

Note that the singlet state is perfectly anti-correlated in every basis (X, Y, Z, etc.). That is, if each qubit is measured in the same basis, then one will be measured as 0 while the other will be measured as 1.

Compare QFT to the case of Bertlmann’s socks, which John Bell recounts:

The philosopher in the street, who has not suffered a course in quantum mechanics, is quite unimpressed by Einstein-Podolsky-Rosen correlations. He can point to many examples of similar correlations in everyday life. The case of Bertlmann’s socks is often cited. Dr. Bertlmann likes to wear two socks of different colours. Which colour he will have on a given foot on a given day is quite unpredictable. But when you see (Fig. 1) that the first sock is pink you can be already sure that the second sock will not be pink. Observation of the first, and experience of Bertlmann, gives immediate information about the second. There is no accounting for tastes, but apart from that there is no mystery here. And is not the EPR business just the same? — Bertlmann's socks and the nature of reality - John Bell, 1981

Observing one sock to be pink doesn't cause the other sock to not be pink. Instead the common cause of the sock anti-correlation is Bertlmann's initial choice. That's analogous to QFT, where the SPDC process is the common cause of the photon anti-correlation. In both cases, locality is maintained.

The key difference is that the observation of Bertlmann's socks is explained by the pre-measurement colors of the socks. Whereas the observation of the entangled qubits can't be explained by pre-measurement values, as shown by Bell's Theorem. Which is why quantum interpretations become relevant (note the Local dynamics column).

The correlated events or actions can be because of a common cause. — universeness

If locality is the case, then the common cause of the entangled particles' correlation is their initial preparation (see spontaneous parametric down-conversion).

I think the use of the term correlation for quantum entanglement is a wise use but really just indicates that the detailed nature of the relationship is not yet well understood. — universeness

:up:

:up: But if the two items 1 light year apart are entangled, then surely "However, if the events are causally connected, precedence order is preserved in all frames of reference." From the wikipedia article: https://en.wikipedia.org/wiki/Relativity_of_simultaneity applies? — universeness

The particle measurement events aren't causally connected (i.e., correlation is not causation). So the precedence order need not be preserved in all frames of reference.

In quantum field theory, observables of events with a spacelike relationship, "elsewhere", have to commute, so the order of observations or measurements of such observables do not impact each other. — Causality (physics) - Wikipedia

My understanding is that Einstein's famous "spooky action at a distance" concern was about something going faster than the speed of light. — Art48

Yes, that was his concern (which, more generally, was about wavefunction collapse). However QM neither specifies nor requires non-locality. Further, locality is one of the axioms of quantum field theory.

Locality is one of the axioms of relativistic quantum field theory, as required for causality. The formalization of locality in this case is as follows: if there are two observables, each localized within two distinct spacetime regions which happen to be at a spacelike separation from each other, the observables must commute. Alternatively, a solution to the field equations is local if the underlying equations are either Lorentz invariant or, more generally, generally covariant or locally Lorentz invariant. — Principle of locality - Wikipedia

We have quantum entanglement, which says that signals can travel faster than light. — Art48

No, quantum entanglement says measurements will be correlated - a very different thing. As physicist Asher Peres noted, "relativistic quantum field theory is manifestly local." (longer quote here).

[From Physics Stack Exchange] Entanglement between two qubits means that if a measurement is made on one of them, the other one is decided instantaneously.

This is true, but this does not allow for faster than light communication. If you have one qubit with you and I have one qubit with me and you make a measurement on your qubit, that will mean my qubit is decided . But how does that send any signal ?

Later on, when I make a measurement on my qubit, I get a measurement, just as I would have got some measurement had you not measured first. There is no way for me to know that I got this measurement after you had measured yours or before you have measured yours. Hence, no signal can be sent faster than light using entanglement — universeness

Yes though the term "instantaneously" can be misleading. The other qubit has to be measured and the results compared. As Peres also noted, "unperformed experiments have no results" (see counterfactual definiteness).

I tried to understand quantum entanglement using my own following thought experiment:
If the distance between us was 1 light year. Then I could send you a message that says.
Measure your entangled particle the moment you receive this message.
I would then have to know very precisely (I think) when to measure my item just before you do, 1 light year away. I would then send a message to you 1 light year away, asking you what your measurement was. If I had measured a 1 then you will have read a 0.
But it will take 2 years after the actual event, for me to confirm this, so no signal is travelling faster than light in this thought experiment.

I don't know if this is correct but it's what I understand as quantum entanglement at the moment. — universeness

That's correct. But note that, per relativity of simultaneity, the order of the measurements can potentially differ in each particle's reference frame. The point is that each measurement is local. One measurement doesn't cause or influence the other measurement.

It's a subject I haven't studied beyond the basics of time dilation. — jgill

OK. Time dilation and length contraction are related in the expected way. If you travel half the distance (i.e., at 87% of the speed of light), then it takes half the time.

You take my simple thought experiment (satire) far too literally as an exercise in relativity. — jgill

Just grounding the thought experiment in some real numbers - there needs to be Lorentz symmetry between the observers. Weren't you also asking a question?

Let me clarify by a simple thought experiment:

I start out driving along a straight road to a destination 1,000 miles away. To an external observer I drive at a steady 100 mph, with the distance between my front bumper and my destination shrinking at 100 mph. — jgill

At that speed, there is no appreciable difference between the distance for you and the stationary observer. Your velocity would need to be six million times faster (about 87% of the speed of light) to contract the distance for you to 500 miles.

However, I was unaware that clocks tick slower in my car than outside stationary clocks, — jgill

Your clock ticks at the same rate for you as it always has. But you measure the stationary clock as slower. The stationary observer also measures your clock as slower.

so at the end of one hour I pass the 200 mile road post, and think, wow, I am going really fast, twice the speed limit! But that’s not possible. — jgill

You are going six million times the speed limit. But the road posts were calibrated for essentially stationary observers. So the post says 200 miles when you have only travelled 100 miles.

So, my destination is approaching my car at that speed. But that’s not possible either. So I surmise that I am driving at the speed limit and the distance between my car and my destination is shrinking that fast as well. Hence, the rate of change of the distance between car and destination is changing more than I originally assumed. — jgill

In your reference frame, you are at rest. So the destination is approaching you at 0.87c.

To an external observer the actual space between vehicle and destination is not contracting, but the distance between them is. It's a playground for metrics. — jgill

Spacetime stays the same (the spacetime interval is invariant). In that sense it's like looking at a round table from an angle - it appears oval, or contracted along one dimension. But it's more than perceptual, since each observer measures the other as contracted and, indeed, you travel only 500 miles to reach your destination.

The selected quotes from the above physicists concern infinity, which I did not mention in my comment. I did reference infinite time.
My comment concerned chatGPT's response to a reference to an 'edge', which implies a spatial boundary with matter, light, and stars on one side and nothing on the other. He does say that such a thing isn't well defined. Again, I don't disagree with any of his responses.
There are models of finite size universes, but most/all(??) of them curve in on themselves, much like Earth having finite surface area without anywhere having an edge. It was the reference to that edge at which I balked. — noAxioms

:up:

I think ChatGPT meant "edge" in the higher dimensional sense, where the finite/infinite distinction is relevant. We can say that the Earth has no edge on the 2D surface. Whereas the edge, or boundary, in 3D is defined by the surface (i.e., ~6,000km from the center of the Earth).

OK, this answer implies that there is a meaningful edge to the universe, which is not part of any accepted theory I've seen. — noAxioms

True, though neither is it ruled out. As physicists' Sabine Hossenfelder and Max Tegmark note:

Whenever you try to measure something infinite, the best you can do in practice is to say it’s larger than something finite that you have measured. But to show that it was really infinite you would have to show the result was larger than anything you could possibly have measured. And there’s no experiment that can show that. So, infinity is not real in the scientific sense.

Nevertheless, physicists use infinity all the time. Take for example the size of the universe. In most contemporary models, the universe is infinitely large. But this is a statement about a mathematical property of these models. The part of the universe that we can actually observe only has a finite size.
...
But maybe using infinity and zero in physics brings in mistakes because these assumptions are not only not scientifically justified, they are not scientifically justifiable. And this may play a role in our understanding of the cosmos or quantum mechanics. This is why some physicists, like George Ellis, Tim Palmer, and Nicolas Gisin have argued that we should be formulating physics without using infinities or infinitely precise numbers. — Is Infinity Real? - Sabine Hossenfelder

Let’s face it: Despite their seductive allure, we have no direct observational evidence for either the infinitely big or the infinitely small.
...
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

My understanding is when an object approaches the speed of light from the point of view of an outside viewer the object contracts in the direction of the path?

If an object could actually reach the speed of light would the object become 2 dimensional from everyone else's perspective?

Is there any hypothetical way an object can travel in all directions at the same time? If that were so would it not be expanding? And if it were doing that at the speed of light would it not also be becoming 0 dimensional? — TiredThinker

I put your question to OpenAI's ChatGPT. I've also included some of my own questions (based on the above exchange between @staticphoton and @noAxioms). To the best of my knowledge, the responses are correct.

According to the theory of relativity, as an object approaches the speed of light, it will appear to contract in the direction of its motion. This is known as length contraction. However, the object itself does not actually shrink; it is only its appearance that changes from the perspective of an outside observer.

If an object were to reach the speed of light, it would not become two-dimensional from the perspective of outside observers. This is because the speed of light is a limiting speed; it is impossible for any object to actually reach the speed of light, so the question of what would happen if an object did is purely hypothetical.

It is also not possible for an object to travel in all directions at the same time, since this would violate the laws of physics. In order for an object to move, it must have a definite direction and speed. An object that is expanding, such as the universe, is not moving in any particular direction, but rather is increasing in size in all directions simultaneously.

As for the question of an object becoming zero-dimensional, this is also not possible according to the laws of physics. An object must have a non-zero size in order to exist, so it is not possible for it to become zero-dimensional. — ChatGPT

Isn't a photon an object?

A photon is not an object in the traditional sense of the word. A photon is a type of elementary particle, which is a fundamental building block of matter. Unlike objects, which have a definite size and shape, photons do not have any physical form or structure. They are characterized by their energy and momentum, and they can travel through space at the speed of light. Because of their unique properties, photons are sometimes described as "particles of light." — ChatGPT

Can a photon travel to the edge of the universe?

It is impossible for a photon, or any other form of matter or energy, to reach the edge of the universe. This is because the universe is expanding, and the space between galaxies is getting larger all the time. As a result, the distance to the edge of the universe is constantly increasing, so it is not possible for anything to reach it. Additionally, the concept of the "edge" of the universe is not well-defined, since the universe is thought to be infinite in size. — ChatGPT

How about in the reference frame of the photon?

In the reference frame of a photon, time does not pass, so the concept of the photon reaching the edge of the universe is not well-defined. In the reference frame of a photon, the universe appears to be a sphere with a radius of zero, and the photon is always at the center of the sphere. This is because the speed of light is constant in all reference frames, so from the perspective of the photon, it is always at the same point in space, no matter how far it has traveled. — ChatGPT

What is the status of treating common sensical language as the correct interpretation as philosophy done correctly. Anyone? — Shawn

Ordinary language is not the last word: in principle it can everywhere be supplemented and improved upon and superseded. Only remember, it is the first word. — J. L. Austin

On second thought, however, I can't see how an example could be constructed without a false belief, so I am very sceptical of the possibility. There are some attempts, but they haven't convinced me. — Ludwig V

That's how I see it as well.

In short, it seems to me that Gettier case ought to be possible. Perhaps the real Gettier problem is why it is so hard to develop one that commands general agreement or to articulate a general solution. — Ludwig V

I think part of the problem is that there are many different tracks along which disagreement and misunderstanding can occur and so care is needed to properly distinguish and relate them. Some of the philosophical and linguistic issues the Gettier problem raises are:

• What conception of knowledge are we talking about? Everyday, specialized, idiosyncratic?
• Can knowledge be decomposed at all? JTB, JTB+, knowledge-first?
• What is truth?
• What is justification? Certain, pragmatic, contextual?
• Relation to nearby epistemic puzzles? Lottery paradox, Harman-Vogel paradox.
• What points-of-view are applicable? Omniscient, stipulative, relative, subjective/objective?
• What is belief? Representational/functional, internal/external, de dicto/de re?
• How are perceptual terms used? Factive, sense-data, other philosophical usages?

It's true that further investigations can involve mistakes. But if we are being strict about what we will accept as believable then we should investigate as far as, and in every more thorough way imaginable, and only commit to believing when all those possibilities are exhausted. It's also true that even then we can be mistaken, but at least our beliefs would then be properly justified.

We can always resort to entertaining something for pragmatic reasons without committing to belief if we realize that our investigations have not been or cannot be, for practical reasons, adequate. So, for example, I see something moving which I think is a sheep, but there is a boundary fence that prevents me from getting close enough to definitely confirm it. — Janus

:up:

As Feynman once noted, “The first principle is that you must not fool yourself — and you are the easiest person to fool.”

Gettier cases rely on the various circumstances not being known to anyone, and so will in real life always exist unknown to anyone. As soon as they become known, they can be resolved, so I can’t see that they can be very important. Some people worry about this, but that's only because they can imagine something that's not known to anyone. That's not real life. — Ludwig V

I think that's right. The main point, as I see it, is to understand what it is that we do and why.

I'm going to take a break, but I really think we should be able to construct a clear case, roughly along the lines above, of Gettier case without a false lemma. — Srap Tasmaner

I'm not yet convinced! If the children understood that the teacher could include non-capital cities in his experiment, then I don't think the children were justified in believing that Topeka is the capital of Kansas (assuming they didn't already know it).

Whereas if the children believed (wrongly) that the teacher could only include capital cities, then that would be a false lemma. Even if, as it turned out, only capital cities were included.

It's like playing a game of chess, and finding out later that certain house rules were in effect that you didn't know about and were never utilized in the game. You "deduced" that you were in checkmate, and you were even according to the house rules, but there were more potential moves available than you thought.

If the farmer saw a cow, there is a specific cow that he saw. If the cow that he doesn’t know about is the one that establishes the truth, then he didn’t know there was a cow in the field. — Ludwig V

In this case, it's the cow that he saw that establishes his conclusion that there's a cow in the field. He is mistaken about which cow he saw, but that doesn't undermine his conclusion.

There's a sense in which we're constructing the argument from premises to conclusion. As long as we can provide a straight-forward argument that doesn't depend on false lemmas, then I think we're generally willing to grant the farmer knowledge.

I’m not happy with the causal theory of perception (though I’m not up to date with more recent ideas about it.) because what we see is so heavily dependent on interpretation, which doesn’t fit happily with causality. — Ludwig V

In the presented case, the farmer does misidentify the cow as Daisy, and so thinks that Daisy was the cause of his perception. But that doesn't present a problem for a causal explanation. It only means that people can sometimes be mistaken about what the causal factors are. In this case, the farmer's misinterpretation of what he saw didn't preclude him from knowing there was a cow in the field. And we know this because we can provide the correct causal explanation that demonstrates this (i.e., that he did see a cow).

Which, again, is why the farmer won't feel nearly such a fool if it turns out to be Clarabelle, even if he's very surprised, because he will still have gotten a lot right.

(I was going to head in a completely different direction, so I'll wait to see what people think of this before trotting out alternative analyses.) — Srap Tasmaner

Seems right to me.

If I see something in field that I think is a sheep, am I justified in believing it is a sheep if I don't take the trouble to move closer and examine it to see if it really is a sheep, or shout to make it move or whatever? It might not be what people ordinarily do, but is that any justification for failing to investigate, and for forming an insipid conception of justification itself? — Janus

I think as a general rule, if you "couldn't too easily be mistaken" (Williamson's phrase) then you are justified in believing it.

Whether that's the case in a particular sheep scenario depends on how well-placed you are. Justification is a pragmatic standard - it has to be both useful (not too easy to be mistaken) and obtainable (infallibility is not required).

However it seems to me that even if seeing the cloth is accepted as justification for believing there is a sheep in the field, it could only be justification for believing that the apparent "sheep" (the cloth) was in the field, not some other unseen sheep that just happened to be there unbeknownst to me. — Janus

That's right. So that's the "no false lemmas" solution to Gettier problems. The farmer justifiably and correctly believed there was a sheep in the field. But because his belief was based on the false lemma that he was seeing a sheep, he thus didn't know that there was a sheep in the field.

As I say above I would go further and say it is an unjustified false belief due to lack of proper investigation, because I have no business believing a cloth is a sheep, if I'm not close enough to it to be sure, or if I haven't seen it moving around and grazing like a sheep. — Janus

OK. But more thorough investigations can involve mistakes. For example, suppose the farmer thought he saw a sheep moving and grazing, but it was a goat (or a robot).

In the original version of the story, it's a bit of cloth that is causally related to the farmer's implicitly disjunctive belief and another disjunct is true. What's different here from the case above where one cow is mistaken for another — and so there's acceptable disjunctive knowledge — is that you cannot see that a bit of cloth is a cow, because it isn't. We are relying on the seeing being factive, and that's already expressed as predication; what causation gives us is an explanation for the acceptability of the predication: you can see that something is a cow only if it is a cow. Which I hope is another way of saying that some cow ought to be causally involved in your formation of cow beliefs. — Srap Tasmaner

:up:

I would just add, though, that the causal connection may not always be sufficient for knowledge. Consider the fake barn scenario. In that case, the traveler in fake barn country does see an actual barn, so the appropriate causal connection is present. But he doesn't know it because he was lucky. The false lemma in this case is that he implicitly assumes that this region is like any other where fake barns are a rarity.

Agreement in philosophy is not easy to achieve, which makes it all the more satisfactory. — Ludwig V

:smile:

As Williamson notes, "Knowledge doesn’t require infallibility. What it requires is that, in the situation, you couldn't too easily have been mistaken."
— Andrew M

All right. But when you find you are mistaken, you need to withdraw the claim to know. — Ludwig V

Indeed. The claim can no longer be justifiably held at that point since knowledge does require truth and the claim in question is now known to be false.

Which is the Harman-Vogel paradox that Ludwig V referred to. Jennifer Nagel has a useful survey of some of the responses (contextualism, relativism, interest-relative invariantism, error theory) and her own solution (dual-process theory) in "The Psychological Basis of the Harman-Vogel Paradox".
— Andrew M

I’m not sure whether to classify the classic Gettier cases as variants of that paradox or a completely different variety. But I am sure that this paradox is much more difficult and more important than the Gettier cases. I have looked at some of what Jennifer Nagel has written about this. I didn’t find any of the theories particularly appealing. I’m certain it deserves treatment separate from the Gettier cases. — Ludwig V

Yes, I agree they deserve separate treatments. In my view, contextualism is a satisfactory response to both.

I find Lewis's contextualism a pretty obscure doctrine, so I'm not ready to go there yet, and I have more reading to do before taking on contextualism in general. Your view seems to be some sort of hybrid, in which knowledge is still a sort of justified belief, but what counts as justification is context-dependent. (Usually contextualism passes right by justification.) — Srap Tasmaner

Yes, my view perhaps differs from Lewis' in that regard. In the case of the clock example, I know it's 3pm as long as I see that the clock says 3pm and the clock is working properly. Now if I was asked whether I knew that the clock definitely hadn't stopped 5 minutes ago, then I don't know that. But wouldn't it then follow that I don't know the time? [*]

My view is that different standards of justification are being applied here. I know the time according to a pragmatic standard (the clock was working and I looked). As Williamson notes, "Knowledge doesn’t require infallibility. What it requires is that, in the situation, you couldn't too easily have been mistaken."

The clock question switches the context and applies a higher standard (for the moment anyway). That is, I don't know that the clock didn't stop 5 minutes ago and so don't know the time per that higher standard. I assume it didn't, but that's not the same thing. But I'm also not merely assuming it's 3pm. I did look at the clock which is all that is ordinarily expected. (Though if our lives depended on getting the time correct in some context, more might be expected - thus raising the standard in that context.)

Must the farmer forevermore wonder whether the bright spot is a cow or an old shirt? Because we know he was mistaken once? We might well ask, but Lewis specifically does not make such demands on the farmer, who either will or won't. This is a puzzling theory, that the less imaginative you are the more you know. — Srap Tasmaner

No he shouldn't do that. Being more imaginative doesn't change the fact that he ordinarily knows there's a cow there. However if the shirt situation commonly occurred, that would violate Williamson's requirement above - that you can't too easily have been mistaken. This is exploited by the "fake barn" Gettier case - what is normally a low probability case is instead a high probability case in that region. One doesn't know they have seen a barn in that region unless they look more closely.

--

[*] Which is the Harman-Vogel paradox that @Ludwig V referred to. Jennifer Nagel has a useful survey of some of the responses (contextualism, relativism, interest-relative invariantism, error theory) and her own solution (dual-process theory) in "The Psychological Basis of the Harman-Vogel Paradox".

I was aware that not everyone agrees with "no false lemmas". I confess that I don't know what the full definition of a lemma would be so I'm not in a position to argue with them. For the sake of brevity, I ignored them. The "apparent dog" is not an impressive counter-example. An apparent dog is not a dog. One might argue that a robot dog is a kind of dog, but that would blow the point of the story, so we don't need to worry about that. — Ludwig V

A lemma, here, is a premise of one's purported knowledge. So the stated premise in the case of the robot dog was that James had observed an apparent dog. Framed that way, it's a true premise, so avoids the "no false lemmas" condition. However I think that framing is problematic and agree that it's not an impressive counter-example.

If justification and truth run on separate tracks, then justification can sometimes lead, quite reasonably, to falsehood, just as we can sometimes hold true beliefs by luck. (Lotteries provide the clearest examples for both: you can pick the winning number, without justification, and you can only be justified in believing that you didn't, given the odds, but you can't know it.) — Srap Tasmaner

Yes.

"No false lemmas," by stipulating their conjunction, doesn't really address the main issue: either the true, justified lemma is knowledge, or it should face a Gettier case of its own — that is, you will be lucky that your premise is true. (If it's knowledge, then we've taken a step toward Williamson's E = K, the idea that rational beliefs are based on knowledge; but to claim that knowledge must be based on knowledge is either empty — because of course we'll take valid inference to be knowledge-preserving — or circular. If there's a third option, it's pretty subtle, but maybe there is.) — Srap Tasmaner

I think it's a reasonable view that the lemma be knowledge (which admittedly is a higher standard than simply truth), but it does need to be contextualized. That is, what counts as knowledge depends on the relevant standard in the particular context.

So the lottery example makes a particular set of possibilities salient. The belief that one will lose is justified in one sense (i.e., highly likely to be correct), but not another (i.e., it's not a valid inference). But it's worth noting that there are other less obvious ways things can go wrong (or right). Maybe Alice bought all the tickets, but then the lottery was cancelled. Or maybe Bob bribes someone and "wins" on that basis.

It's like the the coin flip that is purportedly 50/50 odds of heads or tails, but instead lands on its edge.

I'll admit, though, that it does seem to help. In Russell's example, checking the time from a clock that's stopped, had you looked a minute earlier or later, you would have formed a false belief, so you were lucky to have looked when you did. Now suppose that the clock was working and had the correct time, but stopped right after you looked; now I think we want to say you do have knowledge even though a minute later you would have formed a false belief. You were genuinely lucky in looking while the clock was still such that it was knowledge imparting. — Srap Tasmaner

Yes, that's exactly the issue as I see it. More below.

So what's changed? If you look a minute later, we're exactly in Russell's scenario; a minute earlier, and you're fine. What if we compress things: suppose the clock stopped this time yesterday, briefly surges into life as you approach, just long enough to tell the right time for a minute or so, and then fails again. Now your window of luck is a range of a minute or so — too early or too late is still Russell, but for a brief span, the clock is knowledge imparting. Does that sound right? It sounds a bit dodgier now; you have been nearly as lucky as in Russell's scenario. The clock starting again feels wrong; had it started a minute earlier it would carry on being ahead until it failed, later and it would remain behind. What's missing is the clock actually being set; if a worker had just gotten the clock to work, and set the right time, you would again be acceptably lucky to look while it's keeping the correct time, even if it only did so for a minute before the worker cursed and set to work again.

To say that the clock has been set properly is to say that the time it displays is not only true, but justified, I suppose. But we can keep pushing the problem of luck back into these ceteris paribus conditions, which will grow without bound. Was the worker going by his own watch? What if his watch only happened to have the correct time? We're either going to continue demanding that truth and justification stay conjoined, or we're going to allow them to separate at some point, and that's the point at which Gettier will take hold.

Perhaps though what we're seeing here is that Gettier is the inevitable result of treating beliefs as atomic, and that the revenge cases are indicating that our beliefs never confront reality singly but as a whole, the Quinean view, I guess. — Srap Tasmaner

Yes. So I think what is important here is context. In one sense every part of the world connects to every other part however you carve it up, even if only indirectly. But it's not very useful that we should have to account for everything in order to know anything at all. So we take a slice that is, in some practical and reasonable sense, separable and that is what justification applies to. That might mean keeping all possibilities of a lottery together as inseparable (since they are salient), but not all the ways a clock can go wrong. However, if one focuses on that aspect, as you have done above, then the boundaries of what counts may move or be contestable, at least for the moment. But they will probably move back again when one's focus changes to something else. Consider the coin toss example above. We don't want to miss the forest for the trees by over-analyzing it. (Though experience also counts here - we don't want to continually have financial crises, wars and pandemics and have everyone always say, "Well, who could've known?". They aren't black swans.)

So a particular belief may or may not be justified, depending on where one sets the contextual boundaries. But the clock time is either correct or not independent of those standards. That is, truth keeps us connected to the world and keeps us honest.

It's not clear to me that a knowledge-first view especially helps with these issues. There's still the question of whether you knew the time or not, and what counts as evidence.

However, this “no false lemmas” proposal is not successful in general.
— SEP

That's at least some places to start if you're sympathetic to the "no false lemmas" response. — Srap Tasmaner

I regard the "no false lemmas" condition as essentially correct. The criticisms are really around what counts as a lemma. But if one is to construct a Gettier case, it requires the failed knowledge to depend in some relevant way on something false.

To take the robot dog example from SEP, James thought he observed a dog and consequently concluded that there is a dog in the field. His conclusion was correct, and justified, but not knowledge because the lemma that he observed a dog was false. He in fact didn't.

The SEP analysis frames it as observing an "apparent" dog, which is why they think it's a counterexample to the "no false lemmas" condition.

There is nothing about the definitions of "addition", or "subtraction" which requires that the result be other than the starting number. — Metaphysician Undercover

That's correct. There is also nothing about the definition of "negation" which requires that the result be other than the starting number.

"Negation" is defined as producing a statement other than the one which is negated. — Metaphysician Undercover

You're equivocating. The subject is numbers, not statements.

You provided a definition of "additive inverse", not of "opposite", nor of "negation". — Metaphysician Undercover

Negation, as a mathematical operation, is the additive inverse.

And, as I've told you already, your quote only demonstrates that mathematics uses these terms in a way which is inconsistent with other fields of study, like philosophy and logic. — Metaphysician Undercover

No, it does not. Different fields have their own specialized terms. Recognize the context and equivocation can be avoided.

In fact, I see now that there is inconsistency within the quoted paragraph itself. It says: "For a real number, it reverses its sign". And it also says: "Zero is the additive inverse of itself." Since zero is a real number then it is an exception to the stated rule for real numbers, therefore the inconsistency inheres within your definition. It is self-contradicting, stating a rule then a contradicting rule. — Metaphysician Undercover

That is not part of the definition, that is commentary. To negate a number is to subtract it from zero. It's very simple.

That follows from the group-theoretic definition which @Real Gone Cat kindly provided earlier. The real numbers (and, separately, the integers) are a group under addition, where 0 is the identity element and 0 is its own inverse.

Which is to say, the negation of zero is zero. If you disagree, then you shouldn't trust a calculator since it treats -0 as a valid operation.

Notice, negation takes the proposition to "another proposition". There is no exception, which would allow that a negated proposition could remain the same, as you propose with zero. — Metaphysician Undercover

As noted above, a number is not a proposition.

Do you also hold that adding zero to a number cannot be called "addition" because the number is the same before and after?

Or that dividing a number by one cannot be called "division" because the number is the same before and after?
— Andrew M

Why bring up things which are not comparable? — Metaphysician Undercover

They're comparable because in each case the number remains the same. On that basis you reject that a negation has occurred but, apparently, still accept that an addition and a division has occurred. Which seems to be an arbitrary conclusion.

Yet "negation", defined as zero minus a number, can be just that.
— Andrew M

Well, I've never seen "negation" defined as " zero minus a number". Care to share where you got that one from? Zero minus a number clearly does not negate the number, as negating a number gives zero. So I think you are really stretching for straws now Andrew. — Metaphysician Undercover

I've linked to the definition several times now. Here it is again with the relevant parts bolded.

In mathematics, the additive inverse of a number a is the number that, when added to a, yields zero. This number is also known as the opposite (number),[1] sign change,[2] and negation.[3] For a real number, it reverses its sign: the additive inverse (opposite number) of a positive number is negative, and the additive inverse of a negative number is positive. Zero is the additive inverse of itself.
...
Conversely, additive inverse can be thought of as subtraction from zero:

−a = 0 − a.

Hence, unary minus sign notation can be seen as a shorthand for subtraction (with the "0" symbol omitted), although in a correct typography, there should be no space after unary "−". — Additive inverse - Wikipedia

Thus to negate a number is to subtract that number from zero.

For example, to negate 2 is to subtract 2 from 0 which is -2. Conversely, to negate -2 is to subtract -2 from 0 which is 2.

Similarly, to negate 0 is to subtract 0 from 0 which is 0.

The point is that this cannot be called a "negation". If the thing, zero, is the very same prior to, and after, the proposed "negation" then there has been no negation. — Metaphysician Undercover

Do you also hold that adding zero to a number cannot be called "addition" because the number is the same before and after?

Or that dividing a number by one cannot be called "division" because the number is the same before and after?

"Negation", by definition, creates a statement which is distinct from that which is negated. There cannot be a "negation" with the negation being the very same statement as that which was negated. This cannot be called a "negation". — Metaphysician Undercover

Yet "negation", defined as zero minus a number, can be just that. Just as addition that includes zero and negative numbers is more general than addition of only positive numbers.