Exploring the Meaning of Ontology: Easy for Kids, Hard for Quantum Physicists

[Demystifier]

Ontology is the easiest and the hardest concept in the field of quantum foundations.

It is the easiest because even a child can understand it. When a child asks: "What is the world made of?", she asks about ontology. When you answer: "It's made of atoms" and when she asks "What does the atom look like?", she asks about ontology again. A child feels the concept of ontology in her bones, even if she never heard of that word. (An alternative word with a similar but not identical meaning is "reality".)

At the same time, the concept of ontology is the hardest concept because many mature physicists don't get it. When a child asks: "What does the atom look like?", a mature physicist will often give an answer that makes no sense to a child. That's because the child asks about ontology, while the mature physicist gives an answer that has not much to do with ontology. The concept of ontology is hard to define precisely in terms of other concepts with which physicists are familiar. Many physicists cannot get the meaning of that word even at an intuitive level. Unlike children, many mature physicists don't feel this concept in their bones. When they were children they would probably have no problems with understanding it, but now their brain does no longer work the same way it worked in the past. Now they can easily understand concepts they had no chance to understand when they were kids, but at the same time, some concepts that would be easy back then now look like a total gibberish. Ontology is one such concept.

Of course, that's not true for all physicists. Typically, physicists who like interpretations of QM such as Bohmian mechanics, GRW or many worlds have no problems with understanding the concept of ontology. They feel it in their bones, just like children. On the other hand, physicists who like interpretations from the Copenhagen/orthodox spectrum often have hard problems with understanding what the word "ontology" means. I have tried several times to explain them the meaning of the word "ontology", but without much success.

The question for everybody: How to explain the meaning of the word "ontology" such that even a mature orthodox quantum physicist can understand it?

 

[AndreiB]

The question for everybody: How to explain the meaning of the word "ontology" such that even a mature orthodox quantum physicist can understand it?

Ontology refers to those objects/entities that are postulated to exist by a certain theory.

In Newtonian mechanics you have a 3D (absolute) Euclidean space, time (also absolute). Part of the ontology are also particles (defined by their mass, location in the absolute space and their absolute velocity) and forces.

Anything else can be derived from those concepts.

 

[Morbert]

I think the point of contention between realists and antirealists is the notion of thoroughgoing intelligibility. There are everyday intuitions we take for granted. E.g. We all carry a naive realism, where what we see is more or a less a direct representation. This naive realism is at the heart of the child's question "what does X look like". X is made thoroughly intelligible through our sense of it. It might be the case that the extent to which "The Real" is intelligible is limited to how The Real make contact our experimental probing from our sense world, as opposed to some all-encompassing primitive ontology of enumerable things and properties.

 

[gentzen]

The orthodox quantum physicist is serious, so serious. First he needs to learn to laugh about ontology.
Meinong's Jungle and the quest for the married bachelor might be a good start. But honestly, I think the typical child will miss most of the jokes, at least I would have. But when I read Quine's On What There Is (1948), I couldn't stop laughing (and still laugh) when I read stuff like:

Wyman’s slum of possibles is a breeding ground for disorderly elements. Take, for instance, the possible fat man in that doorway; and, again, the possible bald man in that doorway. Are they the same possible man, or two possible men? How do we decide? How many possible men are there in that doorway? Are there more possible thin ones than fat ones? How many of them are alike? Or would their being alike make them one? Are no two possible things alike? Is this the same as saying that it is impossible for two things to be alike? Or, finally, is the concept of identity simply inapplicable to unactualized possibles? But what sense can be found in talking of entities which cannot meaningfully be said to be identical with themselves and distinct from one another?

 

[martinbn]

I completely disagreewith one part of what you said. I think that it is precisely the interpretation/foundation of QT people that don't know what ontology is. They use the term for things that cannot be ontological. And when you point it out, they say "Ah, well, by ontology I actually mean this." Just like with the terms "locality", " interaction", and some others.

I agree that the rest of the physicists are very sloppy (as usual) when it comes to ontology, or they don't even bother with ontology.

 

[AlexCaledin]

Well, the world is made of all the classically observable things/events and the certain objective math correlating them... The objective math can be formulated this way and that, using various mathematical objects.

 

[Quantumental]

Great topic. The DNA of ontology is the distinction between what is and what can be conceived. Ontology is binary, either it is or it isn't. Either the world is made up of X, Y, Z or it isn't. Ontology, as you alluded to is literally the thing that distinguishes real from conceptual. We could all imagine virtually anything, but explaining how your 'virtual anything' is real, you automatically infer ontology. I guess a good 'common sense' equivalence would be that of perceived sound vs soundwaves. Sound as we know it exist in our qualia - which is often wrong compared to reality, the structure of sound waves however do exist whether anything is there to interpret it or not. That is the ontology of sound.

 

[vanhees71]

Well, if ontology is about what Kant called "Ding an sich" (I don't know, how to translate Kant into English), then it's something not subject to the natural sciences, because the natural sciences are about nature as can be objectively observed. Objectively means the observations must be reproducible and independent from the individual making the expression.

Take, e.g., classical electrodynamics and the electromagnetic field. Is the classical electromagnetic field ontic, providing an "ontology of light"? I don't know, because if one thinks about it, it's just a mathematical framework describing the phenomena of emission of "light" (e.g., the Sun as a thermal light source) and its registration at some distant place. We never directly observe the field but just its action on some matter. Usually it's the photoelectric effect which provides a signal in a detector (photoplate, CCD cam, the retina of our eyes etc). So "is light really" the electromagnetic field or is it the photo electron ejected from matter which then leads to a "signal" we can preceive?

I think an answer to this question is pretty irrelevant from the point of view of the natural sciences. It's simply easier to talk about the electromagnetic field about some "thing" with a kinematics and dynamics in its own right as if it were something like a substance though for a substance it doesn't have a typical "substance-like" (charge-like) conserved quantity. It basically has been introduced by Faraday (based on his experience with experiments and observations) and then brought in a mathematical form by Maxwell to have a concise way to talk about these electromagnetic phenomena.

 

[EPR]

The question for everybody: How to explain the meaning of the word "ontology" such that even a mature orthodox quantum physicist can understand it?

They should switch camps to MWI or BM.

Too bad the CI is probably right, though. You get sort of classical "particles" behavior whenever the particles can be detected(straight line trajectories in the Double slit experiment after the detector). If there is no way for them to be detected in any practical sense, they always behave quantum mechanically(delocalized, spread out, interfering with themselves...). What does this mean for the bottle of shampoo in my bathroom? Honestly, I don't know and I can't tell. I expect it to be there when I look and it's there when I look for it. It's like this classical paradigm is too profound for most physicists to discern its due importance.

 

[gentzen]

Well, if ontology is about what Kant called "Ding an sich" (I don't know, how to translate Kant into English), then ...

I hope that nobody here will try to claim that ontology has anything to do with that.

Take, e.g., classical electrodynamics and the electromagnetic field. Is the classical electromagnetic field ontic, providing an "ontology of light"?

I would say yes, it does.

I don't know, because if one thinks about it, it's just a mathematical framework describing ...

A mere framework would not yet be enough for me. There should also be objects/entities where identity/equality has a clear unambiguous meaning. Maybe exact equality is too much, some way to quantify whether two objects are nearly identical or not could be sufficient.

I think an answer to this question is pretty irrelevant from the point of view of the natural sciences. It's simply easier to talk about the electromagnetic field about some "thing" with a kinematics and dynamics in its own right ...

The stuff you can observe are typically local events. Postulating locally defined object to "explain" those observations is not yet very interesting. More interesting is the question whether you can also find something globally defined that is locally identical/equal to such locally defined object, and how "complicated" such a globally defined thing has to be. Whether this is relevant from the point of view of the natural sciences is not totally clear. Typically the existence of such a globally defined thing is taken to be a proof of consistency, but that is treacherous. (For example, the existence of the natural numbers does not prove that Peano arithmetic is consistent.) There might be better ways to establish consistency and clarify its meaning.

 

[Fred Wright]

My dictionary gives two distinct meanings to the word ontology:

• 1.
  the branch of metaphysics dealing with the nature of being.
• 2.
  a set of concepts and categories in a subject area or domain that shows their properties and the relations between them.
  I think it's a mistake to try to integrate definition #1 with definition #2. This could lead down a thorny and uncomfortable path. For example, the concept of "intelligent design" in cosmology and biology. As a physicist, I wouldn't try to gain knowledge of my subject area by appealing to my knowledge of metaphysics derived from my study of religious scripture.
  

 

[Killtech]

It is the easiest because even a child can understand it. When a child asks: "What is the world made of?", she asks about ontology. When you answer: "It's made of atoms" and when she asks "What does the atom look like?", she asks about ontology again. A child feels the concept of ontology in her bones, even if she never heard of that word. (An alternative word with a similar but not identical meaning is "reality".)

Often is it the things most seemingly obvious things that turn out to be the most complex. Being an autist i can tell you most people attribute this description to many many highly complex interacting non linear systems which full description is still beyond our current science - i.e. social interaction, common sense and what not. And there are people that do have to learn all that the hard way...

But learning it this way, you learn to understand the system slowly in a conscious rational way to the point where where you can start doing further reaching conclusions which people with a natural intuition are blind. Because in fact their thought process is governed subconsciously without them being able understanding how their perception works - and because it seems obvious they never bother to think about it more.

The issue with ontology is that it fails to produce something mathematically tangible and a lot of words are thrown around without making any progress. I had made a lot of thoughts myself on things that only later i learned were related. We really need a Bourbaki style mathematical toolbox to be able to properly formulate these kind of questions. So ask yourself, can you mathematically define what you are looking for?

 

[gentzen]

The issue with ontology is that it fails to produce something mathematically tangible and a lot of words are thrown around without making any progress. I had made a lot of thoughts myself on things that only later i learned were related. We really need a Bourbaki style mathematical toolbox to be able to properly formulate these kind of questions. So ask yourself, can you mathematically define what you are looking for?

Well, I believe some mathematicians like Harvey Friedman, Steve Simpson, ... or Walter Dean at least had fun to investigate connections between ontology, reverse mathematics, and Computability Theory and Foundations of Mathematics. And that fun also included some philosophical conclusions:

Some philosophical observations and guarded conclusions:

• existence simpliciter vs conditional existence
• consistency => existence ?
• ontological commitment de dicto and de re

 

[Killtech]

Well, I believe some mathematicians like Harvey Friedman, Steve Simpson, ... or Walter Dean at least had fun to investigate connections between ontology, reverse mathematics, and Computability Theory and Foundations of Mathematics. And that fun also included some philosophical conclusions:

Hehe, nice. I have to browse it through.

Though, when i look at @Demystifier putting a statement like "What does the atom look like?". Frome there i can see that this is looking for something quite specific yet without a formal definition of what exactly. I can try reverse engineer that question and see that this is asking for something very different from the mathematical meaning of existence. Let's call it "physical existence", something for which a proper definition is missing.

Now things come into existence physically by interaction - only this way we can become aware of their presence. and anything that interacts affects the observation somehow, which we can put as a generalized concept of how things "look like".

So the question is really asking to identify and point out all the information in a theory that is driving the observation. That information can be very practically seen as "real" since any implementation/simulation of the observables will have to have this information one way or another. As in a computer implementation will have to store this information (or a bijective transformation thereorof) in order to be able to correctly simulate the desired behavior. Take for example the screen in front of you: even if we would live in a artificial reality like in the Matrix films, that screen would still exist in that sense, albeit it's wouldn't be made of real matter but rather a charge distribution in transistors on a server that simulates it. Either way the information representing it has to be there in one form or another and that's what makes it "real" perhaps? In this sense i would prefer to call it irreducible.

The opposite of this are artefact information. They are optional and specific to a certain descriptions needed for technical reasons or otherwise. A pendulum moving only on a 2d subsurface has only 2 degrees of freedom, so describing it with 3 instead of 2 coordinates adds one additional information that doesn't drive observation but instead requires a symmetry / conserved quantity to mark it as redundant. If the entire world would consist only of that one pendulum, would the 3rd dimension be even real? Anyhow, this type of information i prefer to call reducible (and Lagrangian equations of second kind are an example to get rid of excess artifact information/dimensions).

You can put it like this: the irreducible information represent the equivalence class of all possible equivalent theories/interpretations (in terms of producing identical observations) while things like symmetries are specific to a theory or rather the technically framework used by a theory.

 

[Demystifier]

We really need a Bourbaki style mathematical toolbox to be able to properly formulate these kind of questions. So ask yourself, can you mathematically define what you are looking for?

Good point!

 

[vanhees71]

I hope that nobody here will try to claim that ontology has anything to do with that.I would say yes, it does.A mere framework would not yet be enough for me. There should also be objects/entities where identity/equality has a clear unambiguous meaning. Maybe exact equality is too much, some way to quantify whether two objects are nearly identical or not could be sufficient.The stuff you can observe are typically local events. Postulating locally defined object to "explain" those observations is not yet very interesting. More interesting is the question whether you can also find something globally defined that is locally identical/equal to such locally defined object, and how "complicated" such a globally defined thing has to be. Whether this is relevant from the point of view of the natural sciences is not totally clear. Typically the existence of such a globally defined thing is taken to be a proof of consistency, but that is treacherous. (For example, the existence of the natural numbers does not prove that Peano arithmetic is consistent.) There might be better ways to establish consistency and clarify its meaning.

As GR (to some extent already SR) teaches us, all we can observe are local events (coincidences of space-time points). It took quite some time until this was established in the 8-10-year struggle to establish a relativistic theory of gravitation by Einstein with his final version of general relativity (cf. the (in)famous "hole argument", which is still discussed in some philosophical works but completely irrelevant in contemporary use of GR in astrophysics and cosmology).

The field concept is thus the preferred way to describe relativistic physics. Even the apparently simpler point-particle concept in classical relativity is highly problematic. All you can consistently define are non-interacting "free" particles. There's no consistent classical theory of interacting point particles. Also on the QT level the only FAPP working concept is the perturbative treatment of local relativistic QFTs.

 

[Lord Jestocost]

The question for everybody: How to explain the meaning of the word "ontology" such that even a mature orthodox quantum physicist can understand it?

There is no need for an ontology in orthodox quantum theory. Jean Bricmont in his book review “Looking for a quantum ontology”( Metascience (2011) 20:103–106) about 'Bohmian mechanics: The physics and mathematics of quantum theory' by Detlev Dürr and Stefan Teufel:

“To explain the problem, consider first classical mechanics. In this theory, the notion of ‘force’ (acting instantaneously throughout the universe) is also obscure. Nevertheless, there are particles in the universe, on which the forces act—which determines their motion. Similarly, in classical electromagnetism, the notion of waves propagating in vacuum is obscure, but again, the waves act on particles and guide their motion. Similar remarks hold for the curved space–time of General Relativity. In all those theories, there is an ontology, to use the expression of Dürr and Teufel, namely something that exists independently of any human observation or even independently of the existence of mankind itself and whose evolution is described by the laws of physics.

There is nothing of the sort in ordinary quantum mechanics. Indeed, in the latter, the abstract vector called the wave-function has no meaning whatsoever, except that it enters into an algorithm that predicts (very accurately) ‘results of measurements’. There is no ontology in ordinary quantum mechanics—there is nothing ‘out there’ that the theory speaks about. Note that this problem has nothing to do with the issue of determinism: one could very well imagine a physical theory whose most fundamental equations are stochastic; it would still be a physical theory about something (particles, waves, whatever). The problem does not come from realism either, at least not as it is usually meant philosophically. In ordinary quantum mechanics (except in the most crazy versions of it), there is something outside of our minds—the measuring devices—and they are in states that are perfectly definite ‘after measurements’. It is just that there is nothing else; in particular nothing physical, that these measuring devices are made of. Or rather, there may be something ‘out there’ but, like the Ding an sich, it is radically unknowable.” [italics in original, LJ]

EDIT: Bricmont is a proponent of the “de Broglie–Bohm pilot-wave theory”

 

[AndreiB]

There is no need for an ontology in orthodox quantum theory.

Of course there is. Any explanation requires a set of primitive objects. Any phenomenon must be explained in terms of those primitive objects. You cannot start with nothing.

Orthodox QM does have an ontology. You need to postulate 3D space and time, otherwise you cannot specify how the system is prepared and measured. You need to postulate that suitable instruments exist, otherwise Born's postulate is meaningless and so on. Or, you may postulate something else and derive the instruments from it.

Nima Arkani does not like spacetime either. But he still needs to postulate something from which spacetime emerges. He still needs an ontology.

There is nothing of the sort in ordinary quantum mechanics. Indeed, in the latter, the abstract vector called the wave-function has no meaning whatsoever, except that it enters into an algorithm that predicts (very accurately) ‘results of measurements’.

Not all mathematical objects (in this case the wave-function) need to be part of the ontology.

There is no ontology in ordinary quantum mechanics—there is nothing ‘out there’ that the theory speaks about.

I disagree. Space and time are implicitly required and also the instruments involved in the preparation/measurement procedures.

In ordinary quantum mechanics (except in the most crazy versions of it), there is something outside of our minds—the measuring devices—and they are in states that are perfectly definite ‘after measurements’.

Exactly.

It is just that there is nothing else;

This does not follow.

in particular nothing physical, that these measuring devices are made of.

If those devices are part of the ontology (primitives), then they are not made from anything else.
If those devices are described in terms of something else, like electrons and quarks, then electrons and quarks are part of the ontology, and the instruments are emergent.

Or rather, there may be something ‘out there’ but, like the Ding an sich, it is radically unknowable.

QM does not say that.

 

[pinball1970]

Easily one of the most confusing threads I have read on physics forums and that is up against some pretty stiff competition I can tell you.

 

[Morbert]

Of course there is. Any explanation requires a set of primitive objects. Any phenomenon must be explained in terms of those primitive objects. You cannot start with nothing.

Orthodox QM does have an ontology. You need to postulate 3D space and time, otherwise you cannot specify how the system is prepared and measured. You need to postulate that suitable instruments exist, otherwise Born's postulate is meaningless and so on. Or, you may postulate something else and derive the instruments from it.

Nima Arkani does not like spacetime either. But he still needs to postulate something from which spacetime emerges. He still needs an ontology.

I think there are crossed wires here. What you are arguing against is a quantum nihilism where quantum mechanics can be premised in some possible world where nothing exists, instruments or otherwise.

Instead when people say ontology does not enter into orthodox QM, they mean orthodox QM makes assertions about phenomena rather than noumena. QM is still of course premised in some actually-existing reality.

 

[AndreiB]

Instead when people say ontology does not enter into orthodox QM, they mean orthodox QM makes assertions about phenomena rather than noumena. QM is still of course premised in some actually-existing reality.

In this case, they use "ontology" in a restricted sense. For a QBist there are no noumena, but I think QBism still has an ontology. Agents exist, their observations (experiences) exist. This is QBism's ontology.

 

[vanhees71]

What ontology does Qbism have? They claim there's meaning in a probability for a single event. I never could get what that meaning should be. I always thought an ontic interpretation assumes more than the existence of observations, i.e., there should be more meaning in a quantum state than a way to calculate probabilities, but precisely this is at odds with the way how the formalism is successfully related to nature as observed in real-world labs. For me some years ago, after reading Ballentine's RMP paper and then his textbook, I came to the conclusion that the minimal statistical interpretation is all that's needed to use QT as a physical theory and which is most consistent with the usual practice of using it to describe the objective facts, and I still think that measurement results are objective facts about Nature and not some "arbitrary construct" or something like that.

 

[Demystifier]

What ontology does Qbism have? They claim there's meaning in a probability for a single event. I never could get what that meaning should be. I always thought an ontic interpretation assumes more than the existence of observations, i.e., there should be more meaning in a quantum state than a way to calculate probabilities, but precisely this is at odds with the way how the formalism is successfully related to nature as observed in real-world labs. For me some years ago, after reading Ballentine's RMP paper and then his textbook, I came to the conclusion that the minimal statistical interpretation is all that's needed to use QT as a physical theory and which is most consistent with the usual practice of using it to describe the objective facts, and I still think that measurement results are objective facts about Nature and not some "arbitrary construct" or something like that.

If I understood you correctly, you simultaneously hold the following 4 beliefs:
1. Probability has no meaning for a single event.
2. Wave function has a meaning for a single event.
3. The only meaning of wave function is to determine the probability.
4. The 3 statements above are mutually consistent.
Do I miss something?

 

[vanhees71]

No, that's pretty much what I think in a nutshell, although it sounds inconsistent put in this ultracondensed form. First of all, of course the 2nd point must read

2. The quantum state, represented by a statistical operator, has a meaning for a single event.

It is important to make this apparently selfcontradictory scheme consistent to note that the meaning of the state is twofold: On one hand it describes a preparation procedure (or rather an equivalence class of preparation procedures) for a single system. A preparation procedure must be reproducible such that you can prepare stochastically independent ensembles with sufficient accuracy. The meaning of the state then is to provide the probabilities for the outcome of measurements, which can be tested on such prepared ensembles in the usual statistical sense.

 

[PeterDonis]

First of all, of course the 2nd point must read

2. The quantum state, represented by a statistical operator, has a meaning for a single event.

But if the meaning of the state is to provide probabilities for measurement outcomes, then it can't have a meaning for a single event, because you can't measure probabilities from a single event. You need to have an ensemble of outcomes from an ensemble of identically prepared systems.

If, OTOH, you want to interpret the quantum state as describing the preparation procedure used for a single run of the experiment, and having meaning for that single event in that sense, then that has nothing to do with probabilities and the 3rd point would be false.

 

[vanhees71]

The point is that the quantum state has both meanings: It's on the one hand the formal description of a preparation procedure (referring to individual systems) but on the other implies only the description of probabilistic properties of the system, i.e., the probabilities for the outcome of measurements on the so prepared system. To test it, there's no other way than to prepare a sufficiently large ensemble (sufficiently large depending on the significance level you want achieve) and evaluate the corresponding statistics for this test.

 

[PeterDonis]

The point is that the quantum state has both meanings

Which means that point 3 in what @Demystifier posted is false. Determining the probability is not the only meaning of the quantum state.

 

[vanhees71]

You are right, one must be very careful with the concise formulation. I'm not sure how to put it in such a general short way without being imprecise.

Consider the example for a pure state, which means preparation of a most-determined state in QT. A direct preparation procedure is to measure a complete set of compatible observables and use a filter to select only the system to be prepared having determined values for this complete set of compatible observables. Even in this "most complete preparation" the meaning of the state is still "probabilistic and only probabilistic" for any other observables that are not compatible with the chosen complete set of compatible observables.

This of course holds true for any other possible preparation procedure, also for "mixed states". That's how I understand the 3rd statement.

 

[Killtech]

What ontology does Qbism have? They claim there's meaning in a probability for a single event. I never could get what that meaning should be. I always thought an ontic interpretation assumes more than the existence of observations, i.e., there should be more meaning in a quantum state than a way to calculate probabilities, but precisely this is at odds with the way how the formalism is successfully related to nature as observed in real-world labs.

probability for a single event is total nonsense... if one were talking about Kolmogorov probabilites. Quantum probability gives up that idea by breaking the very thing in classical probability theory that prevents that to happen. If you allow the information contained in a probability distribution to effectively couple to physical interaction and have a uniquely distinguishible effect on outcomes... that's what you get. That's really the difference. When you want to use the word "probability" in QT, you have to let go of your Kolmogorovian interpretation of it. I still don't see it as a reasonable choice and would have preferred to formulate it within classical probability theory framework to prevent such weirdness in terminology and confusion but whatever.

 

[physika]

always behave quantum mechanically(delocalized, spread out, interfering with themselves...)

But exists

 

[martinbn]

If I understood you correctly, you simultaneously hold the following 4 beliefs:
1. Probability has no meaning for a single event.
2. Wave function has a meaning for a single event.
3. The only meaning of wave function is to determine the probability.
4. The 3 statements above are mutually consistent.
Do I miss something?

2. Is unclear to me. What does it mean?

 

[Demystifier]

2. Is unclear to me. What does it mean?

I don't know, ask @vanhees71!

 

[Lord Jestocost]

I always thought an ontic interpretation assumes more than the existence of observations...

Indeed. Ontic interpretations have to fulfill a twofold task:

1) A claim concerning the existence of the “stuff” which physics is about
2) A claim concerning the character of its existence.

Statements that some “stuff” exists are meaningless phrases when not complemented with a statement how that “stuff“ exists. Orthodox quantum mechanics is silent about point 2, thus not belonging to what is denoted ontic interpretations. Orthodox quantum mechanics has never denied that something “exists” or – so to speak – that there is an “out there”. That was and is a fairy tale still spooking around in the realm of folk science.

 

[vanhees71]

probability for a single event is total nonsense... if one were talking about Kolmogorov probabilites. Quantum probability gives up that idea by breaking the very thing in classical probability theory that prevents that to happen. If you allow the information contained in a probability distribution to effectively couple to physical interaction and have a uniquely distinguishible effect on outcomes... that's what you get. That's really the difference. When you want to use the word "probability" in QT, you have to let go of your Kolmogorovian interpretation of it. I still don't see it as a reasonable choice and would have preferred to formulate it within classical probability theory framework to prevent such weirdness in terminology and confusion but whatever.

I have no clue what you are talking about. Take the most simple example. Say we have a neutron's spin prepared in the $\sigma_z=+1/2$ eigenstate, $\hat{\rho}=|1/2 \rangle \langle 1/2|$. Now use a Stern-Gerlach apparatus to measure $\sigma_x$. All I know from knowing that the neutron is prepared in the state $\hat{\rho}$ are the probabilities for the two possible outcomes of my measurment, $\sigma_x=+1/2$ or $\sigma_x=-1/2$ being $P_{\sigma_x=1/2}=P_{\sigma_x=-1/2}=1/2$. For such a physically meaningful experiment the quantum probabilities can be interpreted in the Kolmogorovian sense, but it doesn't so much matter which axiomatic system for probability theory you use. It's rather a question what it means for physics to say the probability for getting the one or the other result is 1/2. For me it doesn't tell you anything for just a single measurement. To experimentally test whether the prediction for the $\sigma_x$-measurement is correct, I simply have to prepare many neutrons in the state $\hat{\rho}$, measure $\sigma_x$ and count how many times I get either result using the usual statistical analysis methods to provide a significance for the validity or invalidity of the predicted probabilities. I never understood, what the Qbists (or even classical Bayesianists) mean when they say probabilities would have some meaning for a single event.

 

[vanhees71]

2. Is unclear to me. What does it mean?

I've explained how I understand this statement above: States (statistical operators; wave functions are only special cases representing pure states in the usual sense as they define the corresponding statistical operator or equivalently a ray in Hilbert space) are, on the one hand, associated with an equivalence class of preparation procedures and as such refer to a single system. On the other hand the preparation in a state only implies probabilistic properties for the system, i.e., it provides the probabilities for the outcome of the measurement of any observable of the system. In my opintion, this can be tested only on an ensemble of equally prepared systems. I could never make sense of Bayesianistic claims that a probability has any meaning for a single event.