What does it take to solve the measurement problem? (new paper published)

[PeterDonis]

It provides that the Bohmian particle cannot enter more than one branch of the wave function

Ah, I see.

 

[Demystifier]

In the Bohmian interpretation, that's true, as I've already said. And since that is your preferred interpretation, I can see why it seems this way to you. But there are other interpretations.

Can you name one interpretation that does not solve the Hyperion problem? I think they all do, whatever Hossenfelder says.

 

[Fra]

I would not state it this way, because if we believe that QM can't explain the observed motion of Hyperion, and by extension of any system that shows classical dynamics beyond the Ehrenfest time, then why would we describe what we are looking for to take its place as "solving the measurement problem", since the measurement problem itself is a problem of QM, and if we accept the first part of her argument, we are discarding QM?

As I understood it, this is one the reason that pure "interpretations" alone are getting us nowhere. I agree on that.

Some sort of modification or revision of QM seems required? That does not mean we literally throw it away and forget about it. Just like QM should have a correspondence limit to CM in the massive objects. The Revised measurement theory should also have a correspondence to QM as it stands for the case of small subsystems and short timescales (where what is short and small is of course a relative notion). That the timescale is important for the paradigm of QM I think it clear also if you consider the processing of inferring the "timeless laws". The timeless laws, are only true for a timescale shorter than their inference time. Any other scenario seems to force us into a cosmological perspective (from the agent).

/Fredrik

 

[Morbert]

See also one of Sabine Hossenfelder's blog entries:
https://backreaction.blogspot.com/2022/05/chaos-real-problem-with-quantum.html

I don't understand her problem with the textbook response. Given some initial state that includes both Hyperion and its environment, the linear evolution of this state in Hilbert state will predict a chaotic evolution of Hyperion in phase space that lasts longer than 20 years. She likens this to a model of a die that gives the right average but the wrong relative frequencies (e.g. predicting the die will return 106 approximately half the time). What is the Hyperion analogue to the 106 case?

Consider Hyperion $H$, its environment $E$, and a device $D$ on Earth that observes Hyperion's evolution and records one of two ouctomes:"Hyperion is chaotic" or "Hyperion is not chaotic". The initial state of such a system is $$\rho_H\otimes\rho_E\otimes\rho_D$$Let's say the system first evolves for 100 years, such that Hyperion and her environment (but not the device) entangle, resulting in the new state $$\rho^{'}_{H+E}\otimes\rho_D$$Then, physicists observe Hyperion using the device over some time interval (say another 100 years) such that the three systems are entagled in some final state$$\rho^{''}_{H+E+D}$$Classically, we would expect the device to record an outcome "Hyperion is chaotic", and indeed that is what QM would presumably predict with probability close to 1. I.e. if the pointer state "Hyperion is chaotic" has some projector $R_c$, then $\mathrm{Tr}\left[R_c\rho^{''}_{H+E+D}\right] \approx 1$

 

[atyy]

Hossenfelder cites Zurek and Berry references which solve the Hyperion problem with decoherence. Then she rejects the solutions, because they only produce entangled states. But then if you don't look at Hyperion, quantum mechanics doesn't say that it's there, so there is no chaotic motion unless one looks at it. If one looks at it, then one would have measurement and collapse (or whatever one wants to call it). So the answer in the standard interpretation is we don't make any statement as to whether Hyperion is there or not when we don't look at it. So in the end, the Zurek and Berry references should solve the problem.

Zurek: https://arxiv.org/abs/quant-ph/0105127

Berry: https://michaelberryphysics.files.wordpress.com/2013/07/berry337.pdf

 

[Lord Jestocost]

Summary: preprint by Jonte Hance and Sabine Hossenfelder on the measurement problem and quantum foundations

Yesterday Jonte Hance and Sabine Hossenfelder published this preprint on the arXiv: https://arxiv.org/abs/2206.10445

What does it take to solve the measurement problem?​

Berthold-Georg Englert in "On Quantum Theory" (The European Physical Journal D, 2013 - Springer):

"One preexisting concept of quantum theory is the event, such as the emission of a photon by an atom, the radioactive decay of a nucleus, or the ionization of a molecule in a bubble chamber. The formalism of quantum theory has the power to predict the probabilities that the events occur, whereby Born’s rule [4] is the link between formalism and phenomenon. But an answer to the question Why are there events? cannot be given by quantum theory.
. . . . . .
Fifth, since neither decoherence nor any other mechanism select one particular outcome (see Sec. 8), the whole “measurement problem” reduces to the question Why is there one specific outcome? which is asking Why are there randomly realized events? in the particular context considered. This harkens back to Sec. 1, where we noted that quantum theory cannot give an answer.
In summary, then, the alleged “measurement problem” does not exist as a problem of quantum theory. Those who want to pursue the question Why are there events? must seek the answer elsewhere." [bold by LJ]

 

[Fra]

Berthold-Georg Englert in "On Quantum Theory" (The European Physical Journal D, 2013 - Springer):

"One preexisting concept of quantum theory is the event, such as the emission of a photon by an atom, the radioactive decay of a nucleus, or the ionization of a molecule in a bubble chamber. The formalism of quantum theory has the power to predict the probabilities that the events occur, whereby Born’s rule [4] is the link between formalism and phenomenon. But an answer to the question Why are there events? cannot be given by quantum theory.
. . . . . .
Fifth, since neither decoherence nor any other mechanism select one particular outcome (see Sec. 8), the whole “measurement problem” reduces to the question Why is there one specific outcome? which is asking Why are there randomly realized events? in the particular context considered. This harkens back to Sec. 1, where we noted that quantum theory cannot give an answer.
In summary, then, the alleged “measurement problem” does not exist as a problem of quantum theory. Those who want to pursue the question Why are there events? must seek the answer elsewhere." [bold by LJ]

The perspective in that paper IMO misses a whole important facet - which at least for me, and from the interacting agent perspective - is the most important one: Unification of interactions and measurements, for the purpose of understanding unification of forces and the nature of causality.

We can say whatever we want, but we still do now have a consistent conceptually coherent unification of QM and GR, and for some, the mesurement problem is most likely entanglet with that problem. Even the conceptual understanding of all other forces WITHOUT gravity, is not satisfactory I think. We have finetuning that you may or may not think is a problem, but I think the fine tuning is "apparent" because we do not fully understand thinks satisfactory (we can only describe it). If one thinks there is no meaning in asking why, that is a personal choice, but I think there is more to understand.

The measurement problem is IMO related to the causality in the following way. Most would agree that an observers action would be depend on what it knows and has at hand, ie. a kind of locality principle. But as any subsystem of the universe (wether classical or not) should by the principle of observer democracy be a valid obserever. Then by equivalence, the rules of inference and choice of actions of such observer, must be indistinguishalbe from the laws of physics and it's causal principles?

This is IMO at the heart of the measurment problem, and if solve it likely will have implications for understanding how the interactions and it's phenomenology are unified, and to see the naturality of things we today see require finetuning because we have learned to describe things, and now understand the full origin of all symmetries.

/Fredrik

 

[bhobba]

That's not her argument. Her argument is:
The long term motion of Hyperion is in agreement with classical chaotic mechanics. Thus to explain it by quantum mechanics we need to show that quantum mechanics reproduces the classical motion for these long time scales, which has not been done so far. This, in turn, is related to the measurement problem.

I just saw Hossnefelder's video, and that was my first reaction - quantum mechanics should reproduce the classical motion over long-time scales. I didn't realise it hadn't been proven yet. Interesting.

Thanks
Bill

 

[bhobba]

In summary, then, the alleged “measurement problem” does not exist as a problem of quantum theory. Those who want to pursue the question Why are there events? must seek the answer elsewhere." [bold by LJ]

Or why when we measure position we get one answer. I think the construction of our measuring apparatus and the definition of position might have something to do with it. The answer is not in QM but in the questions we ask. When not measured we cant say anything - but when measured we certainly can.

Thanks
Bill

 

[Demystifier]

I just saw Hossnefelder's video, and that was my first reaction - quantum mechanics should reproduce the classical motion over long-time scales. I didn't realise it hadn't been proven yet.

The correct statement is that it hasn't been proven yet in an interpretation independent way. Different interpretations reproduce classical motion in different ways, but without an interpretation one cannot get it.

 

[vanhees71]

Then you say that the classical limit is unphysical? Any physically observable phenomenon must be, of course, independent of the interpretation of QT used.

For the most simple case of a single particle in non-relativistic QM, you get the classical limit by the eikonal approximation of the (time-dependent) Schrödinger equation. In the path-integral formulation that's equivalent to the saddle-point approximation and is valid when the typical values of the action around the classical trajectory under investigation are very large compared to $\hbar$. I think there's nothing interpretation-dependent in this argument.

 

[A. Neumaier]

The correct statement is that it hasn't been proven yet in an interpretation independent way.

It (Namely that ''quantum mechanics should reproduce the classical motion over long-time scales'') also hasn't been proven yet in an interpretation dependent way, thus the additional phrase only obscures matters. If you object, please point to a proof.

 

[A. Neumaier]

For the most simple case of a single particle in non-relativistic QM, you get the classical limit by the eikonal approximation of the (time-dependent) Schrödinger equation. In the path-integral formulation that's equivalent to the saddle-point approximation and is valid when the typical values of the action around the classical trajectory under investigation are very large compared to $\hbar$. I think there's nothing interpretation-dependent in this argument.

Yes, and the eikonal approximation is valid only at short times. Nobody so far proved something interesting for very long times, at the constant physical value of $\hbar$.

 

[Demystifier]

It (Namely that ''quantum mechanics should reproduce the classical motion over long-time scales'') also hasn't been proven yet in an interpretation dependent way, thusd the additional phrase only obscures matters. If you object, please point to a proof.

By "proven" I didn't mean in the strict mathematical sense. Different interpretations explain (rather than prove in the strict mathematical sense) it, but it cannot be explained in interpretation independent way.

 

[Demystifier]

Then you say that the classical limit is unphysical? Any physically observable phenomenon must be, of course, independent of the interpretation of QT used.

The classical limit is physical, but we don't fully understand how it arises. That's one of the reasons why we have different interpretations, to explain the classical limit.

For the most simple case of a single particle in non-relativistic QM, you get the classical limit by the eikonal approximation of the (time-dependent) Schrödinger equation. In the path-integral formulation that's equivalent to the saddle-point approximation and is valid when the typical values of the action around the classical trajectory under investigation are very large compared to $\hbar$. I think there's nothing interpretation-dependent in this argument.

This is only part of the story. Decoherence also plays an important role in the classical limit, while the argument above does not involve decoherence.

 

[gentzen]

It (Namely that ''quantum mechanics should reproduce the classical motion over long-time scales'') also hasn't been proven yet in an interpretation dependent way, thusd the additional phrase only obscures matters. If you object, please point to a proof.

I currently do understand how preparation and "interaction" of sufficiently small systems (certainly not the whole universe) can be modeled within Bohmian mechanics. The way how to model measurement is not fully clear to me. (What is unclear to me is that the individual trajectory is not (supposed to be) observable, but maybe "many particle statistics" of trajectories are "supposed to be" observable.)

But even if this would be clarified (for example in a separate thread, or by a reference to some paper explaining this), I guess it would still not be "sufficient proof" for you.

 

[A. Neumaier]

By "proven" I didn't mean in the strict mathematical sense. Different interpretations explain (rather than prove in the strict mathematical sense) it, but it cannot be explained in interpretation independent way.

''proofs'' in the sense of a formal limit $\hbar\to 0$ don't mean much for long times; the time limit for which these limits are valid are of the order of a power of $\hbar$ (times context-dependent constants that producing the right units), and are microscopically small for the physical value of $\hbar$.

This is a manifestation of the fact that different limits (here $\hbar\to 0$ and $t\to\infty$) do not commute in general.

In this sort of questions, this is of primary physical relevance, not only a mathematical detail. Sabine Hossenfelder referred to the real problem, not to the superficial cure.

 

[zekise]

There is no such thing as the measurement problem. This is the creation of some very dated and wrong ideas, picked up by science writers to create some drama in their profession.

In fact observer A can take a measurement of a particle and the particle wave function would not "collapse"!

Imagine we have two observers A and B. A is coherent w.r.t. the environment E and the observer B, and observer B is embedded in E (entangled with E).

A particle arrives in a superposition of states for both A and B. A measures the particle and it "collapses" (an unfortunate word) for A. But the particle has NOT collapsed for B.

The state of the particle depends on who is asking or measuring. There is no such thing as an absolute wave function. A wave function is always between two (or more) parties. Just like velocity. You can't have V(a) the absolute velocity of object a. You can only have V(a, b).

This is the Relational Interpretation of Carlo Rovelli (1994).

Sabine Hossenfelder is still stuck with the measurement problem. Instead of making videos on 5G cell phones, geothermal energy, social matters, she needs to upload a new release of physics.

 

[bhobba]

There is no such thing as the measurement problem.

To me, like probability, it is just an update of knowledge. But it is a subtle issue to do with the ontological status of the quantum state. I am not ready to dismiss it as a total non-issue, I really like Sabine and watch all her videos but must admit she is a bit fixated on it IMHO.

I think Bell's Theorem is a much more important issue to do with properties between observations. Accepting locality does mean quantum objects do not have properties (except the state) between observations, which is a rather startling statement about how the world works. Our world seems to be created by interactions between quantum objects rather than the objects themselves. I think they are real in the sense they are part of a reality independent from us, but not in the sense they have properties all the time (only when measured). It fits in nicely with the QFT view of things IMHO, as it should.

Thanks
Bill

 

[A. Neumaier]

There is no such thing as the measurement problem. This is the creation of some very dated and wrong ideas

The experts in the field don't agree with you. Have you read Schlosshauer's book on decoherence?

 

[bhobba]

The experts in the field don't agree with you. Have you read Schlosshauer's book on decoherence?

Excellent book. He does delve deeply into the issue.

Thanks
Bill

 

[Fra]

There is no such thing as the measurement problem. This is the creation of some very dated and wrong ideas, picked up by science writers to create some drama in their profession.

In fact observer A can take a measurement of a particle and the particle wave function would not "collapse"!

Imagine we have two observers A and B. A is coherent w.r.t. the environment E and the observer B, and observer B is embedded in E (entangled with E).

A particle arrives in a superposition of states for both A and B. A measures the particle and it "collapses" (an unfortunate word) for A. But the particle has NOT collapsed for B.

The state of the particle depends on who is asking or measuring. There is no such thing as an absolute wave function. A wave function is always between two (or more) parties. Just like velocity. You can't have V(a) the absolute velocity of object a. You can only have V(a, b).

This is the Relational Interpretation of Carlo Rovelli (1994).

One of Rovelli's sound points is that there are no absolute observations, he even goes on to acqknowledge that there are no absolute relations between observations, it should take a third observer to assess it. But onfortunately at some point he claims that communication between observers follows the rules of QM. So he ends up explaining nothing IMO. I think the reasons is that while he at the same time wants to make a nice interpretation, he does not wish to CHANGE the theory, so his solution is conservative. And I think that is a mistake.

Even if the collapse is relative, and I agree there, that is not the real problem. The problem is to consider what happens if one if observers is participating in the interaction, and is makde both a quantum system from the perspective of one observer, and at the same time connecting to the firm classical background (where the commuting information is encoded) relative to another observer. This creates strange things, that requires that we need to EXPLAIN the hamiltonian in terms of infromation updates, and vice versa. This is for me the heart of the measurement problem, that is not yet solved, and i think it's rooted deeply. Decoherence solves nothing of this.

/Fredrik

 

[PeterDonis]

One of Rovelli's sound points is that there are no absolute observations

I'm not sure this is a sound point. Taken as it is stated, it denies the possibility of irreversible observer-independent experimental results, and without those, we have nothing on which to build a theory.

Another way of putting this point is that, when we start considering scenarios in which observers are supposed to be modeled using QM, we have a serious problem: any such QM model will have to include the possibility of reversing any observation (because time evolution in QM is unitary and any unitary operation can be reversed). But that amounts to allowing the possibility of reversing decoherence, and again, once you allow that, you have no irreversible observer-independent experimental results, and thus nothing on which to build a theory.

In short, it seems to me that this kind of "relational" viewpoint undermines the possibility of doing physics at all.

 

[bhobba]

It fits in nicely with the QFT view of things IMHO, as it should.

Rodney Brooks wrote an article describing this in more detail. I dont always agree with Rodney - eg his idea physicists have ignored QFT or even Schwingers great contributions (I really like Schwingers EM textbook personally) were ignored - they certainly were not. Be that as it may he does make a resonable effort describing what I was getting at::
https://www.quantum-field-theory.net/quantum-field-theory-a-solution-to-the-measurement-problem/

Problems still remain, but objects not having properties until measured is understood better. Everything is a field. Properties emerge when fields interact, otherwise they are a field of operators.

Thanks
Bill

 

[Fra]

I'm not sure this is a sound point. Taken as it is stated, it denies the possibility of irreversible observer-independent experimental results, and without those, we have nothing on which to build a theory.

Another way of putting this point is that, when we start considering scenarios in which observers are supposed to be modeled using QM, we have a serious problem: any such QM model will have to include the possibility of reversing any observation (because time evolution in QM is unitary and any unitary operation can be reversed). But that amounts to allowing the possibility of reversing decoherence, and again, once you allow that, you have no irreversible observer-independent experimental results, and thus nothing on which to build a theory.

In short, it seems to me that this kind of "relational" viewpoint undermines the possibility of doing physics at all.

I get why it can seem this way. And it certainly complicates things. But the same can be said qbout science. How can we do science if we are never sure about anything? Here the answer is corroboration.

Conceptually in anyalogy in the relational view, "corroboration" in the lack of perfect objectivity is replaced by negotiation among fellow observers. A middle path is formed from negotiating everbodies collective observations. And the most extreme outliers are destabilized. (Analogous to falsified). Meaning beeing so fatally wrong that it cant be adjusted by revising the state. The whole statespace needs to change.

/Fredrik

 

[PeterDonis]

How can we do science if we are never sure about anything?

We don't have to be "sure" to make predictions and decide whether or not to act on them. You can do that based on probabilities--how likely is it that this number we just calculated from our scientific theory is correct? And it doesn't have to be exactly correct; it just has to be close enough for whatever purpose we are using it for.

Here the answer is corroboration.

Of course this helps in collecting data on how accurate a scientific theory's predictions are; it's a lot easier to get a large amount of data to work with if you have multiple people doing it.

However, I'm not sure I would describe this as "negotiation".

 

[Fra]

However, I'm not sure I would describe this as "negotiation".

Fair enough. That choice is word isnt meant to play down science and suggets it's just politics. I meant it a also deeper way where reality is emerget.

But I agree this is concetpually strange and strips even more away from "realism" that already is done, and its not without problems. It's not a route for those, that already have problems with the lack of realism in standard QM. This is making all this worse - but with other pros IMO.

(It's related to what I labelled "observer democracy", as opposed to "observer equivalence". In theory building, the things that in the latter view as used as a constraint, are supposedly emergent in the former view)

/Fredrik

 

[physika]

isn't Hossenfelder on the verge of crackpot soon?

The main sources which are to be discussed in this thread is an arxiv manuscript and a blogpost. I thought only peer-review published work were allowed on this forum? I am just trying to understand

https://iopscience.iop.org/article/10.1088/2399-6528/ac96cf#jpcoac96cfs5-4

 

[Morbert]

Just to re-emphasise an issue I have. From the paper:

It is sometimes questioned whether the Collapse Postulate is actually necessary (e.g. in [6]). Without it, quantum mechanics would still correctly predict average values for large numbers of repetitions of the same experiment. This is the statistical interpretation suggested by Ballentine [7].

However, we do not merely observe averages of many experiments: we also observe the outcomes of individual experiments. And we know from observations that the outcome of an experiment is never a superposition of detector eigenstates, nor is it ever a mixed state (whatever that would look like)—a detector either detects a particle or it doesn't, but not both. As Maudlin put it [2], 'it is a plain physical fact that some individual cats are alive and some dead' (emphasis original). Without the Collapse Postulate, the mathematical machinery of quantum mechanics just does not describe this aspect of physical reality correctly.

The bit in bold could be interpreted two ways:
i) Without the Collapse Postulate, the mathematical machinery of quantum mechanics does not describe this aspect of physical reality.

ii) Without the Collapse Postulate, the mathematical machinery of quantum mechanics incorrectly describes this aspect of physical reality.

The latter would mean the measurement problem really is a problem, in the sense that it is a point of incorrectness.

The former is a better reading of interpretations like those presented by Ballentine: QM is correct everywhere in its domain, and it is not a problem that QM returns probabilities for possibilities rather than a definite actuality.

 

[vanhees71]

Just to re-emphasise an issue I have. From the paper:The bit in bold could be interpreted two ways:
i) Without the Collapse Postulate, the mathematical machinery of quantum mechanics does not describe this aspect of physical reality.

There cannot be a generally valid collapse postulate, because it depends on your experimental setup, what happens to the measured object. E.g., if you detect a photon with a photo detector you use the photoelectric effect, i.e., the photon is absorbed by the detector and for sure not in an eigenstate of the measured quantity (e.g., the polarization in a given direction). Of course there are (approximations of) ideal von Neumann "filter measurements", e.g., using a polarization filter, which lets through only photons with linear polarization in a given direction. Then the projection postulate holds, and you have a kind of "collapse of the state".

ii) Without the Collapse Postulate, the mathematical machinery of quantum mechanics incorrectly describes this aspect of physical reality.

The latter would mean the measurement problem really is a problem, in the sense that it is a point of incorrectness.

The former is a better reading of interpretations like those presented by Ballentine: QM is correct everywhere in its domain, and it is not a problem that QM returns probabilities for possibilities rather than a definite actuality.

I don't know, what you mean here. There is no problem with filter measurements nor with other kinds of experiments. The "proof" is simple: QT works with great success whereever it is applied!

 

[A. Neumaier]

There cannot be a generally valid collapse postulate, because it depends on your experimental setup, what happens to the measured object.

There can be, and in fact there is:

The dependence of the collapse on the setup is correctly accounted for by the notion of a quantum instrument (or quantum operation, or quantum channel), which generalizes the Heisenberg collapse in the same way as the POVM concept generalizes the Born rule.

The quantum instrument formalism is routinely taught in quantum information theory. For example, the well-known textbook

• M.A. Nielsen and I.L, Chuang, Quantum computation and quantum information: 10th Anniversary Edition, Cambridge Univ. Press, Cambridge 2011.

introduces them (in Section 2.2.3 on quantum measurement, without introducing a name for the concept) even before defining the traditional projective measurements.

I also discuss them in Section 6.6 of my paper

• Quantum tomography explains quantum mechanics.

 

[gentzen]

(in Section 2.2.3 on quantum measurement, without introducing a name for the concept)

See also section 8.2.4 (because it "explains something about the interpretational background" of that concept) ...

 

[Morbert]

There cannot be a generally valid collapse postulate, because it depends on your experimental setup, what happens to the measured object. E.g., if you detect a photon with a photo detector you use the photoelectric effect, i.e., the photon is absorbed by the detector and for sure not in an eigenstate of the measured quantity (e.g., the polarization in a given direction). Of course there are (approximations of) ideal von Neumann "filter measurements", e.g., using a polarization filter, which lets through only photons with linear polarization in a given direction. Then the projection postulate holds, and you have a kind of "collapse of the state".

Adding to the point made by A. Neumaier above, we can also recover a collapse postulate if we generalise our description of the experiment to possible histories $\{C_\alpha\}$ of the joint system (measured system + instrument). Collapse is the change $\rho \rightarrow \frac{C_\alpha \rho C^\dagger_\alpha}{\mathrm{tr}C_\alpha \rho C^\dagger_\alpha}$

https://arxiv.org/abs/gr-qc/9210010

I don't know, what you mean here. There is no problem with filter measurements nor with other kinds of experiments. The "proof" is simple: QT works with great success whereever it is applied!

What Hossenfelder is arguing (unsuccessfully imo) is that, without a collapse postulate, QM's inability to account for the realisation of one possibility over others is a point of incorrectness of the theory.

 

[vanhees71]

What is meant by "QM's inability to account for the realization of one possibility over others"? QT describes, in accordance with all observations, the probabilities for the outcome of any possible measurement, given the state of the system. Which outcome is realized in each individual measurement is inherently random. That's an observed elementary fact about how Nature behaves, and this inherent randomness in Nature was a big problem for physicists used to the classical, deterministic picture of classical physics, and the quest for finding some deterministic description of the random behavior was in vain until today. So far nobody could find any deterministic description of this random behavior. Of course, you cannot exclude the possibility that we simple haven't been clever enough to find such a description, but according to today's knowledge there's not the slightest hint that there exists one.

 

[Morbert]

Actually, to her credit, she clarifies in the very next sentence, the ambiguity I raised. I should have been more patient.

"This means quantum mechanics without the Collapse Postulate is not wrong, but it describes less of what we observe. The Collapse Postulate is hence useful, and part of the axioms because it increases the explanatory power of the theory. It cannot simply be discarded."