Epistemic Horizons: This Sentence is 1/√2(|True> + |False>) by Jochen Szangolies

Markus Mueller

Dear Jochen,

thank you so much for your detailed and very illuminating answer! And thanks so much also for the links, in particular to Baez' work -- this is highly appreciated!

I feel like I would need to dive deeper into category theory to continue the proper discussion. Nonetheless, perhaps one follow-up question. If I understand the notion of "not being Cartesian closed" well enough, then this property also applies to a structure much more mundane than Hilbert space quantum mechanics: probability distributions.

In fact, since you point out the no-cloning theorem, there is a general no-cloning theorem for probability distributions (and more general probabilistic theories), see e.g. here:

https://arxiv.org/abs/quant-ph/0611295v1

In other words, you cannot copy (clone) probability distributions (trying to do so will introduce correlations, i.e. simply broadcast). Therefore, I would be surprised if the structural property that you point out was in some specific way part of the reason "why" we should expect quantum effects.

However, all that I'm writing here comes with the grain of salt that I don't have much background in category theory.

Best,

Markus

Gemma De las Cuevas

Dear Jochen,

Thank you so much for writing this essay! I enjoyed it very much. I especially enjoyed the idea of an epistemic horizon, and of a potential link between mathematical undecidability and physical unknowability, as we know it in quantum physics. As a very rough summary, would it be fair to say that you are applying the limitations obtained by self-reference and negation in a clever way in order to derive some of the epistemic limitations characteristic of quantum mechanics?

I was also wondering how your view point relates to the principle "universality everywhere"; in particular, to the quantum notions of universality in computation, spin models, etc (as mentioned in my essay). My impression was always that these quantum notions would suffer from the same limitations as their classical counterparts. Perhaps we are just applying the paradox of self-reference and negation in different ways?

Finally, I was wondering about your opinion of this work , in particular in relation to your work.

Thanks again, and best regards,

Gemma

Jochen Szangolies

Dear Gemma,

I'm glad you found something worth your while in my essay! Your summary, I think, is accurate: in the general sense, there exists a bound on the information obtainable about any given system (which, in the full sense, requires the appeal to Chaitin's quantified incompleteness theorem given in the Foundations-paper), and once one has reached that limit, 'old' information must become obsolete---as it does when we try to expand our horizon by walking (on the spherical Earth, or, well, any planet will do) west, losing sight of what lies to the east.

I'm not completely sure I get the gist of your question regarding quantum computation (etc.) right. Are you asking whether my proposal implies that quantum computers should be capable of beyond-Turing computation? If so, then the answer is kind of a 'it depends': any functional model of quantum computation will be able to compute only those functions that a classical Turing machine can compute. But still, if quantum mechanics is genuinely (algorithmically or Martin-Löf) random, then obviously, we can use quantum resources to do something no classical machine can do, namely, output a genuinely random number! Hence, as Feynman put it, "it is impossible to represent the results of

quantum mechanics with a classical universal device."

So in a sense, we need to be careful with our definitions, here---any way of implementing a finite, fixed procedure, whether classically or with quantum resources, will yield a device with no more power than a classical Turing machine (regarding the class of functions that can be computed, if not the complexity hierarchy). The reason for this is that simply outputting something random does not avail any kind of 'useful' hypercomputation, because one could always eliminate the randomness by taking a majority vote (provided the probability of being correct is greater than 1/2), and hence, do the same with a deterministic machine.

In a way, the story with quantum mechanics and computability is like that with non-signalling: it somehow seems like the classical constraint ought to be violated---but then, the quantum just stops short a hair's breadth of actually breaking through.

As for the Deutsch et al. paper, I can't really offer an exhaustive analysis. I'm somewhat sympathetic to the notion of 'empiricizing' mathematics (indeed, Chaitin has made the point that the undecidability results mean that at least to a certain degree, there are mathematical facts 'true for no reason'), but I think that notions such as those in Putnam's famous paper 'Is Logic Empirical?' go a step too far. In a way, different logics are like different Turing machines---you can do all the same thing using a computer based on a three-valued logic (the Russian 'Setun' being an example) as you can do using one based on the familiar binary logic, so in that sense, there can't be any empirical fact about which logic is 'better'.

But again, I'm not familiar enough with the paper to really offer a balanced view.

Cheers

Jochen

Yutaka Shikano

Dear Jochen,

Very interesting. Great to hear from the young talented student like you. I would like to ask you the following point. According to the PBR theorem,

In conclusion, we have presented a no-go theorem, which - modulo assumptions - shows that models in which the quantum state is interpreted as mere information about an objective physical state of a system cannot reproduce the predictions of quantum theory. The result is in the same spirit as Bell's theorem, which states that no local theory can reproduce the predictions of quantum theory.

From your viewpoint, 'epistemic horizon', what do you deal with this theorem?

Best wishes,

Yutaka

Jochen Szangolies

Dear Yutaka,

thanks for your comment! And I'll take both the 'young' and the 'talented' as compliments...

As for the PBR theorem, it essentially states that there is no more fundamental description than quantum theory that bears the same relation to it as classical mechanics does to quantum mechanics---i. e. quantum mechanics can't be the statistical version of some underlying, more definite theory.

In that sense, it's very well in line with my result---which essentially states that there is no more fundamental description than that given by the quantum formalism; this is, so to speak, all the description that's possible. Whether that means that's all there is, or whether, as in QBism, there is an epistemic element to this description, is something that, I think, is an open question as of now.

Does this make sense to you?

Cheers

Jochen

Yutaka Shikano

Dear Jochen,

Thank you so much for your clear answer.

This was made clear. This open question will be solved in your future research.

Best wishes,

Yutaka

Jochen Szangolies

Dear Marcus,

good to hear I could add some clarification! I have to admit, I'm myself insufficiently familiar with category theory to really get into the thick of it---it's too vast a subject for me to really get a general overview over, without expending equally vast amounts of time.

Therefore, I'm not sure, offhand, how to answer your question. However, it seems to me that any assignment of classical probabilities must include the case of certainty---that is, of stipulating for each observable, whether a system possess it, or fails to. (As in the vertices of the CHSH-polytope.) In my framework, this assignment isn't possible, so it seems to me that we can't be left with 'just' classical probability distributions.

But I will have to think about this some more. I have wondered about the role of the classical no-cloning theorem, in particular in light of the Clifton-Bub-Halvorson theorem (I know this also includes a 'no bit commitment'-requirement, but in the end, that just tells you that at least some of the entangled states of the resulting algebra must be physically realizable, I think).

And in a sense, no-cloning is just the measurement problem: if you could clone, you could make a perfect measurement; and if you could make a perfect measurement of every state, you could clone. So if the no-cloning theorem in classical probability theory has the same significance, then why isn't there an equivalent measurement problem? (Or is there?)

Anyway, this seems an interesting line of thought to pursue, thanks for suggesting it!

Cheers

Jochen

Jochen Szangolies

Dear Luca,

thanks for your generous comment. You ask some good questions, and I'll do my best to do them justice.

First of all, the point where quantum physics enters into the picture isn't one of 'smallness', exactly, as it's often glossed, but it is one where the amount of information you have about a system nearly or completely exhausts the amount of information it's possible to acquire about that system. In other words: if you know the state of a system only in its gross properties, you will not notice any quantum mechanical behavior; and in everyday experience, we only ever know a tiny amount about anything we engage with.

Think about the properties we typically know about a chair---approximate size, weight, shape---in comparison with the complete specification of each of its constituent atoms. The former will be perfectly sufficient for a classical description, yielding accurate predictions; only if we really had access to something approaching the full microstate would we come close to exhausting the information available about a chair, and thus, notice its quantum character. As this is generally only possible for systems with very few characteristics, and these tend to be submicroscopic, quantum theory tends to be glossed as a theory of 'the small'.

Hence, that there is an approximate classical description of macroscopic systems, as long as we don't know very much about them---as long as we don't exhaust the information available due to the 'finiteness' principle---is a consequence of the approach.

Your second point is more difficult to answer precisely, so I'll wave my hands around a little. In a sense, what I'm proposing is that the epistemic horizon is a metatheoretic principle: it's a boundary on what's possible to grasp of the world in terms of a model thereof. Hence, it's not quite the theory that tells us what we can know, but rather, the act of theorizing. This is a little Kantian in spirit: how we view the world is not just a flat consequence of the way the world is, but equally, a consequence of our viewing of it. (This is perhaps explained a bit better, from a different angle, in my contribution to last year's contest.)

Does this make sense to you?

The part you quote from your essay seems certainly not too far from my own views. I will have a look, and try and see whether I find something interesting to say; thanks for highlighting it.

Cheers

Jochen

Jochen Szangolies

Dear Hippolyte,

thanks for your comments! I think we've got a bit of a common direction in our thinking---the 'Laws of Form' has long held some intrigue for me, but I was never quite able to come to any definite conclusions about it. (Perhaps you know the work of Louis Kauffmann, who if I remember correctly has also proposed some interesting connection between the paradoxes of 'reentrant' forms, complex numbers, and the passage of time---perhaps in the paper on 'Imaginary Values in Mathematical Logic'.)

As for the introduction of an 'indeterminate' logical value, this alone probably won't solve Gödelian paradoxes---you can appeal to the 'strengthened liar', the sentence 'this sentence is false or meaningless', which is either true, or not; if it is true, then it must be false or meaningless, and if it is not true, then it is either false of meaningless, hence true. (That's why you also can't get out of trouble postulating 'null' results for measurements as a way out.) Superposition then can't be thought of as another truth value to be added, but rather, the absence of any definite truth value.

The connection between self-referential paradoxes and temporal paradoxes is an interesting one, but I haven't yet found much time (irony?) to spend on exploring it. In a sense, the two most discussed paradoxes---the grandfather paradox and the unproven theorem---bear a close connection to the self-negating Gödel sentence, and the self-affirming Henkin sentence: one eliminating the conditions of its own determinateness, the other creating them.

But as I said, beyond such generalities, I don't have much to offer. But I'll try and spent a little time thinking about this, if I come up with anything worthwhile, I'll make sure to let you know.

Cheers

Jochen

Luca Valeri

Hi Jochen,

Thanks for your reply. However I still don't get where your Finiteness and Extensibility principles enter your prove of the impossibility of Assumption 1.

Do you think your principles are connected to the impossibility of copying the whole information in quantum systems? So in your chair example, I was thinking, the information of weight, form, size etc. actually already exhausts all the properties, that make a chair a chair. What it makes classical is, that this information is available abundantly/redundantly, whereas this is not the case for quantum objects.

The reason for the the necessity of such epistemic restriction remains unclear. And might be not further justifiable than by the empirical evidence.

However the justification of why such epistemic consideration should have an effect on the ontology, cannot be in my opinion, by the way we (only can) view the world. This makes the whole picture a bit to anthropocentric. Don't you think? The objective quantum mechanical phenomena like super conduction, stability of atoms, etc. cannot be because of epistemic limits in the knowability the underlying world.

In my essay I probe the possibility, that the underlying objective 'reality' is emergent from relations between emergent objects themselves. Something like this could justify an epistemic impact on the underlying structure. I'll be curious on your comments on my essay.

Luca

Satyavarapu Gupta

Dear Jochen,

Yes probably you are correct, Godel theorem may not be applicable to Cosmology.

I hope you will have CRITICAL examination of my essay... "A properly deciding, Computing and Predicting new theory's Philosophy"..... ASAP

Best Regards

=snp

Harrison Crecraft

Hi Jochen,

Thank you for your response. "Your notions regarding---if I interpret you correctly---an inherent thermal 'noise' making the acquiring of perfect information about a system impossible remind me of Nelsonian stochastic mechanics. Is there a connection?"

Your interpretation is not quite right at a subtle but fundamental level. The idea of thermal noise and stochastic mechanics implicitly assumes random fluctuations of precise coordinates, but precise coordinates are definable only with respect to an assumed ambient temperature of absolute zero. Perfect information for a system contextually defined at a positive ambient temperature is complete. There is no randomness in the actual contextual state. Randomness only comes in during irreversible transition from a metastable state to a more stable (higher entropy) state. As long as a state exists, there is no randomness and the state evolves deterministically.

Harrison

Jochen Szangolies

Dear Luca,

if I understand you correctly, I think you've gotten something a little mixed up---Finiteness and Extensibility are the starting points for the reconstruction of quantum mechanics in many recent attempts to justify the formalism from first principles (the way the invariance of the speed of light and the relativity principle are for special relativity). However, that just invites the question---but why are we limited in the amount of information we can obtain about a physical system?

That's the question I'm trying to answer---in other words, Finiteness and Extensibility are the output of my approach, they're what I'm arguing must hold, due to the application of Lawvere's theorem to the notion of measurement. That these principles hold is then equivalent to Assumption 1 being false---there isn't a function f(n,k) such that it yields a value for every state and measurement. There are some measurements on certain states such that it doesn't yield a value (Finiteness, although not quite---you need the argument from the Foundations paper based on Chaitin's theorem for that), and for these, we will learn new information upon measurement (Extensibility).

This doesn't really impinge on the objectivity of quantum phenomena, by the way. My proposal of a relative realism---which I don't really develop in the essay, admittedly---assigns values only to those measurements where f(n,k) yields a value, but that's a perfectly objective statement: in a given state, only those properties where the measurement outcomes can be predicted with certainty actually have definite values.

You can, of course, also interpret this as a subjectivist stance---i. e. claim that there's some real values out there, but our descriptions can't include them. But that's an additional interpretational commitment, nothing that's forced on us by my argument.

Cheers

Jochen

Michael Kewming

Hi Jochen,

Thanks for the really interesting essay! Your introduction to the effects of Finitness and Extensibility was very intuitive and how they may be used to understand Heisenberg's uncertainty principle. I think there could be a very close connection between the amount of energy required to extract information perfect information about the system---I'm thinking squeezed states---and the finitness principle. A perfectly localised measurement requires an infinite amount of energy over an infinitesimally short period of time.

I guess my question is, do you suppose that the epistemic horizon is a physical horizon? However, we might be wading into that age old debate of ontological vs epistemic interpretations of quantum physics!

In any case, it was a terrific essay and I rated it highly! I hope you get a chance to take a look at mine. We certainly have overlap in our ideas, although I fall down on the opposite side of you conclusions if you argue it a thermodynamic angle.

All the best!

Michael

Jochen Szangolies

Dear Michael,

thanks for your comment! I'm glad you found my intuitive approach to deriving the Heisenberg uncertainty relation approachable. It can be made more rigorous---for a start in that direction, I refer to the Found. Phys.-paper---but I think this is a virtue of this particular approach: it makes an otherwise 'mysterious' phenomenon somewhat more easily palatable, by connecting it with statements that have a readily appreciable intuitive import.

Thinking about the energy needed to extract the information is an interesting direction. In some sense, there ought to be a relation there---thinking about this in terms of energy and time, rather than momentum and position. But I don't have a clear intuition there yet.

You're right to note that this sort of picture sort of straddles the epistemic/ontic divide. In a way, I'm not so sure it's good to think of these as rigidly distinct---certainly, in some sense at least, what we know about something is not something removed, off in some Cartesian realm of 'thinking stuff', from the physical world: the stuff in our brains is ultimately physical, itself. Hence, what we can know, and how we know it, is in the end also a question of what there is, i. e. what sort of stuff supports our knowledge. There's too much of the old 'detached observer' still lingering in this picture.

Regardless, if you're so inclined, I think it's perfectly well possible to interpret my proposal in epistemic as well as ontic ways. This is, to my way of thinking, an issue that only further argument will be able to settle. So, perhaps it's a topic for another contest!

Cheers, and thanks again for your kind words

Jochen

Luca Valeri

Hi Jochen,

Yes. I really got it mixed up. That is why I could not find the principles in the prove.

Thanks for the clarification and also for reading and commenting on my essay.

Good luck on the competition with your great essay and you certainly get the price for the best title.

Luca

[deleted]

Dear Jochen,

I am curious about entanglement. What do you think of this assumption of mine:

When we break one rod, no matter how far we move them, we will later easily find that those two parts are the same rod. A possible analogy with electrons is: No matter how much the electrons are the same, they differ minimally in mass, say only for 10 ^ -60 part of the mass, but so that every two separated electrons form wholes with identical masses.

Regards,

Branko

Jochen Szangolies

Dear Branko,

thanks for your comment. I think that your proposal for entanglement would run into trouble with established quantum mechanics, however: for one, quantum particles must be exactly identical---otherwise, quantum statistics would come out wrong, which would lead to easily detectable disagreement with experiment.

But the more important part is that your proposal is essentially a local hidden variable theory---particles have additional properties, not obvious to ordinary measurements, that are responsible for their correlations. But this is just the sort of thing Bell's theorem shows not to be possible---the argument in my essay tells you why: there would then be a probability distribution (whether we know it, or not) for the hidden variables to take on specific values; but this, alone, is enough for all Bell inequalities to be perfectly obeyed. Hence, violation of Bell inequalities tells you that such a picture won't work.

Does this make sense to you?

Cheers

Jochen

Rafael Alves Batista

Dear Jochen,

I really enjoyed reading your essay.

I particularly like one of your conclusions: "The epistemic horizons the pure mathematician and the experimental physicist find delimiting their perspectives are not separated, but instead, derive from a common thread". I reach similar conclusions in my own essay (https://fqxi.org/community/forum/topic/3523).

One thing that I'm still thinking about after reading it, is whether finiteness and extensibility aren't really incompatible. You argue that they are not, based on the collapse of the wavefunction. However, even if the collapse erases old information, current information still takes space. Therefore, if one invokes finiteness, there is a limit to extensibility. Within the information-theoretic framework I presented in my essay, this follows naturally regardless of the 'overwriting' of obsolete information.

Maybe I'm missing something, so I should probably take a look at your other publications on the topic. Anyways, congratulations for your excellent essay.

All the best,

Rafael

Jochen Szangolies

Dear Rafael,

thanks for having a look at my essay. I will try to find some time to read yours.

As for finiteness and extensibility, perhaps it helps to consider Spekkens' toy model, which demonstrates many of the features of quantum mechanics. The basic idea there is the 'knowledge balance principle': "the number of questions about the physical state of a system that are answered must always be equal to the number that are unanswered in a state of maximal knowledge."

In other words, if your have one bit of information about the state of the system, you must also lack one bit of information that could be gained by additional measurement. But if you perform that measurement, then you'd have two bits of knowledge, and no further knowledge could be obtained; hence, to make things come out right with the knowledge balance principle, the state of the system must change so that the previous knowledge no longer applies.

It's similar with my model. You can think of this as trying to localize a system further within its state space: some amount of knowledge will allow you to perform this localization to a certain degree of precision. If only 'finiteness' were true, then well, that might just be it: you've localized the system as well as it's possible to localize it.

But 'extensibility' implies that you can obtain additional information. For instance, if the system (in state space) is localized to some degree along one axis, you can try to increase its localization along that axis by making a more highly fine-grained measurement; but to cope with the finiteness-requirement, its localization must consequently decrease along another axis. You can visualize this like squeezing a bubble, or a squishy ball: the volume (the total localization/total information you have) stays the same (equal to a power of Planck's constant), but the shape will deform, yielding information gain in one property, compensated by loss along another.

Does this make matters more clear?

Cheers,

Jochen

Dean Rickles

Dear Jochen,

Very enjoyable. I think it might be my favourite so far. Ingenious in many ways.

I especially liked the linking between superposition and the liar paradox situations on p. 4. In the other direction, that is a very nice way to understand such self-referential statements: "This sentence is false" is indeed in a kind of superposition.

I agree with the conclusion concerning the "common thread behind mathematical undecidability and physical unknowability", which is not the usual one mentioned (and frowned upon).

I also like the epistemic horizon concept very much. I note also that this allows you to connect in a more natural way uncertainty and Godel's theorem - also often frowned upon as one of the many Godel-based overreaches. However, of course, in viewing QM in these terms you are seemingly committing yourself to an epistemic view of quantum uncertainty. Is that correct?

Also, Wheeler's idea to use the undecidable propositions was not a quantum principle itself, but something deeper: it was to be the stuff of his pre-geometry explaining "why the quantum?" as well as "why spacetime" and "why existence?". He didn't stay with this idea very long... [I mention this because Wheeler uses the same term "quantum principle" in a different sense, so it might be worth disambiguating your constructive sense from his.]

Good luck!

Best

Dean

Jochen Szangolies

Dear Dean,

thanks for your very kind words! I'm happy you got some enjoyment out of my essay.

As for the epistemic vs. ontic question, I'm kinda torn about that. I'm not sure the issue is best framed in these terms---after all, what we know, and can know, depends always on what there is---items of knowledge are, in whatever oblique a way, elements of reality, and even at least supervenient on physical matters of fact, provided one tends towards a materialist metaphysics. So what we call 'epistemic' in that sense is really just a particular way of looking at what's there.

In the other direction, we have of course no unvarnished access to what's out there in the world---whether it's behind the veils of Maya or lurking in Kantian noumena, we (re-)construct the world by means of the phenomena, which are all we can directly access. So what we can know depends on what there is; and what we consider there to be depends on how it is present to us. I'm not sure, then, there's really a hard-and-fast dividing line to be drawn---it may, perhaps, be rather a matter of method: we study the world in an epistemic or ontological way.

So maybe it should not come as too much of a surprise that nobody has yet managed to unscramble the quantum omelette---and perhaps, it's a mistake to try, because the world itself is a mixture of the objective and the subjective. So quantum theory, in my view, can tell us something about what is---for whatever that is, it's something that admits a quantum description at least in some sense---and about what we can know---what information we can hope to obtain about the world.

Hence, I'm not sure if whether the wave function is out there in the world, or in our heads, ultimately makes much of a difference; what's in our heads is in the world, too, after all.

I appreciate the disambiguation regarding Wheeler's 'quantum principle'---but I'd like a bit of elaboration (or perhaps, a pointer to the relevant literature). I don't really know about the connection Wheeler drew between undecidability and space-time (vaguely, in the back of my head, it seems I remember something about building space-time out of a logical calculus, or something along those lines), so I'd love to understand this better!

Cheers

Jochen

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