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

Dear Markus,

thanks for reading my essay, and for commenting!

You are quite right in your observation that my argument, basically, is just equal to Cantor's regarding the fact that the powerset of a set necessarily has a greater cardinality than the set itself, and hence, that there can be no bijection between the two. This is a very familiar fact to us, today, but still, depending on the context, has quite nontrivial implications---the fact that there are uncomputable functions, or indeed, undecidable statements in any sufficiently expressive theories, follow exactly the same fold.

It's not quite right to say that this applies equally well to a quantum world, however. The reason for this is that the basic underlying structure---Lawvere's fixed-point theorem---works in the setting of Cartesian closed categories; the category Hilb that has Hilbert spaces as its objects, and linear operators as its morphisms is not Cartesian closed, however. Baez has provided an interesting discussion on how it's exactly this categorical difference that underlies most of the 'weirdness' of quantum theory.

In particular, the absence of a cloning operation means that the diagonalization doesn't go through---you can't, in a sense, feed the system back the information about the system. So in that sense, my argument entails that sets aren't a good setting for a physical theory, as you run into the paradoxical, and you have to adduce extra structure (by a deformation of the algebra of observables) to avoid this---which leads to something like phase-space quantization. Or, alternatively, you can start out with a categorical setting where you get this structure for free---leading to something like Hilb.

Bohmian mechanics, by the way, isn't a counterexample---indeed, I think it supports my argumentation (this is discussed more in depth in the Foundations of Physics-article). In the end, it comes down to the fact that every function---including noncomputable ones---can be represented by means of a finite algorithm, augmented with an infinite string of random digits (every set is reducible to a random set by the Kucera-Gacs theorem). In general, thus, every measurement outcome in Bohmian mechanics is a function of the entire random initial conditions---which must fit the equilibrium hypothesis to give rise to quantum predictions. (Indeed, if the generation of measurement outcomes in Bohmian mechanics were computable, that would lead to exploitable nonlocal signalling.)

Indeed, that's to me at least a suggestive way of forming the connection to quantum mechanics: a noncomputable function (or sequences) can be 'computed' in different ways---one, with a finite algorithm with interspersed random events, two, with a finite algorithm that reads out a fixed algorithmically random number, three, with an interleaving process computing every possible sequence. These correspond to the major interpretations of quantum mechanics---something like a Copenhagen collapse process, with the algorithm being the Schrödinger dynamics (von Neumann's 'process II'), and random events yielding the 'collapse' ('process I'), a Bohmian nonlocal hidden-variable approach, and a kind of many worlds theory.

That said, I view this as very much a sketch of a theory---perhaps itself a kind of toy theory. To me, it seems a promising avenue to investigate, but I have no illusions about having painted any sort of complete picture at all. I ride roughshod over many subtleties, as you note; and there are several additional open questions. Some of this is treated more carefully in the Foundations of Physics-paper (which also properly cites the work by Hoehn and Wever---well, not quite properly, since I call him Höhn!), where I am also more cautious about some of my claims. There, also an argument based on Chaitin's incompleteness theorem, that doesn't boil down to 'mere diagonalization', is included.

Thanks, again, for taking the time to read and comment on my essay. I would very much enjoy continuing this discussion---since I work on this mostly in isolation, there's a high danger of getting lost down blind alleys, so I welcome any reality check on my ideas. So any and all criticism is greatly appreciated!

Cheers

Jochen

Hmm, I have problems getting my comments to post. Initially, I got a 'post is waiting for moderation' or something like that, then I had apparently gotten logged out. I will wait for a while whether the comment appears, and if it doesn't, type a new one sometime later.

Dear Marcus,

I've decided to try again submitting my comment, as long as my reply is still fresh on my mind.

First of all, thank you for your comments, and criticism! I work on this topic largely in isolation, so it's good to have a little reality check now and then, to be kept on track, and not loose myself down blind alleys. Therefore, I hope to keep this discussion going, in some form!

Now, to try and answer some of your concerns. You're of course perfectly right to point out that my argument really doesn't do more than point out that the powerset of the set of states can't be put into one-to-one correspondence with the states themselves---a fact of course long familiar, thanks to Cantor. But that doesn't mean it can't have subtle consequences---essentially, the existence of uncomputable functions, and the undecidability of certain propositions, all boil down to the same phenomenon.

This was worked out by Lawvere, who first exhibited the fixed-point theorem that underlies the different realizations of unpredictability, undecidability, and so on. Within the preconditions of this theorem also lies an answer to your objection that the same should be possible in quantum- and even post-quantum worlds: the theorem's setting is that of Cartesian closed categories (such as Set, with sets as objects and maps between them as morphisms). In particular, in these categories, there exists a natural copying operation---which is basically what makes the diagonalization-argument possible, by 'feeding' the information contained in the system back to the system itself (think about the halting-checker examining its own source-code).

Of course, this isn't possible in quantum theory, due to the absence of a cloning operation---which, in category-theoretic terms, means that the category Hilb with Hilbert spaces as objects and bounded linear operators as morphisms isn't Cartesian closed. John Baez has pointed out that much of the 'weirdness' of quantum mechanics boils down to this fact.

So in this sense, my argument can be read as saying that Set isn't a good arena for a physical theory, for to avoid it lapsing into paradox, you have to adduce extra structure---corresponding to the *-deformation of the algebra of observables that essentially leads to deformation quantization (not that I'm claiming to have the complete picture there, mind). On the other hand, you can directly work in a setting---such as Hilb---where these problems don't arise.

As to Bohmian mechanics, as I also argue in some more detail in the Foundations of Physics-paper, I think it's not a counterexample to my ideas, but in fact, very well in line with them---Bohmian mechanics, to reproduce the quantum predictions, essentially needs to be seeded with an initial random configuration (conforming to the 'quantum equilibrium hypothesis'). Its nonlocality means that essentially every measurement outcome is a function of this random seed (and not just of some finite portion thereof confined to the past light-cone, say). But every function (including non-computable functions) can be decomposed into a finite algorithm and an infinite, algorithmically random seed (this is just the Kucera-Gacs theorem that every set is reducible to a random one). Consequently, one could always interpret the 'computation' of a non-computable function as a finite algorithm seeded with an infinite random initial string---which then is what I would say Bohmian mechanics boils down to.

Besides, one can show that every model in which non-local correlations are generated in a deterministic way must either be uncomputable, or will lead to exploitable signalling.

Furthermore, there are (at least) two more ways to interpret the 'computation' of a non-computable function (or sequence). One is that every now and then, genuinely random events occur---that is, an algorithmic 'process II' is interspersed with 'process I' random occurrences. The other is simply to compute all possible sequences, in an interleaving manner---leading to a sort of many-worlds picture. Hence, the attempts to make sense of quantum mechanics at least suggestively map to the attempts to make sense of the non-computable. But this is of course merely heuristic.

However, you are right to point out that I ride roughshod over many subtleties that need to be addressed, eventually. Personally, I consider this to be more of a sketch, than a full-fledged theory---a direction that I find suitably promising to explore (and hey, two other essays in this contest directly reference my work, so that's something at least!---or, of course, it could just mean that I've managed to lead others down the same blind alley I'm lost in. Hmm, that's not as cheerful a thought...). I am somewhat more careful, both in pointing out the preliminary nature of my investigations and in trying to make them more robust, in the Foundations of Physics-paper; in particular, there, I also present an argument based on Chaitin's incompleteness and algorithmic complexity that doesn't boil down to 'mere diagonalization'. (I also properly cite the work by Hoehn and Wever---or rather, almost properly, as I spelled his name 'Höhn' by mistake!)

Anyway, I think this was most of what I originally intended to post. I would like to thank you again for engaging with my ideas---ideas that grow for too long in the shade away from the light of others' examination tend not to bear fruit; one sometimes needs the expertise of others to know where to cut, where to graft, and where to nurture a tender sapling.

I hope this one will post!

Cheers

Jochen

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

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

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

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

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