Four Verses from the Dàodéjīng by Jochen Szangolies

@All, I'm sorry if I'm a bit tardy in my replies---work has been keeping me a little busy lately. This should clear up in February, though, so I hope to be quicker with my answers then.

In the meantime, why don't you have a look at the discussion of my essay over at the fantastic blog on all things consciousness, Conscious Entities where Peter Hankins provides his insightful commentary on the issues I discuss in the essay!

Hi Luca,

thanks for your nice comments (in particular in regard to the Mary's room argument)! As for your skepticism, I think there's many different ways of approaching this question. One, for instance, is that I think Gödel's two incompleteness theorems establish a limitation of the human capacity to formalize mathematics, not a limitation on mathematics itself, as it's usually phrased---in other words, it's not that 'mathematics is incomplete', but rather, that the axiom systems human mathematicians can formulate---being necessarily finitely specifiable, effective, and so on, which ultimately boils down to being computable---don't suffice to capture all of mathematics.

But to say that this implies the existence of something beyond our modeling capacity would require a belief in the preexistence of mathematical objects, i.e. some form of Platonism---which is not something I myself would be willing to commit to.

But things get clearer once one things instead of abstract formal systems in terms of computers. The halting problem is an instance of the same phenomenon that also gives rise to incompleteness (Lawvere's fixed-point theorem, which Noson Yanofsky, who also has an entry in this contest, discusses in a very accessible way here), and the two are equivalent in some ways. Now, with the math we know how to use, we can't in general predict whether a given TM halts; nevertheless, one might justifiably hold that there's still a fact of the matter regarding whether it will or won't. But Turing machines are themselves abstract entities, and would require an infinite tape or something equivalent to build; so again, it's not immediately clear if that has any real-world significance.

So let's now turn to physics. You mention Breuer's results, which I think are very interesting; but there's many more indications of a connection between QM and logical independence (in fact, John Wheeler managed to get himself thrown out of Gödel's office for suggesting a connection). For instance, Caslav Brukner and colleagues could show that if they encode a set of axioms into a quantum state, measurements related to propositions that are not derivable from those axioms yield a random outcome. However, this is not specific to Gödel's incompleteness, as the axiom systems were too simple for Gödel's construction to apply---Gödel's theorems then simply mean that such phenomena never 'go away', so to speak.

A more qualitative line of reasoning is the following: you can represent any non-computable function with a computation plus a source of randomness; consequently, if we are beings using computational methods to model a non-computable world, we should expect the world to look like a computation with intermittent random events. That's of course exactly what we have in QM. An obvious counterexample here might seem to be Bohmian mechanics: but actually, it turns out that a model that reproduces quantum mechanics by introducing nonlocal influences (as BM does) cannot be computable---otherwise, one could use these influences to transmit information, contradicting the tenets of special relativity.

There are many more results pointing in a similar direction scattered about the literature (and of course, there's my own argument that I cite in the essay); but I think it's as yet unclear what the picture is they paint. Zwick in the 1970s, and later on Peres and Zurek, proposed that QM's "inability to completely describe the measurement process appears to be not a flaw of the theory but a logical necessity which is analogous to Gödel's undecidability theorem". I think that it's again as with the incompleteness theorems in mathematics: they're not a limitation on mathematics itself, but on human mathematicians; likewise, the limitation we see is not one on physics, but on physicists: using computable models, which we must if I'm right, to model a non-computable world, will inevitable lead to something like a quantum description.

But this is an argument that needs to be made much more carefully, and which I'm in the agonizingly slow process of working out. We'll see what comes of it!

Dear Heinrich,

thanks for your interesting comments! I'll have to have a look at your 'fundamental universe'. And you're right, I did read Gabriel's "Warum es die Welt nicht gibt" exactly because I also perceived some kinship in his ideas, but found it ultimately rather disappointing.

Regarding models and strong emergence, well, it depends what stance you take. The traditional view would be that models possess some immediate correspondence with the things they model, at least up to some suitable approximation, and that thus we should consider the entities they posit, and the explanations they provide, as giving us some insight into what really goes on in the world. Here, I think, strong emergence is troubling: in some sense, there would be certain things that violate Leibniz' principle of sufficient reason---facts about the world that would obtain without any answer to the question of why they should.

In some sense, though, this is also true in classical reductionist ideas---after all, if everything can be reduced to some set of base facts, then those facts themselves admit no further justification. From this point of view, strong emergence maybe doesn't seem that much worse---it merely adds additional 'fundamental' facts that obtain at some higher level of coarse-graining, for instance.

But in my view, all models are inaccurate to some degree---every model is incomplete by necessity, just as all axiomatic systems (of sufficient power) are. The world admits no more of a single model than mathematics admits of a single axiom system.

Are then facts that fail to fall under the purview of a given model strongly emergent? From the point of the view of the model, you might say so: after all, within that model, there is no way of reducing them to fundamental facts; they're true for no reason.

But this tells us something about the model, in my opinion, not something about the world. In other words, it's our problem, not that of the world---just as the fact that there is no single axiom system for all of mathematics ultimately may be considered a limitation of human mathematicians; if we weren't limited to finitely specifiable, effective systems, these problems wouldn't exist. So I think that if my thoughts entail strong emergence in some sense, then one also should think that mathematics contains strong emergence. But then it seems to me that's just fine, after all, mathematicians have still been able to make great strides in mathematical understanding post Gödel. (In that manner I, too, am optimistic about the progress of science---our understanding of the world will continually increase, but I don't think it will come to a true end point, although for all practical purposes, it eventually may.)

(By the way, have you seen Sabine Hossenfelder's take on strong emergence in this contest? I think it's a clever idea, although ultimately, I don't believe it really buys the necessary elbow room for things like free will.)

You're right also to point out the difference between consciousness, the sense of self, and so on. I wasn't intending to suggest an equivalence here, but I think there is a sort of progression---in order to be properly called 'conscious', you need to have a sense of self, and in order to have that, you need to be able to conceive of yourself as an entity distinct from the rest of the world, and in order to do so, you need to be able to model the world as sort of a container with you in it. Each lower rung is necessary, but not sufficient, for the next one. (As for literature on the subject of consciousness, I can't really claim to be an expert, but if you haven't read it, I still think that Jaegwon Kim's 'Philosophy of Mind' is one of the best overviews of the subject, and contains many references for further digging into the topics you find particularly interesting.)

There's also an issue regarding what constitutes a model that you highlight. In some sense, a brick is a model for our solar system---we could use it, say, in a model of the nearest couple of star systems. That empty beer bottle over there is Alpha Centauri, the football is Epsilon Eridani, and so on.

Undoubtedly, however, it would make for a very bad model of the solar system. That's because only its most elementary structure---the quantity, its being 'one thing'---is represented. So it's not clear in how far my talk about 'the world without models' is itself something of a model.

In particular, one also may talk about 'the round square cupola of Berkeley college', but does that necessitate a model of it? Can one model impossible objects? I think that one instead has models of its parts, or understands its properties individually, without really being able to put things together---so one knows (by model) what 'round' means, what 'square' means, what 'cupola' means and so on; but one can't really put these together, since that would be an impossible object. I think one understands 'the world without models' in a similar way.

Grüße zurück aus Köln,

Jochen

Dear Narendra,

thanks for your comment. My essay isn't really all that concerned with quantum mechanics, but regarding your question, well, it depends. QM is certainly a different kind of theory than classical mechanics---after all, it generally can only give probabilistic predictions, while classical mechanics comes with a (to some) reassuring certainty (at least in principle).

But that doesn't mean it's not a full fledged theory. If by this you mean a mathematical apparatus, and a way to connect the mathematics with observation (sometimes called the 'minimal interpretation' that makes the difference between physical theories and purely mathematical frameworks), then QM is just as reputable a theory as classical mechanics.

If you want further to know 'what it all means', however, quantum mechanics seems much less amenable to intuition. But is that the fault of the theory, or of the human mind that isn't used to it? I don't have an answer.

Dear Terry,

I have to admit that I chose the title in order to add a little shock value---so I'm at least somewhat glad to see it worked in your case! You seem to have recovered OK, though. ;)

Thank you for persisting through the shock, and your comments on my essay. I agree that what I'm hinting at is just the tip of the iceberg, and I'd give several non-essential organs in order to take a look at the whole edifice.

Using compressibility/Kolmogorov complexity in order to qualify more 'fundamental' levels of description is a worthwhile project, I think. I wonder, are you familiar with the story of Leibniz and the inkblots? Chaitin likes to tell it. Basically, Leibniz noted that if one randomly splatters ink on a page, then there won't be any description of the pattern they make that's much shorter than just noting every individual stain; but if there is a law behind their distribution, then that law will make for a much more concise description. It's almost all in there, nearly 300 years ago!

Anyway, sorry for taking so long to reply; I'll have a look at your essay, maybe I'll find the time to comment there, too.

Cheers,

Jochen

Thanks, Marc, for pointing this out! Although I'm not sure how this happened, since I just copied and pasted the link... Anyway, for anybody interested, with any luck clicking here should get you to Peter's discussion of my essay.

Dear Avtar,

thank you for your comment! I'm glad you found something you agree with in my essay. I'll have a look at yours---maybe I'll find something to agree with, too!