Undecidability and unpredictability: not limitations, but triumphs of science by Markus P Mueller

Dear Markus,

I'm happy to see you've found the time to enter an essay into this year's competition. Like last year's, it's a highly intriguing and insightful piece, and I'll spend some quality time studying it further. I like the way you frame the discussion: undecidability and related phenomena need not fuel defeatist stances, you enlist them in an optimist message about our ability of getting to know what's there to know at all.

This bears some similarity to what Karl Svozil has called the 'third path' in dealing with these notions: most of the time, the issue is either neglected as an artifact, an esoteric mathematical rumination not more relevant to the (physical) world than the precise number of angels on the head of a pin, or on the other extreme, enlisted into grim pronouncements of our doomed quest for knowledge. Trying to find a third (or perhaps 'middle') way of taking it seriously, even as the starting point of some constructive challenge to outdated ideas that force us to engage with some core assumptions is a welcome change of pace.

Before I go into deeper discussion, however, I'd like to ask a question, if I may. The sort of structural realism you champion derives from a venerable tradition, stretching back at least to William James, Bertrand Russell, and Arthur Eddington. But it seems to me that a challenge raised against such views by the mathematician Max Newman (known these days generally simply as 'Newman's objection') hasn't really found a good answer. That challenge essentially consists in pointing out that if everything you know about some domain is the relational structure it fulfills, then really, all you can answer are questions of cardinality---for if there's the right number of elements, any structure will fit, so to speak.

Do you think that's a problem for your view? If not, how do you propose it is overcome, or why doesn't it apply?

Best of luck in the contest!

Cheers

Jochen

"... there is a widespread view of quantum physics which regards its statistical character as a symptom of incompleteness ..." Is "quantum physics" a well-defined concept? Is "spacetime" a well-defined concept? Is a "potential infinity" a well-defined concept? Consider the following hypothesis: For every positive integer n, if nature contains n bits of information then nature contains n+1 bits of information. I suggest that there is empirical evidence that the preceding hypothesis is false. Why do I suggest this? I say that dark-matter-compensation-constant = (3.9±.5) ^ 10^-5 ... contrary to the widespread belief that dark-matter-compensation-constant = 0. If dark-matter-compensation-constant were equal to 0, it seems to me unlikely that Wolfram's cosmological automaton would work. I conjecture that the 4 ultra-precise gyroscopes of Gravity Probe B worked correctly. Why do the string theorists think that Milgrom's MOND is wrong? First, they refuse to carefully study the empirical evidence. Second, they realize that string theory with the infinite nature hypothesis makes any string theoretical explanation of MOND extraordinarily complicated and rather dubious in terms of their paradigm.

What might be wrong with Kroupa's analysis?

Kroupa, Pavel. "The dark matter crisis: falsification of the current standard model of cosmology." Publications of the Astronomical Society of Australia 29, no. 4 (2012): 395-433.

arXiv preprint

Dear Markus Müller,

interesting essay, but too post-modern for my taste: never call a problem a problem, at most a challenge or ideally, make it into the solution.

Heinz

Dear Markus,

thanks for your reply. I think you're right regarding OSR and the Newman Objection, at least as it's usually conceived. I'm not totally convinced by the argument, though. To me, there seems to be a threat of a priori considered to be distinct structures collapsing onto one another---any claim to theory A giving the 'right structure' can be challenged by some theory B yielding the same predictions. Essentially, you can mirror one theory within another---roughly, via something like a Gödel numbering, but more explicitly, if you have a theory which allows for universal computation, you can just 'code up' a simulation of whatever other theory you want to say gives the 'right structure' within the former.

Say you've got a fully worked-out version of string theory, and want to claim that hence, that gives you the full 'real structure' of the world, and with that, everything that can be said about it (in the sense of everything there is to be said). But then I come along, and encode string theory into some electromagnetic field configuration that corresponds to a computer's memory containing a program that simulates string theory with some initial conditions, and derive all the same predictions from (presumably rather complicated) calculations using nothing but Maxwell's equations---in what sense, then, is the structure of the world that of string theory, and not that of Maxwell's electromagnetism? (Apologies, by the way, if you address this in your essay, unfortunately I still haven't gotten to digesting it fully.)

Of course, there's going to be more structure than just the Maxwell equations, corresponding to some complicated initial state, but there's presumably some initial state in the string theoretic description too, plus there will be very many programs on many different computational architectures---many different field configurations---leading to the same predictions.

But I'd need some time to make this thought more precise. I think I'm otherwise quite happy with accepting OSR as a live option---I've in a sense chosen the other way out: rather than conclude that structure is all there is, I presume that what there is will always outrun our ability to fully describe (which description, I agree, is essentially structural), largely motivated by something like the above worry on structural underdetermination (regarding the question of implementing a computation).

I'll hold off on further comments until I've gone through your essay. I can say already, however, that it's excellent, and deserves to go far in this contest.

As a final remark, I've found many articles of Svozil highly illuminating on this topic. I think he's in Vienna, too, no (albeit at the TU)? Maybe you should strike up a discussion and see if you find some common ground!

Cheers

Jochen

Dear Professor Mueller :

I like your views on the question of Undecidability, Uncomputability, and Unpredictability:

Our only limitation is data; if we have the data then the proof is possible. If the data is not available then the proof is uncomputable. The undecidability question becomes whether or not we have all the data. Finally, unless we are clairvoyant we have no observational data about the future, consequently any prediction made is a projection of history and subject to falsification.

Just giving a name to something is not data. Richard Feynman: The Difference Between Knowing the Name of Something and Knowing Something.

A nonanswer may be proof that the data does not exist or one does not have all the data or that the data is unrecognized.

It is proposed that any evidence describing the Big Bang is beyond science's reach and yet this essay [entered January 18th below] "Common 3D Physics Depicts Universe Emerging From Chaos" presents a plausible description with plenty of replicable evidence.

Respectfully,

Charles Sven

Dear Markus,

thank you for this well-argued essay, which I enjoyed very much. You managed to give a new perspective to a notorious analogy between incompleteness in maths and undecidability in modern physics.

I particularly liked your statement against a sort of diffused Platonism (expecially among theoretical physicists and some mathematicians): "we may believe that there is something called "the natural numbers", N, a well-established "thing" (after all, formalized as a set) that somehow "sits there", waiting for our mathematical tools to discover all of its properties and to prove all of its true theorems".

One comment that I should perhaps like to make, is that while I am in principle very sympathetic with this idea, I always find a bit disappointing how vaguely structural realism is spelled out in the philosophical literature. For it remains vague enough to accomodate many views which perhaps would not naturally go hand in hand. So, also in your essay, even if you indeed took a "structural" standpoint throughout all of it, I found the connection in section IV a bit unsharp. But it may well be that it is me who always struggles understanding ontic structural realism in a non superficial way.

Anyways, great essay and best of luck for the contest!

Cheers,

Flavio

Dear Markus,

I've finally gotten round to giving your essay the reading it deserves. I think it's in some ways very close to my own thinking, and a few years back, I would've heartily endorsed all of its claims---indeed, in my entry to the 'It from Bit'-FOXi contest, I expressed similar reservations against 'thingism': "The world is then not something comprised, at the very bottom, of things, but rather, it is given by a web of relations."

Back then, I appealed to a notion of 'relative facts' to encompass roughly what you call 'differentiations' of a structure---the idea being that the questions not answered by a given structure have answers only relative to other events, sort of like an electron's state is 'spin up' relatively to a measurement apparatus' registering the appropriate measurement, whereas to you, I gather, the 'spin up'-value would be a further differentiation of the structure giving the system's state.

One worry, to me, is then how differentiation happens. If we, say, make a measurement on a quantum system, to stay with the example, does this then entail a literal further specification of the state---i. e. does the structure go from S in which the question of the electron's spin is indeterminable to S' in which it has a definite answer? If so, that seems hard to square with a purely structural reading---for if there were some further structure that determines what value is provided, then we simply didn't have the full structure to begin with (something which will be hard to square with the various limitative theorems of QM).

But on the other hand, if differentiation happens essentially randomly, then that process does not have a formulation in terms of structure---so in a way, considering this as a structural view is like having a 'computational' universe that every now and then has to look at an oracle to draw a genuinely random bit from a hat.

The third alternative would be to go to a kind of 'many worlds'-view, where all differentiations already exist---something like the 'relative facts'-version. But it seems questionable whether that's then still a 'structural' view---after all, if we have some structure S, allowing for differentiations S' and S'', and we say that those differentiations are already out there---say, again, in the two distinct possibilities of an electron's spin value in a superposition---then these differentiations, from the point of view of us who only have the description according to S, take the role of the 'things' that the structure is a structure off, and an experiment will tell us which it is---although of course, it will yield both answers.

Regardless, I think your criticism of 'thingism' is apt, on the whole---I've been trying to steer a kind of 'middle way', having become skeptical that pure structure can yield enough of a world to ground our experiences, but all my conclusions there are very tentative. I don't believe that there's a world of 'fundamental' facts that's simply given to us, a container of stuff to discover; but I'm also not sure that the mere specification of relations does not leave it just all empty, so to speak.

Anyway, I think this is an intriguing topic to explore. You may actually find some congenial notions in my essay this year, and the Found. Phys. article that spawned it---while I agree that Gödel's results, as such, don't have applicability to the physical world, they, together with a great many other similar ideas---the unsolvability of the halting problem, Russell's paradox, and others---are really a manifestation of a more general structure, captured in a fixed point theorem due to F. W. Lawvere. This I try and connect with the measurement process in quantum mechanics, in a way which seems very apt to your idea of 'differentiation'. I view it slightly differently, based on additional considerations, but I believe the framework fits.

Cheers

Jochen

Dear Markus,

I fail to see the relevance of Turing/Godel to the accessibility of the Platonic world. Turing only showed that any single, finite machine would necessarily have a blind-spot. It tells us nothing with regard to the decidability of "whether machine M halts on input I".

As to

"[...] the history of successful novel prediction science is the most compelling evidence for some form of realism, but [...] the history of ontological discontinuity across theory change makes standard scientific realism indefensible."

one could argue that the rational way to make progress is to `rewrite the history' of theory using a single, common ontology. This is my ambitious approach with respect to its chances of success I'm fairly optimistic by now

https://arxiv.org/pdf/1201.5281.pdf

Best,

Yehonatan

We are not on the same page...

What I want to say is that Turing's (and Godel's) result expresses a limitation of machines - systems which can be realized even within a classical, objective ontology.

In contrast, Bell's result expresses a limitation on the ontology: If particles were machines ("robots" in Bell's words) then his inequality would need to be satisfied (under reasonable assumptions). In my essay I define the notion of a "non-machine" to overcome Bell's limitation within a definite ontology. I argue that this new category of physical systems, which is mandated not only by Bell but even by classical electrodynamics(!) is the more generic, with machines being in some sense an `uninteresting' private case thereof.

I'm very sympathetic to your project in re.f [30]. Being under the spell of Hofstadter and Penrose in my high-school years, I also believed that was the right way to proceed. But I have since discovered the limitations of a system called Yehonatan Knoll, and that system, if it is to produce any real progress in physics, must stay as close as possible to a `pedestrian' objective ontology :)

Dear Marcus,

I have you to thank for engaging with my comments in such an open way! These contests are always at their best when they stimulate frank exchanges on views that may go slightly beyond what one would normally put into a journal article or the like.

And I'd very much like to come for a visit to Vienna---although thinking about traveling plans seems almost frivolous these days. Perhaps one lesson we could take away from the present situation is that we should try to create more and better avenues for online exchange of views---something like virtual research/discussion groups, where people interested in some topic can congregate, discuss with either live-sessions or in a chat/forum based manner, exchange drafts for articles/request comments and the like. Could be as simple as a Teams channel, or something like that.

But back to the things themselves, as Husserl said---or to their absence, as it were. Regarding the 'modding out' of equivalences between structures, I'm afraid that this might leave us with altogether too little in the way of substance to account for the world and out experience within it---if we agree that in my example, electromagnetism and string theory yield in some sense the same structure, then one could also draw in all manner of 'different' systems---say, for example, the three body problem is at least conjectured to be computationally universal, so you could encode the requisite data into its initial configuration, and then just let Newton's laws do the rest. Or, of course, any other theory that allows for universal computation.

So fine, one might try to say that then, most of the structure is in the initial condition---the program, so to speak. But this, too, is far from unique: you can instantiate the three body problem with all manner of initial states, implementing different Turing machines that then instantiate the requisite computation from different initial conditions. In each case, that would add some constant number of bits to the length of the initial program, corresponding to the specification of a Turing machine to be simulated.

So suppose that you have an initial condition for the three body problem that can be specified using n bits, such that the resulting system implements the 'structure of the world' in some sense, by essentially implementing some Turing machine T executing the n bit program. Then, it seems to me you could find a TM T' such that it takes at most n O(1) bits input to implement the same program, with the O(1) factor corresponding to T' simulating T. But then, have you really said more about the world other than 'it contains at least n bit of information' if you specify its structure in this way?

I think this is essentially the Newman problem again. In some sense, all 'universal' structures---structures corresponding to theories allowing for universal computation---are equivalent: whatever you can describe using one, you can describe using another, with at most some constant overhead.

Maybe one could try to argue for parsimony, and single out that structure which yields the most compact specification---which has the problem that the question which one this is will be undecidable, due to the uncomputability of Kolmogorov complexity. Or, one could try to adapt the various attempts at solving the threat of trivializing computationalism---because that's essentially the same problem, again: virtually every system can, naively, be viewed as instantiating virtually every computation. There are, I think, certain avenues regarding dispositional/counterfactual/causal accounts of computation that one could pursue, in order to arrive at a notion of isomorphism between structures that's coarse-grained enough to allow for the identification of 'obviously identical' structures, without being so coarse-grained as to trivially identify virtually all structures with one another. I'm not sure if that'll work, but, with a more careful fleshing out of the notion of structure, I think there's at least a few avenues to explore here.

Regarding the concrete application towards quantum mechanics, I think I understand your proposal somewhat better now; but if the structure, as such, only accounts for the correlations in measured data, then how are concrete measurement outcomes accounted for? Pre-measurement, only a certain probability distribution over outcomes exists, but post-measurement, we at least seem to have observed one definite outcome.

Now, I suppose one way to account for this without appealing to some sort of coming-into-being of a more differentiated structure is to appeal to a sort of facts-as-relations account: before the measurement, relative to the '|ready>' state of the detector, the (say) qubit is in an equal superposition; after the measurement, relative to the '|detected 1>'-state of the detector, the qubit is in the state |1>, and relative to the '|detected 0>' state of the detector, it's in the state |0>. We think about this as moving from a superposition to a definite state, but this is really thinking as if we could hold the state of the detector fixed---but thinking about this as a relation between the detector and the qubit, the three propositions 'relative to |ready>, the qubit is |0> |1>', 'relative to |detected 1>, the qubit is |1>', and 'relative to |detected 0>, the qubit is |0>'---which are not actually in conflict at all, and hence, can well be part of a consistent structure.

As for my 'middle way' between the naive box-of-things view of the world, and the---to my way of thinking---somewhat too rarefied view of relata-less relations, I don't really go into that in the present essay, but the germ of the idea---which is still pretty much all I've got---is in my entry into last year's contest, where I essentially propose that there are fundamental facts only relative to a certain perspective on the world, or a certain way to split the world into distinct subsets, or sub-objects, or perhaps, subject and object. It's a bit of a tightrope walk, and I'm far from certain something like that can be made to work at all, but not really liking to drop to either side, it's kinda all I got.

I hope you manage to meet all your deadlines!

Cheers

Jochen

Dear Marcus,

I think the sort of explication of structure you're looking for might work along the lines developed by Lutz in 'Newman's Objection is Dead; Long Live Newman's Objection!'. Lutz essentially points out that there's no real way out of Newman's objection if the usual, Ramsey-sentence based notion of structure is used, but argues that this notion is insufficient anyway, and proposes to use an isomorphism-based notion: some sets A and B provided with relations R and S respectively have the same structure if and only if there is a one-to-one relation between A and B such that for all elements a1 and a2 of A such that a1Ra2, they are mapped to elements b1 and b2 of B such that b1Sb2.

I haven't gone through the complete paper yet, but so far, it seems promising.

Cheers

Jochen