Undecidability and indeterminism by Klaas Landsman

Dear Klaas,

major parts of the physical literature, beginning with Newton, can be taken as a reference, if one is prepared to take off one's empirically-tinted glasses. In the scholium of the Principia Newton makes a pretty clear case for an idealistic view of physics:

"Wherefore relative quantities [time, place, space and motion] are not the quantities themselves, whose names they bear.....And the expressions [time, place, space and motion] will be unusual, and purely mathematical, if the measured quantities themselves are meant." So, Newton's space and time, for instance, are not meant to be isomorphic with the everyday use of these notions and he stresses that the notions of physics (=mathematics) must not be confounded with the notions of empirical quantities [vulgaribus mensuris], which I have radicalized to the idea that natural language and physical language can only be Absolutely non-contradictory if they are incommensurable, complementary or orthogonal.

Further, one still finds very empiricism-critical approaches to relativity and quantum mechanics in the early papers of Einstein (1905) and Heisenberg (1925), respectively. Both make very clear that they speak of KINEMATIC representations, not of PHYSICAL theories. The fusion of the 'two languages' and hence the Unanschaulichkeit of modern physics begins with the coup of the Göttingen school of mathematics...

Just in case you are interested in the basics of my thinking (and happen to speak German), google for "Wird die Wissenschaft zum Feind des Wissens?" or "Das Ding an sich", both posted on Philosophie.ch

Heinz

Dear professor, Landsmann,

You write

"...it may be helpful to note that in classical coin tossing the role of the hidden state is also played by the initial conditions (cf.Diaconis & Skyrms, 2018 Chapter 1, Appendix 2). The 50-50 chances (allegedly) making the coin fair are obtained by averaging over the initial conditions, i.e., by sampling. By my arguments, this sampling cannot be deterministic, for otherwise the outcome sequences appropriate to a fair coin would not obtain: it must be done in a genuinely random way. This is impossible classically, so that (unless they have a quantum-mechanical seed) fair classical coins do not exist..."

And, concluding you, state:

"I would personally expect that a valid theory of the Planck scale (including quantum gravity or string theory, though these words are misleading here), far from assuming the Born rule and the rest of quantum mechanics (as these theories normally do), would derive quantum mechanics as an emergent theory (instead, the opposite seems to be the majority goal, i.e. deriving gravity as an emergent phenomenon from quantum theory). Thus quantum mechanics would typically be a limiting case of something else, which would, then, also render the Born rule valid in some limit only, rather than absolutely."

My humble self is interested to see that we can indeed reduce this whole classical/quantum divide to an intuitive picture -- when we think of the self-referencing state (typically every mind/observer) as, by you, own "quantum mechanical seed".

That is, we can approach quantum mechanics from the perspective that it is fundamentally about self-reference such that the observer is by definition the norm/normal e.g. the unit or constant refractive index by which we are at any instance describing our cosmology. Modelled as Gödel's self-referencing state, every mind simply is thus the quantum of own observables (typically the probe energy or beat frequency or quantum vacuum) if the Landauer limit.

This will be just in the exact same way that Planck's quantum (v = E/h) is that norm/normal by which in spectroscopy we are attempting to describe the applicable black-body radiation as a unique arrow of time (the black body spectrum).

In this sense every mind would be as the phase constant own Kolmogorov incompressibility (own energy/time or wave/corpuscular uncertainty threshold); just what the constant refractive index is to all observable dispersion relations of light i.e. to all self-referential splitting of light into constructive versus destructive interference; perhaps your binary string.

Hoping that you can take a look at how I grapple with this vision (being an editor, please don't let poor wordings distract you).

Chidi Idika (forum topic: 3531)

Sorry about the typo in the name. Meant to write Landsman.

You are right. Bear with me, sir.

And when you find the time please do take a peak I'm sure will find the connection interesting.

Dear Klaas,

Please do not bother to react to my question above; I also know what is like to have to try to reply to all comments while teaching. Moreover, I posted the question rather for your consideration and did not expect an answer, but should have made that explicit. The reason for posting the question was that, personally, I think it is interesting to see what bringing the idea of atomism to its logical completion looks like.

Best wishes,

Vesselin

Hi well , like I said your essay is a good essay with a good ranking of all the different thinkers and a mix of their ideas about the computabilities, the randomness, the mathematics, we see a very good knowledge of all their works, Solomonoff, Kolmogorov, Chaitin, Svozil ,Downey, Hirschfeldt, Born, Bell, Godel, Hilbert,Durr, Goldstein, Zanghi, Hooft,and others that I forget to name.

So indeed you have well learnt their works and you play with all this, but I see in all humility maybe a big problem , you consider these strings and correlted bits and a kind of wave pilot, so it is correlated with these 1D main strings at this planck scale and the 1D main field form this cosmology if I can say, you search the good road to correlate the universe, the quantum mechanics and the quantum computing in trying to consider the good randomness, series finite and infinites in a kind of universal converging partition, but for me if the main essence is made of particles instead of fields like in the strings, branes, Mtheory , so you are not going to converge with this universal foundamental mechanics if I can say.

We cannot affirm that all is made of fields and philosophically the same, the strings consider a 1D main field at this planck scale and extradimensions but we cannot affirm that it is the truth to explain our geometries, topologies, matters and properties.

My model considers 3D spheres coded but I am not here to speak about this, we are not here to prove philosophically who is right about these foundamental mathematical and physical objects and the correlated philosophy of this universe, we cannot affirm simply. Like I told your essay shows that you have well studied all these works and that you perceive the generality, it is important. But if I can , if your strings generally are not correct , so never you shall explain the quantum computing , I recognise good mathematical Tools with these strings to rank the fields, but witten has created a prison and I beleive that many confound his field medal wich is a very good mathematical work with his theory of strings. I doubt that this universe is only made of photons and that they oscillate due to strings to imply the physicality. Witten and Einstein even if the GR and SR are correct have created a prison for the thinkers. So the landscape is complex and we cannot affirm the main cause and the philosophy.

You can utilise all what you want for a general universal landscape, if the foundamental objects and the philosophy general are not correct, so just a part is correct but not the generality. And the non commutativity is not the problem but the foundamental Tools , objects and the general philosophy yes. I consider like I told you these 3D spheres in a kind of superfluid gravitational aether, space. And 3 main primordial coded series, one for the space and two fuels, the photons and the cold dark matter and when they merge they create these topologies, geometries, matters and properties with fields.

he wave particles duality is respected also because all is in contact when we consider specific finite primordial coded series of 3D spheres. I formalise all this with these mathematical Tools, an intrinsic ricc flow, the Hamilton Ricci flow, an assymetri Ricci flow that I have invented to explain the unique things, the lie derivatives, the lie algebras, the lie groups, the topological and euclidian spaces, the Clifford algebras, the poincare conjecture, the deformations of spheres also. Who is right ? me or the strings theorists ? we don t know and nobody can affirm but when I see the nature, it seems evident that these spheres, spheroids, ellipsoids,....are the choice of this universe, why I don t know but this geometry is different, it is the perfect equilibrium of forces and has no angle and can create all geometries.

The big problem is philosophical in fact. It is not easy I know to change a road of thoughts for the thinkers but the doubt is the real torch of generalists, if they are persuaded about unknowns , so it is a problem of Vanity for me, in all case these strings and these 3D spheres can converge because they oscillate also and are in motions, and in contact due to this universal superfluidity of the space.

All this to tell you that for a quantum computer , we must absolutelly utilise the good foundamental objects , if not we cannot reach it, the aim is not to play with all the mathematical Tools invented by all our best thinkers in maths about this randomness, the infinities, the real infinity, the finite series, the Waves, fields... the aim is to find the good universal partition with the foundamental mathematical and physical objects and their universal philosophy.

Regards

My comments are based on the text presented in the essay, without a consideration to any work cited, and my limited knowledge of the subject.

1. "For a fair coin flip the probability of each string Пѓ within 2^|Пѓ| is 2в€'|Пѓ|, so that an apparently "random" string like Пѓ = 0011010101110100 is as probable as a "deterministic" string like Пѓ = 111111111111111. Therefore, whatever its definition, the "randomness" of a string cannot be defined via its probability."

I am not sure why the string 111111111111111 is referred to as deterministic? If we consider the fair coin flip generated strings then this is as probable as 0011010101110100 as you stated, but if we consider 111111111111111 to be deterministic for its given generator mechanism produces only 1s, then all sequences are to be deterministic to be the outcome of their respective generator mechanisms, not a result of fair coin flips. I presume the difference in string length is typographical (16 vs. 15 bits), but that is intended, then I missed the point. Indeed the Kolmogorov complexity of a given string 111111111111111 may be very low, but the Kolmogorov complexity of the process of generation (coin flip) has to have infinite detail to generate truly random sequence of any length. Moreover, complexity of even 0011010101110100 can be as low as one bit if we compare against the pre-set base sequence of 0011010101110100. That is the complexity depends on the basis of generation or comparison.

Furthermore, if we leave the definition of randomness of a process on interpretation, then by ignoring the generator mechanism any physical process can be interpreted to have a degree of randomness.

But, if we consider the underlying reality of quantum system to be analog or continuous with infinite possible configuration and require transitions to be always in quanta depending on the gate with discrete allowances presented by the observing system, then the process will have genuine randomness as discussed in my essay -- Mother of All Existence.

2. "Theorem 3.2 With respect to P^в€ћ almost every outcome sequence x в€€ 2^N is 1-random."

Quite a few places, 2^N notation has used the space of integers in the power rather than a number N, which I could not follow if that was intended.

Rajiv

The problem is complex and mainly philosophical. Why the majority of thinkers now consider that all comes from the fields ? it is due to the strings theory in fact and Witten, they consider that we have only photons due to an infinite heat and that this thing oscillates these photons with strings at this planck scale in 1D connected with a 1D main Cosmic field. And after with the geometrical algebras they create extradimensions towards 11D. and they explain the emergent topologies, geometries and matters, I can respect these strings and recognise good mathematical plays to rank these fields but the generality for me is totally false, because we have probably coded particles and 3 main finite series of particles coded merging in a superfluid to create these emergent topologies, geometries, matters and properties of Waves, fields and particles. I consider a pure 3D at all scales. In fact it is logic that these particles are the main essence , we cannot have fields without particles because without motions, we have particles but no field, not the opposite, this strings theory and general relativity have really created a prison, and it is difficult now to discuss with these strings theorists, they cannot change their line of reasonings, they are persuaded. But these strings have a big problem philosophical ,they don t explain the evolution...

Dr Landesman -

A brilliant, erudite essay. Unfortunately I do not have the expertise to follow the mathematical reasoning and I am puzzled about the epistemological significance of the findings. I was left "in the weeds" so to speak. But I do agree that physical determinism has been shown to be false - although that result is not provable.

I have made a more ambitious effort to bridge the gap between the quantum and mathematical challenges using a three-worlds framework. I'd appreciate your review if you have the time.

Cheers - George Gantz: The Door That Has No Key: https://fqxi.org/community/forum/topic/3494

I really enjoyed this essay. It seems to me that shifting the focus from individual experiments to sequences of outcomes is a very important insight and a step that quantum foundations certainly ought to take - after all, the experimental evidence for quantum mechanics comes from relative frequencies in sequences of measurements, not single-shot experiments, so it's likely that our interpretational difficulties come in part from trying to apply a theory defined over sequences to individual probabilistic events.

That said, I didn't entirely understand the motivation for imposing the requirement of 'typicality of almost every outcome sequence of an infinitely repeated fair quantum coin flip.' Given that all the outcome sequences we have actually examined are finite, and that in any case typicality of an observed sequence can't be proved, the experimental evidence cannot justify the claim that outcome sequences must be typical. This claim, as I understand it, comes from assuming that the standard quantum-mechanical representation (based on the Born rule and the combination of independent systems by taking tensor products of the underlying Hilbert spaces or operator algebras) is the correct mathematical representation to obtain the probabilities for outcome sequences even in the infinite limit. Would the proponents of the de Broglie-Bohm and t'Hooft models not simply deny this claim, arguing that there is no reason that their models need to be compatible with quantum mechanics in an untestable limit? Requiring compatibility with the predictions of quantum mechanics even for unphysical examples seems to be assigning too much weight to the mathematical structure which we happen to have chosen to represent the observed physical results.

Dear Prof. Landsman,

You essay is brilliant, very important for closing loopholes and final ending with determinism in QM. The question now is where the fundamental indeterminism or randomness plus undecidability comes from. You state that theory of the Planck scale would derive QM as emergent, then hopefully we may know. Such theory, or an ultimate theory of everything should also be able to tell emerging 'why' and from 'what' and this 'what' should be then kind of irreducible or self-referential. Such topics are obviously rather mind-blowing. In my essay I am focusing on the 'why' and 'what' and it turns out it is uncomputable sequences which have curious relation with nothingness. Material in my essay is rather compressed and concepts are not stated precisely but I hope the essence can be grasped.

Best regards,

Irek

Lol, are you just good in studying the works of known thinkers and mix their ideas or are you also able to go deeper in philosophy and link the generality of this universe, and of course without Vanity because we know that this parameter decreases the complementarity and evolution of the theoretical sciences Community, too much forget to change their line of resoning the most of the time due to the paramter cited , I repeat but a few number are able to create new innovative theories linking the maths, physics and philosophy, so it is probably the reason why they fear to discuss because they are probably not made to go deeper or to discuss about these things, so they don t develop these innovant theories :) spherically yours

Dr. Landsman,

Not to detract in any way from the QM arguments you elucidate (much of it above my paygrade) but when it comes to the mathematical construct of Bell's Theory it seems to me that it assumes a disconnect from the physical in the first statement. A line segment of indeterminate scale with incrementals between -1 & +1 would not be physically capable of embedding correlations on three vectors of R3. And for R3 to become embedded in S3, it would have to evolve into R4 and undergo one more half pi rotation for the three right hand axes of R4 to coincide with the three RH (assuming RH initial designation) axes in S3 without one axis becoming a pseudovector.

What ever the physical reality of the entanglement correlations might be, and which none can adequately describe beyond endless debate, technologically it matters naught whether deterministic theories or non-local theories are the more correct. What matters is that whom ever gets a functional system up and running first will corner the global market of electronic transfer of funds. And with that the preferred national currency will quickly become the global reserve currency for the coming century. Simple political science. Technology is becoming more an essential commodity than oil.

Best wishes, your essay seems to me to be a tour de force of many painstaking years. jrc

Fervent congratulations, Klaas. I have been a passionate follower of your work since I saw you give a talk in Oxford in the early 2000s. You may recall that my focus is on algebraic QFT: algebraic QM with, as we might say, arbitrary subdivisions allowed.

On page 8, you suggest a dilemma, "The sampling is [or is not] provided by the hidden variable theory".

I suggest that there is, at least, a trilemma, including "The sampling is provided by the design and construction of the experiment."

That is, we search for materials and devices that generate signals that are not constant and are not obviously periodic, both of which would be not fit for purpose, but instead that exhibit thermodynamic transitions that are in some sense random or very nearly so, between metastable and more stable states. Given that search succeeds, we design electronic circuitry and computer software that recognizes and records details of those thermodynamic transitions as events. Then we put such devices as we construct into different surroundings and examine the statistics of events and how well they match quantum or random field models for them. Such transitions, at MegaHertz rates, are not determined by consciousness, but they are determined by the possibility of agency: a scientist has to be able to choose to build one apparatus rather than another (whether such a choice as that is determined or not we might not be able to determine.)

To put the argument above rather more classically, when we want something close to a random sequence, we do not toss a featureless sphere, we toss a coin that has identifiably different sides. "An algebraic approach to Koopman classical mechanics", in Annals of Physics 2020, suggests this amongst other considerations.

In such a short article very little can be said, but I also wondered how you would tackle statistical decision and parameter estimation problems such as "when will I accept that this coin is biased and by how much will I change the parameter away from ½?" We do make such decisions, and of course there's a huge literature on how to make them, both for classical and quantum states, even though we know that more experience will almost surely make us change our minds.

I'm always a little graceless in this kind of comment. Sorry! I hope you will forgive me, but in any case thank you again for such clarity.