Dear Eckard,

thank you for your comment. Regarding quantum computing, to me, this is a 'the proof is in the pudding'-kind of case. Either it will work, which will then legitimize at least some of the foundational principles at the heart of quantum mechanics; or it won't, in which case, we'll likely have learned something excitingly new about the world. Since that appears to be a win-win situation, I don't see why one should not continue to put in the effort.

The reasoning in my article has important similarities to that of Cantor, so if you reject the latter, I can see you having some trepidation regarding the former; however, it does not seem that you have the same reservations against the Gödelian argument, which is in the end yet another example of the same technique (diagonalization, or more generally, the application of Lawvere's theorem). Or do you see an essential difference?

Cheers

Jochen

Dear Peter,

thank you for the kind comments! 'Flawless' is high praise indeed, though I myself keep going back and thinking of ways I could've done better, or be more clear about. But if it all hangs together in the end, I'm willing to be content.

Your project seems engaged in questioning the foundations of logic---revoking the law of the excluded middle invokes comparison with dialetheism, and of course, quantum mechanics has itself been argued to lead to similar revisions, see Reichenbach's three-valued logic, and the more familiar von Neumann/Birkhoff logic. Although of course, in a sense, trying to make logic empirical, or at least, renege on it due to experiment, is sort of putting the cart before the horse.

I will try to get to your essay, in the hope of understanding the rest of your comment better. However, due to recent circumstances, my time for this has rather been slashed, so it might be a while, I'm afraid.

Cheers

Jochen

Hi Eckard, I can understand what you say, but let s go deeper in philosphy and about our physicality. What is the main cause of our physicality and how we must consider this infity and these infinities and our finite series. Can we really understand this universal distribution at this moment, we can take all the past thinkers having worked about this, that will not change our limitations due to a lack of knowledges generally speaking.

We don t know the main philosophical cause of this universe, I consider an infinite eternal consciousness beyond this physicality and this thing that we cannot define is so Deep and so far of our understanding. This infinity , the real infinity has created a physicality with a system in evolution with informations, particles and Waves and we see that this physicality is under a kind of universal partition where the numbers, the particles and Waves create this physicality and its topologies, geometries, paproperties of matters. We see these infinities appearing everywhere like mathematical Tools and physical ones but they don t explain this infinity really, they are just like Tools simply inside this physicality.

The philosphy and ontology appear indeed but we are limited simply, we cannot affirm because it is far of our understanding. It d be odd to pretend the opposite, nobody can prove what is this non physical infinity. We can analyse all what we want inside this physicality with the maths, numbers, and physics , that will not change our limitations, nobody can prove philosophically the orogin of this universe, the same for the main codes at this planck scales or the foundamental objects , all what we can is to study and improve our limited knowledges inside this physicality. The aim is to accept these limitations I beleive and respect simply the pure determinism inside this physicality.

Regards

Dear Jochen,

I see it as horse first, as the problems in logic ('paradox') and Philosophy were worse than physics, needing resolving by checking starting assumptions. How do we imagine Aristotle dreampt those up anyway! We now have far better information than he did, but the foundational issues took a long time to dig down to. The sound consequences of the proposed revisions alone seem to confirm veracity.

BUT the most important thing for you to study is the apparent physical solution to QM you suggest your'e looking for on page 1. It's verified by computer, but I trust a good well informed brain more! You didn't get to it last year so I suggest I'm owed a priority look!

Ridiculous Simplicity fqXi 2018.

I do hope your problems aren't family

Very best. Stay safe.

Peter

"respect simply the pure determinism"

I agree if determinism is not meant as the wrong belief that anything can be reduced to laws. I rather trust in causality rather than such demon even in cases of chaos.

Is there any reason to question truly fundamental logic?

Wasn't Leibniz correct and wrong at a time when he introduced the useful in mathematics quantity of being relative "infinite"?Just the name infinity for it was inappropriate. We may often calculate as if it was identical with the property

oo 1 = oo.

In general, I dislike attempts to question basic logic. Euclid's point corresponds to the irrefutable idea of endless divisibility. Equality of numbers implies the TND up to infinite acuity. That's why assumption of real numbers contradicts to Hausdorff's dots at zero.

Eckard

Dear Jochen,

Thank you kindly for your comment on mine. My reply is here;

Jochen, Thanks, My mentor Freeman Dyson agreed, ANY advancement means all OTHERS will "feel sort of lost", also Lorentz, Feynman etc. And yes I also studied logic & philosophy, both in crisis! Yes I pack a lot in, testing conventional thinkers, but all refs are given.

You wrongly infer I suggest loosing "the absolute identity of quantum particles.", I just suggest they can have different polar axis angles, except when 'paired', but I DO challenge that only a "statistical approach to QM", can work, & show how we can "do better" as Bell suggested! Shocking? Tes. But seems also true (I cited the verification plot). That's what I'd like you to test.

I hope you get a mo as it may be rather important to advancement.

Very Best

Peter

I've learned that when seeking to explain "existence" using "diagonals" it's more practical (than "uncountable sets") to consider one zero-origin number line from opposite side of a professor's closed window; so I disagree with the foundational math in the title as suggests classical arithmetic like "squaring negatives" and "prioritizing zero". Other than that, I believe one square root of three disproves zero-evidence-cubed, as suggested in my essay, and in your essay here:

"The above has the form of a diagonal argument. Diagonalization was first introduced by Cantor in his famous proof of the existence of uncountable sets, and lies at the heart of Godel's (first) incompleteness theorem, the undecidability of the halting problem, and many others. "

It looks like I'll be graded as a 1/10 hack but there is at least one good Cantor quote in mine. Help yourself to naming rights on "definition" vs "infinition". I liked another essay I read better for the simple reason it wasn't about being right it was about including our own pretending of "general authority"; otherwise this essay would be perfect so I'll give it a fair 9/10.

    Dear Jochen,

    What I see is not a difference between Cantor who died in a madhouse, perhaps because his trust in his own point set theory was shuttered by König, and Lawvere who is hopefully still very healthy. I am rather concerned with differences between reality and mathematical models. Many years ago I got health problems because mathematicians rejected "for mathematical reason" my argument that future data are not yet directly available by means of measurement in reality.

    Meanwhile, mandatory mathematics seems to require sophisticated efforts to fabricate new definitions of good old ideal notions like point and line. Why?

    As I tried to exemplify in my current essay, mathematical pragmatism (calculate as if ...) must not be one to one translated into a universal tool for description of reality. Are orthogonal quantum states still required if I am correct concerning Fourier?

    My hint to Kadin's extraordinary courage was meant as a challenge: How to apply your theory?

    Eckard

    Dear Daniel,

    thanks for your comment. I'm glad you found something to like about my essay, and thanks for the generous score. I'm not sure I'd say my essay is about 'being right'; I think about it as an 'essay' in the original sense of the word---it's an attempt, something that may well fail. So it's less 'this is how it is', and more 'wouldn't it be neat if this were how it is'.

    Anyway, thanks for reading; I'll have a look at your essay soon.

    Cheers

    Jochen

    Dear Eckard,

    I can certainly understand being weary about whether mathematics and physics is always in the sort of correspondence many physicists take it to be; mathematical theories may be as spectacularly intricate and aesthetically pleasing as they like, that still doesn't mean there's something in nature that they describe. There can sometimes be cases of 'rigor mortis': a focus on mathematical formalism to the detriment of the underlying physics.

    However, to me, mathematics is first and foremost a tool; properly applied, it forces us to be clear and consistent in our reasoning, which unaided reason often is not. In that, it is not different from other tools of scientific inquiry---the microscope is superior to the naked eye in resolving miniscule details, but still, the microscope needs to be pointed at the right spot to tell us what we need to know.

    I haven't yet gotten to reading Kadin's essay. I'm skeptical, as you'd perhaps expect, of any proposed return to classical physics---there are arguments that are very simple (I give one in my essay, regarding the existence of a joint probability distribution) that would seem to prohibit something like that. But I'll keep an open mind.

    Cheers

    Jochen

    Dear Jochen,

    Thank you very much for this essay it was a real pleasure to read. Very well written and thought provoking too.

    Pardon my slow-mindedness but I would like to grasp the essence of the argument for the non existence of f(n,k) i.e. what are some necessary preconditions for it to hold.

    - Is the claim valid even if there are finitely many states?

    - If k belongs to a finite interval of integers then I could build a finite set of experiments that would create a f(n,k) table for all possible values of 1 and -1 for each state (like when designing a truth table). In that case the mg operation must bring back an existing row since all rows would have already been exhausted by my truth table.

    Many thanks for your help on this.

    Best,

    Fabien.

      Hi Eckard,

      Sometimes you know , it is not necessary to extrapolate or discuss about evident truths. It is maybe a lost of time and we are not here to see who is the most rational, deterministic or logic, all rational thinkers respect only the pure determinism and it is not necessary to extrapolate the causality , because the causality in maths can imply confusions sometimes, so I insist only on this pure dterminism if I can say.

      We search answers and all assumptions are assumptions if they are not proved simply by experiments or proved mathematical Tools. Is it necessary to discourse about so evident things, I don t Think.

      I repeat but we shall can never understand this real infinity beyond this physicality, we cannot define it , and inside the physicality , even if we can rank these infinities, we are limited still and Always , the same for our finite series . We can utilise all the best mathematical Tools that we want and take all the best past mathematicains or actual ones around a table, that will not change our limitations even with the best Tools utilised. We name this the real humility in fact because we are simply Youngs at this universal scale simply. It d be very vanitious to consider the opposite.

      We can even take the set theory or others, that will not change our limitations about these finite series, the infinities inside the physicality or this philosophiocal infinity. The zero is not the problems , nor the numbers and their distributions and the correlations with the foundamental objects, we don t know what they are and why they are and how they are distributed really. We just know a small part of this universal partition. You can take all what you want like series and harmonical partitions towards the zero and the infinity or infinities, that will not change this truth of limitations even in considering the speeds and the physicality . Alkl this becomes philosophical.

      So if I can Eckard , explain me your general philosophy about this universe and its origin, what do you consider like main cause ? and what do you consider also like main foundamental objects, because it seems necessary to better calculate this physicality and its numbers, particles and fields .

      Best Regards

      Hi Jochen,

      Thank you for your well-written and engaging essay. I agree 100% with your closing thought for the need of a relative realism, except that I would express it as a contextual realism, since context encompasses more than 4D (or 3D+1) inertial reference frames.

      You correctly assert that the key attributes of physical reality needed to explain quantum phenomena must include 1) finiteness and 2) extensibility. If physical context includes a non-zero ambient temperature, finiteness and extensibility are immediately accommodated. Absolute zero is an idealization that does not exist in reality, and the universe as a whole has an ambient temperature equal to its 2.7 K cosmic microwave background. For a contextual reality, a finite ambient temperature means that space itself is not infinitely continuous and position-momentum information is finite. A decreasing ambient temperature fine-grains space and creates new information to be discovered. In addition, when perfect measurement is defined with respect to a system's actual and objective context, empirical classicality is restored.

      In my essay I analyze the deeply held assumptions that have led to the widely-held conviction that objective physical reality is non-contextual. I develop a conceptual model of physical reality that is objective and contextual, that is fully compatible with empirical models of physics, and that eliminates quantum paradoxes. I would welcome your thoughts on it.

      Harrison Crecraft

        Dear Jochen,

        Thank you for your deep constructive response to my comment.

        You write:

        "... and on the other, has so far failed to produce any large-scale consensus."

        I believe that the FQXi contests are a good "field" for the start of the Big Brainstorming. Look, ideas are already the tenth contests, and physicists and mathematicians have not yet found consensus on the two main fundamental issues for physics, mathematics and cognition in general, which Carlo Rovelli writes about in Physics Needs Philosophy / Philosophy Needs Physics : "What is space?", "What is time?" And today, the time is very worrying for all of Humanity, and we must learn to find consensus on all issues. Especially on the basics of knowledge. Here, just Philosophy, "mother of all sciences" comes to the rescue .. Recall Hegel: "The owl of Minerva begins its flight only at dusk"...

        "... start with some reasonable assumptions and inferences about ontological matters, and see whether the quantum formalism can be reconstructed from there --- the project of finding a foundational principle for quantum mechanics."

        I believe that the search for the "fundamental principle" is necessary not only for quantum mechanics, but for knowledge in general.

        In an interview with mathematician and mathematical physicist Ludwig Faddeev ( in the journal "EXPERT" (2007), entitled "The equation of the evil spirit" it is written: «Academician Ludwig Faddeev believes that today mathematical rigor is more important than physical intuition and it is thanks to mathematics that a "unified theory of everything" will be built.

        The long-standing debate of scientists about what is more important - mathematical rigor or physical meaning, a correctly solved equation or an intuitive understanding of a natural phenomenon, continued throughout the 20th century, but at some time physicists seemed to win in it: Einstein as the creator of a special and general theory of relativity is better known to the average man than Poincare or Hilbert, Schrödinger is more popular than Weil, and Landau is more popular than Bogolyubov. But in recent decades, the situation began to change: it turned out that successful mathematical techniques have not just technical significance, but deep physical meaning. Mathematical intuition in solving increasingly complex physical problems may be more important than physical. And this caused a noticeable irritation of many great physicists. In the second half of the 20th century, a new generation of scientists appeared who could no longer be called pure physicists or mathematicians. Ludwig Faddeev is one of them. After graduating from the Physics Department of Leningrad University, he gained worldwide fame as a man who, together with his student Viktor Popov, solved the most complicated mathematical problems of the Yang - Mills theory, which later formed the basis of the theory of superstrings. The effects that were discovered were called "Faddeev-Popov spirits" and under this name entered all modern textbooks of theoretical physics. Faddeev is convinced that just as physics solved all the theoretical problems of chemistry, thereby "closing" chemistry, so mathematics will create a "unified theory of everything" and "close" physics. Faddeev is convinced that just as physics solved all the theoretical problems of chemistry, thereby "closing" chemistry, so mathematics will create a "unified theory of everything" and "close" physics. "

        But can mathematics, the "language of Nature" "close biology? ... Big doubts ... Big questions .. Everywhere is the problem of the ontological basis of knowledge. I believe that there will be Pavel Florensky: "We repeat: worldunderstanding is spaceunderstanding."

        I am very concerned that the crisis of understanding in the fundamentals of knowledge has spread to global society ... Let's begin this brainstorming session ... I score the highest rating for your constructiveness and the ideas of the "epistemic horizons". Please look at my ontological ideas and give critical comments .. .

        With kind regards, Vladimir

        Dear Fabien,

        thanks for your comment. I'm glad you found something to enjoy about my essay!

        Regarding your questions, I think the most common take would be that, like the halting problem, these difficulties don't occur for finite/discrete systems. So, for instance, the halting problem for finite state automata is commonly said to be decidable---by simple brute simulation, if need be.

        However, that's not actually a difference to the situation with Turing machines: you need to appeal to a computational system with fundamentally greater computational capacities to solve the halting problem of one with lesser power---you need a system with more states than a given FSA to solve that FSA's halting problem, or a system capable of performing an 'infinite' amount of computational steps to decide a TM's halting problem. Such an augmented system will, however, have itself another halting problem it---or systems of its class---can't solve, and needs a system of fundamentally greater power to solve.

        So the situation in the finite and continuous cases isn't really that different; the only distinction is that our intuition does not balk at imagining a system that has more states than a given system with finitely many states, the way it does at imagining one with a 'transfinite' number of states. But the problem only iterates: the halting problem of a system with finitely many states can be solved by one with more states, but what of that system? Either, you have an infinite number of system with a higher amount of states---but then, why disallow infinitely many states in the first place, as the total computational capacity of that collection will equal any Turing machine?

        Or, you have some finite system that's at the top of the hierarchy. Then, its halting problem will not be solvable by any concrete system, and the same problems will persist.

        This argument can be made more carefully, but in essence, I don't think that it suffices to merely restrict allowable systems to those with finitely many states.

        Another possibility would be to invert this reasoning: as the argument I present seems to imply the sort of 'deterministic evolution interspersed with random events' we actually observe, we could take that observation as evidence for the applicability of the argument.

        Does this address your question?

        Cheers

        Jochen

        Dear Jochen,

        I loved your essay, very interesting approach to reconstruct quantum mechanics from first principles. I like the epistemic horizons idea, and the diagonal argument making connection with undecidability. Very nice interpretation of Wheeler's suggestion that you mention in the conclusion. I also liked that you try to get close to the classical ideal of local realism by using what you define as relative realism. In this respect, you may find interesting, I hope, a result that QM can be formulated on the 3d space or 4d spacetime, rather than on the configuration space, and especially that this gives a local dynamics as long as there is no collapse (possibility that I think is still on the table, as explained in the attached pdf).

        Cheers,

        CristiAttachment #1: post-determined-block-universe.pdf

          Dear Jochen,

          Thank you for your reply. That there are hierarchies of halting problems was insightful.

          With regards to my initial question, I suppose I was mostly confused by the leap from the two states of coin tossing to infinitely many states at the core of any diagonalisation argument.

          As I said, I am just trying to restate what is actually being stated in your proof and in particular what are the necessary assumptions for the proof to hold. That the system can be in infinitely many states appears as such an assumption.

          That the system can be in infinitely many states does appear actually reasonable a priori, for we never know what "the" state of a system is before we measure it in some way.

          I actually think that the coin example could be used to go beyond the two state system. In fact m1 would still be "Head is showing once the coin has landed" but the state could be something much more complicated linked to the initial setting of the experiment...or to the mechanical state of the coin.

          Then I believe what your proof is saying is that:

          "Given that a system can be in infinitely many states, there necessarily exist states such that the outcome of some measurements on those states is undetermined"

          If that is the case, different physical theories can choose to work only with subsets of the states the system can be in (in the same way that one can decide to work with integers instead of the reals...in some sense, as long as this subclass of states forms a closed set under some dynamical rules and chosen operations, then it seems fine). Classical physics restricts the states to those that are determinable while QM embraces both kinds of states.

          Is that a fair restatement of what you are saying?

          Many thanks.

          Best,

          Fabien

          Dear Fabien,

          you're correct that in the essay as it is, the infinite number of states is an assumption. I made this somewhat more explicit in the Foundations of Physics article preceding this work, by introducing the notion of 'program world': physical states encode initial data, and programs take these as input, to spit out a measurement result. As there is a denumerably infinite quantity of programs, we get infinitely many possible measurements.

          Classical physics essentially emerges from this using coarse-graining: that is, once you're unable to distinguish between certain states, and thus, have only extracted an amount of information about the system that's sufficiently below the finiteness bound, you won't notice any quantum effects. So yes, this is, I think, in some sense what you're saying: you lump all the real numbers in some interval together, and don't distinguish between elements of the resulting set.

          Cheers

          Jochen

          Dear Harrison,

          thank you for your comment. I'm happy to see that some of my ideas seem to resonate with you.

          It's interesting you should mention contextuality---my earliest work in quantum mechanics was on that notion, and in a sense, you're right, you can think of a sort of 'contextual reference frame' in analogy to (perhaps generalization of) spatiotemporal reference frames. It wasn't primarily relativity theory I had in mind with the notion of relative realism, however, but merely the idea that we can attach the label 'real' to certain events only as relative to others---for instance, the electron's spin value being 'up' could be thought of as a claim about 'reality' only relatively to the measurement apparatus showing an 'up'-reading.

          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?

          I will have a look at your essay---I hope I'll soon find the time to give it a good reading.

          Cheers

          Jochen