Dear Flavio, Very interesting and groundbreaking essay, well informed by history, philosophy, physics, and mathematics (in the Newtonian tradition criticized here for other reasons!). It seems to me that we are only at the beginning of dealing with the drawbacks of the use of the real numbers in physics, and similar idealizations which have led to the conclusion that determinism itself is an idealization, even or especially in classical physics (which is what I take to be the main message of this essay). My own hunch is that intuitionistic and constructive mathematics may provide a way out, although, as Hilbert feared, this means we are driven out of Cantor's Paradise and we have to start all over again. In view of the tremendous success of even classical physics (think of putting men on the moon) this might be too much to ask, so there should be some result to the effect that physics based on the real numbers gives valid results with high probability (from the point of view of the new physics based on finite approximations), or so. Alternatively, think of results (due to Gödel and others) that theorems of classical mathematics are valid even intuitionistically if they are replaced by versions that are classically equivalent but intuitionistically different (typically by adding a double negation). In this spirit, results of classical physics based on the real numbers should be replaced by results that are empirically equivalent but logically different in your system, and provable in that system.

Best wishes, Klaas

    Dear Roger,

    thanks for reading my essay and pointing out the small inaccuracy and the typo.

    Best,

    Flavio

    Dear Steve (if I may),

    very many thanks for your kind words!

    best wishes,

    Flavio

    Yes of course, you are welcome also , I loved your essay, it is one of my favorites.

    best regards

    Dear Klaas,

    thanks, I really appreciate your kind comments. I totally agree that we are just now scratching the surface of problems that have been either not recognized, or deliberately ignored, putting them under the carpet for ages. I think that we will soon reach a critical mass, though, of people that recognize this issues in a non trivial way. I believe we should rediscover the eneasyness of some eminent scholars of the past, as I try to do in my essay by pointing out some historical arguments. Fore instance, only today I found a very interesting document; Thomas Kuhn interviewed a student of Boltzmann, who reported: " there was in Boltzmann's ideas some anticipation of quantum theory [...]. He had from the beginning the idea that the space of phases must be fundamentally quantized".

    Moreover, you are completely right that we should seek new mathematical ways to model our physics. And constructive mathematics, perhaps intuitivism, seems most promising.

    All the best,

    Flavio

    I really enjoyed this essay. The insight into the importance of infinite precision to classical physics is a really valuable one and I'm very interested in your work on an indeterministic formulation of classical physics.

    I did have some questions about the overall motivation for this work. As I understand it, your argument is that real numbers cannot be physically meaningful because they contain an infinite amount of information. This is supposed to be a problem because it would violate the Bekinstein bound; but the Bekinstein bound comes from GR and/or quantum physics, so is there any reason to think it should hold in classical physics?

    Alternatively, it supposed to be a problem because 'physical systems have finite size' and this implies a limit on density of information - but do physical systems have finite size in classical physics? Perhaps there is a coherent interpretation of classical physics where the world is constituted of pointlike particles?

    These questions are linked to a larger questions about what an 'interpretation' of classical physics is supposed to do. When we try to `interpret' quantum mechanics, I take it that we are making hypotheses about what the actual reality underlying quantum mechanics might be like. But we don't need to do this in the case of classical mechanics, since the reality underlying classical mechanics is understood to be quantum mechanics. So when we ask whether classical mechanics is deterministic, we're not asking about whether reality is deterministic, rather I guess we're asking whether the view that classical physics is deterministic was a coherent one - but in that case it seems unfair to make an argument based on facts that are not inherent in classical physics? Does it make sense to try to 'interpret' classical physics from the point of view of a modern physicist?

      Dear Flavio,

      Re "Information is a well defined mathematical concept (defined by Shannon and others) and I used it in that sense in my essay", Author Flavio Del Santo replied on Mar. 11, 2020 @ 20:58 GMT:

      Despite its usefulness in physics' mathematical calculations, I would think that "Shannon information" does not get to the essence of what information is. Because "Shannon information" is about the probability or surprisal value of information: it is not the actual information.

      So, I would think that information is a general term, where all actual information is representable as category names and associated numbers, or representable as category names and associated TRUE/ FALSE values. "Shannon information" is just one such category and quantity of actual information.

      And, I would think that information is information because it always exists in context/ relationship to other categories of information. One can only build information out of existing information, and "Shannon information" is built out of existing information.

      I know that there are plenty of people that, like you, seem to consider that "Shannon information" IS information. But I would like to know how physics and philosophy justifies this redefinition of the meaning of the word information, this takeover of the meaning of the word information, by "Shannon information".

      Dear Flavio,

      I have come up with my own little collapse model, which came to me when pondering collapse within the context of your essay. I think the pictures speak for themselves:

      https://vixra.org/abs/2003.0385

      Does this remind you of anything in quantum physics? I'm asking as a novice.

      - Shawn

      Dear Dr. Flavio Del Santo,

      I really enjoyed reading your essay. Your wonderful words in the beginning .....................One specific story that seems to have crystallized among practitioners is that classical physics (i.e., Newton's mechanics and Maxwell's electrodynamics) would allow, in principle, to predict everything with certainty. The standard story continues by telling that the foundations of such theory are perfectly well understood and free of any interpretational issues. In particular, it is widely accepted that classical physics categorically entails a deterministic worldview.

      Indeed, due to the tremendous predictive success of Newtonian physics (in particular in celestial mechanics), it became customary to conceive an in principle limitless predictability of the physical phenomena that would faithfully reflect the fact that our Universe is governed by determinism. .................

      I want to say few words about "Dynamic Universe Model". It is a singularity free N-Body problem solution uses NEWTONIAN PHYSICS and IS free of any interpretational issues,and it got very good predictive success IN YOUR WORDS...

      For further details have a look at my essay please.

      "A properly deciding, Computing and Predicting new theory's Philosophy"

      Best Regards

      =snp.gupta

        Hi Emily, thank you for your appreciation and your comments.

        The Bekenstein bound, indeed, brings along a huge amount of theoretical background from GR. I always mention this just to prove that there exist formal arguments in support of what I want to tell, but, my (and Gisin's) ideas are more intuitive and fundamental. My main argument is not as formal, but perhaps can be conceived as an operational approach that takes as a primitive the Landauer's principle.

        You ask: "Does it make sense to try to 'interpret' classical physics from the point of view of a modern physicist?" I think this is a fair criticism. My original motivation, was that of a quantum physicist. Indeed, my whole research program is devoted to the understanding of what the differences between classical and quantum physics boil down to. Read any textbook maual, encyclopedia entry, or review article on QM and you will see that they introduce the first conceptual difference of quantum mechanics was to introduce indeterminism (and it is historically is quite true that the physicist at that time focused a great deal on this). So, showing that good old, familiar, harmless classical physics could perhaps be seen already as deterministc let us rethink the scope of the conceptual gap between classical and quantum physics. Moreover, as gisin points out in his motivations (https://arxiv.org/abs/1803.06824) classical physics is seen as the paradigm of the perfect scientific explanation, the one to which all theories should strive for. Our contribution, I believe, scales down thongs a litte.

        Thank you for your kind message. I will have a look at your essay and comment if I have something pertinent to say about it.

        Best wisehs,

        Flavio

        Dear Flavio,

        Thanks for a thought-provoking and well-written essay. We need to question conventional "truths" once in a while, such that classical physics is self-evidently deterministic. However, I think you play down the difference between classical and quantum physics too much when it comes to determinism.

        It is true that we can evolve ensembles R in phase space to E(R) according to the classical equations of motion, but we do not have to. The naked classical equations happily eat single points P and spit out evolved points E(P) in a completely deterministic manner. E(R) equals the union of the evolution E(P) of all the points P that define R.

        In contrast, we must evolve ensembles R to E(R) in quantum physics because the quantum of action defines a minimum ensemble area, as you discuss in relation to Fig. 2. It is not possible in quantum mechanics to "decompose" the evolution of R to the union of the evolution of all points P in R. Interference prevents such point-wise, deterministic evolution.

        Another way to put is to say that if R = R1 U R2, then E(R) = E(R1) U E(R2) in classical physics, but not necessarily in quantum physics.

        I guess my point is that while it is true that both classical and quantum physics allow an indeterministic interpretation, only classical physics allows a deterministic one. (I am aware that Bohmians and MWIers would not agree.)

          I have rated the Essay not so well, because I strongly disagree, that Science as presented in the Essay is inherently undecidable. The model of reality called "classical science" is deterministic within the model. If it would be indeterministic, then e.g. the Navier-Stokes equations would have such property. But latter is not discovered yet. The quantum science consists of two fundamental notions: observer and nature, and how they co-relate. Yes, the indeterminism comes into reality through freewill of observer, but the nature itself -which is subject of Physics- is perfectly deterministic even while talking about Quantum Physics.

            It is quite amazing that rating "not so well" is for you a 1. It seems from yoyr superficial and cofused comment that you did not even passed the abstract. In front of such an intelleftual honesty I don't think is even worth replying.

            Dear Flavio,

            great essay with many good arguments, i like it. As i just wrote on Klaas Landsman's page, 1-randomness isn't that shocking since it must also obey some logical rules. True enough, since if we presuppose some generic randomness at the level of QM, this randomness nonetheless has to obey the statistics of QM.

            If you like, check out my own essay. I would be happy if you would leave a comment there.

              Dear Stefan,

              thank you very much for appreciating my essay.

              I will surely read your essay asap and comment on your page.

              All the best,

              Flavio

              Per,

              I agree that quantum physics and classical one are not quite the same. But not necessarily for the resons you explained. You clearly expounded a text-book (or "orthodox" as it were) interepretation of these two theories.

              There is, indeed, a kind of indeterminism that the violation of Bell's inequalities entails, which cannot be retrieved in classical physics. However, notice that appealing to the uncertainty principle is not enough, because this means that you believe unconditionally that QM is a complete theory. You might not like Bohm's hidden variable model (I am also not a great fan for the same reasons Eintein was not) but it does convincingly demonstrates that the uncertainty principle could be merely epistemic and not fundametal, if QM is completed with hidden variables. In this sense, Classical and quantum physics can be both INTERPRETED in aither deterministic and indeterministic fashions.

              Dear Flavio,

              I wanted to compare the mathematical structure of classical and quantum evolution equations, to see what kind of interpretations they allow. That is, I wanted to "ground" the discussion in the formalism.

              From that perspective, my last paragraph was ill-chosen and obscure. I agree with what you write in response.

              I think we need to highlight interference as a distinguishing feature of quantum evolution equations, in addition to the uncertainty principle and the quantum of action. Interference is the reason why we cannot "decompose" the evolution of an ensemble in phase space in quantum mechanics to the evolution of its subsets, or to the deterministic evolution of its points. And that's the reason, as I see it, why quantum mechancis cannot be interpreted deterministically - in a certain sense.

              Trying to be more precise what I mean by this, I'm talking about the standard postulates of quantum mechanics (thus excluding the additional stuff in Bohmian mechanics), and the evolution of observables, or points in phase space. (In contrast, the determinism in MWI resides in the wave function, that is, in the deterministic evolution of points in Hilbert space.)

              The standard example is, of course, the double-slit experiment. You cannot "decompose" the evolution of the setup to the alternatives R1 and R2 where the particle goes through the left and right slit, respectively. Then you lose the interference pattern, which exludes som final particle destinations that are allowed in the "decomposed" evolution.

              6 days later

              Dear Flavio,

              You conclude the essay with the phrase:

              "We can only have the certainty that the future of the battle between determinism and indeterminism is open, too."

              But do we need a "battle" of determinism and indeterminism? ... I believe that the dialectic of "coincidence of opposites" and the most extreme ontology overcome the need for this epistemological "battle". It is time to complete the Big Ontological Revolution in the philosophical basis of Science, begun by Planck and Einstein. And the first step is the rejection of the unclear and unclear concept of "classic", which introduced maximum uncertainty primarily in the understanding of "space / absolute space".

              "We repeat: worldunderstanding is spaceunderstanding." (Pavel Florensky).

              The first most important problem for fundamental science is the problem of the ontological basification of mathematics, which means knowledge in general. Do not be afraid of dialectics and understand the dialectics of Nature, whose language is mathematics. To understand is to "grasp the structure" (G. Gutner "Ontology of mathematical discourse"). "Grasp" primordial generating structure.

              With kind regards, Vladimir

              Hi Flavio,

              I like your essay a lot -- well argued and well written! Very nice food for thought!

              Perhaps two remarks / questions:

              1. I suppose that your main hypothesis is quite independent of the specific construction of FIQs? Probability theory would probably give you many alternative ways to implement finite information quantities, without spoiling your conceptual conclusions?

              2. Your remarks at the end of Section IV are somewhat disjoint from the rest. Isn't this all about the interpretation of conditional probability, P(A|B), as what you can *learn* about A given what you know about B, not as causal influence? I didn't really get the point of this part.

              Otherwise I enjoyed reading it a lot!

              Best,

              Markus