Indeterminism, causality and information: Has physics ever been deterministic? by Flavio Del Santo

Hi Flavio,

Thanks for the really well written essay. The principle of infinite precision is super useful in highlight the tension, either ontologically or epistemically, of formal axiomatic systems and physics. I'd never read Max Born's essay that you quote on page 2, but its an excellent reference!

I completely sympathise with your idea of finite information quantities are loved the reference to Gisin's recent work. As you mentioned, Landauer has written extensively on this---I lent heavily on Landauer's perspectives in my essay.

You wrote that

``In this view, the orthodox interpretation of classical physics can be regarded as a deterministic completion of an indeterministic model, in terms of hidden variables''.

The finite information quantities seem to suggest that indeterministic models are fundamental. For example, a lot of powerful claims can be made from statistical physics or thermodynamics. Would these kinds of ideas occupy a more fundamental, rather than derivative role, in a theory of finite information quantities?

Again, thanks for the great essay! It was a pleasure to read!

Cheers,

Michael

Dear Flavio,

The refusal of actual infinities goes back to Aristotle, and those who followed always thought of measurement in terms of rational numbers with finite representations. And as the inverse square laws were established, classical determinism came to be understood in the same way, with laws restricted to quadratic equations, or conic sections, with rational solutions.

When Poincare went beyond that to transversals, and the symbolic sequences that constrain them, he was exploring a new physics of chaotic dynamics, beyond what was possible for Laplace and classical rationalism. Those *infinite sequences give you the Kolmogarov-Sinai entropy, as a measure of entropy production, and thereby unpredictability. The axioms for probability of both Kolmogarov and Karl Popper both assume infinite sequence representations.

Of course there is also the classical rationalism of games of chance, working from finite binomial distributions, but there measurement with complete precision becomes possible, and Maxwell's Demon is very much in business. If you insist on viewing reality as a computer it becomes hackable.

Dear Flavio,

during the contest, i advanced a lot in trying to explain the meaning of pi=1. It ist like a meditation between two competitors.. the small child and the old theoretical physicist. Hard stuff to explain that all our pocet calculaters ar "fake news".

As my native language is german it is hard for me to explain in english.

I tried here to explain the final theory of everything in quantum-biology (finalazing the ideas of A Turing)

For some people of course this is "shocking". : https://www.youtube.com/watch?v=EIBjt_5TU0U

(Language should be understood also by "non-scientist")

All good wishes.

Manfred

Dear Flavio,

Your essay gave me an idea. If you have time, I wish to know what you think.

Indeterminism leads to irreversibility, which leads to entropy. A true reversible system would require infinite information density. Because we have the arrow of time that is entropy, we cannot be in a deterministic University, since a net increase in entropy requires irreversibility. It does not matter if this is a quantum or classical system.

Thank you,

Jeff Schmitz

Dear Flavio,

Thank you for your very interesting essay. Your angle of attack is quite original, and I completely agree with your argument that indeterminism is a matter of interpretation, and can both be chosen for quantum or classical theory. Indeed Laplace's demon is often considered as being exorcised by quantum theory, but its existence is already facing difficulties by the only fact that if the demon is not external to the Universe, then it might predict its own behaviour, which might lead to logical inconsistencies.

Are you familiar with Breuer's theorem ? It reminds me of your principle of infinite precision. Breuer, analysing the measurement problem as emerging from self-reference, has shown that every observer cannot distinguish the states of a system in which it is contained, irrespective of the nature of the system (classical or quantum) and of the time evolution (deterministic are stochastic). In a paper entitled "Quantum meausurement and Gödel's proof", analyzing Maxwell's demon, Zwick also presents, as you do, a mesurement problem for classical mechanics : "To avoid infinite regress in the description of measurement, and paradoxes either of self-reference, one must assume that at some point, the perturbation by the measurement can be ignored, or introduce a statistical postulate as an arbitrary addition to the dynamics. If neither is allowed, on can simply accept the need for the two-levelled theory." Do you think that this might be in line with your essay ?

The "future undecidability" that is described in Gisin's final quote of your essay is an old problem of classical logic, that goes back to Aristotle and "Sea battle tomorrow", his introduction of the notion of contingency. This problem of future contingents was especially studied by the scholastics, who were trying to conciliate Aristotle's logic and the Biblical narratives. Ernst Specker, who was one of the father of the theorem showing "quantum contextuality", was motivated by these scholastic study of "Infuturabilien" when he found a result that will become later the Kochen-Specker theorem.

If you ever find some time to do it, I will be glad to have your feedback on my essay, that defends an indeterministic interpretation of quantum theory based on contextuality and an analysis of the measurement problem as self-reference (analysis that I believe could be extended to classical mechanics, an incompatibility between absolute universality, measurement as a meta-theoretical process and full measurability).

Best,

Hippolyte

You are absolutely correct that my "criticisms are based on misconceptions."

You wrote an interesting and well-reasoned essay based on the (legitimate) premise that the existence of what cannot be determined amounts to a refutation of determinism. Subsequently, since some things are indeterminate then these things cannot be predicted, which is technically correct. I simply made the case that this does not refute determinism unless, as you postulated, these things are inherently indeterminate and thus determined by "nothing".

It certainly appears that the concepts of what is (ontology) and what can be known (epistemology) are inextricably linked and their exact fundamental difference ineluctably fuzzy.

I wish you well in the contest.

Dear Flavio,

what an interesting topic. I was happy to see, you have a contribution in the contest. The problem of the origin of randomness, which is not merely epistemic is a big puzzle to me. The investigation of indeterministic classical physics is interesting in order to see, where the structural/conceptual differences are between classical and quantum physics. And to recognize that it is not the randomness itself nor the collapse part of the measurement problem - which I am not so sure if this is a real problem - that make the difference.

However, I am not sure you can reach with your approach beyond an epistemic randomness (hidden variable theory). If epistemic limits should have an influence on the ontological status of things - and I would be willing to follow that - that would need a good explanation. As I - and I think most physicist - are trained to accept Laplace's view and in fact imagine the underlying world to be like that. I call this in my essay 'simplistic realism'.

In my essay I study the conceptual structure of scientific theories - specifically of physics. From there one additional randomness might occur, which I would like to share. In the view I put forward quantities and objects are only definable within closed systems, i.e. if systems and objects are separable from the environment. Imagine if forces outside the system would constantly be shaking the system around, never could any law or concept manifest in that system. But separability necessarily is always only an approximation. Hence the deterministic laws hold only approximately. The environment would create small random fluctuations impeding strictly deterministic laws. (May one could connect this to Tejinder's theory in this contest.)

I do not think this explains true randomness. Nor do I think, it is in conflict with classical physics. That's fine for me. The goal in my essay was a bit to reconcile our realistic imagination of the world with our epistemic necessities.

Luca

Hi Flavio,

I take it this excellent essay more or less summarizes your development of Gisin's application of constructive maths to classical mechanics? I'd be interested to know how you might see Brouwer's intuitionistic philosophy in relation to this mathematically indeterminate approach.

As I understand it, his intuitionism was derived from Kantian intuition (anschauung, intuitus) as intuiting/apprehending/perceiving the forms of sensibility, space, and time given in empirical (phenomenal) experience. From that perspective we intuit/perceive phenomenal patterns in our empirical experience of the world (thus information is physical!), and the constructive mathematics is based on that empirically intuited pattern perception. The formal, intersubjective communication of these empirical patterns (or information) is effected in that constructive mathematics, for which classical mechanics thus becomes necessarily indeterminate, at least from this intuitionist (also phenomenological) perspective.

What is objectively real in this sense are the phenomenal patterns themselves (or real patterns cf. Dennett) as given in empirical/phenomenal experience, rather than say, the Laplace demon's idealized external world of point particles with infinitely precise, initial physical conditions. Does this mean intuitionism, in your view, must reject the notion of a classical world defined as 'objective external reality' in favour of the actual empirical experience of such a merely potential reality? Or can potentia remain an idealized unobservable continuum from which our discrete actualitas emerges?

Best regards,

Malcolm Riddoch

Je suit, nous sommes Wigner!

Dear Flavio,

Congratulations on the well-organized essay on the new notation of the randomness. As you visited my essay, I really enjoyed learning this. In the past essay contest, I wrote the essay on information to be finally published in the book. On this essay, the FIQ is not discussed. However, one-bit is not enough to well defined. Therefore, we seem to need more than two bits on the well-defined FIQ.

Also, from your perspective, how do you understand the integrated information theory (IIT)? Since mathematical formulation of IIT, the conditional probability related to the certain randomness is implicitly assumed. What do you think?

Best wishes,

Yutaka

Flavio - Thanks. A coherent, well-written essay bringing a key, and incorrect, premise of classical physics to light. Your discussion and critique of the principle of infinite precision is impressive and should, if widely read, put the notion of classical determinism to bed permanently.

Where I was disappointed however, was in the very limited perspective your essay gives to the much broader epistemological issues of incompleteness and undecideability. These issues also point to invalid premises in our conceptual views and understanding of the world. I've taken a stab at these broader issues and would be very grateful if you gave my essay a look.

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

Dear Flavio,

It appears that the following differences in ideas on determinism and on information may require clarification.

1. Principle of infinite precision: Ontological - there exists an actual value of every physical quantity, with its infinite determined digits (in any arbitrary numerical base)... It is only when its formalism is complemented with this principle that classical physics becomes deterministic.

An argument may be constructed against infinite precision describable as sequence of infinite digits. Value of pi in, say, binary digits does have infinite expansion, but it is just a point on a real line. And in the units of pi, its value is just 1. So, a system may have a state that corresponds to pi, infinite digits are not needed to construct infinite precision. What can be said instead is that not all points on real line may be traversed by the state of a physical system. Irrationality of numbers is relative to unit of measures as all datum. Also, one has to give a mechanism by which indeterminism can be realized in physical systems. Both the points are dealt with in my essay (Mother of all existence).

2. Though following Rovelli, you have discussed quantization of space, yet a distinction is warranted. Usually, we do not interpret classical dynamical expressions as quantized, it does not necessarily mean that one cannot think of deterministic quantized changes in observable spaces. This statement has nothing to do with quantum physics of superposition and entanglement -- quantization can exist in classical domain too, e.g., at Planck's scale as you refer. Since we do not have quantization limited precise measuring instruments the possibility of quantized determinism does exist -- this is only for arguments sake. Then all measures get translated into integers, which will avoid the requirement of infinite precision. It is like spring loaded switch, which can be either on or off only by classical function, or checker board like time and space. The determinism must be killed with some other arguments. Therefore, the statement, "This clearly shows that the principle of infinite precision is a necessary condition for determinism", does not hold. The classical quantization seems to directly oppose the statement, "as soon as one realizes that the mathematical real numbers are not really real, i.e. they have no physical significance, then one concludes that classical physics is not deterministic."

3. You state -- This view goes under the name of Landauer's principle, in short, "information is physical". In Ref. [13], Gisin gave sound arguments to support the claim that "a finite volume of space cannot contain more than a finite amount of information".

Landauer's principle refers to how all information are represented by states of matter, referring to what they mean or express. But in Gisin's view information is reduced to quantity of information in bits, losing the reference to the meaning. This is an example of why we have not been able to construct mechanism of processing of semantics of information as brain does. Moreover, physicists' interpretation of information content of a system being its own state description causes so many issues with the reality of information. If it was to be so, then no matter what information processing results from interaction, an information can never be anything but the description of physical state. This is how physicists have artificially created a barrier between this interpretation of information and what a physical device like brain does in dealing with the semantics of information. Instead, the reality of information relates to what an observable state of a system causally correlates with, as dealt with in my essay. A single elemental state of a system may represent the information of very high level structured and abstract semantics if processing is organized in modular hierarchy as is also evident from neuronal processing in the brain. It is bizarre that physicists are blinded to this apparent reality.

It is because of Gisin's like interpretation that requires information to have certain amount of physical space. Moreover, such interpretation also runs in opposition to the fact that even in artificially designed devices, information is assigned to and coded by the states of registers (systems), not to and by the registers themselves. Physicists and computer scientists have hijacked the term information to mean amount of information measurable in bits leaving the most apparent phenomena of all to us humans, semantic processing in the brain unresolved.

Rajiv

Wonderful essay, Flavio! Probably the best I've read so far, in fact (and I've read a lot...). Well written, full of lovely ideas about determinism and indeterminism in physics, and a clever take on the central question of the essay contest.

I still wonder whether classical physics "is" deterministic, a la the views expressed by Boltzmann and Exner you quoted at the beginning of your essay. But I guess this is a non-question, since physics isn't classical anyway. And even if it were classical, (i) it's hard to imagine a way to experimentally distinguish between determinism and indeterminism, and (ii) whether such a description is useful in a world where measurements provide only finite information is another question all together. You seem to take the view that this indeterminate view is more useful, which I agree with.

Your essay also makes me wonder about the possible different interpretations of classical mechanics. I see you've already thought about this a bit (e.g. you mention in section 3C a classical analogue to objective collapse models, and you mention in footnote 8 a possible analogue to the Everettian interpretation). Has someone written about this? If not, we should write an article about it! There are already many different mathematical formulations of classical mechanics (e.g. the amusing Koopman-von Neumann formulation, involving a Hilbert space and operators); perhaps the issue of the different interpretations has been overlooked.

Some miscellaneous comments/questions:

In a classical world, what prevents knowing an arbitrarily large number of digits of some observable? If I take more and more time and measure more and more carefully, can't I measure more and more accurately, in principle?

Stability property of measurement: I understand what you're getting at, but can't consecutive measurements have slightly different values? I guess it depends on what you mean by a digit being "determined". Determined in the sense of being reliably to known to be some value? As you know, there is uncertainty in later digits of some measurements (and less trivial measurements may have more than one uncertain digit, in practice).

Also, how does this uncertainty affect ideas about the arrow of time? Stochastic theories are not time reversible, in a certain technical sense.

John

P.S. A book that was recommended to me a few months ago may be up your alley: "Reductionism, Emergence and Levels of Reality" by Chibbaro et al.

Dear Flavio

Thanks for your wonderful essay that deals quite beautifully with several delicious subtleties!

I do have a few questions/comments. (Perhaps if I were less busy, I would find answers by studying your references, but, sadly, that will not be possible for me any time soon.)

1) (Perhaps important) You have, I believe, misidentified "real numbers" as the culprit in your "principle of infinite precision." I take "infinite precision" to refer to the precision of the limit of a hypothetical infinite string of ever-more precise measurements. If your spacetime is modeled by any topological space that is dense, you will have the potential for an "infinite precision" problem. It springs from spacetime being dense, rather than from it being continuous. This means that as far as the principle of infinite precision is concerned, rational numbers have the same potential as real numbers. For example, you can build your "Figure 1" argument equally well on a small piece of the rational number line.

Perhaps it's important to add that it is clear that none of the above impacts your nicely constructed, alternative model of classical mechanics. So perhaps it's of no real consequence.

2) (Probably quite minor) Near the top of page 5 you say (statement 1) "Note again that without REAL NUMBERS, one cannot any longer uphold determinism." Even if we change that to (statement 1a) "... without INFINITE PRECISION, one cannot...." it appears to be incorrect, and it is certainly not justified (or even broached) by anything you say earlier in the essay. Everything earlier was more along the lines of the inverse (statement 2) ".....WITH real numbers one cannot avoid determinism." And in addition, statement 2 is also all you need or use for everything later in the essay. So statement 1 appears to be just an odd extraneous claim that appears to be of no importance for your essay. Or am I missing something?

3) (I seem to be lost, here.) You cite Landauer's Principle but, I guess I'm confused. I see nothing there to justify singling out information from any other real world concept that can also be treated mathematically. To be sure, one can make a case against Platonic Ideals "existing" in any meaningful way, but is the statement "information is physical" in some way independent of the more pedestrian idea that "all circles, all parallel lines, .... all geometry is physical" or "all counting is physical" or, etc...??? It seems almost capricious to single out information. OK. What am I missing?

John S