The Ultimate Physics of Number Counting by Andreas Martin Lisewski

Tobias,

thanks for your questions.

First of all, in hyperset theory, the membership relation can well be symmetric (A is member of B, and B is a member of A), and this fact is being used for the bisimulation with the proximity relation. The proposed bisimulation between both Kripke structures is therefore well defined.

Secondly, you are right that some parts of this essay are available as an arXiv preprint but they have not been published. I have taken the opportunity of the essay contest to rewrite, shorten, carve out and further develop those original ideas. In fact, the main idea about the fundamental physical nature of natural numbers was not presented explicitly in the old preprint text.

I appreciate your interest, thank you.

wow, that was an instant reply, great!

So do you mean that the hyperset U discussed in your essay does have a symmetric membership relation? How is U defined it all? Is it defined via its membership graph, which in turn is taken to coincide with the proximity relation? Then the bisimulation principle trivially holds by definition of U.

I have started to study your preprint arXiv:quant-ph/0412047 and find myself having trouble parsing some of the statements and extracting their precise meaning. For example on the bottom of page 21, you state that phi, phi' and psi, psi' are elements of Kripke structures, i.e. possible worlds. But then you also use them as arguments of value assignment functions, just like in section 2 where phi stands for a formula of modal logic. So, what is the intended meaning of these symbols, and what does part (a) of the definition of bisimulation (p. 21) actually state? Just trying to understand...

If I get this right, the evolution of the system is governed by transition probabilities to specified basis vectors. In other words, the system is described by a Markov chain? This sounds a lot like a non-contextual non-local hidden variable model.

Finally, what is the physical motivation behind the choice of self-test operator on page 35? IMHO the Euclidean distance matrix, and therefore also the lambda_i and the self-test operator, depend on the chosen embedding of the tree metric into l_1.

Tobias,

I am glad we can discuss this.

First, what you might call "trivial" is a central result in hyperset theory, i.e. that every graph depicts, up to bisimulation equivalence, a set. Thus, in my work, a central point is to identify the tree graph of structural unfolding with the proximity relation of experimental outcomes in the universal quantum system.

Second, every possible world in a Kripke structure (a graph node) represents a modal sentence. In my preprint you are referring to, as a general rule, \varphi stand for modal sentences in set theory and \psi stand for their bisimilar counterparts in the preferred basis.

Third, with regard to the embedding, every tree metric can be uniquely transformed into an ultrametric, and any ultrametric can be isometrically embedded into l_2. The resulting distance metric (self-test) in l_2 then becomes independent of the emdedding vectors (see, e.g. the work of Deza and Laurent, ref. [10] in preprint). The old preprint you refer to does not explicitly mention this.

I am not sure if I understand your Markov chain remark.

Thanks und viele Gruesse nach Bonn,

Andreas

danke! I hope you don't mind that I keep questioning your work, being a generally extremely skeptical person towards my own and other people's ideas. Overall, I find the paper pretty hard to read since the distinction between postulates, derived results and their proofs is hard to make out. So far I have not been able to spot anything which resembles a non-trivial proof.

Regarding the first point: in my understanding, the principle that every graph depicts, up to bisimulation equivalence, a set, is not a result of hyperset theory, but an axiom (strong antifoundation axiom, "SAFA", in Baltag's own words). That's why I called it "trivial".

Regarding the second point: I don't understand what you mean by the statement that "every possible world [...] represents a modal sentence". A valuation assignment associates to each world and each modal sentence a truth value. But that's the only relation I see between the two concepts. Could you explain? And what does part (a) in your definition of bisimulation mean then?

Regarding the third point: what does it mean to "transform" a tree metric into an ultrametric? Does it have something to do with balancing of trees? Also, you seem to be saying that you take this ultrametric, isometrically embed it into l_2, and then take the distance matrix. What's the point of using an embedding? Couldn't you just take the distance matrix of the ultrametric, which is obviously the same as the distance matrix in the embedding? Also, I still haven't been able to see a physical justification for the specific choice of self-test operator.

About the Markov Chain remark: what I meant is that at every stage alpha of the unfolding, the universe is in some state psi^alpha. Can I interpret alpha as time? The transition probabilities to another state at the next stage only depend on the information which is available at stage alpha and the previous stages. In this sense, the states of the universe form a Markov chain. On the other hand, if psi^alpha is the "wavefunction of the universe" (another dubious concept), shouldn't it follow unitary deterministic evolution?

Finally, another concrete mathematical question: in section 6.5, you mention the splitting of the proximity relation into a positive and a negative part. I assume that the reflexivity relations of the form xPx are taken to lie in the positive part? You implicitly use this towards the end of that section. More importantly, there are many choices for splitting a symmetric relation into two antisymmetric ones -- how do you know which one to choose? For example when all elements of a set are related (the complete graph), you could split it into two total orders, or you could split it into two directed graphs which contains cycles.

Having many other things to do, I will unfortunately probably not be able to study your paper in any more detail. Although I found some of the observations made quite interesting, I still can't decide whether it's anything more than pseudoscience. Have you tried to publish it?

Tobias,

Yes, you are right and I was inaccurate in my language when I called the AFA a result (in the sense of a theorem). What I meant, however, was the observation that every rooted graph depicts a set is a generalization of conventional set theory and thus a "conceptual result" with non-trivial implications. To call it "trivial" is misleading; in the same sense it would be misleading to call, say, the Axiom of Choice a triviality.

A Kripke structure of modal logic can be represented as a graph where the accessibility relation is formed by its edges, the possible worlds are the nodes, and, on top of that, a truth assignment function. This defines a one-to-one relationship between the necessity and possibility modalities in modal logic. I guess this is quite basic.

Third, by "transform" I meant to use a mathematical formula in the following sense: Let Dij be a finite tree metric with a root r, then the transformation

D*ij = c 1/2 (Dii - Dir - Drj)

defines an ultrametric, which can be embedded isometrically in the Hilbert space l2. See, e.g. Bandelt. Recognition of tree metrics. SIAM Journal on Algebraic Discrete Methods, 3(1):1-6, 1990; or Deza and Laurent. Applications of cut polyhedra. Journal of Computational and Applied Mathematics, 55:191-216, 1994. Again, I pointed out the last paper already to you. The self-test of the universe emerges entirely from a Bisimulation principle for universal unfolding.

Four, it is not a Markov chain, because the current state alpha (yes, you can call it time, exotime as defined in the essay) depends on all previous states. This is nicely visualized in Figure 1, where the current state is composed of the well-ordered ordinal of all predecessor ordinals. Also, why should unitarity requires determinism?

Fifth, the split you refer to is obsolete. Therefore the present essay is an advancement of the old preprint, where the reflexivity of the proximity relation is directly reflected in non-wellfounded sets.

Regarding publication, I have once sent it to Int J Theo Phys. The editor and two reviewers were overall positive, but the old manuscript was not accepted at that time and I had not the patience to address the reviewers concerns accordingly. Also, Metod Saniga invited me to talk about it at the 2005 "Endophysics, Time, Quantum and the Subjective" workshop at the ZiF Uni Bielefeld, but I had to decline due to time conflicts. For your information, here are the two reviewer reports from IJTP:

> Review 1: The paper is imaginative and provocative, and the questions addressed are interesting. The author mixes the mathematical structure of quantum theory with the conceptual structure of classical physics. For example, he speaks of the universe as "taking place" in a Hilbert space, while quantum theory uses a Hilbert space to represent interactions between a system and an observer and never speaks of the physics as "taking place" in the Hilbert space. We never see wave functions in th laboratory the way we see planets in the sky, but only events for which wave functions give probabilities. As a result of his re-interpretation, the delicate connection to experiment that is established in quantum theory is lost. This is not inevitable in the problem, and possibly a small reformulation can make a big

difference. For example, one way to express quantum theory that works is as a theory of transition amplitudes for transition experiments; and there are many others. The author seems to use none of them, but invents his own. This is OK if he also provides its experimental meaning. Without that the work is just a mathematical structure. The key question is: How does one go from his quantum

description of the universe to a prediction for quantum experiments in that universe?

> Review 2. A courageous and competent work. It would be appropriate, also, to credit Finsler for pioneering non-well-founded set theory. I recommend publication.

Thanks again and, as a friendly advice, be careful with the word "trivial".

A minor correction: The formula in my previous post should read

D*ij = c 1/2 (Dij - Dir - Drj)

with c = max(Dij}.

One more explanation as you said " I don't understand what you mean by the statement that "every possible world [...] represents a modal sentence"."

In structural set theory, every node (possible world) in the graph representing a Kripke structure represents a set and every set is satisfied by a modal sentence (formula) in modal logic , i.e for all sets A there exists a modal sentence \varphi such that A |= \varphi.

Thank you for your extensive explanations, this has cleared up a lot! Now I find myself not being able to resist another reply ;)

First, good, I agree about AFA and that it shouldn't be called "trivial". From your paper I just had the impression that you claimed to have a derivation of this, therefore the misunderstanding, sorry.

Second, I know what a Kripke structure is. And certainly yes, at any node there is a sentence turning true at that node; for example, any tautology will do. This doesn't answer my question, but I can see the direction and will have to do some more reading and think it through.

Third, thank you for the explanation of "transforming". I don't doubt that one can then isometrically embed this into Hilbert space; I just don't see the point of doing it.

Fourth, unitary evolution is deterministic in the sense that if you know the exact state at the present time, you can know with certainty the state at any future time. So if we apply quantum theory to the whole universe as a closed system, it becomes deterministic. If my understanding is correct, this is not the case in your model. Then, in particular, your model should have somewhat different physics than ordinary quantum theory. Are they different observationally? (There is good reason to be suspicious about any theory which assigns a state to the whole universe, but that is a different story.)

Fifth, fine, this is a nice simplification.

It was very interesting to read the referee reports!

Thanks Tobias. One more response. The embedding of the tree metric into the Hilbert space has the purpose of obtaining pointer states, i.e. the elements of the preferred basis, as the eigenvectors of the euclidean distance matrix in the Hilbert space (the self-test).

One point regarding time. The ordinal alpha is the exotime, or stage time, in the sense described in the essay and introduced also by Jaroszkiewicz, Bucchieri, among others. This is not the local (internal) time in quantum theory which appears in the Schroedinger equation. Physical time is therefore two-fold: it has a discrete stage character and it is a local, continuous parameter in unitary dynamics. Both characters of time are radically different, of course. I recommend further reading about "endophysics" which opens this new dimension of physical time.

I think, a the very least, the essay write-up and the discussion here have already motivated me again to write up these ideas in a shorter, concise scientific manuscript. I'll keep you updated on the results, if you wish.

Good luck also with your ideas!

I am the editor of the "Markov Chain Universe" website and would like to comment on the above discussion from the context of Markov Chain Universe theory. (If you are curious what it is, please google the term).

The flow of time represents the transitions of the universe from one state to another state. The future of potential states is determined completely by the current existing state of the universe (and there is no secret hidden markings somewhere outside of the current state dictating its future trajectory). Hence, the universe is a Markov Chain.

It unfolds through a time history where the past is distinguishable from the future by a set of distinctly available states. The states of the future are not the same available states as the past. This is the primary distinguishing feature of the passage of time- movement into a new state space.

Quantum theory does indeed prescribe a specific deterministic movement through time to the future- excluding the "decoherence" which is physically observed that places the outcome of the future into only one of the possible sets of that deterministic evolution of the probability function of quantum theory.

The exciting aspect of Markov Chain Universe theory is that some states are lower probability than others. It appears that living organisms themselves are lower probability structures propagating themselves through time.

In any event, whether or not you accept that you do have to accept the universe behaves as a markov chain, providing you describe the set of states of the system coherently with regards to what is observed by physical law.