Dear Alex

I read your extraordinary essay. The coincidences with what I am trying to propose are so shocking that my legs are trembling. I try to explained why, I think you have the information I am missing. The discrete models that you are proposing are just a partial description of the order I am trying to find, that it explains why we see the properties of the classical world we see. Particularly, the fact of no locality in my approach is expressed by the fact that when we collapse to a classical world we are taking a generic ultrafilter on the order topology which is a global fact. The way you relate the nodes of the graph is just the structure of the order, for example, the Continuum Hypothesis example shows that if the order is not choosen right, we don't get the result in the classical world, this is what it is happening in your models. When you say how we should iterate your models what are you doing is describing how the order topology should behave locally, i.e. you are choosing the ultrafilter.

Finally your concern about the logic we should use it is missing something. We already use classical logic to describe and model quantum reality, it is what the classical approach does, but we can't understand very well quantum reality. What you do is construct your model by try and error and you try to explain why some iterating model gives the result you are looking for and others not. Why I am trying to say is that, if we introduce non classical logics, in my case a intuitionist one, we can explain these phenomena perfectly.

I would like to hear your opinions again.

J.Benavides

John,

To one with a meager understanding of higher math, your argument is esoteric and difficult to comprehend.

Mathematical models certainly help to understand the parts of reality we are modelling but I can't see them as a substitute for reality, and that reality takes on the models characteristics.

I contend that we can't truly know reality but that simulations can aid in the effort, but my deficiencies in math may taint my opinion.

    Dear James

    Thank you for your comments.

    I hope you will give me another chance and you will try to read my essay again. I understand your concern about the role of mathematics. Mathematics can become dangerous if they become the objective, and their beauty, an intrinsic justification. This for example is what it is happening with string theory and topological quantum field theory. What I am trying to propose is different because the motivation is to understand better reality, in this sense mathematics could be the key, as the curvature was the Key that Einstein used to create General Relativity. I know that what I am proposing sounds very weird because it have never been used in physics before, like Lorentzian manifolds where never used before Einstein, but I think I am giving strong arguments to justify my choice I think what I exposed is what we are missing and the main reason because the unification program has failed.

    J. Benavides

    7 days later

    John

    I found your essay highly intuitive and logical, even as one who'se looked elsewhere but maths for logical solutions. You said; "If we can get a model of quantum reality in this context, for example taking as the base space some kind of causal set, we will have a complete description of the hardware."

    I have tried using what we CAN be reasonably sure of at the small scale, to pattern match with the classical, and derived what looks worryingly like a toe, and which I can't disprove. I hope you may rad my essay and can follow the logic. Yours deserves a higher rating and I'll oblige. I hope you may feel the same of mine, though poles apart in viewpoint we observe the same reality.

    Best wishes

    Peter

      Dear Peter

      Thank you for your comments.

      Reading your essay and also reading the essays which propose a digital model based on a discrete lattice and computation, I see that the discrete features that you describe together with the discrete features of the digital-computer approach are converging with my ideas but in my context can be explained in more deep level as the construction of a variable-set structure on a partial order. It would be interesting if you try to considerate your approach from this point of view, to see if you find new interesting explanations or motivations.

      J. B.

      • [deleted]

      • Post #19

      Hi to both of you,

      I agrre with Eckard, the maths are cool and essential but when the physicality want be explained, of course the rationalism of our continuity and our discreteness become so important.

      The topos of Mr Baez is for computing and its sortings, for the physicality it's an other story.All that to say that in fact we can't confound a simulation on computer and on the other side the real physical dynamic in 3 Dimensions. The maths imply so many confusions about our physicality, a time travel, the external cause of mass as higgs, the multiverses, the strings and this and that...all that is a pure irony for the rationalism and its axiomatization of physics and its pure laws.

      The computing is a tool, an of course we can invent topologies or the topology...that is the question.

      This difference is essential at my humble opinion.

      Regards

      Steve

      Dear John,

      Your essay interested me with its desire to explain reality through a pure analysis of quantum behavior by eliminating remaining classical concepts left in the theory. That seems like a logical and perhaps necessary step if it is to produce a satisfactory unifying theory. The introduction mentions gravity but it is not mentioned again in the rest of the essay. Is gravity expected to be explained by the history of the causal sets? Would that be the same as considering gravity as something we cannot evaluate in the non-collapsed quantum state, and gravity only appears in the collapsed humanly visible state?

      I see your essay making a valid logical argument which makes it very interesting. My own approach is very similar in requiring the theory to be explained simply in terms of its core assumptions. It is interesting how our core assumptions are different, but I want to say I highly value your ideas.

      Kind regards, Russell Jurgensen

        Dear Russel

        I appreciate very much your comments. About your question concerning gravity, You have expressed very well my idea of how gravity should appear in this context. I think gravity is just a consequence of the internal structure of quantum reality. In other words to obtain a satisfactory model unifying QM and GR, we should treat gravity and our perception of classical reality as an emergent phenomena in the sense of my essay, i.e as a classical logic structure arising from a non-classical logic ground model. An inevitable conclusion will be that gravity is not a fundamental force but just the result of how the interaction of matter and the other forces create our classical perception of spacetime ruled by the Einstein's equations.

        Dear Florin

        I had the opportunity to read your essay of the previous contest, I enjoyed very much and I am very flattered by your comment thanks.

        J.B.

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        • Post #24

        Dear John,

        I too am flattered by your comments. I was pondering for some time questions along the lines of your essay and your essay resonated strongly with what I am thinking. Hope you will win a prize.

        Interesting essay John. I appreciate in particular the emphasis you give to emergence, and emergence in computation. Also, the closing quote by Deutsch shines. I did not know it, and I am very pleased by his mentioning, among the deepest explanatory theories, (i) quantum theory, (iii) evolution theory for living organisms, and (iii) theory of computation, side by side. Surprisingly, Relativity is not in the group of four. Why do you think he excluded it?

        I am worried/confused by the fact that the set P at the right of your central Figure 1 can be equated to two very different things such as (i) 'a space of boolean algebras whcih represent history propositions...' , or (ii) just a causal set modeling spacetime. I agree that 'richness can become an enemy', as you write -- and I could add 'meta-theories eventually need instantiation', or 'the devil is in the details', etc...

        Finally, you write that 'The duality between discrete and continuum is just one more of the misunderstandings caused by a classical logic reasoning'. I am not sure I can retain a clear and strong argument explaining why this would be the case, after reading your essay. Does it mean that I have missed your main point?

        Tommaso

          Dear Tommaso

          Thank you for reading my essay. I think Deutsch does not mention general relativity just because, he is assuming that soon or later we will understand it within the quantum formalism. On the other hand, which kind of order or topological space could be appropriate to describe quantum reality in the sense I propose is something that I don't know yet. I think it depends on the approach you choose to interpret the classical formalism, i.e if we used the classical Copenhagen approach the more natural order is the Boolean algebra that I mention, but I am more interested in the digital approach that you propose because a lattice-order simplifies considerably the models I have proposed.

          Finally discreteness is not fundamental, because discreteness on quantum mechanics is mainly related with the measurement, which is the tool that make work the classical logic approach to model quantum reality. You are right this is a more deep issue that I haven't clarified on the essay.

          Regards,

          J.B.

          You have written an interesting essay. I give Topos theory some discussion in my essay

          http://fqxi.org/community/forum/topic/810

          However, that is more mentioned with respect to the Zariski topology.

          I am not entirely sure what the continuum hypothesis אּ_1 = 2^{אּ_0} has to do with physics. The Cohen Bernay theorem indicates this is consistent with the ZF set theory by Godel's theorem. I am somewhat familiar with these developments, but they are not entirely my area of expertise. Your paper gets a good thumbs up from me.

          Cheers LC

            Dear Lawrence

            Many thanks for your comments. I think that topos theory will play a fundamental role on physics in the next years, unfortunately Isham's work have not received the attention It deserves. This is mainly because the language of Category theory is still too abstract and difficult to handle. On the other hand, the importance of Cohen's forcing resides in how this technique allow to relate two models ruled by different logics, unfortunately the categorical tools of topos theory blur this fact, and because of this, many people working on topos theory have not noticed this important feature. What I am doing is trying to simplify this approach based on the importance of how a structure of variable sets can connect two theories ruled by different logics like quantum mechanics and general relativity.

            Regards,

            J. Benavides

            • [deleted]

            • Post #29

            Hello John,

            I make references to Topos theory, largely with connection to the algebraic or projective varieties with Zariski topology. This is the basis of what might be called "pre-topos" theory. It is a form of pre-sheaf construction, which can be used to build a sheaf theory. One main interest is in a twistor geometry with E_6 subroup (or E_6xE_8) with a sheaf or pre-sheaf construction for twistor geometry.

            There is a relationship between lightcones and Heisenberg groups within this Kleinian quotient system. I tend to see this as a precursor for the far more generalized system you present in your paper. I am also interested in the prospect for monster-moonshine structure, which is are projective varieties in 26 dimensions (eg the bosonic string) with Lorentzian structure. These projective varieties form the pre-sheaf construction for topos or grothendieck-Etale structure.

            The space of lightlike geodesics is a set of invariants and then due to a stabilizer on O(n,2), so the space of lightlike curves L_n is identified with the quotient O(n,2)/P, where P is a subgroup defined the quotient between a subgroup with a Zariski topology, or a Borel subgroup, and the main group G = O(n,2). This quotient G/P is a projective algebraic variety, or flag manifold and P is a parabolic subgroup. The natural embedding of a group H - -> G composed with the projective variety G - ->G/P is an isomorphism between the H and G/P. This is then a semi-direct product G = P x| H. For the G any GL(n) the parabolic group is a subgroup of upper triangular matrices. An example of such a matrix with real valued elements is the Heisenberg group of 3x3 matrices (sorry for the inconvenient representation, but I have bad luck with these html-TeX systems)

            |1 & a & b|

            |0 & 1 & c|

            |0 & 0 & 1|

            which may be extended to n-dimensional systems to form the 2n+1 dimensional Heisenberg group H_n of n + 2 entries

            |1 & a & b|

            |0 & I_n & c|

            |0 & 0 & 1|

            where for O(n,2) the Heisenberg group is H_{2n+3}. The elements a and c are then n+2 dimensional row and column vectors of O(n,2). These are Borel groups, which emerge from the quotient space AdS_n/Γ, where the discrete group Γ is a manifestation of the Calabi-Yau 3-cycle, and which as it turns out gives an integer partition for the set of quantum states in the AdS spacetime. So both spacetime and quantum structure as we know them are emergent.

            This of course can exist in more general setting, which is the type of construction you are presenting with the continuum hypothesis. I will not write that with alephs, for the Unicode representation does not work well. This suggests that the universe (the system of multi-cosmologies or multiverse) has this underlying system of topos between different structures. The system above illustrates how lightcones and Heisenberg groups emerge from the same quotient structure, where the topology indicates this is a form of topoi.

            Cheers LC

              Dear Lawrence

              What I have in mind is very different and is more close to the use of topos made by Isham and not in the context of twistor theory or string theory, I am sure there could be some connections but the approach, that I am looking to propose, is very different. Particularly I am trying to understand emergence, from a different perspective that cannot be based on models based on classical set theory like Heisenberg groups or lightcones on any kind of classical manifold. For example a first step would be just to redefine the measurement postulate of the classical formulation of QM, in terms of the collapse of a structure of variable sets, on a presheave on a poset, to the classical Hilbert structure. I wonder If the emergence you describe can be put in these terms.

              Regards,

              J. B.

              John,

              I find your idea that theory limits what we 'see' of Nature very interesting and also very relevant to the raging debates going on. Let me add my voice to that debate and point you to a result along the same lines in my essay.

              I show that Planck's constant is a necessary boundary to our ability to 'see' the Universe because h is the 'measurement standard' that defines Kelvin temperature. The entire theoretical regiment through which we measure, observe and understand the Universe breaths existence into h. Planck's constant is the 'focal point' to the physics we have created beyond which we cannot 'see' the world. It is NOT a necessary fact of Nature!

              Just the other day I posted a very short paper, "If the speed of light is constant, then light is a wave", that mathematically proves that light must be a wave.

              I look forward to your comments and your support for these significant results.

              Best wishes,

              Constantinos

                • [deleted]

                • Post #32

                Dear John,

                My approach is a bit more "conservative" I suppose. I think that this sort of topos construction may pertain on a deeper layer than what I am working on. I happen to be more focused on hypothetical connections with physics we understand currently. I would say a relationship between what I am doing and the sort of topos you advocate is similar to the relationship between pre-sheaf and sheaf theory.

                Cheers LC

                Dear Constantinos

                What you are saying about Planck constant is very interesting. I am looking for this kind of arguments to understand better what kind of limitations or features are closely related to the character of our theories related with classical logic. I will read your paper about the wave character of light I hope to give you some comments soon.

                Regards,

                J. Benavides