AM I IN THE ANALOG OR DISCRETE PARTY?

Dear visitor,

I think that this contest stimulated in all of us the curiosity about the orientations of each of the participants: "is he or she on the discrete side, or on the continuous side?".

So I will try to clarify my position, which is somehow ambivalent.

At the present time, I put my hopes in a continuous spacetime, on which continuous fields "grow". This may be obvious from my essay, in which I stated that the solution I propose to the singularities requires a continuous spacetime. (Of course, I may be wrong and the spacetime be discrete.)

On the other hand, I do not exclude the possibility that, even in the conditions of continuous spacetime and fields, the world still may be discrete. I illustrate this with the example of vector graphics format in computer graphics. This type of format allows infinite resolution, but in the same time it is digital. Digital does not necessarily means pixelated, so discrete information still may describe a continuous spacetime.

Similarly, continuous spacetimes endowed with continuous fields may very well be describable by digital information. After all, all books on continuous mathematics and physics can be scanned into a computer.

I argued for the continuity of spacetime and fields, but I do not exclude the possibility that all the information contained in these fields can be compressed in a digital format. I wish I could write about this too in my essay, but I need to do more research in this direction.

Best regards,

Cristi

    Dear Cristi,

    Yes, this is a nice question! Choose your side!

    My side is that reality is continuous, but it looks discrete sometimes because it has a differential structure which is scale-dependent. Then, making an experiment amounts to draw a map of the part of reality at one scale, to another part of reality, at another scale. There is a mathematical limit of the precision any such map could have.

    Physicists somehow are lost in another dream. Despite many claims that physics so inspired modern mathematics, in fact since some decades this is only very limited so, while the most dynamic mathematical fields have little to do with physics, but more with computer science or (less now but more in the future) biology. Or just pure curiosity.

    AM I IN THE ANALOG OR DISCRETE PARTY?

    Part 2

    Here is a link to some older work I did, named World Theory. It is a mathematical framework (based on sheaf theory) - a general mathematical structure which speaks about any possible world consisting in space, time, matter and laws of nature. It can be particularized to obtain many of our current theories in the foundational physics. In other words, to make abstraction of the particular solution we adopt, and to say the most general things we can say about the world. The intention was to write the laws in the most general possible form, so that we can compare them, and see which principles really contradict each other and which can be reconciliated on a higher level of generality. It was not a unified theory, just a unified framework.

    The mathematical structure defined there can be particularized to most of the continuous, and of the discrete theories which are currently researched. In other words, two theories about space, time and matter, one which is discrete and another which is continuous, are both particular instantiation of the mathematical structure I named there "world". The matter is, in all cases, a section in a sheaf over space and time, and this notion works with both continuous and discrete spacetimes, as the examples I gave there show.

    Although it would have been appropriate for the theme, because it brings under the same umbrella discrete and continuous, in the present essay I did not pursue this idea. The reason is that I spoke about the World Theory in my FQXi essay about time, Flowing with a Frozen River. There I used World Theory to discuss time, determinism and causality.

    Best regards,

    Cristi

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      • Post #18

      Dear Cristi,

      Thank you for clarifying your position. When I read your essay, I was under the impression that you were saying that space and fields are divisible ad infinitum, and thus continuous.

      Some of your older work implies that Nature is simultaneously continuous and discrete, and that was also my essay's point.

      It seems we have many similar ideas, and I need to read your older work.

      Have Fun!

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      • Post #19

      Dear Cristi,

      In connection with your just stated position, can you comment on the following opinion of Schrödinger (which I quote on p.1 of my essay:

      "If you envisage the development of physics in the last half-century, you get the impression that the discontinuous aspect of nature has been forced upon us very much against our will. We seemed to feel quite happy with the continuum. Max Planck was seriously frightened by the idea of a discontinuous exchange of energy ... Twenty-five years later the inventors of wave mechanics indulged for some time in the fond hope that they have paved the way of return to a classical continuous description, but again the hope was deceptive. Nature herself seemed to reject continuous description...

      The observed facts (about particles and light and all sorts of radiation and their mutual interaction) appear to be repugnant to the classical ideal of continuous description in space and time. ... So the facts of observation are irreconcilable with a continuous description in space and time."

      Did we learned anything fundamentally new which might have changed his opinion?

      Dear Ray,

      thank you for the feedback. You understood well what I said in my essay: my solution to the singularities in general relativity requires spacetime and fields to be divisible ad infinitum.

      In World Theory I define a mathematical structure named "world", which describe possible matter fields over possible spacetimes, subject to possible laws, "possible" in the mathematical sense. Known theories in physics, discrete or continuous, are find to be particular cases of this structure, in the same way as the definition of group applies to both discrete and continuous groups. But World Theory does not make any implications about the real world, it is just a metatheoretical framework.

      But there is enough room for discrete structures in continuous theories, and I will return to this subject soon.

      Best regards,

      Cristi

      Dear Lev,

      Schrödinger and Einstein are two of my favorites. They both suggested at some point that nature may be fundamentally discrete and combinatorial. Yet, their most astonishing results are based on the continuum.

      Einstein: Special and General Theories of Relativity are based on continuum. His explanation on the photoelectric effect shows indeed that there is something fundamentally discrete about photons. But what is that discrete aspect?

      Schrödinger: his equation, based on continuum, answered Einstein's question and provided the mathematical formalism of de Broglie's wave mechanics. From an equation based on continuum, we can obtain discrete sets of eigenstates. There is no contradiction: discrete emerged from continuous. So, Schrödinger's "own child" said that the continuum is fundamental.

      Einstein tried to unify the forces by using continuous means. He did not succeeded, but he was able to build, with Rosen, (The particle problem in the general theory of relativity) a topological model of a charged particle. Again, discrete emerged from continuous.

      Einstein, in his pursuit for local realism, supported de Broglie's pilot wave theory. On this basis, he rejected Schrödinger's idea (also based on continuum) that the wavefunction's square is the charge density of the electron (interpreted by Born as the probability density). Schrödinger realized later that, in fact, the entanglement was the main enemy of his idea, because the wavefunction's square cannot be interpreted as charge density for entangled particles. The entanglement rejected the locality, but not the continuum, from which it was initially derived.

      Schrödinger and Einstein were not comfortable with the idea that the particle is a singularity of de Broglie's pilot wave, so they started to hope for a discrete solution. Which they couldn't provide.

      So, to your question:

      "Did we learned anything fundamentally new which might have changed his opinion?"

      I would answer: "Did we learn anything which might have confirmed his opinion?"

      Best regards,

      Cristi

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      • Post #22

      Cristi,

      Schrödinger would not have changes his opinion.

      Now, to your question: "Did we learn anything which might have confirmed his opinion?"

      In order to learn something that would confirm his opinion, as always, we need to get out of the old "kitchen", build a new one, and to verify the predictions of this fundamentally new formalism. We need to look at the reality through new discrete "glasses", which we have not had: we don't even know what these "glasses" look like. That is why, I believe, we need to focus our efforts on trying to construct true "discrete" models of reality, not the handicraft models we call discrete. As you know from my essay, I am suggesting we need to construct a fundamentally new formal language that would elucidate the presently nebulous concept of discreteness.

      AM I IN THE CONTINUOUS OR DISCRETE PARTY?

      Does it matter?

      My personal opinion is not that important. I tried to prove something, and only the arguments should be important. My personal opinion can be a complementary information, which helps to put the things in a larger picture, but what matters is what I can support with arguments, what I can prove.

      In this line, I would like to say that I am glad to see at this contest such a wide diversity of opinions. The Nature is one, indeed, but we are far from knowing how she really is. So we try to guess the principles guiding her. So far, they are incomplete, and although they complement one another, they are in contradiction. I have no intention to take a side and claim that this is the truth, because I don't know the truth. I am just happy to see each effort, and each progress made by us in various directions. Am I proposing a continuous theory? Sure, but this shouldn't blind me against the discrete approaches. So, if you see me liking your discrete explanation, you should not conclude that I should not like a continuous explanation as well. Important is to progress, to add a new viewpoint, to solve a problem. Hence, we should encourage all good ideas which solve, or have the potential to solve a problem, to enrich our understanding, to widen our vision.

      Good luck to all in this contest

      Cristi

        Dear Lev,

        I am glad that you and others are trying to construct this new vision. I sincerely believe that this is a good thing, and that it will enhance our understanding. And I also sincerely believe that there should remain some of us to work in the "old kitchen" as well, because we need food until the new kitchen is ready.

        Schrödinger needs to prove his opinions like anybody else. And he succeeded very well to do this for many of his opinions. Now, if in the quote you gave, he states that space and time are fundamentally discrete, but he never provided evidence for his statement, I am free to adhere to his opinion or not. But just because he said so, it is not enough. I prefer to adhere to Schrödinger's opinions which he managed to prove, and which incidentally opposed the one you quoted.

        Now, this doesn't mean that eventually it will not turn out that the spacetime is discrete. I don't know. When the new kitchen is ready, I would like to be invited at dinner.

        Best regards,

        Cristi

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        • Post #25

        Dear Cristi,

        Thanks for the good wishes! My best wishes to you too.

        Just one little note. You know, of course, that at least 99.99% of researchers prefer to work in the old kitchen: it is so much more comfortable, but ... our scientific intuition, as was Schrödinger's, should not be biased by such comfort. ;-)

        Besides, all the comfort may turn out to be illusory, and so in the long run, the research life might turn out to be wasted (which wouldn't be so comfortable after all). ;-) The latter possibility has always been my greatest fear.

        Dear Lev,

        I invite you to visit my "kitchen" and taste my recipe. Please don't judge me only for using honey instead of sugar for this recipe; taste instead the cookie. Aren't the 99.99% you mention said that this meal cannot be cooked? How many of them do you see in my kitchen, cooking the same recipe?

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        • Post #27

        Dear Dr. Stoica,

        I enjoyed reading your essay before you lost me in mathematical technicalities. Nevertheless, before you launch into your thesis, you ask the following 3 questions, which I hope to answer:

        1) Is reality discrete or continuous? At least 'matter' and 'radiation' are discrete.

        2) (Was) it possible to find the answer by experiment? With respect to 'matter,' no, because we did not know what to look for.

        3) Or at least from theoretical arguments? Yes.

        You also point out that the reason we cannot decide between the continuous or discrete is that: "(T)he theories we know so far don't seem to make use in an essential, irreducible way of the discrete or continuous nature of the reality they propose." And further: "In addition, this theory should be mathematically and logically consistent, very well corroborated by experiments, and as simple as possible."

        You then proceed to make your case for your 'version of General Relativity.'

        While I am unable to mathematically argue the merits of your thesis, in terms of foundations it would be pointless to do so, as I show in my essay. Furthermore, the derived foundations in my essay are "mathematically and logically consistent, very well corroborated by experiments, and as simple as possible."

        I merely wish to bring this to your attention, for unless I am fooling myself the derived foundations leave no room for debate.

        Kind regards,

        Robert

        P.S - If I am fooling myself, could you please leave me a post to let me know.

        Dear Cristi,

        Here is a concrete mathematical question. It may be trivial, or not.

        Given a semi-riemannian metric g with its covariant version curvature

        tensor field R, is there (locally, around a point, or generically, like almost everywhere) another, non degenerate metric g', with the same curvature field R?

        Marius

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        • Post #29

        Cristi: I'm even happier that you like my essay after reading yours. I appreciated your resolution of the singularity and information paradox -- I found myself wondering why nobody had thought of your degenerate metric idea. I want to look further into your smooth quantum mechanics by way of your paper. Anyway, I am glad to find new and innovative arguments for (fundamentally) continuous nature that supplements the work by Zeh et al., so thank you for that, and good luck!

          Dear Karl,

          Thank you for your appreciation. Yes, when I initially considered to apply degenerate metrics to singularities I though it will be simple. It turned out that it was not that simple, because of the divergences and other problems which occur by the normal methods. The hardest part was to find a way to avoid them.

          Good luck to you too,

          Cristi

          Dear Marius,

          sorry, I just saw your post. Nice question, and far from trivial. And I think important, because we may need to know how to obtain the metric from the stress-energy tensor. I don't know how to answer it. Searching the net I found this article, this and this, which seem to show that in general the solution is unique, up to one arbitrary scalar. I don't know if it applies to all cases, and if they apply to the singular case as well.

          Best regards,

          Cristi

          Dear Marius,

          indeed, it is hard to choose a side. I agree with you that there is a limit in the precision, this is why I think it is hard to decide :)

          My personal view is that the continuum is a differentiable manifold, and many of the discrete aspects occur from its topological properties.

          Best regards,

          Cristi

          "AM I IN THE CONTINUOUS OR DISCRETE PARTY?"

          There is obviously both the continuous and the discrete in reality. The empirical evidence for this is obviously that we are all able to see bits and pieces of reality somehow and some say they can divide the bits and pieces to infinitely ever-increasing entropy.

          It does not really take peculiar minds to see the obvious. But understanding the obvious takes peculiar minds, because pointing out the obvious for peculiar minds to understand is quite difficult.

          The difficulty comes because of the unclear picture of the foundational ideas. Once the foundational ideas are clarified, eveyone could readily adapt their interpretations and most everyone will see that we have been describing the same thing, albeit unclearly.

          The answer to "what of reality is continuous and what is discrete" can readily be clarified if we take the analysis down to the fundamentals.

          Perhaps you will agree with me regarding what is continuous and what is discrete in reality if you read my essay... http://www.fqxi.org/community/forum/topic/835

            Dear Christi,

            I don't exactly know how to put this. But what makes the "degenerate metrics" degenerate - what is the cause of the process? Does the degeneration bring about "particles" or "voids"?

            Just a thought...

            Castel