Excellent paper Phil,

I like the way you would re-frame our treatment of the vacuum, and I agree that a multiplicity or landscape of vacuum state solutions is a feature of a large class of quantum gravity candidate theories, not only Strings. I like the way you informed us about how the concept of an algebra can be generalized, and about the free Lie algebra - which is new to me, but obviously significant. Also; I like that you weave in higher category theory, as I savvy that the category theoretic framework can subsume a lot of other Maths.

Well done!

Jonathan

    Thanks Jonathan and good to see you in the contest again. I will be looking at your essay soon.

    Phil

    Dear Philip,

    In Grothendieck's 'dessins d'enfants' all is about a two-generator free group, as I am rediscovering step by step with Magma, motivated by the application to quantum observables and contextuality. Reading your wide range excellent essay, I start to be convinced that playing with free algebra over the appropriate rings, new territories of understanding may be reached.

    As you refer to topics also investigated in my essay, e.g. Moonshine, modularity and groups, I suspect you may be interested to read me. I would love to have some comments from your side.

    Best,

    Michel

      Michel, thanks for your feedback. I am glad to see you back here. Your areas of maths are very closely linked to my ideas although I do not knoe them as well as you do. If my talk of free algebras gels with your thinking then I am encouraged by that.

      I will of course be reading your essay shortly

      Dear Philip Gibbs,

      "You talked about nature of physical laws and mathematical structures and then metaphysical structures to guide us in the development of physical theories. why we exist and why laws of physics are so steeped in mathematical abstraction.Uncovering the meta-laws is now the most important goal in our quest to understand the universe.

      You also talked about referring Wigner that We don't know why complex numbers originally formulated in analysis and algebra became so useful in number theory and physics?

      The deepest questions we can ask about existence are "How do we exist?" and "Why are things as they are?"The Mathematical Universe Hypotheses tells us that all logical possibilities are equal. It does not require a magic spell to bring one chosen system of equations into reality.

      We will discover more about the relationships between algebra and geometry that determine the emergence of space and time in a universe governed by the laws of energy and entropy that are needed for life to evolve."

      Let me quote Swami Vivekananda who had addressed World's Parliament of Religion held at Chicago in 1893. Swamiji was a man of higher consciousness.The great scientist Nicholas Tesla and many others were deeply influenced by him and used to take guidance from him by attending his lectures. He hinted at scientific theories decade before Albert Einstein formulated his relativity theories.

      So, what did Swamiji say?

      He made statements like, "Take

      anything before you, the most material

      thing--take one of the most material

      sciences, as chemistry or physics, astronomy

      or biology--study it, push the study forward

      and forward, and the gross forms will begin

      to melt and become finer and finer, until

      they come to a point where you are bound to

      make a tremendous leap from these material

      things into the immaterial. The gross melts

      into the fine, physics into metaphysics, in

      every department of knowledge."[Please refer to the attached file for his detailed view on Cosmology,Universe,Matter,Existence.

      Before asking the question why do I exist, one needs to be able to answer Who am 'I' and what is 'I' ? paradox of Self-Consciousness Albert Einstein rightly put(please refer to my essay for his quote) that the separation of human from the Universe as different entity is optical delusion and deep lack of consciousness. MUH,ERH claiming about "physical laws independent of human" & mathematical structures should explore what is human or say 'I'(for that human) ? Who creates mathematical structures used as language for physical reality because every language has certain structures behind its existence. If it asks the existence of mathematical equations,but what governs the physics of equation itself ? If Wigner said that complex number is advanced concept ? Complex numbers have been so useful in number theory to Quantum physics? Why so effective? Eugen Merzbacher in his QM book took the most general structure of wave and showed that in order to satisfy the condition that physical characteristics of the wave should remain invariant under the displacement in space-time directions, the parameter comes out to be'i'(complex number. Here is the laws of invariance behind complex number that makes it so useful . Why we defined (-)*(-)=(); why not ()*()=(-) also? It is this laws of invariance behind the mathematical structures, which makes it compatible in different physical and mathematical scenarios.

      In context of Skolem's paradox - "A particular model fails to accurately capture every feature of the reality of which it is a model. A mathematical model of a physical theory, for instance, may contain only real numbers and sets of real numbers, even though the theory itself concerns, say, subatomic particles and regions of space-time. Similarly, a tabletop model of the solar system will get some things right about the solar system while getting other things quite wrong. So, for instance, it may get the relative sizes of the planets right while getting their absolute sizes (or even their proportional sizes) wrong"

      Why so? Its not mathematics describing physics rather the laws of invariance behind mathematical structures(whether discovered /invented) describes the laws of invariance behind physical reality.

      Mathematical Structure Hypothesis(which I have propounded) states that they both having no independent existence because they both originate from Vibration. So,its not that laws of energy and entropy governs only physics but also the mathematical abstractness,which can be seen in context of Poincare & Geometrization conjecture.

      Anyway, you have written a great essay.

      Regards,

      Pankaj ManiAttachment #1: vivekanada_universe.pdf

      Dear Pankaj Mani, thank you for your detailed analysis.

      I agree that the question about conciousness is very relevant to this topic. There is a lot to say about it and I could not possibly have fitted the subject into the space for this essay. What I hope is that a future essay topic will ask that question so that I can write about it at length. Meanwhile I am always glad to see other people bringing it up.

      regards

      Phil

      Hi Philip,

      Last essay contest I had a system that created QM from just pure random numbers" reality is a mathematical structure". This year's essay has much more astonishing results and I have put in the links (at the end of the sections) to the JavaScripts program, which I am sure you have no problem with. Although I know you are a busy man.

      Philip, I am relying on you since I don't seem to have too many customers here. It is ironic that often people say "show me the math" and cut the word salad, and now when I show it, they don't want to bother and it seems like they are saying "bring on the word salad"!

      Essay

      Thanks and good luck.

      Dear Philip,

      Your statement, 'SUSY is a natural consequence of string theory and would account for fine-tuning of the Higgs mechanism', indicates the imperativeness for the string theory to be modified. Thus we may think of another set of principles with strings, in that, field with matter is ascribed as single unit of eigen-rotational string-segment. Thus the Univacuum and the Multiplicity of the vacuum, that are descriptive with Space time foam and Spin-foams by String Vaccuua, is differently interpreted with an alternative paradigm of Universe; that is used for a comparative analysis in my essay, ' Before the Primordial Geometric origin: The Mysterious connection between Physics and Mathematics'. Hope you may enjoy in reading that.

      With best wishes,

      Jayakar

      Dear Philip,

      Thank you for your essay, excellent as always, stimulating reflection. But it is not about praising. Let us start a discussion.

      I do not agree with your interpretation of Tegmark's Mathematical Universe Hypothesis... You argue that "such ideas are about concepts beyond our ordinary experience for which we do not have predefined words. To think about them we can only use metaphors with meaning that we understand within our own limits." Really the MUH is about to find the mathematical structure (or structures) isomorphic to the world we observe (the empirical domain) and finding a description (language) expressible in a form that is well-defined also according to non-human sentient entities (say aliens or future supercomputers) that lack the common understanding of concepts that we humans have evolved. In MUH not all mathematical structures exist as physical reality. Only these that we can embrace with our empirical domain.

      If we want to know the geometrical structures describing the observed reality we cannot invent them. We can only discover what geometries are possible in 3+1 dimensional spacetime. With helping hand of Perelman, that proved the geometrization conjecture in 2003, we know for sure all geometries that are possible in this case. Starting from the conjecture we do not need more than these (complicated enough) structures to describe all observed reality. So I conclude that dimensions higher than observed 3+1 are INVENTED not DISCOVERED.

      I argue that we can find "connections between subjects that had previously seemed unrelated". I mean Thurston geometries in connection with matter and fundamental interactions. So I have coined the related name: Geometrical Universe Hypothesis. Thanks to the correspondence rule, that is a real paradigm shift, the geometrization conjecture becomes the first theorem in physics. Moreover it promises universality. There is naturally a metric associated with each Thurston geometry. Let us remember that Perelman proved the geometrization conjecture using Ricci flow with surgery. The constant curvature geometries (S3, H3, E3) arise as steady states of the Ricci flow, the other five geometries arise naturally where the dynamics of the Ricci flow is more complicated and where topological changes (like neck pinching or surgery) happen. Thurston geometries with Ricci flow and surgery make the spacetime devoid of singularities and we naturally get the symmetries.

      Do we need new data or new experiments to find details of GUH? Only to confirm its predictive power concerning that five more exotic geometrical structures that remain to be uncovered in nature.

      Maybe GUH is too good to be true? Or maybe it seems too simple to be acceptable (simple if one can comprehend the Perelman's proof!)? In details this is really complicated and at the moment it is only an sketch. The examples of that complexity you can find in Torsten Asselmayer-Maluga's publications e.g. "How to include fermions into General relativity by exotic smoothness" http://arxiv.org/abs/1502.02087

      There are also publications on how the Ricci flow can pass through singularities and continue on a new manifold! E.g. The Kahler-Ricci flow through singularities, Jian Song, Gang Tian, http://arxiv.org/abs/0909.4898v1

      The final question: Is there any possibility to get an experimental confirmation of the statement that space and time are emergent? You confirmed that no one knows from what and Matrix-theory, the amplituhedron and LQG are purely theoretical works.

      If you are interested you can find details of GUH in my essay.

      Sorry for my excessive self-confidence. I would really appreciate your criticism. Thank you.

      Jacek

        Hello Philip

        I was able to read your essay despite the fact that most of the concepts you deal with either in physics or mathematics were too technical or specialized for me to understand the points you are making. Is it possible to describe what you mean by universality in a simple way or by analogy?

        Nevertheless the essay is lucid and well-written and I read it through. It left me with the hope or rather belief that beyond all the disparate phenomena and complicated theories there is a breathtakingly simple unity.

        That is what I have argued in this year's essay, and this time did not have the aid of the elephant as I too did in a past essay, but of the amazingly 'smart' slime mold that can solve mazes. Amazing world!

        Best wishes

        Vladimir

          Vladimir, these issues are technically difficult for everybody and we can only move forward by pulling together and throwing in good ideas. I am glad to see you back to do that again. I too find the subjects difficult so I pick out the bits I understand and try to see some of the picture take shape from those.

          Thanks for your comments, I will probably be nicking your slime mould idea next time.

          Jacel, thanks for your interesting points. We aeem to take different philosophical views. You prefer geometry and I prefer algebra for one thing. Of course I cannot say that I am right and you are wrong. That remains to be seen, or perhaps the truth will be neither or both. All we can do is each take our ideas to their conclusions and see which result fits nature.

          "Is there any possibility to get an experimental confirmation of the statement that space and time are emergent? "

          That could only be done directly by an experiment so cataclismic that it tears space and time apart. However, indirectly we can find the right TOE and use other more doable experiments to check it. The answer will then come indirectly from that TOE. We will never be totally sure that the TOE is completely correct unless we tests everything it can predict including things like the emergence of space and time, but I hope that once we have the answer it will be something sufficiently convincing.

          Thank you Philip for the response.

          I agree that we will never be totally sure that the theory is completely correct unless we test everything it can predict. However I would not include the emergence of space and time (this just seems to me too speculative). I guess that if we uncovered in nature these five exotic Riemannian manifolds: S2 ﾃ-- R, H2 ﾃ-- R, SL(2, R), Nil and Solv geometry, it would be convincing enough that the geometrization conjecture is the first physical theorem. And the theorem we cannot falsify. But, apparently, at the moment we have to wait. But not so long. As I have mentioned, Torsten Asselmayer-Maluga's works are my hope to show the direction for future experiments. He now works on NIL, SOLV and SL2.

          Best,

          Jacek

          Dear Dr. Gibbs,

          I think my earlier post has gone unnoticed because somebody inadvertently posted a new comment as a reply to my post. So excuse me, I am posting it again, especially because I claim myself to be an independent researcher and find solace in the free-for-all 'VIXRA'. I express my sincere thanks to you for providing an asylum for people like me.

          The first essay I read was yours. Instantly, I identified you as a 'mathematicalist' trying to impose the rule of mathematics in the domain of physics. The way you have written, however, is impressive that any one reluctant will jump into the 'mathematicalist' wagon. That I think is the beauty of the mathematics-oriented thinking coming from somebody who knows the intricacies of both mathematics and physics.

          Your statement "geometry is an angel and algebra is a demon .......... the signs are that the devil rules at the deepest levels of existence" is thought provoking. Can I say that the rules of mathematics are essentially algebraic, and geometry just represents its emergent structures. Then the universality of mathematics is in its rules, not in its structures. Regarding the question, whether mathematics is invented or discovered, I think the rules are discovered, but the structures are invented. For example, in chess the properties of the pieces are invented, but the emergence follows mathematical rules and the overall structure of the game is thus invented. Starting with another set of arbitrary properties, you will obtain a different structure.

          Again I would like to quote another statement "the theory you get by recursively iterating quantisation should be unique" . Without referring to existing Quantum Mechanics, your statement can be construed to be implying that fundamental particles, just because they are quanta, may be obeying a unique law, which is universal. Does it simply mean that starting from qunatised entities, you cannot have an infinite number of emergent structures?

          The 'physicalist' idea that I propose in my essay is this: physics decides the properties, mathematics decides the rules. For the given properties, mathematics decides the emergent structures; for that emergent structures, physics again decides the emergent properties, and so on. Thus, the equations are mathematical, but the variables are physical. Starting from a finite number of variables having finite properties, the number of variables will soon come to the minimum that further emergent structures are impossible. That final structure is the physical world that we observe.

          Somewhere above, you have stated that 'physics emerges from mathematics'. This I think tantamount to saying that 'the physical world emerges from mathematics'. Or, given the basic properties of matter, mathematics decides the final emergent structure. That way, I will have to call you a physicalist. Kindly go through my essay: A physicalist interpretation of the relation between Physics and Mathematics. I would be awaiting for your comments.

            Hi Philip,

            I still wait for your next replies to my previous message, especially about the Many-worlds interpretation.

            Recently I wrote this general overview of how I see things going in this contest, with a list of essays I found best. I also included there a list of essays criticizing the mathematical universe hypothesis which you defend. So I look forward to your comments on these essays.

            I also included some not so rejoicing stuff there. I am sorry for this and I wish I did not have to do that, but when I see a nonsense growing too big I cannot go on very long as if it did not exist. So I am also curious to have your view on that. Thanks for your understanding.

              Ho Jose, I am sorry for not replying earlier. I was busy due to unrelated events and am now back.

              I think your assessment of my position is fair and correct. I see mathematics as the realm of logical possibilities and physics as the realisation of those possibilties. From this point of view it obviously makes sense to start with the possibilities and then move on to the realisation.

              However, this is just one philosophical position of many and I always recognise that other ways of looking at it can be equally valid. The test of meta-physical ideas is how they lead to more physical theories and from there to experiment. I have described in my essay how I related my philosophy to physics via multiple quantisation, algebraic geometry etc. It will be a long time before we know how it all pans out.

              The multiple quantisation idea has been around for a while. I speculatd about the uniqueness of iterated quantisation a few years back at http://arxiv.org/abs/hep-th/9603165 My view of that has not changed but it has developed. I now think that the structure this gives us has to be seen as a system of meta-laws rather than a specific physical theory inside a particular geometry like our spacetime that we can relate to directly through experiment. Physics as we know it is just one solution of these meta-laws. In act you have to go through a cascade of refined solutions to get to the end. So in one sense the end-result of iterated quantisation is unique but the physical theory is not. Perhaps you can realate this to your way of seeing things.

              I look forward to reading your essay

              Hi Sylvain, it is a nice idea to try and label all the essays with different isms (as on your linked page) so long as you do not take it too seriously.

              I think you may also be trying to take the rating too seriously. In my post at the top I say the rating and prizes don't matter to me. What I have found each time is that I start near the top, then later I drop down. You see there are just as many people who hate viXra as people who love it but they arrive later :-)

              The reason I keep coming back to the contest is that it is good to formulate thoughts and get some feedback. The FQXi essay contests are very good for that.

              I want to respond to some of the unanswered questions here as follows.

              I see mathematics as something that comes before physics but I don't see mathematics as a platonic realm. Mathematics is just the study of logical possibilities and our physical experience is just a stream of those logical possibilities being played out. We don't need to explain existence and reality any more than that. Our intuition may demand a causal and structural explanation for why it all happens but that is just part of our psychological makeup and has no answer. However, we do need to explain why the laws of physics follow certain mathematical rules.

              When people started doing mathematics they were interested in counting and measurement. There was no mystery about why the mathematics was effective in physics, because it was derived from it. But then mathematicians saw interesting logical structures that did not have obvious applications, such as prime numbers. Mathematics took on a life of its own.

              Logical possibilities include stuff that is very interesting to mathematicians and stuff that is less interesting. The interesting stuff is characterised by its universality. It is applicable to a range of problems. Mathematicians are delighted when something they formulated for one problem turns out to be useful for another. They get a sense that those logical structures are discovered while others are merely invented. This is what distinguishes pure mathematics from other intellectual endeavours such as art and literature where we consider things to be created rather than discovered. All these things are logical possibilities but the mathematically interesting structures are more universal. They would probably be discovered by an alien race of mathematicians no matter what point they started from.

              Already there seems to be some mysteriousness about this universality. Why is it there? Some people are not convinced. They see no mystery yet, so let's look further.

              As pure mathematicians continued to study these objects for their own sake without any remaining interest in physics they went beyond the naturally occurring logical structures. They discovered the mathematics of complex numbers, non-Euclidean geometry and higher dimensional spaces. They did not expect these things to be useful to physicists but later they were. Already the unreasonable effectiveness of mathematics seemed mysterious to Wigner, but some people still shrugged their shoulders. They can say that these things were still inspired by physical ideas originally or that the universe is obviously going to be mathematical so of course these things are going to be useful.

              For me the real clincher came when string theory was used to prove the monstrous moonshine conjectures. These were mysterious problems connecting areas of number theory and algebraic geometry that nobody would have expected to be connected to the real world. String theory has not yet been shown to be real physics but it was certainly discovered by physicists generalising the framework of quantum field theory with the goal of forming new physical theories. The distance across mathematics spanned by string theory and monstrous moonshine could not have been greater, and yet they turned out to be deeply connected in an unavoidable way.

              This is no longer just a simple matter of the unreasonable effectiveness of mathematics in physics. It is also about the unreasonable effectiveness of physics in mathematics. The mystery is deep and cannot be shrugged off. It demands an explanation.

              In my opinion, that explanation will come from a better understanding of universality and how it emerges in complex systems. It must be a self-organised structure embedded in the complex system of logical possibilities and their interrelations. It may be characterised by scale-free networks, self-similar fractal structures, path integrals over the grand ensemble of algorithms, iterated quantisation, n-category theory, symmetries etc. Our experience then unfolds according to laws of physics that form as a hierarchy of solutions derived from these universal systems. That is how our universe is put together and it explains the mysterious links that bind mathematics and physics together because this universality is ultimately what both mathematicians and physicists are drawn towards for different reasons.

              Hi Philip,

              I agree with Ken Seto about the need for concentrating more on physical models. I have witnessed their potential in my own research and theory (TOEBI). Also, don't be too judgemental towards people who think that the contemporary physics has gone into woods for a long time ;-) I'm one of those people.

              Anyway, you are a solid performer as usually, thanks for your essay! Check out my essay, it's a good example of a more physical theory.