Philip,

You have an amazing ability to "defuse" situations. Ideas that I tend to reject rather strongly when described by others seem much more reasonable when you discuss them. That is a real talent, in my opinion. I also like that, when discussing some of the farther out ideas you note that "we should not get carried away by thinking they are less speculative or more testable than they really are."

You have a most impressive grasp of the modern range of math and physics, and seem very comfortable moving over this range. Thank you for doing so in your excellent essay.

Your metaphor of chess playing aliens is a very clever way of contrasting 'invent' versus 'discover' math. And you seem to have split the baby when you caution not to worry about 'exist' [Tegmark's MUH] and then describe the 'ontology' in the next paragraph. You certainly have an impressive graphical representation of a metaphorical chart of mathematical ontology. It is quite a universe, isn't it?

I also appreciated your discussion of symmetry, generally conceived as the key principle that determines laws of physics. As I doubt this, I was happy to learn that "there is a growing movement... that thinks symmetry is not so fundamental." I tend to agree with you that "the fundamental principle that determines the laws of physics is universality, not symmetry." I have significant problems with the Holographic Principle, or more specifically with the "black hole information loss puzzle", but your analysis in terms of symmetry in terms of one degree of symmetry for every field variable is interesting.

If I understand you correctly I agree that "the structure that emerges from universality is also a mathematical structure in its own right." And that it is self-referential and recursive. And I need to reflect more on your idea of recursively iterating quantization. Finally you note that Tegmark's mathematical universe hypothesis tells us that all logical possibilities are equal. This seems to imply that there is more than one completely self consistent logical possibility [for the universe]. I doubt this.

In other words, as we've come to expect from you, a first-class essay! I invite you to read and comment on my essay.

Finally, having last year posted three papers on viXra, I wish to join many others in thanking you for creating that system.

My best regards,

Edwin Eugene Klingman

Eugene, thanks for going through my essay in such detail.

It is certainly the case that symmetry is being dissed from all sides at the moment. The importance of symmetry was drummed into me as part of my education and I have taken it too much for granted, so I am pleased that there is now some opposition to the idea. As well as the reference to string theorists in my essay you can find this in John Horgan's latest interview with Lee Smolin http://blogs.scientificamerican.com/cross-check/2015/01/04/troublemaker-lee-smolin-questions-if-physics-laws-are-timeless/

This forces me to question why I think symmetry is so important. Of course symmetry has been central to 20th century physics and is supported by experiment but that does not necessarily mean that there is more of it to be found. My main remaining justification is the argument I gave about holography, so if you have a different solution to the information loss puzzle I cannot hope to persuade you. Well we have to explore all the possibilities and each physicist will take one route, so this is good.

When I talk about different logical possibilities I think that different solutions to the laws of physics from different starting conditions are different logical pssibilities. I also think that your experience of this universe is a different logical possibility from my experience. I think that different laws of physics are no different from these different solutions. They are all different solutions of some higher meta-laws. Of course this is just one point of view.

I will be reading your essay soon

    I think this idea of different possibilities/solutions is the key element in the new paradigm shift. In the past people thought there would be one nice quantum field theory that would tell us everything about particle physics and GR and all the parameters of physics could be derived from first principles, then these laws would have different solutions depending on different initial conditions, perhaps there would even be just one unique possibility for the initial conditions.

    Now the view has changed so that the laws of physics are not thought ot be so fixed. For string theorists there is M-theory from which all other string theories can be derived as solutions with different vacua, then our universe is also just a solution which has this vacuum state. Lee Smolin also thinks this way when he talks about cosmic evolution. I take this further because I think that even M-theory is just one solution in a more general system of meta-laws which I imagine might be something like a free weak omega category. This is just an algebraic structure, perhaps the most general possible algebraic structure so that all other algebras are images of this structure under homomorhisms. This is how "solutions" are realised. To make this idea work you need to be able to project those algebraic structures onto geometry which is where the maths of algebraic geomerty come in to play. The pay-off for this way of thinking is that if all logical possibilities are included in the laws of physics then you dont need a magic spell to decide what the laws of physics become reality.

    Gary, thanks for your comments. I agree that knowing how to ask the questions is an important goal. Until we know that we are all just stabbing in the dark, but if we keep trying one of us may hit something.

    I noted your comment to Klingman's essay. Those are the two popular views of math and physics.

    For me the math and physics emerge together and are the same physical reality. Therefore, properties of math can be used to suggest the physics of reality. The difficult things of math can also imply things that don't exist in physical reality such as mapping math and infinity. So the quantum math (not real) of Bell is incorrect which is shown by the de Broglie-Bohm interpretation that suggests the ``hidden variables''. There are few papers written on this, I am aware of only one that takes this issue head-on that references earlier work that have been ignored. Klingman's paper uses this to highlight how this works.

    For me the prime thing to understand is the double--slit experiment with the Afshar's experiments of which--way and single photon interference. This experiment is the key to understanding the world of the small. That was the subject of my previous paper on photon interference and current effort on the single photon interference. Newtonian mechanics must apply to create the wave (Bohm's weakness) and direct the particle.

      John thanks for reading my essay. I am still looking at yours.

      I an sure there will be many other ways of thinking about what came first in this contest besides the three you have highlighted here. My view is in sync with that of Max Tegmark who proposed that the existence of mathematics came first. I just see that as a tautological statement that there are logical possibilities for our world or experience and that these can be explored and understood using mathematics. The main idea I add is that of universality, i.e. that the uppermost meta-laws of physics come about because of a principle of universality or self-organisation within the "Ultimate Ensemble" as Tegmark calls it. Our universe is just one solution of these meta-laws which are much more general. (By the way I first wrote about these ideas in 1996 before I had read Tegmark's paper which appeared in 1997. the first piece I wrote about it can be found at http://www.karlin.mff.cuni.cz/~motl/Gibbs/tot.htm)

      However, if I was cornered I would admit that the difference between starting with maths, physics or both together could just be a matter of philosophical interpretation. I just find it nicer to start from the mathematical possibilities because it leaves me with a sense that the existence of physical reality and the laws of physics are derived without a requirement for any unexplained process of design or arbitrary principle.

      What really matters is how we can use these meta-physical ideas to say something about what the laws of physics are like. For you this involves an analysis of the two slit experiment and for me it is about the structure of physical law in terms of multiple quantisation, complete symmetry, algebraic geometry and the emergence of space, time, causality and the universe as we know it.

      Hi Philip,

      You essay was interesting. You do point out the apparent direction of physics that research usually pushes us into new areas of mathematics. Some eagerly embrace this and others have to be dragged kicking and screaming into it.

      I do think there are possible trends into different mathematics. I would have to say that this is probably in the direction of category theory, motives and "magma," and with that into HOTT, or HOmotopic Type Theory. This approach would reduce the salient mathematically based observables into topological invariants. The two slit experiment is a form of homotopy, and this further is expressable as a logic switching theory.

      I don't know if I will write an essay for this iteration. I do have a couple of ideas, but the whole business seems rather futile to me right now.

      Cheers LC

        An earlier comment sent me to your comment on Klingman's essay, in which I noted "For you the physical world comes first and is unique while mathematics emerges in its many forms. For me the mathematical world is a unique structure from which many possible physical realities emerge." Whereas I couldn't find a point in your paper on which I felt a wish to hang my hat, those two sentences lead me to ask how does mathematics emerge? What kind of detector do we use and what kind of signal processing do we use to allow "mathematics" to "emerge"? It seems that we use the same body/senses and brain/mind to discover/invent physical structure as we use to discover/invent mathematical structure. An empiricist such as myself might think that the two are names for parts of the same thing, our evolving attempts to model, describe and to control some aspects of our continued existence (and that discover/invent presents a delicate demarcation problem), at progressive levels of abstraction.

        I feel unable to engage with this FQXi topic in an essay because however much I have thought about it I have far too little knowledge of the literature, which I know, however, to be voluminous. So a teeny comment is as far as I will go.

        Another earlier comment leads me to discover that you are the founder of viXra, for which Kudos.

          Lawrence, its good to see you are looking at the essays. I hope you will decide to write one too.

          I think the idea of something like a magma as a starting point is very powerful. A magma is just a very general algebra with a binary operation, by imposing other conditions you get loops, semi-groups, groups etc. I see this as a model for what I referred to as the "cascade of solutions" in my essay. You start with something like a free magma, each time you impose new conditions you are defining a homomorphism onto a more restricted structure such as a free group. Further homomorphisms give you specific groups. You can get any group this way, or any algebra if you start from something sufficiently general like an n-category.

          Since I noticed that a free lie-algebra has a structure like discrete closed and open strings that can be mapped to string states in continuous spaces using iterated integrations I have been keen on this as a way to see things. Ultimately the cascade leads to something like M-theory from which known string theories can be derived by compactification. What is compactification? It is just a process of identifying points on a manifold so it is a continuation of the algebraic process of setting identities to map a free algebra to more specific examples by taking it modulo some expressions, i.e. by mapping with a homomorphism. So starting with a general algebraic structure you can see how it could lead to a specific universe through a cascade of "solutions" which are just homomorphisms. In a category the homomorphisms are already built in as morphisms so a very large general category or n-category is a natural multiverse of related universes. That is how I see it at least.

          I have tried to limit the amount of mathematics in my essay because I want to reach a wide audiance and in past essays the mathematical details have not hit home. With so many essays to get through people want an easy read. Your essays have always been heavy on the maths so that may limit their appeal, although you have still done well.

          I have lost interest in the prizes especially since you now need a second prize to get membership, but the essays and discussions have always helped me map out my ideas a little further each time so I keep doing them.

          Hello Peter, I think if people allowed "too little knowledge of the literature" to stand in their way there would be a lot fewer essays written for these contests, and that includes mine.

          I am not sure about how mathematics might emerge either from physics or from nothing. Klingman has some nice ideas about emergence from physics using pattern regognition of whatever. My current view is that mathematics is just the structure of all logical possibilities so it does not have to emerge from something else. It is about what can be rather than what is.

          I am therefore more interested in how physics can emerge from mathematics. I had another idea about how I might tackle this question which was a little different from what I eventually went with in my essay. I was going to write about what might happen if there were only mathematicians and no physicists. How many ideas from physics would they invent without any input from the real world. You can imagine that they even have no direct contact with the physical world. They could just be brains in a vat left to ponder on logical problems. It may even be possible one day to see this happen using artificial intelligence.

          To be more specific we might program an AI system using Sparse Acataleptic Bayesian Inference algorithms to solve integer diophantine equations. It would try to classify solutions to as wide a range of possible equations as it could. Initially it would have just the definition of integers and polynomial equations to work with but would use heuristic methods to find new definitions to help it solve problems and find logical proofs, just as mathematicians do. Real mathematicians have of course had the benefit of knowledge about the physical universe to inspire the use of real numbers and geometry to solve this kind of problem but they have also invented new concepts such as complex numbers and quaternions from scratch which were only later known to be useful in physics. I think an isolated AI program if it is sufficiently good would do the same thing. Diophantine equations are very rich in terms of the kind of mathematical tools are required to solve even simple cases. The AI system would have to invent rationals then real numbers and even geometric ideas. It would probably also realise that to find new ideas it has to explore a wider field of mathematical concepts and would need to get a feel for what is interesting enough. I am confident that it would discover all the concepts used in mathematical physics just to use them to solve diophantine equations. If you are skeptical you should remember that string theory has already been used to solve the Monstrous Moonshine conjectures which came from problems in number theory.

          If this project could be carried out in practice it would be proof that physics can emerge from just mathematics. Unfortunately I made up the term "Sparse Acataleptic Bayesian Inference" and nobody really knows how to do it yet.

          It would be interesting to see the foundations of mathematics necessary for physics reduced to groups, groupoids, monoids and categories such as motives in a Grothendiecke type of system. I have I think found a possible route towards this using discrete systems.

          There are various entanglement schemes, and the GHZ entanglement is 1/8 supersymmetric. There are bipartite and tripartite entanglements that are ½ and ¼ supersymmetric. This means that on BPS black holes these entanglements of states associated with the BPS charges have this number of supersymmetric generators that are unbroken. The algebraic geometry of these entanglements involves a quotient homology of projective varieties. These are systems between the Hilbert space and a projective Hilbert space with the geometric phase as the fibration. This system is categorically the same as a quotient homology on the moduli space for quaternionic bundles, such as with SO(4), or SO(3,1) in the hyperbolic case. The isometries for the this system is SO(4,2) and the moduli is AdS_5 ~ SO(4,2)/SO(4,1).

          The two quotient systems are given by discrete groups. In the case of the AdS_5 the quotients are Kleinian groups, which are quotients with a discrete group, such as an elementary Z_n ~ Z/nZ, or a more complex polytope group. For the case of projective varieties on the Hilbert space these are a system of discrete orbits that have a discrete geometric phase ~ e^{nEt}. The two orbit spaces are I think categorically equivalent.

          The various quotient groups correspond to cobordants of one dimension lower.

          For instance, with AdS_5 there is a boundary spacetime, and for the quotient group on the moduli AdS_5 defines two boundaries with different spacetimes that may have different topologies. The equivalency between quantum projective varieties and the Kleinian orbit space of different spacetime topologies connects topology changes with different quantum states or sets of quantum states.

          I think these correspondence goes beyond one particular type of entanglement. There is a whole algebraic category of entanglements by Micheal Duff and his research partners. This algebraic system of entanglements is connected to this structure of quotient homologies and algebraic varieties. The categorical equivalency with the AdS_5 is a surprising aspect that I have suspect exists with respect to the mathematics of four manifolds as found by Atiyah, Donaldson, Freed, Singer, Uhlenbeck and others. The moduli space when reduced to a finite group is equivalent to the orbit spaces of quantum states with a discrete structure. This structure is given by the Kirwan polyhedra of holomorphic coadjoint orbits. This is categorically equivalent to the quotient of the moduli space or AdS_5 with a discrete or Kleinian group.

          The theory does of course connect with Raamsdonk's observation that entanglement can be converted to geometric content. In particular the entanglement of a quantum system with states associated with gravitation is equivalent to the entanglement of that system with the stretched horizon of a black hole or similar system. The "large N limit" means the set of states entangled with the gravitational system become entangled with a system that has a coarse grained structure, such as how an event horizon has lots of Planck area units that states can be shuffled around in.

          By doing this I think we can reduce physics to certain topological invariants, and all of physics can be reduced to a homotopy theory of logic gates. I expect in time to see physics rely upon mathematics that is less motivated by concerns with infinite and infinitesimal elements or sets, and more motivated by discrete structures. For now the more traditional geometric interpretation of things is a necessary aspect to how these are derived, but in time these things may be of less importance.

          LC

          Philip,

          A very nice essay. I am also intrigued by the multiplicity of landscape solutions, and the thought that a priori we can only assume that all consistent vacuum solutions have equal probability of being correct. Such indetermination may not be an obstacle for progress though. We can certainly define the maths that use indeterminates and at some level, that might be enough. Curious as to what you think of just letting go of the idea that we will ever know the exact vacuum state.

          Best,

          Harlan

            Dear Sir,

            You have raised some very important questions that can be answered only if we think out of the box. The problem is that we collect lots of data and without proper examination, reject most (like at LHC) that could have given us equally plausible ideas about the natural laws. Secondly, we follow the beaten path without reviewing it and reconciling the apparent contradictions that are being increasingly observed. However, your essay provoked us to expand a few thoughts.

            Even though the viability of the loop quantum gravity is questionable, one of its predicted scenarios is the big bounce. If we replace the big bang with the big bounce, add to it the laws of thermodynamics and some ancient ideas about time, we get a totally different picture.

            Let us start from the last. Time is the ordered interval of events, which are measurement of observables at various coordinates. There is a fixed pattern of all events. These are: being (situation leading to its creation), becoming (its creation itself), (growth due to addition of other particles/events), transformation (as a result), transmutation (due to the same effect - incompatible/excess addition), destruction (change of form as a consequence) to start a new chain. Since galactic blue-shift has been observed putting a question mark to dark energy concepts, let us assume a steady state universe, where everything follows this pattern. Everything is measured/perceived through the radiation it emits - thus, through thermodynamic processes. Condition of maximum entropy is the final stage of the cycle. Then, in the Universal scale, big bounce will be the beginning of a cycle. At that stage, it will be only creation through redistribution. There is the universal space and universal energy, but no one to perceive or measure. The one energy is all pervasive. The emergent energies can be different, local or unknown. Structure formation being an event, must have followed the beginning of the cycle. Since space is the base and interval of structures, space as we know it, must be an emergent property after time. But how did it all start?

            If you look at motion and action, you will find that action is momentary, but it creates a pair of equal and oppositely directed inertia that create local disturbances to create composite and differential inertia that tends to restore equilibrium in a multiple reaction mode. On the other hand, motion is mechanical - it perpetually responds to density fluctuation in all sorts of manners: energy, material density, air density, charge density, etc, created by all sorts of manners including heat (electric), cold (magnetic), etc. Anything subject to strong interaction has the capacity of confining motion. It generates inertia that also acts mechanically till local equilibrium is restored (weak interaction). This is followed by redistribution (electromagnetic). But action is different. It is induced by a conscious agent that breaks the stability or equilibrium. Thus, at the creation event, inherent instability of the conscious system of the universal observer starts the process by creating a perturbation. Some may question this as religious belief. But can quantum theory survive without observer?

            We have a fully developed theory that explains many things. In our essay, we have discussed the Wigner's view of unreasonable effectiveness of mathematics and Gödel's incompleteness theorems as well as Einstein's formulations.

            Regards,

            basudeba

            Philip,

            Thanks, for sharing your ideas. It is nice to see so many different points of views regarding this topic.

            Best Regards,

            -D.C.Adams

            Professor Gibbs,

            The new paradigm shift (from one Universe to multiple Universes) is hard even on Western thought. At first I was reticent but then... . It is actually amazing when one thinks about it compared to the concept of the one singular Universe our egocentric minds have somewhat logically been led to believe in (hard to let go). There are hints to the Meta-Laws. Frank Wilczsek stated something to the order that why are the gauge coupling forces unifying (GUT except gravity) at one point in our Universe (I am assuming SUSY makes the unification more exact)? Thinking about this, is it probable that most successful Universes has this form of unification of gauge forces more or less in some tolerant area near a unifying energy (Planck energy type) with gravity? This cannot be attributed to coincidence. You stated in your essay, "It is known that the combination of quantum theory and general relativity imposes tough constraints on the possible range of consistent space-time models." Also, the GUT point being (how much?)variable with other Universe formations this does rule out 'fine tuning" and maybe one should not look at just one number (say the Higgs boson mass) and say it has some 'unnaturalness' because it looks random and especially 'fine tuned'. It is best just to look at the GUT points in any formed successful Universe that has a 4D space-time. It may be hard to trust just a few numbers whereas the GUT points are more of a gestalt of what is going on. Why 4D? It is well know that 4D manifolds are the most interesting manifolds/topologies in mathematics. Even more so than higher dimension topologies. Recently, the Triangulation Conjecture was disproven. Coverings of simplices (triangles or tetrahedrons) cannot completely cover higher dimension topologies (past the 4D) based on some simple rules. This leaves higher dimension coverings 'foamy' or full of holes. Makes one wonder whether this means that higher dimensional Universes can exist in a 'physical' sense. I am not sure that 'all solutions exists'. It is bound to be super variegated (though with 57 varieties ;)). I am thinking that one should not even consider 'fine tuning' anymore but to look at the tolerance or range of solutions in the hierarchal space as a mathematical structure that can eventually be computed in a Scientific framework. All other Universe that have the 4D space-times would somehow have a computational relation to each other (a dictionary?). The GUT point would also be a measure of how fast nuclear rates (and chemistries) and gravity combine to create the Universe. In some Universes (even if 4D) that stars would burn too fast or gravitational collapse only produces black holes where life could not have enough time to form or exist. Or the GUT point is such that stars cannot ignite and that Universe is a cold dead world with not much going on. Where there is no GUT point for a Universe it perhaps collapses to nothing. I think a lot of people misunderstand the Multiverse especially when it is hyped to the point that suggests 'all outcomes are possible' somewhere in a Universe we will never know. And there is no parallel person that is me somewhere else with a different set of beliefs or lifestyle as that is hogwash. I think that the other Universes have some sort of 4D path integral sensibility which produces some similar outcomes but not the same or nearly the same outcomes found in the other Universes. Finally, here is a direct quote from a white paper "Higher-Order Intersections in Low-Dimensional Topology" by Conant, Schneiderman and Teichner, "The Whitney move, sometimes also called the Whitney trick, remains a primary tool for turning algebraic information (counting double points)into geometric information (existence of embeddings). It was successfully used in classification of manifolds of dimension > 4 specifically in Smale's celebrated h-cobordism theorem (implying the Poincare conjecture) and the surgery theory of Kervaire-Milnor-Browder-Novikov-Wall. The failure of the Whitney move in dimension 4 is the main cause that, even today , there is no classification of 4-dimensional manifolds in sight." I quoted this to make the point that there is a lot of work to be done and that perhaps the unreasonable effectiveness of math in physics is really related to the 4D space-time issue of physical-ness.

            Philip,

            Your mathematical description of doing frontier theoretical physics is noted. However I believe that there is a need to change our emphasis from the mathematical development to the physical model development.

            In the past 100 years theoretical physics and cosmology developments have been conducted almost exclusively on a mathematical basis, leading to non-physical objects or processes such as fields, space-time, curvature in space-time, time dilation, length contraction, virtual particles, action at a distance, curled-up dimensions, Entanglement, Dark Energy, Dark Matter....etc. I believe that these abstract mathematical objects are different aspects of one physical model of our universe. Therefore I urge that we devote more efforts on the physical model development.

            Regards,

            Ken Seto

            Sorry for not responding to comments for a while. I will get back to it at some point soon.

            I am what you call an amateur scientist. In 2001 I discovered what I thought something significant. I submitted a paper to the "American Physical Society, Physical Review D", put up a web site, published a book: nothing. I discovered FQXi and thought it great. I have submitted contest entries with the simple hope that someone would review my material. Now, I have found viXra. The purpose of this note is to express extreme thanks for your involvement in that device. Now, I will continue to read your essay.

            Thank you for your well written essay.

            I like your idea showing the symmetry between the complexity of mathematics and physics, going hand by hand. I am not sure they are attracted towards a point of universality. It is an interesting subject.