David. I share your enthusiasm for the monster group. Its relation to physics via the Leech lattice is very intriguing. However, I am looking for a group with one dimension for every degree of freedom in physics. According to our current view of field theory there are several field variables at every point in spacetime, so I require an infinite dimensional Lie algebra with one dimension for each one. This itself is not so outlandish. In gauge theory there is an independent gauge group at each point in spacetime. These generate the gauge symmetry. However the gauge field has four of these at every point so there are not enough for my theory. If the final theory is something like string theory then it takes even more variables to describe the state space, so the symmetry has to be even larger. I don't know anyone who shares a belief in this idea but I bet that when they realise it is right they will say they knew it all along.

The monster group is the largest sporadic finite simple group. It has a huge number of elements, but it is tiny compared to the invariance group of any gauge symmetry. It is really answering a different sort of question to the one I am thinking of. However, the intricacy of its structure is remarkable. I would not be surprised if it has a part to play. In contrast, the groups that I consider most fundamental are a little boring. These are the free groups where you just multiply and invert group elements without imposing any structure apart from associativity

Hi Lawrence. Yes the Lebesgue universal covering problem is in the same class as the moving sofa problem. They are both geometric optimisation problems like a minimax problem in game theory. The interest is in that simple questions lead to complex, but comprehensible answers. It seems to be the nature of the type of problem that this happens. The state of the physical vacuum is also an optimisation problem, but we don't know the question we have to answer yet. What we can say is that the complexity of the vacuum could emerge from a much simpler starting point.

I am sure you are aware that the problem of classifying pentagonal tessellations with a single (non-regular) pentagonal shape was solved this year Pentagonal Tilings but there are still lots of other problems to solve in this area.

If we are instructed by your essay we may stray from scientific methods. Why should governments and universities fund endeavors if nature is relative to the view of each observer? They can just say "tens of thousands of people have been trained in the sciences and you guys are more confused than you were a hundred years ago". But you say some very important things. There is a structure to consciousness and it draws on information. The information it uses is not always relative. For example most believe that atoms and the quarks they contain are the same every time they are measured. Electronic structure may be probabilistic and complex but it is consistent. I believe you would say that some symmetry causes this. I couldn't tell for sure but you might also believe that the structure of consciousness may cause this consistency. This is a very productive line of thought. I associate what MIT calls the unitary operator 1=exp(iet/H)*exp(-iet/H) with the structure of consciousness. My essay deals with the quark "quantum circles" that this operator describes. The quantum circles can be either information based or real time and energy based. The operator and its quarks are a symmetry. The consciousness that contains quarks, atoms and electronic structure has access to consistent information that it can shape into what it pleases including relative thoughts.

Thank you again for creating viXra (but I notice that you are able to use arxiv). My December 2017 paper vixra: Information and Reality, viXra:1602.0219v2 follows the line of thought above.

Phillip, I posted the above with too much haste. I re-read it and want to clarify that I meant no disrespect to you or your fine, thought provoking essay. I was reacting to only one of your thoughts, not your essay in general. I am sorry if it came across wrong. I don't know if you saw Tyson's concern that we have some leaders who ignore science as "fake news". I was thinking about damage control if there is criticism that science doesn't appear to be converging.

    I was aware of the Bagina results and Mann, McLoud, Von Derau, but not the complete set by Rao. I suppose I should have done more of a heads up on this, but I have this small stack of drawings with calculations that are almost high school level.

    This seems in some ways a bit similar to the covering problem. If I want to tessellate a plane with pentagons I have to deform them. If I have a dodecahedron I can form a plane by sending a point to infinity and the pentagons are deformed and I have an icosian. This gives metric data near the origin, but far out there is little data. I might think of then piecing icosians together to create a regular pentagonal tessellation of the plane, What then are the deformations necessary? I have to perform transformations on the pentagons, and what are transformations or deformations are required?

    The vacuum is I think a sort of quantum time crystal. Wilzcek worked this up, where there is a periodicity of a discrete system in not just space but time. I make mention of this in my paper that will be coming soon. It is a part of my question with respect to tessellation of space with pentagons.

    Cheers LC

    Hi Gene, thanks for your comments. I have not been following the latest from Tyson, but I did submit an essay to the Global Challenge New Shape essay contest about how we could deal with these kinds of problems. The last thing we need to tackle fake news is fact checking organisation or any brake on free speech on the internet. I think peer review needs to be more open, not less open, but that is another subject. Nothing I have written is meant to be anti-science. When I say that reality is relative to the observer I am not saying we should accept alt-facts. I am talking about observers in different universes.

    You say that I have access to arXiv but actually my access is very limited. I can only post to a small number of categories and my papers have often been moved to different ones. All my submissions are held for moderation. It was better in the past but now I prefer to submit my work to only viXra and researchgate. I have not submitted to arXiv for nearly four years

    I noted your comment to Gene Barbee's comment in Scott Gordon's essay.

    Yes. We independents do it for our amusement. But, if one of us does get the beginning of a TOE a breakthrough, society and science will not know it. Or, will they?

    A breakthrough means a new paradigm which the money people (the powers that be in science funding) wold consider it a challenge to their authority. So, society suffers because an advance means the society gains.

    How does society become aware of the new paradigm? There are so many out there (just look at viXra) and most have almost no data or solved problems let alone predictions made and found.

    I've been published in peer reviewed journal, on arXiv. But Now my model is just too radical (apparently). I no longer try.

    But the model (STOE) has made predictions that were later found, explained problems standard models consider problems, the STOE suggested 2 experiments that were performed and it rejected wave models of light (a photon model produced diffraction and interference). Well, I still talk about it when I get a chance.

    Hodge

      It is going to be very hard to get people to notice a new theory of fundamental physics from an outsider. Sometimes the sociology lines up and there is some short-lived media attention (think Garrett Lisi or Eric Weinstein) but the main problem is that a worthy theory needs to be very complete to be recognised. It may be that what I or you are saying is correct, but it is hard to see that now. It may become clear much later but even then we will get little credit because they will say that we did not have much influence and that is what counts. There is some truth to the "challenge to authority" claim, but a really clear breakthrough would get past that. Some of the essays here are mathematically sophisticated, but their correctness would need to be overwhelmingly obvious in some way to grab immediate attention.

      I now prefer to work on mathematical problems because if you solve an interesting unsolved problem in mathematics it is much more likely to get noticed and appreciated. In physics I just do he FQXi essays because I find that the questions I get help push my ideas along a little each time

      "... a group with one dimension for every degree of freedom in physics ..." If nature is infinite, then it is plausible to assume that physics has infinitely many degrees of freedom. If nature is finite, then nature might have only 78 degrees of freedom. Consider 3 copies of a model of 26-dimenional bosonic string theory, yielding 78 dimensions of bosonic waves. There might be a boson/fermion duality theorem derivable from Wolfram's cosmological automation. There could be 6 "barks" or "big quarks" each carrying a barkload of 12-dimensions of information, yielding 72 dimensions controlled by Fredkin's 6-phase clock, thus 78 dimensions of fermionic information. Each 12-dimensional barkload might represent 4 dimensions of spacetime, 3 dimensions of linear-momentum density, 3 dimensions of angular-momentum density, 1 dimension of quantum-spin density for matter, and 1 dimension of quantum-spin density for antimatter. By redundant representation of information, it might be possible to derive an 11-dimensional model of M-theory and a 12-dimensional model of F-theory -- the idea is that the interior of the multiverse would be 72-dimensional in terms of "barkload" data, and the measurable universes would all be 71-dimensional and located on the boundary of the multiverse.

      Phil,

      "I expect to find this symmetry in a pregeometric meta-law that transcends spacetime."

      That says it pretty well. Like the shape of a Lotus petal bespeaking the whole form of the opening blossom. In spite of the possibility that not even the universe always works perfectly. Merry Christmas and a Happier New Year. jrc

        quote

        The biggest difficulty faced by theoretical physicists of this generation is that positive experimental

        input on physics beyond the standard models is very hard to come by. That situation could change or

        it could continue for much longer. Without empirical data how is it possible to tell if the answer is

        string theory, loop quantum gravity, non-commutative geometry or something else? The theorists

        can still progress by working with the few clues they have, but success will depend on guessing

        correctly the answer to questions like 'what is "fundamental"?' If they don't know then they must be

        prepared to consider different philosophical options, letting the mathematics guide the way until the

        experimental outlook improves. If young researchers are all corralled into one pen it could turn out

        to be in the wrong place. The chances are they are going to be influenced only by the highest profile

        physicists. If those leaders say that symmetry is unimportant because it is emergent or that

        geometry is more fundamental than algebra, other possibilities may be neglected. It appears to me

        that there is a clear program that would combine the ideas of algebraic geometry with quantum field

        theory. It just requires mathematicians and physicists to bring their knowledge together.

        You nailed it !!!!

          Dear Phillip,

          I knew there's a reason I always prioritize reading your contributions to these contests. Excellent work, and you certainly succeeded in your aim of provoking the readers' minds.

          I particularly like this sort of theory-independent view in terms of events: whatever the fundamental theory may turn out to be, it has to have events within it in some form, be those worldlines crossing, particles of all conceivable kinds interacting, string splittings or whatever. So let's not worry about those details for the moment, but rather, think in terms of those events, and the stories that can be told with them.

          One tiny bit of criticism I have is that there's many deep and possibly controversial ideas that aren't developed in the way they deserve (although that is likely owed to the length restrictions, and I'm also aware that this is a criticism you could probably lob straight back at my own essay if/when it gets posted). In particular on topics where I perceive some confluence with my own thinking---like the relative nature of reality, or the idea that in terms of information, 'nothing' and 'everything' are really the same---I would have liked more discussion, just to see how somebody like you develops these notions.

          But these are the complaints of one having been hooked by your ideas, and now finding themselves jonesing for more. Which, as you said, is really all you intended with this essay.

          All the algebraic stuff has re-awoken that curious sense that if you could just take one further step back, you'd just see the big picture pop out. There are so many tantalizing hints and connections, it's hard to believe that there isn't some fundamental story to be told in these terms.

          But I think that's for someone smarter than me to discover. I may get back to meddling with this some day (although my love of the octonions means that I'm skeptical of requiring associativity---alternativity is really all you need!), but for now, I'll concentrate on other matters.

            "The mind itself is not fundamental. Neither are the biological processes by which it works, but the principles of information by which it functions are"

            The central principle of Shannon's Information Theory is that, in order to reduce the length of any transmitted message, to the least possible number of encoded bits, it is imperative that the transmitter never send anything that the receiver already knows. For example, I don't need to keep telling you your name. But everything that you can predict, is a subset of the things you know. It follows, that everything that you can predict, is not even considered to be information, in Shannon's theory. That fundamental reality is enough to make most physicists apoplectic. They are searching for the truth, but as the movie said "You want the truth, you can't handle the truth." Because the truth is, the information content of most physical processes lies almost entirely within the unknown initial conditions, required to solve the equations of mathematical physics, not the long-sought equations themselves. This is what "emergence", emerges from.

            Rob McEachern

              I hope you do get round to submitting an essay this year. There is some overlap between our philosophies which helps me find ways to expand my own viewpoint.

              There are two sides to my essay, the philosophical and the mathematical. On the philosophical side it is partly about finding the right words to express ideas in a way that makes them sound reasonable. I think in terms of a high degree of emergence, so fundamentals must take us away from anything we know in conventional physics. This is bound to be ambitious and speculative to a high degree, but structures like space and time and particles have properties that are too specific to be fundamental in my opinion. Information, events are the relationships between them are much more generic. I think you have a similar view.

              The mathematical side is more important of course. Without mathematics to interpret the philosophy there is no end point. I have some mathematical ability in problem solving and algorithms but the more abstract ideas needed to develop these ideas are outside my comfort zone. I feel like an art critic who can appreciate what is good and can talk about how things should be, but without actually having enough skills and creativity to do it myself.

              I agree that non-associativity is likely to play a part. I see octonions as just one algebraic structure with some nice properties that plays some role in certain possible solutions with good properties. The starting point must be something much more general like free universal algebra or higher category theories. Simple categories are associative, but with higher categories it is more natural to relax and weaken the structure to allow more interesting properties. The identities that define associativity are replaced with isomorphisms. Symmetry always arises as a useful tool in any algebraic structure. For example, if you want to understand octonions you will certainly want to know its automorphism group, and then there are the exceptional Lie algebras up to E8 that are also related to octonions. I think symmetry has to be generalised to supersymmetry, quantum groups, n-groups etc, so there is a long way to go.

              The free Lie-algebra that I discuss in the essay is just a starting point that is simple enough to illustrate my point. It provides the important mapping from algebra to geometric structures using iterated integrals along paths. I suspect that there are generalisations of this where iterated integrals map more general alegbras onto branching networks, like Feynman diagrams. I don't know if I will ever get my mind round it well enough to formulate something that works.

              Philip Gibbs

              I thank you for this interesting article. I have been provoked in my thinking. We should, as you say, regard science as finding better, and better, approximations. You are also right when stating information as fundamental in the field of physics. It is dangerous to listen to only one guru, as you say.

              Best regards ___________________ John-Erik Persson

              Good luck.

                Rob, you are right to highlight the principle of redundant information. Imagine you wanted to send some information into space to tell any aliens something about us. You might send a bitmap photo image for example. To keep the transmission short you could compress the data, but the aliens would not have the decompression algorithm. When data is maximally compressed it becomes a stream of random bits that is impossible to decode without the algorithm. You could send send the algorithm in some uncompressed from, but that is adding extra information. The point is that fully compressed data without redundancy is incomprehensible.

                The information that described the state of the universe is holographic, so it can be represented on a surface. This is the compressed form of the data. What we observe in the bulk volume is an uncompressed form with lots of redundancy in the form of gauge symmetry. In this form it is comprehensible to us. we observe and understand the universe in its expanded version, not the compressed holographic form.

                Phillip & Andrew,

                There is an implicit assumption when depending upon mathematics "to guide the way" for new directions in physics. That assumption is that our current mathematics is adequate to the tasks we attempt to use it for. If it is not, then we will find it very difficult to make much progress. Mathematics likely suffers from the same effect as you describe for physics - the pen and corral situation.

                I will suggest that this is actually the problem physics, which tends to lead other scientific disciplines so all of science, is faced with: The mathematical tools we currently have are not adequate to the task science has put to it.

                The limitations of our mathematical tools might actually be keeping us from seeing aspects of our universe, which would be even more reason to consider fundamental reviews of mathematics and its limitations (especially on how it is applied).

                I believe we will find a guide to a new direction this way.

                Don

                Phillip & Robert,

                There is an interesting assumption in information theory - that there is a limit to what can be compressed or represented by a 'unit' of information. There might be a limit, given today's mathematics, but will that always be the case?

                How efficiently can I represent pi? Using decimal notation, it is an infinite non-repeating sequence. If I use pi as the base of the numeric system, then pi is 1 - possibly a tremendous compression of information, although not without its problems for other values. What if a new numeric system, that used different bases in the same representation of a number were found - might this supplant our current system?

                If context and perspective can make such a difference in the presentation of information, can we be sure that the limitations of our current representational structures will not be radically altered in the future? Is a positional numeric system the optimal way to present the value of pi? Like optimization concerns in general, there might not always be an optimal solution. This could suggest there is no limit to what can be represented as (a unit of) information.

                This also appears to be the implicit assumption of any final Unification Theory - that there is an optimal way (usually assumed to be mathematical) to characterize all phenomena in the universe. If mathematics cannot present an optimal solution then likely neither can physics.

                Don