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  • Post #19

Jonathan, thank you for an enjoyable and enlightening read. I'd like to take your provocative opening sentence seriously and see where it leads. If we had to formalize the operations of arithmetic for Roman numerals, we might develop a set of rules and conditions for letter incrementing/decrementing, substitution, and so on. Even something as basic as multiplication might be so onerous it could only be feasibly done by machine. Then algebra and higher-order functions are needed, and the sets of rules and conditions become still more astronomically complex. To say nothing of what would be required when we need calculus, diff eq's, complex numbers, and so on. We wouldn't even be able to solve for X, because X is already taken by a number! (That's an attempt at humor.) In our bizarro-Roman world, we'd recognize that the math required to calculate the trajectory of a cannonball actually does work, and we might therefore be tempted to say that this math describes how the world functions, that the rules and conditions we've discovered dictate how the particles of the universe behave -- although in reality, calculations with Arabic or binary numerals would be equally effective, only much simpler.

Perhaps we're in a version of this Roman mathematical nightmare today. In our world, there could be a completely undiscovered mathematics that is equally effective at doing for us what higher algebras can do, only in a simpler and more holistic or fundamental manner. In other words the complexity of our higher maths might emerge from the lower maths, in the same way that the vast complexity of the rules and conditions for solving a Roman-numerals differential equation would emerge from the rules and conditions for doing arithmetic in that system. I don't believe we're in a position yet to rule that out.

Of course no one knows what this ultimate fundamental mathematics might be. But the higher maths remind me of the pre-Copernican epicycles, which were useful for human calculations even though they are demonstrably not features of nature. The added entities of our higher maths may only *seem* necessary for our calculations because we don't have the tools to perform the same operations any other way. Applying Occam's razor, we should demand evidence that these maths really are features of the world (and not just useful/predictive to humans), evidence that is commensurate with their complexity, while at the same time looking for a fundamental mathematics. A deeper math could make our higher maths not only unnecessary, it may also lead to discoveries about the evolution mechanisms of life/intentionality/consciousness and so on -- in the way that Copernicus' model led to discoveries about gravitation, which might have been impossible had we insisted on sticking with the Ptolemaic system.

Thanks again for the stimulating essay -- I wish I'd had the opportunity to think about these questions for the Trick or Truth contest, which I sat out. Best of luck!

Karl Coryat

    A cogent argument, Jonathan, but somehow I can't see that Math makes the emergence of the universe and life inevitable. Math certainly helps the dynamics of life be more understandable through modelling and such (which I have done), especially if you are conversant in math, and maybe that's my problem. I can see that nature, for example, wings, flower petals and leave, are mathematically expressed origami-style in their patterns of development. I too worked in an octal system of computers years ago, the Univac 1100, but I can't see that math tells the universe what to create beyond certain mathematical laws of entropy and Dr. England's theory, which I use as an example.

    I am impressed with your presentation, though.

    Jim Hoover

      Johnathan,

      Liked your essay very much. Was wondering how you were going to fit octonions into the mindless, wandering theme the essay subject seemed to imply is how things are, happy to read your rejection of such a notion. It certainly alienated me from participating.

      One does not have to throw out multiplicative commutation and associative laws with octonion algebra. Within its rules are real number math, associative and commutative math in any of the complex subalgebras, generally non-commutative but associative math of the quaternion subalgebras, and generally non-commutative non-associative math of the full octonion algebra. If something described by octonion algebra needs commutative math and/or associative math it is available, if not then this is also available. Octonion algebra tells us where each fits in, and more importantly how these disparate characteristics play well together in the greater whole of physics.

      Back on the IMHO strange notion of wandering towards a goal, or emergence for that matter, to me Octonion Algebra is a well lit highway with numerous road signs that too few people have learned the language used such that they can appreciate their directive nature. I did not title my 2012 fqxi essay "The Algebra of Everything" just because it was a catchy title. It is no accident all of the differential forms in the Octonion equivalent of the stress-energy-momentum tensor are the full complement of my so called "Octonion Algebraic Invariants", nor that Electrodynamics is just a subset of the presentation. Nor is it a coincidence the Lorentz transformation form Electrodynamics requires falls out from the requirement both differentials of any field component transform in kind. The only ones doing aimless wondering towards a goal are the people that can't or refuse to read the road signs.

      Rick

        Dear Jonathan,

        Very interesting, in-depth essay and important findings to research ways to overcome the total crisis of understanding in the fundamental science:

        "Roman numerals carved in stone represent unchanging quantities well, but viewing Mathematics in that way is a mistake, because Math is about how unchanging attributes and quantities come to be that way through a process. Seeing Math as dry - as though it was mindless and lifeless - is the real problem, and the mystery of where evolution comes from will disappear when we realize what Math is at its root, a systematic exploration of features characterizing the laws by which form evolves.

        The unchanging quantities of Math itself include figures like the Mandelbrot Set, E8 and the other exceptional groups, as well as other mathematical invariants. While one could argue that humans constructed these things; it can also be said they were only discovered or always existed - even before the universe had its birth. So it is with even higher orders and levels of Mathematics we have not discovered yet, which the universe is already putting to use. But we are fortunate, at this juncture, to be equipped to learn how Math gives rise to life, in order to foster the evolution of consciousness."

        I believe that the solution to the "hard problem of consciousness" is possible after solving the super hard problem of the foundations of mathematics (knowledge). Mathematicians, physicists and poets should have a single, full of life's meaning, picture of the world. I invite you to read and evaluate my ideas.

        Yours faithfully,

        Vladimir

          Hi Jonathan,

          You present an excellent portrayal of the dynamism of mathematics. I found the emphasis on non-associativity particularly engaging since I bump into that in my unfinished essay about a quaternion derivative. For me, your essay is kind of inspirational, in that there seems to be growing interest in understanding the origin of the non-associative Elephants.

          Colin

            Thanks greatly Karl..

            You write "Perhaps we're in a version of this Roman mathematical nightmare today." and I think it is sadly true. Too many people treat lessons in Math and Physics as concepts etched in stone, rather than something dynamic that becomes more clearly defined as we learn more about things and how they work. I agree that what we are looking for is a deeper Math, but it would appear that we have to be willing to go higher in order to see farther.

            It's rather like Rick says in his comment below, that once you arrive at the octonions - they tell you what to do (so long as you follow the road signs). Michael Goodband illustrated that concept in a comment during a previous contest, showing how the information in John Baez article "The Octonions" could be used to derive the Standard Model particles - in simple and direct steps. What we don't know is how nature did it, but we do know the mathematical infrastructure is already there.

            Thanks again for your comment.

            Regards,

            Jonathan

            I here append Michael's comment..

            Author Michael James Goodband replied on Dec. 7, 2012 @ 16:44 GMT

            I just want to point out that a (S0, S1, S3, S7) universe is not just a proposal. I claim that it is the conclusion when we follow the spirit of Rick's standard of letting the algebras do the talking. However, Rick's choice of listening to the octonions violates the meta-principle of "make no preference". Instead, we should listen to all the normed division algebras R, C, H, O and let the octonions tell us their role. Now as a physicist, I want some experimental facts, but I will only need the Standard Model table for left-handed particles and the mismatch with their right-handed counterparts. table

            I claim that you can read-off the structure of the reality in which these particles exist from the properties of the normed division algebras given in John Baez's paper - attached for easy reference. I really do mean read-off, as in number of calculations = 0. The first things to note are the left and right-handed spinor irreducible representations in 4 (quaternions) and 8 dimensions (octonions) in Table 4 (p161 or pg17 of PDF), and the normed trialities of irreducible representations that give the quaternions and octonions on p162 or pg18 of PDF. Specifically note the triality of the irreducible representations V_8, S_8, S-_8 and their relationship to the Dynkin diagram D4 (on p163 or pg19 of PDF) of Spin(8) described on p162 (pg18 of PDF).

            Now count up the number of particles in the SM table: in each column there are 2 quarks with 3 colours plus a lepton and lepton neutrino, giving 8. This matches the triality of the 8-dim irreducible representations V_8, S_8, S-_8 and so the octonions tells us that they are about the different particle charges: octonions=particle space, just as quatenions=real space. Returning to Table 4, we see that left and right handed spinors only occur for the octonions and quaternions. Now to get the 8-dim octonion spinors to have the spatial handedness of quaternion spinors - and so match the chirality of the SM particle table - the octonion space would need to be mapped to the quaternion space so as to acquire their spatial handedness. This requires first picking out a 3-vector from the octonion space - which is something that has to be done in order to define the cross-product in 7D space residing in the octonions - and then map it to spatial 3-space. This mapping of a 3-vector in the octonion space to 3-space gives the 3 8-dim irreducible representations a spatial chirality, and breaks the symmetry of the octonion space. Thus the normed division algebras have just told us what the Higgs field is really all about.

            Returning to the D4 Dynkin diagram on p163 (pg19 of PDF) to consider what the symmetry breaking for the SM particles must be, using the Dynkin diagrams: SU(2) is 1 node; SU(3) is 2 linked nodes; and SU(4) is 3 linked nodes. If we imagine breaking all the links of D4, we would have a central SU(2) and triality involving an outer SU(2) ~Spin(3), plus a U(1) symmetry between the central and outer SU(2) groups. Now symmetry breaking in a Dynkin diagram involves removing a node, which here would be the central SU(2), leaving intact the triality over the outer SU(2)~Spin(3) and the U(1) symmetry. A 3-dim colour representation - needed to get particles in the 8-dim of V_8, S_8, S-_8 - selects Spin(3) over SU(2), giving the symmetry breaking encoded in the SM particle table as:

            Spin(3)*SU(2)*U(1) -> Spin(3)*U(1)

            The normed division algebras are telling us that the colour group cannot be SU(3), but is actually Spin(3). The condition for getting particle-like objects in any symmetry breaking pattern demands closed spaces, which here means S7 -> S3*S3*S1 and the monopole homotopy group PI6(S2) = Z4*Z3 (S6 being left after the unbroken S1 is put to one side) confirms the triality of 3 families of 4 particles in the SM table. The homotopy group PI7(S3) =Z2 also confirms the chirality of the mapping from the closed octonion space (particle space S7) to the closed quaternion space (closed spatial universe S3). These homotopy groups obviously come from the structure of the corresponding algebras.

            Given the SM particle table, the normed division algebras are screaming out the structure of reality to anyone who is listening. Arguing against this conclusion is not arguing against me, but arguing against the fundamental fermions and the structure of the normed division algebras - that's an argument that's lost before it's even begun. Structure of reality: DONE.

            That's the easy way of how we arrive at what the structure of reality is, now how did the universe arrive at the answer?

            MichaelAttachment #1: baezocto.pdf

            I really appreciate this comment Rick..

            Since I wholeheartedly agree with almost every point you make verbatim, I shall let your words stand as evidence that I backed the right horse (or elephant as the case might be). I see the octonions as doing all of the heavy lifting that gets the ball rolling in the early universe. Since all of the exceptional Lie groups are of the octonion lineage (according to the Cartan-Killing classification), it would be hard for nature to create the array of forms we observe without them.

            So indeed; I'm with you brother. The octonions are a well-lit road and the only ones who see them as aimless are those who have not taken the time to learn enough about them. My conversation with Tevian was a reality check, because I could hardly believe so many really smart people could ignore the need to employ non-associative Maths to crack the deeper problems, or the loud and clear message of the octonions. When he said "I agree with everything you've said up to this point; what's your question?" I knew for sure that I'm not just imagining things.

            All the Best,

            Jonathan

            Many thanks Vladimir,

            I agree with your assessment. The 'hard problem' of consciousness is more easily solved when we do understand the evolutive properties in Math that provide a clear basis for such things. As you implied; Math is about knowledge, or how we can know things, so the two go hand in hand.

            All the Best,

            Jonathan

            This is most appreciated!

            I greatly respect your work Colin, and given your areas of expertise I really value the compliment. To learn that I have inspired someone like yourself is about the best reward I can have. I think there is indeed growing interest in non-associative Maths, and I hope my shining a light on the elephants will wake a few people up, or speed up the process of our collective awakening.

            Who knows? Maybe my curiosity at GR21 got Tevian thinking and he has something really fun cooking. I asked some pointed questions of Gerard 't Hooft at FFP10 and then found a few extra slides in his presentation at FFP11 - addressing the concerns I raised about Lorentz invariance. To a degree; what's needed is for enough of the right questions to be asked, so the experts are forced to think more deeply about certain things that would otherwise escape notice.

            All the Best,

            Jonathan

            Thanks a lot Jim!

            Intriguing comments. I have imagined that origami artists could assist the folks working in Causal Dynamical Triangulations to crack certain standing problems in quantum gravity. But I also read recently that some folks are using Julia sets to create 3-d forms origami style, guided by the mathematical symmetries and scaling rules inherent in such forms. So I think there is a connection.

            My model for entropy is the Mandelbrot Set, which reproduces the Verhulst dynamic along the real axis - when the Real-valued Mandelbrot formula is iterated with a random seed. But each place where the Set folds back on itself becomes a bifurcation point on the adjoining diagram. I first saw this in Peitgen and Richter "The Beauty of Fractals" after it was pointed out by Michel Planat.

            Thanks again!

            All the Best,

            Jonathan

            Yes I agree, and Thank You Lawrence!

            I wanted to avoid giving people Math overload, while teaching them about the benefits of higher-order Maths. I am a real fan of the octonions, where the more I learn about them, the more useful applications I find. I have attached the Baez paper to a comment further down, in reply to Anonymous - who is actually Karl Coryat.

            I started reading your marvelous paper, but I'm also trying to work through the whole stack in chronological order - at least 3 or 4 papers a day. I will make sure I give you a review right away, and I'll likely boost your score a bit with my rating since you merit it.

            All the Best,

            Jonathan

            Invisible trolls would be my guess..

            It appears that some people accord points only to those who agree with their premise, and we are among the ones who do not. I hope you have good luck otherwise, or anyway.

            All the Best,

            Jonathan

            Your notion of maths guiding reality rings true. The octonion algebra is new to me, but seems really complex. It would seem that one can always explain reality by adding hidden dimensions and multiverses and octonions.

            But a simpler reality of just two complex dimensions seems to have all the complexity needed to explain the universe. Isn't simpler better?

              I am putting my response you wrote in my area here. I noticed you had not gotten to it or responded yet. I have have been getting caught up with work after having spent the week down with the flu.

              Thanks for the positive response. I will comment more tomorrow. It is getting a bit late for a long writing session. This is in line with the approach with Raamsdonk that spacetime is built from entanglements. I wrote an answer on stack exchange that connects with this perspective with regards to Hawking radiation.

              The open world emerges from the existence of gauge hair and BPS charge. The hair of the black hole is entangled with particles in a vast number of other black holes in the universe. In the unique situation where there are two black holes maximally entangled one would have a complete Einstein-Rosen bridge connection to the interior of the other black hole in this other world. The openness comes from the fact the spatial surface in region I has an ambiguity with respect to being connected to other cosmology or the black hole interior region. For maximal entangled Bhs one in principle can avoid the singularity and travel around to other worlds.

              I will write more tomorrow. I have been recovering from influenza, and today it the first day I feel not utterly horrible. I am not that familiar with DGP model, but I will see what I can make of it.

              Cheers LC

              PS --- I also noticed something bizarre happened. I was ranked around 17 or so from the top, but a lot of people were I think tanked with a vote of one. Now I am #4. It looks like you went down with that; too bad. I was not me who did that. I would not give a lot of papers the lowest score like that.

              LC

                The person who did this has to be an essay author. I would suspect they are somebody who wrote an essay that appeared in the group Feb 21.

                LC

                Jonathan,

                Is it possible for there to be a non-integer value for the number of dimensions associated with a space? If so, how?

                Trolls? The trolls in "Frozen" were amusing. The trolls in "Trolls" are amusing. These trolls are just small. That makes it easy to hide.

                Best Regards and Good Luck,

                Gary Simpson

                  Interesting question. I have this idea cooking up about Hausdorff dimensions with Ising chains. This might be a way of working a renormalization scaling for estimating the 3-dim Ising problem.

                  LC

                  The correct answer is always the best one Steve...

                  Effective theories need only consider the parameters they incorporate, or attempt to explain. But this is why GR and QM don't agree at the fringes. Explaining the whole universe at once requires one to work in the area of fundamental theories - which is fundamentally more difficult.

                  All the Best,

                  Jonathan