Thanks for your kind remarks Georgina..

I am happy you grasp my main point. Some folks learn a lot of Math, but don't have the sense of such things, possibly because they are stuck on manipulating symbols or finding equations that are easy to work with, while nature ignores our boundaries of convenience. One knowledgeable friend asserts that many scientists simply don't go far enough with the Math to do the job correctly, or stop short of a rigorous treatment with the solution in sight. Borrowing an idealized equation from a textbook, without considering its limitations, won't always allow you to capture nature as it is.

But I agree that it is a waste of time to invoke the octonions for everyday Physics, while the quaternions have great utility - and do capture the needed dynamism to model evolutive properties. However; when one gets to some of the tough problems at the crux of fundamental Physics, as with quantum gravity where one is dealing with the extreme regions where both gravity and QM have a strong effect, is where knowing about the octonions and non-associativity becomes important. And the 8-d octonions are adequate to hold E8, even though its largest regular extent is in 248-d. One can also make a beautiful Zome model in 3-d, which is a projection of the 8-d figure. Pretty cool, huh?

All the Best,

Jonathan

I don't know yet, where still higher dimensions will fit in..

But I think there is a place for them on the spectrum of possibilities, that nature will employ and project down to three dimensions if that's where it's needed. I mainly imagine that there are important things to know about what happens in the higher heavens of Math, but I don't claim to be able to explain their import. There appears to be a telescoping property that I think is inherited from Bott periodicity, where higher order quantities are extensible, but can be reflected down to lower order forms. So some of what happens there can be shown relevant to what happens here.

All the Best,

Jonathan

Thanks Jonathan.

Yes, I completely agree with you that awakening people to part of what we have inside of the box with us is important in the same way that thinking outside the box. What is important in both of the cases is starting from plausible hypotheses and proceeding with mathematical rigor. Your aphorism on elephants is very nice.

I hope that you will have a chance to read our Essay.

Dear Jonathan Dickau

I saw your comments on the essay of K. Willy and I become curious about both essays.

There are different ways of "decoding" the universe; yours and mine are quite different. I read your essay, which is very well written, and you present well your point. I think that you have a promising approach but it is not my line of reasoning, I can only be a curious spectator of your work.

I read that you are part of viXra team; well, I want to congratulate and thank you. I have a paper in viXra with more then 900 downloads in spite of my almost null divulgation of it (vixra.org/abs/1107.0016) ; I receive emails from cosmologists confirming that I found the solution to the cosmological problem, some urging me to publish it in whatever "citable" place. However, as I am not "affiliated" and that solution implies a change of paradigm, hardly could I ever publish it in a relevant scientific journal. I hope that one day you will see viXra entering the history of science because of that paper that could not be published anywhere else.

From the above, you may assume that I am an "amateur". If you are so kind to take a look at my essay, you will understand that I am something else - in spite of our different approaches. And I have something to ask you: a piece of advice.

All the best

Alfredo Gouveia de Oliveira

    Dear Jonathan,

    thanks for the very interesting essay (with a high vote from my side). Your consideration of octonions keep me interested

    again in this interesting topic. I considered the hypercomplex numbers years ago but forgot

    it.

    Some remarks from topology. There is a topological view on these normed algebras (C, H, O).

    The unit elements are spheres: S^1, S^3 and S^7. Now there is a "topological" proof that

    C,H,O are the only normed algebras. If there is a non-zero vector field on the sphere S^1,

    S^3, S^7 then there is a non-zero element of the algebra which can be reversed (allowing

    division). But one needs more: every element of the algebra (except zero of course) must be of this form. But then you need as much as non-zero independent vector fields like the dimension of the sphere. A deep theorem states now that only the S^1, S^3 and S^7 have this property so that the only normed algebras are C, H and O. (BTW I used this result for a no-go theorem in quantum computing, see my paper.)

    Nonassociated algebras are not so special in math. A large class are the Lie algebras (with the Jacobi identity instead of the associative law). But octonions are more than that. This algebra is deeply connected to the exceptional Lie groups (like E_8, E_7 or G_2). In geometry there are special manifolds with a calibration (a linear form with special properties). There, the octonions appear as G_2 structures on 7-dimensional manifolds. M theory as part of string theory used them.

    But you rather ask for low-dimensional (3D, 4D) counterparts. I'm specialized in this dimensions and here it is: a very large class of 3-dimensional manifolds (arithmetic hyperbolic 3-manifolds) is constructed by using the quaternion algebra and its properties are strongly related to (or determined by) this algbera.

    4-dimensional manifolds (simply-connected i.e. where all closed curves can be contracted to a point) are classified by the intersection form: a bilinear form with the intersection number of the embedded surfaces. This bilinear form is taken over the integers. In the classification of these forms, the so-called E_8 lattice played a fundamental role. But this lattice is nothing else then the integer octonions (Kirmse integers). So, your octonions are directly related to the spacetime (seen as 4D manifold) and you don't need any extra dimensions.

    If you like please have a look into my essay where I used topological methods to understand our brain.

    All the best

    Torsten

      4 days later

      Hi,

      Thank you for an interesting article. A central part of my essay is also about the Octonions. However I think our perspectives are different. But, again thank you for some interesting ideas.

      All the best,

      Noson

        This post is full of useful info Torsten!

        I think you are talking about the fact that the Frobenius conjecture is proved, and is therefore a theorem, in your comment "A deep theorem states now that only the S^1, S^3 and S^7 have this property so that the only normed algebras are C, H and O." I used to freak out when I saw that name, close the book or put down the paper, and then say "that's enough for now," but I have since learned that Frobenius was a friend to my cause all along.

        I will follow up on your essay page.

        All the Best,

        Jonathan

        Thank you Noson..

        After reading your essay; I agree our perspectives are quite different, but our views complement each other well - filling in the gaps of what is left unsaid.

        More later,

        Jonathan

        Your comments are most appreciated Alfredo..

        Differences in perspective often bring clarity to a subject, because people tend to get tunnel-vision over time. Something called the Einstellung effect sets in, where people try to adapt their prior assumptions as little as possible, rather than adopting a superior model when it comes along - even if there is a great leap in predictive capability. Sometimes the compelling utility of existing understanding, or even the great investment in effort required to master a subject (as with String Theory), will blur our vision on better alternatives.

        I am also unaffiliated with any academic institution. But I make an effort to appear at conferences and submit my papers for publication. At this point; I get invitations from time to time, some of my prior publications got me on somebody's list of authors in a certain field. But once that happens; you need to keep yourself in the game by participating - submissions to journals, abstracts to conference organizers, proposals for grants, entries to contests, and so on. Only when the right people know who you are can they advance your cause.

        All the Best,

        Jonathan

        I had a deep theorem of Adams in mind (which spheres have a trivial tangent bundle, answer only S^1, S^3 and S^7)

        Frobenius conjecture is more connected with sporadic groups (with many relations to integer octonions).

        More later

        Torsten

        Dear Jonathan J. Dickau,

        Thank you for your comment and the article of A. D. Sakharov, you sent.

        And I very much appreciate that

        "This is a question I addressed in my first ever Gravity Research Foundation essay, just submitted this week. I also employ the metaphor of a sink drain, though it is not my central thesis"

        If it is possible, I would like to read your essay in Gravity Research Foundation.

        Thank you again,

        With Best Regards,

        Ch.Bayarsaikhan

          Nice to read your essay.... and your proposal about small scales...from my poit of view the String proposal of KK spaces (Universes) of 6D is very interesting and suitables... these 6D could be the floor generators of Our Scale Universe, and that could explain why String Theory is good to explain Our Scale Range... because it is composed by very small 6D branes.... Vacuum will be these 6D KK structure.... but inside these KK 6D spaces, other universes could be...

            Dear Jonathan,

            I am going to disappoint you ! I never felt so handicapped perusing any essay as I felt while reading this. I had to consult and learn the mathematical terms afresh. Took the whole day just doing that. Even then my knowledge fell short to understanding dense / cryptic use of mathematics and their implications. So, I am not in a position to add value with my analysis.

            My general style is to comment on the whole of the essay as a theme, whether it is in line with the idea of this essay contest, but I cannot do that in this case. I am reduced to picking only those statements that I could understand, even if it meant taking them out of the context. Your statements are double quoted before my responses.

            "This lends support to the author's idea that nature employs the totality of all Mathematics - discovered and undiscovered - in its handiwork, such that invariant realities in Math spell out their own importance to Physics, and give rise to the universe we see today."

            Not all of known mathematics is useful in describing the physical reality. Furthermore, it does not appear that mathematical methods can have any kind of limits, they can be discovered ad infinitum, irrespective of their applicability in describing the natural function.

            "In my view; the laws of nature arise largely because the Math has its own ideas about what is relevant to Physics, and also engenders the evolution of form that is capable of consciousness and volition. So in this essay; I will spell out, in some measure, what I think Mathematics is telling the universe to create."

            You seem to be ascribing volition to mathematics, as if it has causal power of its own. Could mathematics function on its own, or if material things are required to carry out the function of applicable mathematics?

            "Nature is not limited by our views about what in Math is relevant to physical law." -- Agreeable.

            "If multiplication depends on the order of the elements being multiplied together and even on how they are grouped, then at one fell swoop, geometry enters the calculation in an organic way. The Principle of Indeterminacy could then arise in a natural fashion from relativistic considerations, making quantum theory a consequence of an underlying 8-dimensional hidden-variable process, very much in the flavor of the theories of de Broglie and Bohm." [3] So we see that octonion Math dictates emergence."

            I read it so many times, yet I could not understand any of the assertions, how they followed from non-associative multiplication. Certainly I am not enlightened enough in maths and physics to judge this essay.

            I am sorry, I cannot evaluate.

            Rajiv

            P.S. Previously, the comment appeared under Anonymous.

              Thanks anyway Rajiv!

              In a way, you have paid me a compliment and shown me I am at a crossroads. I felt the same, the first time I read Alain Connes' paper "Noncommutative Geometry Year 2000" I would get a few pages in, get overloaded, then come back again another day until I could get a little further each time. Much of it seemed utterly incomprehensible, but I eventually grasped a few key concepts - due to sheer repetition. Later I learned that Connes advised budding mathematicians to do exactly that, adding that when his brain became full he would recline for a while and nap or lay in reverie while letting the new ideas sink in.

              I am presenting an idea that is foreign to almost everyone literate in Math, which goes against the grain of some of what we are taught early on, and that only a handful of mathematicians are masterful about. The fact that I see it as a key is only that I have focused so intently on certain points of interest for years. No worries!

              All the Best,

              Jonathan

              Dear Jonathan,

              Interesting and original essay! I share your interest in the strange beasts of higher mathematics, the Monster Group, the Mandelbrot Set, E8, etc., even if I am not convinced that we can recover as much fundamental physics from them that you think can be done. But what I fully agree with, having come up with the same incomprehension with my own Mathematical Universe Hypothesis related scenarios, is that most people have a completely inaccurate conception of what math is! I fully agree with you when you say that

              "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."

              Good luck in the contest! Sincerely,

              Marc

                Thanks greatly Marc,

                I appreciate that you share my opinion about the true nature of Math, because it has become hard for me to see Math as dull and lifeless. Instead; I am continually surprised by discoveries of beauty and hidden order in structures I thought I had understood. I thank you for the thoughtful remarks and a fair rating along with. I have been looking forward to reading your essay, and I'll push it up the queue a bit, to get to it sooner.

                All the Best,

                Jonathan

                Dear Jonathan

                I enjoyed your essay despite the fact that it mostly refers to areas of Maths that I only heard about without learning - quaternions, octonians, E8, and the others you are thankfully comfortable with.

                Because of writing my contribution to last years's fqxi contest I have thought a lot about the intimate relationship of physics and mathematics. I concluded that it is not to be wondered that the human mind, having evolved from organisms that themselves evolved evolved closely on the molecular scale with Nature itself, has the capacity to understand that Nature. Also that mathematics itself is firmly rooted in -at least - the model of the Universe I have concocted: concepts like number, dimensions, rotation, geometry all emerge from the lattice of of an evolving cellular automata.

                For the above reason I felt a little uncomfortable with the autonomous almost living powers you almost give to mathematics, although you are of course right in trusting its continuing importance in physics.

                Vladimir

                  Oh and I forgot to mention that I wrote my entire post whilst enjoying listening to Pete Seeger's "At 89" which you contributed to as "choir, chorus, engineer, mixing, vocals"- great! One wants to know more about your other talents and achievements!

                  V

                    Thank you so much Vladimir!

                    Your comments are most welcome, and I look forward to reading your essay. It is a busy day for me, but I expect to get back to reading later today, and I'll have a full comment then.

                    All the Best,

                    Jonathan