Hopf fibrations: yet another resonant bit of maths associated with these dimensions. I'm afraid I don't know anything about their relationship to entanglement theory. Sounds interesting, but like Sherlock Holmes, who did not know the moon revolved around the earth, that fact being no help in his work as a detective, I am shockingly ignorant about a great deal of contemporary physics ideas.

By the way, these sphere fibrations can be extended to lattice theory. I wrote a paper on that back in the age of dinosaurs, but included the idea in the windmill book too. As to those hypothetical denizens of another universe who may extol the virtues of different maths ideas ... I'm not sure I like them very much, or their universe. :)

As to messing up the thread, my first attempt to use some of the suggested tags failed miserably. I applaud you willingness to tackle it at all.

Dear Geoffrey,

Regarding: "The Boltzmann-Mach debate was a mere two lifetimes ago. Have we evolved in the intervening decades? Uh, no. We have not."

I do agree. That is the reason to return to the study of the achievements of the great physicist. In addition to the mentioned two: Newton, Boskovic, Maxwell, Planck ..

But please do not misunderstanding of Lemeitre, Habble and some modern science promoters. You wrote a really good essay.

Regards,

Branko

Dear Geoffrey Dixon,

You wrote: "The sum total of mathematics at its profoundest is an explanation of why only certain mathematical objects are interesting. The sum total of physics is these objects." All real visible objects have surface.

I have concluded from my deep research that Nature must have devised the only permanent real structure of the Universe obtainable for the real Universe existed for millions of years before man and his finite complex informational systems ever appeared on earth. The real physical Universe consists only of one single unified VISIBLE infinite surface occurring eternally in one single infinite dimension that am always illuminated mostly by finite non-surface light.

Joe Fisher, ORCID ID 0000-0003-3988-8687. Unaffiliated

Professor Dixon,

You provide an enjoyable point of view. No one is truly humble, as each of us is the center of the entire universe. We are born knowing everything and spend our lives learning otherwise.

I would ask though, since I put the idea out in my entry, whether zero is the foundation of math?

The flatline from which all features and qualities expand and to which they coalesce. Otherwise it would seem maths exist as some platonic realm, rather then emergent with the features they map.

I extend that out to the proposition that empty space is the physics equivalent of zero. The vacuum that is the metric of C.

It does seem physics would prefer it to be emergent, from geometry, from the Big Bang, from time, etc, but it keeps sitting there quietly in the background and that would seem to be the quality of being foundational.

I like this offering a lot Geoff...

I am in broad agreement with your statements about the manner in which the primacy of Math applies. One of my past FQXi essays argued that the "Totality of Mathematics Shapes Physics" with a similar notion that the division algebras and other prominent or recurrent patterns in Math are naturally selected as relevant.

My essay has yet to post, but I will be sure to give you a high rating once it does. You aptly address the contest question, going to the heart of several questions about what is really fundamental, given certain pairings. I will probably reread once I do return to this, but you make some things very simple and direct, so there is no ambiguity or complication to speak of or complain about. This means you are also speaking at the appropriate technical level for this audience and contest.

Good luck!

Jonathan

    Once my entry does post...

    You will see that I also appreciate the fact that complex numbers are closer to the source than the reals, or share in your thinking on that, and that I have some appreciation for the quaternions and octonions as well. I hope this contest gives your work some visibility.

    All the Best,

    Jonathan

    Merci beaucoup, mais ...

    Let ∆ be the probability of visibility. Let Ω be the probability that it matters. Let ø be the probability that bananas can be grown on the sun. Then we propose:

    ø = ∆Ω.

    Hmm. I really AM a curmudgeon.

    And now, inorder to post this comment, I have to click the box entitled "I'm not a robot". Let ВҐ be the probability I actually am not a robot; then ВҐ

    O.K. Wait...,

    So bananas grow on the sun? I'm just having fun with you Geoff. But my current essay has posted, if you want to take a look. One point of possible interest is how the Mandelbrot Set recreates Cartan's rolling ball analogy for G2, which I mention is also the automorphism group of the octonions.

    In the meanwhile, enjoy the elevation, however brief. It is well-deserved. You address the topic head-on, and I give you kudos for cogent answers.

    All the Best,

    Jonathan

    Hi Geoffrey Dixon

    It is wonderful to meet you an expert Algebra person, especially complex numbers... Very nice. A great deal of modern theoretical physics rests on complex and imaginary algebras.... Dear Geoffrey Dixon.... I want you to have a look in this paper also, where complex numbers are omitted.............. I highly appreciate your essay and request you please spend some of the valuable time on Dynamic Universe Model also and give your some of the valuable & esteemed guidance

    Some of the Main foundational points of Dynamic Universe Model :

    -No Isotropy

    -No Homogeneity

    -No Space-time continuum

    -Non-uniform density of matter, universe is lumpy

    -No singularities

    -No collisions between bodies

    -No blackholes

    -No warm holes

    -No Bigbang

    -No repulsion between distant Galaxies

    -Non-empty Universe

    -No imaginary or negative time axis

    -No imaginary X, Y, Z axes

    -No differential and Integral Equations mathematically

    -No General Relativity and Model does not reduce to GR on any condition

    -No Creation of matter like Bigbang or steady-state models

    -No many mini Bigbangs

    -No Missing Mass / Dark matter

    -No Dark energy

    -No Bigbang generated CMB detected

    -No Multi-verses

    Here:

    -Accelerating Expanding universe with 33% Blue shifted Galaxies

    -Newton's Gravitation law works everywhere in the same way

    -All bodies dynamically moving

    -All bodies move in dynamic Equilibrium

    -Closed universe model no light or bodies will go away from universe

    -Single Universe no baby universes

    -Time is linear as observed on earth, moving forward only

    -Independent x,y,z coordinate axes and Time axis no interdependencies between axes..

    -UGF (Universal Gravitational Force) calculated on every point-mass

    -Tensors (Linear) used for giving UNIQUE solutions for each time step

    -Uses everyday physics as achievable by engineering

    -21000 linear equations are used in an Excel sheet

    -Computerized calculations uses 16 decimal digit accuracy

    -Data mining and data warehousing techniques are used for data extraction from large amounts of data.

    - Many predictions of Dynamic Universe Model came true....Have a look at

    http://vaksdynamicuniversemodel.blogspot.in/p/blog-page_15.html

    I request you to please have a look at my essay also, and give some of your esteemed criticism for your information........

    Dynamic Universe Model says that the energy in the form of electromagnetic radiation passing grazingly near any gravitating mass changes its in frequency and finally will convert into neutrinos (mass). We all know that there is no experiment or quest in this direction. Energy conversion happens from mass to energy with the famous E=mC2, the other side of this conversion was not thought off. This is a new fundamental prediction by Dynamic Universe Model, a foundational quest in the area of Astrophysics and Cosmology.

    In accordance with Dynamic Universe Model frequency shift happens on both the sides of spectrum when any electromagnetic radiation passes grazingly near gravitating mass. With this new verification, we will open a new frontier that will unlock a way for formation of the basis for continual Nucleosynthesis (continuous formation of elements) in our Universe. Amount of frequency shift will depend on relative velocity difference. All the papers of author can be downloaded from "http://vaksdynamicuniversemodel.blogspot.in/ "

    I request you to please post your reply in my essay also, so that I can get an intimation that you replied

    Best

    =snp

      Hello SNPG

      I am retired, and even when not retired was never drawn to gravity theory, as I never felt there was anything I could contribute to such a crowded field. I am like a person who has heard a good song, and even when it is over can not get it out of my head. But I have been hearing that song in my head for over 30 years (RâŠ--CâŠ--HâŠ--O), and I shall likely still be hearing it on my deathbed.

      I wish you luck. But think about how the world works, how it has always worked, and always will work ("always" means as long as our species is here to muck things up), and you should realize you will need much more than luck. But that's ok, as long as the work gives you joy.

      The last time I looked my banana plantation on the surface of the sun was doing quite well. Haven't you noticed the sun is a bit yellow?

      4 days later

      Dear Fellow Essayists

      This will be my final plea for fair treatment.,

      FQXI is clearly seeking to find out if there is a fundamental REALITY.

      Reliable evidence exists that proves that the surface of the earth was formed millions of years before man and his utterly complex finite informational systems ever appeared on that surface. It logically follows that Nature must have permanently devised the only single physical construct of earth allowable.

      All objects, be they solid, liquid, or vaporous have always had a visible surface. This is because the real Universe must consist only of one single unified VISIBLE infinite surface occurring eternally in one single infinite dimension that am always illuminated mostly by finite non-surface light.

      Only the truth can set you free.

      Joe Fisher, Realist

      Geoffrey,

      I enjoyed your essay. This is because I read your https://arxiv.org/abs/1407.4818 paper some years ago. I do have a few questions. In particular is the 128 dimensional T^2 hyperspinor space the same as the E8/SO(16) = 128? The other is that I have written notes or a pre-paper on some work with the Jordan J^3(O). This is I think more general than the Leech lattice, or embeds the Leech lattice. I was wondering if you have done any work on this and automorphism of the FS "monster group."

      I have pondered how it is that spin ½ leads to FD statistics. I have found myself thinking exactly what Feynman responded with, "I can't do it." It does seem plausible that because BE statistics integrates 1/(e^{-Eβ} - 1) into ζ-functions. The FD statistics 1/(e^{-Eβ} + 1) can be thought of as related to the BE with the general form 1/(e^{-Eβ} + e^{iθ}) for θ a phase angle. This is a bit like anionic statistics. It seems in a way this involves some deep relationship with the Riemann zeta function.

      The motivation by mathematics can at times be compelling. I have some resonance with Dirac's call to seek beauty. It is though not clear to me whether mathematics is more fundamental than physics. There was a time when I thought this might be the case. Then as time goes on this seemed difficult to uphold, while on the flip side it appears to be a collapse of objectivity to just assume mathematics is a sort of game or human invention. I am at a stage where I have not the faintest idea what the deep relationship between mathematics and physics is.

      Cheers LC

        Ah yes...

        The bananas must be thriving.

        Warm Regards,

        Jonathan

        Hello again Geoff,

        I wanted to thank you for the thoughtful 'play by play' review of my essay. It is amazing what one can learn seeing what you have written through someone else's eyes. I especially appreciate your catching me on the dodgy usage of the word 'likely' which has no place in academic writing, where the goal is to be crystal clear and mathematically precise.

        As for the Mandelbrot Set; it was the horse I rode in on, as it were. I had a few phone conversations with Ben Mandelbrot more than 30 years ago that greatly encouraged and shaped my learning. The last time we spoke; he called me out of the blue on an Easter Sunday morning. I had worked until midnight, the night before, and decided to sleep in rather than attending a church service - but I got lucky and talked to Ben instead.

        Since then; I've found out M doesn't stand alone, but connects with a number of other mathematical objects - so it was a good place to start me going in a worthwhile direction. I'm glad to have had your musings to refer to, as well, because there does appear to be a special significance to T. As I recall; the Sedenion sphere S15 only has three possible fibrations, S1, S3. & S7 - yielding the C, H, and O algebras. This would seem to indicate they are a foundational trio.

        I thought the comment above was priceless, and I'm very glad to see that your banana plantation is thriving!

        All the Best,

        Jonathan

        "In particular is the 128 dimensional T^2 hyperspinor space the same as the E8/SO(16) = 128?"

        I don't have any reason to believe it is or isn't. Tony Smith thinks so, but I am unwilling to lead the maths. I prefer to be led, even if led astray. At least it won't be my fault.

        "Jordan J^3(O). This is I think more general than the Leech lattice, or embeds the Leech lattice."

        Wilson, at U of London, and I have independently represented the Leech lattice over O3. Anyone with any interest in J^3(O) will naturally point out that there is a nice copy of O3 in J^3(O). I don't know if this is meaningful. If one could show that there was a bilinear of trilinear multiplication on the Leech lattice as a subset of J^3(O) that closed on this subset, then I personally would be hugely interested. But yes, one can stick it into J^3(O), but without some further interesting structure ...

        "I was wondering if you have done any work on this and automorphism of the FS "monster group.""

        No. I of course find the monster group enticing, because it is exceptional, but I haven't got around to looking into it.

        As to your paragraph 2, this leads into the quantum quagmire. If you can find two people who agree on much of anything in that quagmire, let me know. Meanwhile, the level of disagreement means to me that we are not ready to understand at a deep level.

        "... it appears to be a collapse of objectivity to just assume mathematics is a sort of game or human invention. I am at a stage where I have not the faintest idea what the deep relationship between mathematics and physics is."

        This is a hard one, because one has to define what one means by mathematics. I think of it as a thing that is there even in the absence of any intelligent life or any conception of consciousness. Many will likely give this idea a label named after a dead Greek and think they understand. Maybe they do. Anyway, in the presence of human intelligence we have this notion of Ur-mathematics, and we have the human symbols and formalism that is our lens onto this world. So, ignoring the invented formalism, I think of physics - I'm just making this up, but it sounds good to me - anyway, physics crystallizes on the exceptional, generative, and resonant bits of this Ur-maths, for only there is there a structure rich enough to nurture it. And to give rise to us, for what that's worth.

          I replied to this below, but may have failed to make it a reply, as opposed to a comment.

          And again, I check "I'm not a robot", but I likely am.

          Hello Geoffrey,

          I liked the way your essay is written, mainly the introductory part. I believe your essay makes much sense, no one can reach an agreement for fundamental. I also believe that mathematics is more fundamental than physics which I have written very descriptively on my essay.

          I gave a good rating to your essay because I find it unique and interesting and I hope you'd enjoy mine too.

          Kind Regards

          Ajay Pokharel

          Geoffrey Dixon

          Dear Geoffrey,

          I rarely feel so lucky and pleased to read something with which I agree almost completely. Your essay gave me this high satisfaction, so thank you. There may be two points where we slightly diverge, but only as a matter of preference. The first one is that I may be a bit biased towards geometry and consider complex, quaternionic, and octonionic structures as living on real vector spaces. The second one follows from the first one, since the compositions of transformations preserving structures is associative. I fully agree with the paramount role of spinors, but consequently I tend to see them as representations of Clifford algebras (a quite mainstream position among mathematicians). So my views about the Standard Model are shaped by this. Not that I would disagree with you, in fact I think you are right from another perspective. Another consequence of my view is that it kept me away from properly investing time in studying your work, although I I knew about the Dixon algebra and that you made a mathematically beautiful and physically insightful model for leptons and quarks. It's time to fix these lacunae I have and read carefully your writings. Your essay convinced me of this, even though you mentioned your work only incidentally, being focused on answering the question "what is 'fundamental'". I also just included in my paper about the Standard Model based on a Clifford algebra a mention of your model (fortunately my manuscript is still under review). I think your work deserves more attention. What I find intriguing is that, unlike Clifford algebras which are infinitely many, the Dixon algebra is one of a kind. Since I still am a bit biased against nonassociativity, I would like to ask you if you know some physical consequences of this feature of octonions. Congratulations for your excellent essay, and success!

          Best wishes,

          Cristi