Michael,

Thank you for your reply, and the link. I will look at it. Thanks also for the clarification. One may have interpreted your Sept. 20 13:43 comment "pretty maths just is not tempting" as dismissive, not to mention an understatement of my position. The math I present actually is pretty, but also substantive. We both agree it is less than optimal to insert anything "by hand", which is just why my mathematical cover using Octonion Algebra is both substantive and compelling, for as I say over and over again in the essay, you must "let the algebra drive", no insertion is permitted let alone required.

Your statement on Godel's mathematical completeness is precisely why this is a red herring. Beyond this, conceptually we are a species of limited intelligence. Logically, could we possibly be able to grasp the full realization of "completeness" beyond the dull, boring and simplistic notions of integers?

P.S Just watched the Space Shuttle do its fly-over of NASA Ames, from a bridge next to my work. What a magnificent sight!

Rick

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  • Post #73

Rick,

I really had in mind, the so-called "pretty" maths of super-symmetry and string theory, whose "prettiness" has been used as justification for pursuing what has proven to be a pointless course for decades.

I have a preference for the geometric view of the same algebra, as it conceptually gives a big picture view which can guide the choice of direction to go in, and this is what the S7 geometric view of the octonions has done for me. An algebraic view is necessary to guide individual steps, and to conceptually check you don't step in anything nasty ;-), but can it really provide the big picture view of where to go next?

Your response on Godel's incompleteness strikes me as the exact opposite of what I meant. The classical physics theory over the integers is provably mathematically incomplete; the unbridgeable category divide between integer particle numbers and waves makes reaching a wave property from integer particles impossible in classical physics. So when a particle has a wave property, the classical physics theory is scientifically incomplete in the experimental sense that there-exists an observation that cannot be derived within classical physics. It is this specific sense of scientific completeness which can be addressed by dropping the "if and only if" condition of describing particle numbers by integers and instead use a real valued wave-function that incorporates the undecidable wave feature. The result is quantum theory - surely a very un-herring-like outcome?

Deeper philosophical points about "completeness" don't arise for this restricted experimental sense of scientific completeness; I use this term precisely so as to avoid the general notion of "completeness" which seems so vague you wouldn't know it if you found it.

No Space Shuttles out my window, just a buzzard if I'm lucky ;-)

Michael

PS. In my Sept 25 reply to Jonathan I give a view of Joy's frameworks in terms of topological mappings and particles as monopoles if your interested.

8 days later

Hello Rick,

While the story goes that the Octonions are the weird and crazy old uncle whom everyone would like to hide away in their closet or attic; that is only partly true, and misses the point somewhat. Instead; the octonions are like the wealthy and eccentric old uncle, who almost nobody else in the family associates with, because they are intimidated by his freedom and power. Of course; the rest of the family is jealous, but they would never admit it, because then they would end up feeling a little less powerful themselves.

But we would all be a whole lot less wealthy, if it were not for the Octonions - which are the great granddaddy of all the other number types. However; the wonderful properties of those amazing octonions assure that the rest of the family emerges and persists as well. I rated your essay highly, because I believe in what you are saying - and that you said it well. Of course; I fully understand that you can't talk meaningfully about your subject without getting too technical sometimes, which you did, but so it goes.

All the Best,

Jonathan

    Hmm,

    Thinking on the above comment; I imagine you could make a detailed storyline out of it - where, for example, the rest of the family will only approach uncle Octonius in committees of three, but there is always a member of each committee in common. Such a dramatic approach might illustrate the kind of special respect the octonions demand. Just a thought.

    But I like the idea of the Octonions being a wealthy and eccentric uncle (who knows untold secrets) better than a weird and crazy one, any day.

    All the Best,

    Jonathan

    To continue;

    Good old Uncle Octonius was one crafty old genius. The family insisted that the only way anyone in the family could be trusted to meet with him is if they went in committees of three. And of course; he insisted that - in order to be fair - there must be one member in common each time they send a committee. But how did he know that there would be seven committees? That crafty old guy must have planned it that way.

    Of course; the committees are called quaternions - which is short for "It's our turn again." Anyhow; I'm just thinking out loud on your page.

    All the Best,

    Jonathan

    If you do not understand why your rating dropped down. As I found ratings in the contest are calculated in the next way. Suppose your rating is [math]R_1 [/math] and [math]N_1 [/math] was the quantity of people which gave you ratings. Then you have [math]S_1=R_1 N_1 [/math] of points. After it anyone give you [math]dS [/math] of points so you have [math]S_2=S_1+ dS [/math] of points and [math]N_2=N_1+1 [/math] is the common quantity of the people which gave you ratings. At the same time you will have [math]S_2=R_2 N_2 [/math] of points. From here, if you want to be R2 > R1 there must be: [math]S_2/ N_2>S_1/ N_1 [/math] or [math] (S_1+ dS) / (N_1+1) >S_1/ N_1 [/math] or [math] dS >S_1/ N_1 =R_1[/math] In other words if you want to increase rating of anyone you must give him more points [math]dS [/math] then the participant`s rating [math]R_1 [/math] was at the moment you rated him. From here it is seen that in the contest are special rules for ratings. And from here there are misunderstanding of some participants what is happened with their ratings. Moreover since community ratings are hided some participants do not sure how increase ratings of others and gives them maximum 10 points. But in the case the scale from 1 to 10 of points do not work, and some essays are overestimated and some essays are drop down. In my opinion it is a bad problem with this Contest rating process. I hope the FQXI community will change the rating process.

    Sergey Fedosin

    Jonathan,

    One must make an intellectual commitment to reach an appropriate respect for Octonion Algebra and just what it can tell us about physical reality. The premise of this essay contest is there may be fundamental assumptions that may need rethinking. Octonion Algebra tells us the list is long and distinguished, and thus it intimidates many who take this as a fontal assault on their core beliefs. You will have a tough time convincing a Southern Baptist that they should study the Koran. It is just about as bad for Octonion Algebra; so many very intelligent people do not make the necessary open minded intellectual commitment.

    I try my best to present my findings in an approachable way, and in the setting of this essay contest I can only hit the highlights. I believe I did a very good job with my essay given the limited format. For years now I have been both surprised and disappointed that so few people have developed the enthusiasm I have for the potential of this powerful algebra, so the lack of dialog and moderate community rating is nothing new for me. As they say, you can lead a horse to water, but you can't make it drink.

    Rick

    Mr Lockyer,

    a very interesting essay. I remember a few years ago I was looking for math that would allow the multidimensional nature of space to be expressed with ease. I wish I had found Octonion Algebra then, but all I saw was quaternions. I imagine how hard it was to pack it all in 9 pages -- well done! I hope your essay will get the recognition it deserves.

      Ms. Vasilyeva.

      Thank you for your kind words. I must say I enjoyed your essay also. The structure of space is important for us to think about. But we must not over think it, and apply more structure to the fundamental idea of space in favor of structure built on top of space. Potential functions over a simplistic notion of space whose purpose more is to set the algebra, dimensionality and allow us to distinguish one point from another has always resonated with me more than say intrinsic curvature of space itself to describe Gravity. But mathematics is robust enough to permit useful activity either way, perhaps both a blessing and a curse at the same time. The map reality to mathematics is one to many, some choices will be more optimal than others. I tend to look for the choice that provides fewer options while speaking loudly with authority about how nature must be. Octonion Algebra does this quite well in my opinion.

      Good luck with your fine essay,

      Rick

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      • Post #82

      Lisi , Jonathan,Tom ...are stealers ahahahah frustration +lack of general skillings=your comportment.

      conclusion return at school ahahah spherization of balls and spheres ahahahah yes of course band of comics.

      Steve score= 1001

      bad teamscore = 0

      in fact you are not cramped or pinched or troubled just between us. Because frankly , if you think that you are intelligent, let me tell you that frankly your sciences are weak, in a general point of vue of course. Cramped and pinched are weak words for your pseudo team of pseudos gneral scientists. I really suggest that you retrun at school you know. I don't know what are your foundamentals learnt at school during your young education, but frankly here in europe I beleive that our courses are better, but it is just a suggestion of course. Coordonates of nothing badband team. You are ironical Jonathan and Tom.

      Study for your Phd instead of loosing your time with things that you do not understand in fact. Or change of option , take the marketing for example.

      Each person at its place badteam of nothing for nothing.

      I am parano ok ,but I have my reasons.

      Regards

      Mr. Lockyer,

      Congrats to us all on having it over with!

      Thank you for your response. You wrote, " Potential functions over a simplistic notion of space whose purpose more is to set the algebra, dimensionality and allow us to distinguish one point from another has always resonated with me more than say intrinsic curvature of space itself to describe Gravity. "

      -?? did I understand you right?...If it does not curve, what's gravity?

      I am very curious about your take on space. Like, there are many topological, geometrical, algebraical spaces whose characteristics and properties are mathematically defined. There is only one real thing. What do you think defines its characteristics and properties?

      I imagine you must think that space has more dimensions than the observed 3. Otherwise, why Octonian Algebra?

      Ms. Vasilyeva,

      Gravity without curvature? Yes, you do understand me correctly. Einstein showed that gravity *could* in a 4D setting be described by intrinsic curvature. He neither showed that it *must be*, nor did he or anyone else show physical reality *must* be 4D. Think about how gravitation was handled prior to Einstein, with a potential function. As I stated in my essay, integrating this with the charge force in a 4D setting is a bit problematic. I can't say this was Einstein's main motivation for looking for something completely different, but I have never heard a convincing argument for using curvature on a first principle basis.

      If you think about 4D tensors of rank > 1, this is entirely an algebraic structure that adds dimensional count, as it must for say, the 4D EM field tensor, since the electric and magnetic fields have distinctly different component transformation characteristics. We often say they are manifestations of the same thing, but this same thing has more than 4 dimensions to it, or more precisely additional degrees of freedom. The robust character of mathematics gives us more than one way to address the need for additional degrees of freedom. Whether or not you call the additional degrees of freedom "physical dimensions" is a semantics choice.

      When we talk about "observed 3D space", just what does this mean? I think this is typically interpreted as observed with our primary sense of sight and perspective. God gave us this ability to enable us to function in the world long enough to procreate, not to be able to fundamentally observe the nuances of the nature of things. We should not impose three spatial dimensions just because we can "see" it. We should look to span the space in a purely mathematical way. This requires more dimensions, and obviously I think the number is 8.

      We all understand tensors fall short of a full deck of cards. This is why spinors are in fashion. Both are algebraic structures that are not fully general. This may be why they have had their share of success while leaving so much unrevealed, and it looks like the missing pieces must be hand crafted. This is not the case for Octonion Algebra. It is the most general of the division algebras, and as I have stated in the essay, fully descriptive when the concept of algebraic invariance is applied with the proper calculus. Clearly things are not yet theoretically complete, but this is fertile ground.

      Rick

      Hello Rick,

      Good old Uncle Octonius would be proud! Assuming no further vacillation; I wish you the best of luck in the finals. You deserve to be there!

      All the Best,

      Jonathan

      Congratulations, Rick.

      You have brought our crazy old uncle to the party. I hope he unleashes unimaginable havoc.

      Joy

        Thanks Joy,

        I hope to unleash unimaginable clarity, not "unimaginable havoc". But as you know well and most essay authors have stated time and again, it is a tough sell to get acceptance for ideas that suggest they rethink positions long held. This is however what the essay contest is all about.

        All I can do is point out what I have come to know, and ask the reader to decide for themselves if the results are merely coincidences or if they point to fundamental truths that show us a different and better way to proceed.

        Rick

        Mr. Lockyer,

        congratulations on making the list of the finalists! It did not look so when I posted above. I am glad you have made it in the end.

        Thank you for your reply and sorry for the delay. I needed a break from physics and enjoy the cluster of birthdays. So, for you it's all algebra and no curvature at all... I guess my visual approach requires geometry, but I do like it very much that your world is set in 8D.

        Again, congratulations and good luck with the rest of the competition!

        Hello Rick,

        I'm cross-posting this comment I just left on Ben's page, with due respect to John Baez, who is not my cousin in real life, because it deals with octonions.

        A story:

        They say my uncle is crazy, and cousin John tells me some family members wanted to lock old Uncle Octonius up in the attic, but I think he is only eccentric because he's seen the universe, and knows its secrets. For years we thought he just wouldn't associate with the other family members at all, but somehow we worked out how to do it safely. You see; Octonius is very persuasive, and can make people do almost anything - so he can't be trusted, or rather no one person can ever see him alone. And when we send two, they always disagree on what was said. Therefore; we always visit Uncle Octonius in committees of three. But; the first time a group of us visited, he insisted that he must see all the family members - with equal frequency - and that there always be someone in common between any two visits. Luckily; this worked out, because there are seven of us.

        The thing is; Octonius is incredibly wealthy and knows the secrets of the universe, but we were all so afraid of him that we never knew why he seemed so crazy. You see; he always liked to break the laws of algebra - or insist on things being backwards sometimes - whenever we tried to use the associative and commutative rules to simplify expressions for him. But we never understood why that was, until we attempted to rank ourselves - thinking that both the greatest and slightest within our family needed to be included, within each committee, to assure trust. Then Octonius explained that committees follow a rule that is non-commutative, and then if you include everyone at once things become non-associative, because there can be disagreements between members or committees - but there is also a hierarchy or ordering of and within any committee.

        Though we are still not sure we can trust him, Uncle Octonius tells us this is as fair as it can be, and now he is teaching us the secrets of the universe. So who could complain? I'm glad cousin John didn't let the others lock him up in the attic, or we would never have learned of his vast wealth and untold secrets.

        end of story

        Jonathan

          Hi Jonathan,

          Cute story. The only problem I have with it is "You see; he always liked to break the laws of algebra...". Not the case. The general algebraic rule for multiplication e_i * e_j = sum k: p_ijk e_k encompasses commutative, non-commutative, associative and non-associative algebras. It all comes down to the structure constants p_ijk.

          Rick

          Or structure variables p_ijk(x) in my variable torsion picture, at every point x in S^7.

          Excellent!

          You two certainly know how to spice up a story. I like both of your answers. And though it might be tough to work into the story, that's part of what makes our 'crazy' old uncle so interesting. But if we had decided he was crazy, and just left it at that, we never would have learned of his incredible wealth.

          But I think I clearly conveyed that he likes to be in charge, and makes the others do things his way. And as you both just said, in the final analysis he never really broke the law. He just bent the rules, to have a little more play, and ultimately made the others play the game by his rules.

          All the Best,

          Jonathan