Efthimios,

Thanks for reading my essay. Hopefully you gained something useful.

I think that Hamilton's methods have not been satisfactorily applied. My objective with this essay was to present several ideas. I wanted to show that ordinary Calculus could be applied to quaternion functions. That allows four times as much information to be expressed by a given number of symbols. I wanted to show that the resulting kinematics can produce a curved path. That hints at how to treat gravity. And I wanted to show that the Lorentz Transform could be extended into a time quaternion whose vector portion resides in 3-D space. That hints at how to remain consistent with Special Relativity and it offers the possibility of eliminating time as a fourth dimension.

Best Regards and Good Luck,

Gary Simpson

9 days later

Gary,

Your quaternion derivative is very similar to the Gateaux derivative for quaternions according to the Wikipedia article on "quaternionic analysis". If you were unaware of Gateaux, you are to be congratulated for your insight in finding a directionally dependent quaternion derivative. In your case, the direction of the quaternion itself is a natural choice.

The difference is that you take the limit (eq.6) of a ratio, while the Gateaux derivative does not include the denominator. It looks like the direction quaternion in the denominator has to be divided out separately. The wiki article gives the example of q^2.

I have been playing around trying to find another way to produce dq*/dq with no luck yet.

Best regards,

Colin

    Colin,

    I was not aware of Gateaux. It looks like you have become quite interested in quaternions and possible methods associated with them. Excellent. This will be a long and difficult battle.

    Best Regards and Good Luck,

    Gary Simpson

    Dear Gary,

    I just post a reply to your comment on my paper.

    Friday I'll have time to approach your essay before replying here on your own forum.

    Best regards and good look

    Peter

    Gary,

    I enjoyed reading your essay, and felt grateful that you made me think about "the topic" in the context of the history of science. Obviously, this approach ought to inspire at least some of the answers.

    Your method of involving the human element (including the fact that things would have progressed differently had certain individuals known about the work of others - perhaps even more so if not coeval) supports the narrative that the connection between physics and math should not be viewed in isolation from the people actually "doing" the two disciplines. I was not sure if one should interpret this observation to mean that you view mathematics as something that people "develop," rather than something that would have been always out there (somewhere, somehow) even if no human ever existed. Taken to its extreme, this interpretation could imply that the connection between physics and math "resides" in the nature of humanity. It appeared safer not to draw such a conclusion without your blessing.

    In any case, your essay is good work, and deserves a good rating.

    I also wish you Good Luck.

    En

    P.S. I replied to your comment on "my page."

      Dear Gary,

      I just found an essay as interesting as original. Reconstruct a posteriori the continuity between Newton's MATHEMATICAL pioneering work on calculus and its successors, AND THEN deduce the epistemological continuity from Newtons's pioneering work PHYSICAL until "quasi SR theory", what a great idea! All my sincere congratulations! After reading your essay, it is crystal clear: If Newton had been in possession of the necessary mathematical tools, he would have reached the confines of of SR, and probably more adequately than the pre-SR approaches of Lorentz and Poincaré. As you notice it indirectly on page 6, Newton, ignoring the constancy of c for every reference frame and starting subsequently from a pre-SR definition of simultaneity, would not have exceeded effectively your equ. 10, but the SR-framework would have been potentially there.

      You can also do the following overlapping: As everyone knows, Humanity already possessed SR by Maxwell's equations, but without realizing it, and, consequently, without taking offense on the pretty discrepancies between classical dynamics and electromagnetism. According to the current design, it is the need of a new paradigm following the discovery of the constancy of c for every reference frame, which is the origin of the recognition that "SR, by Maxwell's equations, preceded SR". But your essay allows a broader approach of this historical process.

      In my case, your essay reinforces my platonistic convictions, that of course many people cannot share. But personally, I do not see how natural phenomena may at a given moment confirm mathematical potentialities formerly unknown by the discovery of their own consequences, if these mathematical laws and their potential extension were not inherent to the correspondant natural phenomena. But this is another story...

      Congratulations again,

      Best regards,

      Peter

        En,

        Many thanks for taking the time to read and consider my essay. I am pleased that you enjoyed it and that it made you consider the historical sequence of some of our major mathematics. To me, it emphasizes that what you think is influenced by what you already know.

        I had a very pleasant exchange with Akinbo Ojo concerning Zeno's Paradox. My thinking is that Zeno was a missed opportunity. He correctly identified a flaw in his thinking but he was not able to step outside of it to make the next step. If he would have recognized the need for an infinite sum then he would have been one step away from calculus. What would the world be today if the Greeks had calculus 2000 years ago?

        My opinion is that mathematics is a human construction. It is useful in physics to the extent that both math and physics seek truth. It seems that physical truths have mathematical equivalents. We are still struggling with this in the areas of GR and QM.

        Best Regards and Good Luck,

        Gary Simpson

        Peter,

        Many thanks. You understand my intentions exactly I think. How much different would some of our ideas in physics look if they were formulated purely as vector or quaternion representations? The special subsets would be much more clear and lucid I think.

        Best Regards and Good Luck,

        Gary Simpson

        12 days later

        Hi Gary--

        I enjoyed your essay very much. Hypothesizing how Newton could have used quaternions to get to Special Relativity is fantastic. Confession: I'm a huge fan of Newton and, in particular, have enjoyed reading about the development of Calculus (starting, of course, with the Newton-Leibniz blowup).

        I do not claim to be an expert regarding quaternions. So, I was hoping that you might be kind enough to take some extra time to explain how your Eq. 10 would have helped Newton "lay the groundwork for Special Relativity". A few more words might be helpful for those of us who don't have the math at our fingertips. I know how in these essays, with the space and word constraints, it is tough to put in all of the extra explanatory asides and so forth.

        I think that your essay has been undervalued by the community.

        Best regards,

        Bill.

          Bill,

          Many thanks for reading my essay.

          Regarding your question concerning Eq 10. The way that we normally think about distance, velocity, and time would cause Eq 10 to produce a value of 1 with no vector terms. Newton knew of the trigonometric substitution needed to integrate the square root of (1 - u^2). So he would have realized that somehow he could convert the cosine term into sqrt(1 - u^2) and also the sum of the squares of the three sine terms would equal u^2. But he would not know that u^2 = (v/c)^2.

          Essentially my point was that he could have produced a vector transform that looks like the four-vector that people use today in SR. So, when Einstein developed SR, he might have done it differently because he would have already had Eq 10 or something similar and then SR would not have seemed so radical. It would simply have been a question of re-interpreting something that was already known.

          It is a kittle ironic isn't it, that we credit Newton with Calculus but we use Leibnitz's notation?

          Regarding scoring, you are most kind. People who are actually authors and writers say that you should not put much math in an essay because it tends to lose or annoy some of the readers. I choose to ignore this because the message I want to convey is mathematical. In this case, the message is that quaternion functions can be differentiated with respect to quaternion variables exactly the same way that real functions are differentiated with respect to real variables.

          Best Regards and Good Luck,

          Gary Simpson

          Dear Mr. Simpson,

          I thought that your engrossing essay was exceptionally well written and I do hope that it fares well in the competition.

          I think Newton was wrong about abstract gravity; Einstein was wrong about abstract space/time, and Hawking was wrong about the explosive capability of NOTHING.

          All I ask is that you give my essay WHY THE REAL UNIVERSE IS NOT MATHEMATICAL a fair reading and that you allow me to answer any objections you may leave in my comment box about it.

          Joe Fisher

            Hi Gary--

            Thank you for taking the time to provide such a good reply. I now see the connection clearly.

            There are many ironies regarding the Newton-Leibniz "Who Invented Calculus" dispute. We use Leibniz notation--and, yet, we physicists also use Newton's dot symbol for d/dt. What a mish-mash. At least we didn't get stuck with Newton's term "fluxions"!

            I wish you the best of luck. (As for me, I am getting beaten down over at my essay. Yet no one is leaving any negative comments or even questions. I'm beginning to think that there is some algorithm to this scoring system that I'm just not getting!)

            Best regards,

            Bill.

            Bill,

            It sounds like you are being trolled. I got scored by three 1'a and two 2's. It seems to be just part of the system. The key is to get some positive votes.

            Best regards and Good Luck,

            Gary Simpson

            Joe,

            You must have forgotten ... I read and commented on your essay fairly early. I also scored it ... higher than what you've got now but not much.

            Honestly, I had a really hard time understanding what you were trying to say.

            If you are arguing for some kind of mathematical Nihilism then you are incorrect. Abstract ideas allow for easy manipulation of real things. We use those abstract manipulations to design and build roads, bridges, dams, chemical plants, refineries, airplanes, satellites, space probes, computers, cars, electronics, etc. etc. etc. It ain't random. It does work.

            Best Regards and Good Luck,

            Gary Simpson

            Hi Gary--

            Thanks for the system insight. I love the phrase "being trolled". What an apt description. I had not heard that phrase before (which clearly means that I don't get out enough!).

            I have been scored in a similar manner. I won't speak to my essay, but I will speak to yours: Such scoring is simply and purely wrong. To score a "1" or "2" means that one finds the essay in question to be both totally off-point and written in pure gibberish. That is most certainly NOT your essay. Enough said.

            Best regards,

            Bill.

            Gary,

            As I read some of your comments on various essay pages over several days, I observed that you and I share views on more things than would appear judging solely by our respective essays. Hopefully this is not an unwelcome observation.

            Your 3rd paragraph (in your reply to Joe just above) captures the essence of the matter.

            En

            En,

            Many thanks. It is not unwelcome. I am educated as an engineer and that strongly influences my thinking.

            Best Regards and Good Luck,

            Gary Simpson

            Dear Garry D Simpson,

            Thank you for having read my essay. I downloaded your essay weeks before (based on the abstract), but found it be just mathematics. Verifying your arguments require some effort, and so I took the easiest route - avoiding any comments.

            My opinion is that any theory in physics should be verbally explainable, and treating mathematics as the most appropriate language for explaining the physical world is incorrect. Any concept that 'has physical meaning' and that 'does not go against commonsense' will be verbally explainable.

            However, any physical model should be mathematically viable. The simplest and the most suitable mathematical form should be used for this. Even then, venturing into new mathematical ideas, even if complex, is something that has a beauty of its own. You have stated that some original work has been done by you in the field of quaternion functions. I will try to follow your papers in vixra.

              Hi Gary,

              Do you think it is possible that we may be living in a finite and discrete universe that could be described in an informational way? Do you think we could make more progress in our understanding of physics if we looked towards computer programs/simulations, instead of new sets of math equations, for explaining phenomenon? How much complexity do you think is in the universe, and how much of it is compressible?

              Please check out my Digital Physics movie essay if you get the chance.

              Thanks,

              Jon