Calculus - Revision 2.0 by Gary D. Simpson

Gary Simpson

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

Gordon Watson

Gary,

To Philip's list (and to his point) I'd add the δ-function, my favourite non-function!

(But then I am an engineer.)

Gordon Watson: Essay Forum. Essay Only.

William Parsons

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.

Gary Simpson

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

William Parsons

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.

Gary Simpson

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

William Parsons

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.

Joe Fisher

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

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

En Passant

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.

Gary Simpson

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

Jose Koshy

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.

Gary Simpson

Thanks for giving it a read.

I agree completely. Nature should make sense and it should be mathematically viable. The question is ... does nature agree?

Best Regards and Good Luck,

Gary Simpson

Jonathan Khanlian

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

Gordon Watson

Dear Gary,

I've replied to your nice post in my Essay Forum. Points that are relevant to your work here are reproduced below:

4. Geometric Algebra is peeking its head out regarding the beables and their local values.

I am so glad that you see that! Please be the first to help that shy, beautiful (and sometimes tricky) GA out of the closet and work with her in the unified "BT" context proposed in my essay. For I'd love to see elementary GA taught in primary schools: with GA on its way to becoming Nature's local realistic Mathematics.

5. Re GA.

How is your work received within the GA community? Have you any rejections from journals? If so, what do they say? (Write to me privately if you wish.)

Are you familiar with Elio Conte's efforts? For example: Conte, E. (2001). Biquaternion Quantum Mechanics. Bologna, Pitagora Editrice? (Alas, he supports nonlocality!)

How about this Caves, Fuchs, Schack essay [arxiv.org/pdf/quant-ph/0104088v1.pdf] and the view that amplitudes should be complex numbers rather than reals or quaternions?

With best regards, and looking forward to spending time with your important ideas.

Gordon Watson: Essay Forum. Essay Only.

Gary Simpson

Gordon,

Thanks for your comments. It is interesting that you should mention teaching GA in the public schools. That was one of the things I was thinking when I wrote the Conclusions. I think that most of the folks in the essay contest probably took calculus in high school. GA would be a nice addition to that. BUT ... there is no point in doing so until more of Physics and Engineering are formulated using GA. Once the ground work is laid and students are prepared to use GA, I think thee will be a surge in knowledge and understanding.

Regarding reception of my work ... I simply get rejections of course ... No big deal. I post to viXra and participate in essay contests when they look relevant.

Best Regards and Good Luck,

Gary Simpson

Alma Ionescu

Dear Gary,

I enjoyed reading your derivation and very much agree that Newton was very close to the framework of SR, had it not been for his intuition that space needs to be fixed. I noticed you worked during Christmas (given the date on the document), therefore around Newton's birthday. I am sure he would have been proud to know that people have something to say about his work after four hundred years since it was finished.

Wish you best of luck in the contest!

Alma

Gary Simpson

Alma,

Many thanks for taking the time to read my essay. I hope the math was not too troubling.

I did indeed work on it around Christmas. Quaternions make great gifts:-)

It looks like you will make the cut for the finals. Congratulations!

Best Regards and Good Luck,

Gary Simpson

Alma Ionescu

Dear Gary,

The math was not very easy for me since I don't have the right background, but I was able to understand it and your work actually helped my understanding very much. It took me maybe three hours to go through it and make sure I really feel the argument. There were some things that I didn't know about the framework and that I was able to guess because of the clarity of your presentation. You should at least consider to publish it in a pedagogical journal because there are many people out there who would benefit from reading it, especially students.

My sincerest appreciation! :-)

Alma

Gary Simpson

Alma,

Many, many thanks. You have made my effort worthwhile. I am flattered that you would spend so much time understanding the subject matter. So you see, if you can express a function as a function of a quaternion, then it can be integrated an differentiated exactly as though it were a simple real function of a single variable. To me, that is amazing and unexpected.

There is very little possibility that anything that I write will be published in a journal. That is actually the reason why I included the mathematics in this essay. There is no other effective way for me to share the idea. Dr. Gibbs has created a website named viXra.org that allows anyone to post work such as this. I post my works there and I participate in essay contests when possible.

Again, many many thanks.

Best Regards and Good Luck,

Gary Simpson

Jonathan Khanlian

Hi Gary,

Thanks for your response on my post. You said, "I think the universe is finite. I cannot say anything about whether or not it is discrete. I'm not even sure how the word "discrete" would be applied to the universe. Is the universe a discrete solution to a massive system of wave equations? Some people argue that the wave equation for a Bose-Einstein condensate at 2.7 K describes the universe."

Here are my thoughts:

I'm not sure if you could actually have a continuous universe that was finite. I assume you are imaging a continuous universe that is bounded by something, say the observable universe or something like that. Here's a math analogy that might illustrate my point: You might say that the interval of real numbers between 0 and 1 is continuous and finite, but I would say that you have the infinite in the form of the infinitesimal because you implicitly believe in infinite precision non-computable real numbers when you believe in the continuum. Infinite precision non-computable real numbers are what make up the continuum in a mathematical sense. Computable reals which include numbers like pi and e (as well as fractions) have a measure 0.

A discrete universe would rule out a continuous wave, just like a computer couldn't actually contain the infinite amount of information needed to represent every point on a curve, although a computer could contain a finite algorithm (e.g. a wave equation) to generate the wave to any desired level of accuracy... It just can't contain the non-compuable, which is what makes the continuum the continuum.

I'm interest to hear your thoughts on this perspective. If you could post a notice in my forum when you respond so I know when to check back that would be helpful.

Jon

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