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RECENT POSTS IN THIS TOPIC

**Akinbo Ojo**: *on* 6/18/15 at 14:44pm UTC, wrote Hello Gary, Would you mind if I tapped your brain a little? I have a draft...

**Akinbo Ojo**: *on* 5/13/15 at 8:50am UTC, wrote Sorry, in addition seeing your interest in the wave equation, what is your...

**Akinbo Ojo**: *on* 5/13/15 at 8:37am UTC, wrote Thanks Gary. I just posted a reply to your well reasoned comment. All the...

**En Passant**: *on* 4/24/15 at 1:28am UTC, wrote Author En Passant replied on Apr. 24, 2015 @ 01:24 GMT unstub Gary, I...

**Gary Simpson**: *on* 4/22/15 at 21:41pm UTC, wrote John, Many thanks for taking the time to read and study my essay. I hope...

**Gary Simpson**: *on* 4/22/15 at 21:16pm UTC, wrote Many thanks James. I rated your essay near the time I read and commented on...

**Gary Simpson**: *on* 4/22/15 at 21:14pm UTC, wrote It was a pleasure Akinbo. I will periodically visit the forums to see what...

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FQXi FORUM

October 15, 2019

CATEGORY:
Trick or Truth Essay Contest (2015)
[back]

TOPIC: Calculus - Revision 2.0 by Gary D. Simpson [refresh]

TOPIC: Calculus - Revision 2.0 by Gary D. Simpson [refresh]

This text demonstrates that how we think about both Mathematics and Physics can be influenced by the mathematical tools that are available to us. The author attempts to predict what Newton might have thought and done if he had known of the works of Euler and Hamilton and had been familiar with the matrix methods of Linear Algebra. The author shows that Newton would have come very close to Special Relativity.

I was educated as a Chemical Engineer with BS and MS degrees from Texas A&M University. My interests in Physics are the wave equation and Hamilton's quaternions.

Thank you for taking the time to read and consider my essay. I hope that you judge it to have been time well spent.

I was very casual in the presentation of the derivative of a quaternion with respect to a quaternion. For a more complete development of the subject, please refer to the following URL: http://vixra.org/author/gary_d_simpson. The title of that work is Quaternion Dynamics, Part...

view entire post

I was very casual in the presentation of the derivative of a quaternion with respect to a quaternion. For a more complete development of the subject, please refer to the following URL: http://vixra.org/author/gary_d_simpson. The title of that work is Quaternion Dynamics, Part...

view entire post

Gary,

A very interesting essay, concluding with a quaternion representation for time that I will need to think about. I am a fan of Hestenes' writings on Geometric Algebra, and tried to keep his "Spacetime Physics with Geometric Algebra" in mind while I worked through your essay. [particularly his III 'Proper Physics and Space-Time Splits', page 8.] Somewhere along the way I got lost....

view entire post

A very interesting essay, concluding with a quaternion representation for time that I will need to think about. I am a fan of Hestenes' writings on Geometric Algebra, and tried to keep his "Spacetime Physics with Geometric Algebra" in mind while I worked through your essay. [particularly his III 'Proper Physics and Space-Time Splits', page 8.] Somewhere along the way I got lost....

view entire post

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Edwin,

Thank you for the comments. I had hoped that the alternate history vehicle would not be too tacky. I had just completed some work on quaternion functions and derivatives and I thought the subject matter was a good fit with the essay contest. I simply needed a way to convey the main result (that they differentiate exactly like real functions). Who better to convey that than Newton...

view entire post

Thank you for the comments. I had hoped that the alternate history vehicle would not be too tacky. I had just completed some work on quaternion functions and derivatives and I thought the subject matter was a good fit with the essay contest. I simply needed a way to convey the main result (that they differentiate exactly like real functions). Who better to convey that than Newton...

view entire post

Gary, this was a very interesting idea for an essay. I sometimes wonder to what extent pure mathematicians would discover ideas from physics if they had no input from physicists or observation. Complex numbers and quaternions are examples of mathematical ideas that were found to have applications in physics so I think if mathematcians were smart enough and left long enough they would become interested in many structures that were actually first found by physicists, such as Lorentz type groups. Your idea that this could have come about through a study of quaternions is very nice. I think a lot more could be said about what this implies for the relationship between maths and physics.

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Philip,

Many thanks for the thoughts. I think the Mathematicians would be able to eventually figure it all out but it would take longer without input and feedback from Physics. It is curious though ... it seems to me that historically, the mathematicians have always been 50-100 years ahead of the application. When Einstein did GR, Reimann had already done the math. But today it seems like the math is lagging behind. Perhaps grad students in Physics should be encouraged to study math instead?

Regards,

Gary Simpson

Many thanks for the thoughts. I think the Mathematicians would be able to eventually figure it all out but it would take longer without input and feedback from Physics. It is curious though ... it seems to me that historically, the mathematicians have always been 50-100 years ahead of the application. When Einstein did GR, Reimann had already done the math. But today it seems like the math is lagging behind. Perhaps grad students in Physics should be encouraged to study math instead?

Regards,

Gary Simpson

Duh ... you are so right ... and I know how to spell his name correctly too ... I will blame my fingers. Clicked "submit" without a proof-read:-(

Regards,

Gary Simpson

Regards,

Gary Simpson

I must read in detail, but I understand that the quaternion algebra can be used like operator of translation and rotation (I read the use in the spacecraft control); it is completely new to me the use in relativity.

I don't understand – in this moment - if each Lorentz transformation can be a quaternion transformation, but I think can only a boost could be a quaternion transformation (the number of components of the Lorentz transformation are three only for a pure boost).

It is all clear, and the extension of the use of directional derivative to derive the quaternion is interesting.

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I don't understand – in this moment - if each Lorentz transformation can be a quaternion transformation, but I think can only a boost could be a quaternion transformation (the number of components of the Lorentz transformation are three only for a pure boost).

It is all clear, and the extension of the use of directional derivative to derive the quaternion is interesting.

report post as inappropriate

Gary,

I found your formulation of a quaternion derivative intriguing. I recall that with complex numbers the derivative of the conjugate is zero: for a complex number z, dz*/dz = 0. This can be deduced from the Cauchy-Riemann equations for a complex number. Taking the derivative of the square of the norm zz* then gives dzz*/dz = z*.

It turns out things are less complex (if you will excuse a pun) for a quaternion q. I was curious about the derivative of its conjugate, expecting more complexity, but it turns out (thanks Wikipedia) that the conjugate of a quaternion can be expressed entirely using multiplication and addition, q* = -(q + iqi + jqj + kqk) / 2, so that it is easy to see that dq*/dq = 1. Taking the derivative of the square of the quaternion norm qq* gives dqq*/dq = q + q* or twice the real part of q.

Thanks for your essay which helped advance my understanding of quaternions.

Best wishes,

Colin

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I found your formulation of a quaternion derivative intriguing. I recall that with complex numbers the derivative of the conjugate is zero: for a complex number z, dz*/dz = 0. This can be deduced from the Cauchy-Riemann equations for a complex number. Taking the derivative of the square of the norm zz* then gives dzz*/dz = z*.

It turns out things are less complex (if you will excuse a pun) for a quaternion q. I was curious about the derivative of its conjugate, expecting more complexity, but it turns out (thanks Wikipedia) that the conjugate of a quaternion can be expressed entirely using multiplication and addition, q* = -(q + iqi + jqj + kqk) / 2, so that it is easy to see that dq*/dq = 1. Taking the derivative of the square of the quaternion norm qq* gives dqq*/dq = q + q* or twice the real part of q.

Thanks for your essay which helped advance my understanding of quaternions.

Best wishes,

Colin

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Colin,

Many thanks for having a read. Yes, quaternions have some very nice algebraic behaviors. You can sum a conjugate pair to eliminate the vector. You can take the difference between a conjugate pair to produce a vector. You can multiply a conjugate pair to produce a scalar.

It principle, d/dx(uv) = (du/dx)v + u(dv/dx) should be applicable but I have not gone through it in detail and the order of multiplication shoyld matter. My next effort will be developing identities.

Best Regards,

Gary Simpson

Many thanks for having a read. Yes, quaternions have some very nice algebraic behaviors. You can sum a conjugate pair to eliminate the vector. You can take the difference between a conjugate pair to produce a vector. You can multiply a conjugate pair to produce a scalar.

It principle, d/dx(uv) = (du/dx)v + u(dv/dx) should be applicable but I have not gone through it in detail and the order of multiplication shoyld matter. My next effort will be developing identities.

Best Regards,

Gary Simpson

Colin,

Many, many, thanks.

Thinking about your post some more has made me realize something interesting. The product of a conjugate pair is a constant scalar value (it is the sum of the four squares). Therefore, the derivative of Y = (Q*)Q with respect to Q must be zero (dY/dQ = 0). But if I think of this as Y = AQ and take the derivative with respect to Q, the result is dY/dQ = A. So if A = Q* then something looks to be amiss. I've been wondering what to do with Equation 3 in the work and if there are any Eigen-value type problems that need to be identified and resolved. You have given me a big clue.

Thanks again.

Regards,

Gary Simpson

Many, many, thanks.

Thinking about your post some more has made me realize something interesting. The product of a conjugate pair is a constant scalar value (it is the sum of the four squares). Therefore, the derivative of Y = (Q*)Q with respect to Q must be zero (dY/dQ = 0). But if I think of this as Y = AQ and take the derivative with respect to Q, the result is dY/dQ = A. So if A = Q* then something looks to be amiss. I've been wondering what to do with Equation 3 in the work and if there are any Eigen-value type problems that need to be identified and resolved. You have given me a big clue.

Thanks again.

Regards,

Gary Simpson

Dear Mr. Simpson,

Thank you for thoughtfully warning me that I needed to have a “knowledge of quaternian and Linear Algebra” in order to understand your essay. Unfortunately, Newton was wrong about abstract objects having the option of being stationary or in motion, and Einstein was wrong for assuming that it was abstract light that was capable of obtaining constant speed. It is the real surface of all real objects that is in the same constant motion at the same constant speed, and as light does not have a surface, light is the only stationary substance in the real Universe.

Regards,

Joe Fisher

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Thank you for thoughtfully warning me that I needed to have a “knowledge of quaternian and Linear Algebra” in order to understand your essay. Unfortunately, Newton was wrong about abstract objects having the option of being stationary or in motion, and Einstein was wrong for assuming that it was abstract light that was capable of obtaining constant speed. It is the real surface of all real objects that is in the same constant motion at the same constant speed, and as light does not have a surface, light is the only stationary substance in the real Universe.

Regards,

Joe Fisher

report post as inappropriate

Joe,

Many thanks for making the effort. I hope the experience was not too frustrating.

Linear Algebra is not too bad but quaternions were unknown to me prior to three years ago or so. It took a lot of effort on my part to stop the voice in my head from telling me that it makes no sense to add a scalar and a vector. What finally convinced me that it was ok was simply using them for what Hamilton intended ... namely, the ratio between non-collinear vectors.

It is interesting that you believe that light is stationary. In the 2012 FQXI contest, I presented a scalar solution to the wave equation that was precisely that. Also, if you examine Equations 11.2 - 11.4 of this essay and set v = c for any of the velocity components and then apply the Lorentz Transform, the result is that the change in position is zero. Note that I did not say velocity but rather change in position. I'm still pondering the meaning of both of these things.

I'm catching up with my reading and should be able to comment on your work soon.

Best Regards,

Gary Simpson

Many thanks for making the effort. I hope the experience was not too frustrating.

Linear Algebra is not too bad but quaternions were unknown to me prior to three years ago or so. It took a lot of effort on my part to stop the voice in my head from telling me that it makes no sense to add a scalar and a vector. What finally convinced me that it was ok was simply using them for what Hamilton intended ... namely, the ratio between non-collinear vectors.

It is interesting that you believe that light is stationary. In the 2012 FQXI contest, I presented a scalar solution to the wave equation that was precisely that. Also, if you examine Equations 11.2 - 11.4 of this essay and set v = c for any of the velocity components and then apply the Lorentz Transform, the result is that the change in position is zero. Note that I did not say velocity but rather change in position. I'm still pondering the meaning of both of these things.

I'm catching up with my reading and should be able to comment on your work soon.

Best Regards,

Gary Simpson

Dear Gary,

Your essay is superb! I gave it a 10 with only a regret that 11 or 12 are not options. It is worth pointing out that your matrix equation for the inverse of y = Qx has Q^{-1} that is the same form as the electromagnetic tensor. This is why Maxwell formulated electromagnetic theory with quaternions.

If at all possible you might find my essay

http://fqxi.org/community/forum/topic/2320

of interest. I did not make explicit references to quaternions, but they are lurking in the background in the discussion on Bott periodicity.

LC

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Your essay is superb! I gave it a 10 with only a regret that 11 or 12 are not options. It is worth pointing out that your matrix equation for the inverse of y = Qx has Q^{-1} that is the same form as the electromagnetic tensor. This is why Maxwell formulated electromagnetic theory with quaternions.

If at all possible you might find my essay

http://fqxi.org/community/forum/topic/2320

of interest. I did not make explicit references to quaternions, but they are lurking in the background in the discussion on Bott periodicity.

LC

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Lawrence,

Many thanks for the kind words. I am very appreciative of your enthusiasm. If I can supply someone with a new tool to use when attempting to solve some of these difficult problems then I will count myself as fortunate.

Having said that, I should mention that it would be better not to indicate how you might have rated an essay. The administrators at FQXI might construe that to be vote trading and that could be cause for disqualification.

One of my objectives is to get back to Maxwell in quaternion form. I need to spend some time working on identities and simple kinematics prior to that. I also think that it should be possible to formulate and solve a quaternion style wave equation. That should closely resemble Dirac's solution without the need for factorization using Clifford Algebras. Something that puzzles me regarding that work by Dirac is the equation ab = -ba. I understand non-commutation etc ... What seems odd to me is that the first thing that I think of is simply the cross product of two vectors. If a and b are both vectors then (a cross b) = - (b cross a) for any arbitrary vectors. Obviously, I need to study the subject some more.

I have read your essay once. I will need to read it one or two more times before being able to make any meaningful comments. I see from your e-mail address that you are no stranger to quaternions.

Best Regards and Good Luck,

Gary Simpson

Many thanks for the kind words. I am very appreciative of your enthusiasm. If I can supply someone with a new tool to use when attempting to solve some of these difficult problems then I will count myself as fortunate.

Having said that, I should mention that it would be better not to indicate how you might have rated an essay. The administrators at FQXI might construe that to be vote trading and that could be cause for disqualification.

One of my objectives is to get back to Maxwell in quaternion form. I need to spend some time working on identities and simple kinematics prior to that. I also think that it should be possible to formulate and solve a quaternion style wave equation. That should closely resemble Dirac's solution without the need for factorization using Clifford Algebras. Something that puzzles me regarding that work by Dirac is the equation ab = -ba. I understand non-commutation etc ... What seems odd to me is that the first thing that I think of is simply the cross product of two vectors. If a and b are both vectors then (a cross b) = - (b cross a) for any arbitrary vectors. Obviously, I need to study the subject some more.

I have read your essay once. I will need to read it one or two more times before being able to make any meaningful comments. I see from your e-mail address that you are no stranger to quaternions.

Best Regards and Good Luck,

Gary Simpson

The Dirac equation can be looked at as the multiplication of quaternions. The Dirac operator is a quaternion valued set of differential operators and the spinor field is also quaternion valued.

What you might be pondering is the role of forming differential forms from quaternions that are antisymmetric and quantum commutators. One can think of the 1, i, j, k as one-forms that give wedge products that are a generalization of the cross product. I think this is some question with the relationship between the Heisenberg group and quaternions. I think this relationship involves the AdS_5 spacetime.

My email address golden field quaternions refers to the 120 quaternions in the icosian group, that is half of the E8 lattice. That gets into octonions.

Cheers LC

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What you might be pondering is the role of forming differential forms from quaternions that are antisymmetric and quantum commutators. One can think of the 1, i, j, k as one-forms that give wedge products that are a generalization of the cross product. I think this is some question with the relationship between the Heisenberg group and quaternions. I think this relationship involves the AdS_5 spacetime.

My email address golden field quaternions refers to the 120 quaternions in the icosian group, that is half of the E8 lattice. That gets into octonions.

Cheers LC

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Dear Gary D. Simpson

Your essay is an example of how learning enables identifying or finding new problems and persistence leads to proposed solutions. Moreover, the presentation of your ideas was logical and accessible, which is considerate of the reader, likely reflecting a desire that reader can share in the benefit of your hard work.

Does the following question in relation to your essay make sense? Can a space of points described by quaternions each of which isotropically scales by the same increasing (scalar) scale factor model the isotropic cosmological expansion of space?

Regards.

Bob Shour

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Your essay is an example of how learning enables identifying or finding new problems and persistence leads to proposed solutions. Moreover, the presentation of your ideas was logical and accessible, which is considerate of the reader, likely reflecting a desire that reader can share in the benefit of your hard work.

Does the following question in relation to your essay make sense? Can a space of points described by quaternions each of which isotropically scales by the same increasing (scalar) scale factor model the isotropic cosmological expansion of space?

Regards.

Bob Shour

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Dear Bob,

Thank you much for your generous comments. You are most gracious. I try to write as simply as possible with short sentences in a linear, logical sequence with clear endpoints. Sometimes I have to leave an idea dangling so that I can merge it with something else.

Regarding your question, I think that I understand your question and I think that the answer is yes. I'll expand on this a little. If you look at Equation 3 in my essay and you make the vectors collinear, the cross product term becomes zero. Now, you only have to worry about the scalar term. I have done the calculation using 13.8 billion light-years as the radius of the universe. This allows me to predict an expansion rate equal to 70.75 km/sec per Mpc. The observed expansion rate is 67.80 km/sec per Mpc.

So it looks like I can get to within roughly 5% of the accepted value by using a fairly simple analysis. But I think there may be a problem. The way the cross product term is defined, it looks to me like linear velocity and angular velocity are linked (ie, they are not independent of each other). The implication is that if something is far away and moving linearly very fast then the space associated with it must be rotating fast.

Best Regards,

Gary Simpson

Thank you much for your generous comments. You are most gracious. I try to write as simply as possible with short sentences in a linear, logical sequence with clear endpoints. Sometimes I have to leave an idea dangling so that I can merge it with something else.

Regarding your question, I think that I understand your question and I think that the answer is yes. I'll expand on this a little. If you look at Equation 3 in my essay and you make the vectors collinear, the cross product term becomes zero. Now, you only have to worry about the scalar term. I have done the calculation using 13.8 billion light-years as the radius of the universe. This allows me to predict an expansion rate equal to 70.75 km/sec per Mpc. The observed expansion rate is 67.80 km/sec per Mpc.

So it looks like I can get to within roughly 5% of the accepted value by using a fairly simple analysis. But I think there may be a problem. The way the cross product term is defined, it looks to me like linear velocity and angular velocity are linked (ie, they are not independent of each other). The implication is that if something is far away and moving linearly very fast then the space associated with it must be rotating fast.

Best Regards,

Gary Simpson

Dear Gary,

You forewarned that knowledge of quaternions and Linear Algebra would be required to fully grasp your essay. I don't have much knowledge of either but was able to decipher that your essay was trying to bring out something important from the depths using calculus. I am happy that your essay suggests a way of reconciling irreconciliables like Special relativity and the Aether.

My reservation about calculus is how the limit or the infinitesimal can be equated with zero and in the same breadth not zero. This looks like a mathematical trick, even if a useful one. Under a magnifying lens what appears infinitesimal or tending towards zero can be seen to be physically non-zero.

What is your take on these statements about the infinitesimal, dx

dx ~ 0 (~ = indistinguishable from)

neither dx = 0 nor dx ≠ 0

dx^{2} = 0

dx → 0 (→ = becomes vanishingly small)

The statement dx = 0 and dx ≠ 0 cannot both be logically true thus raising suspicion. While not denying the usefulness, suggest to equate dx with the Planck length, 10^{-35}m . You may read my efforts and comment when you have the time.

Regards,

Akinbo

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You forewarned that knowledge of quaternions and Linear Algebra would be required to fully grasp your essay. I don't have much knowledge of either but was able to decipher that your essay was trying to bring out something important from the depths using calculus. I am happy that your essay suggests a way of reconciling irreconciliables like Special relativity and the Aether.

My reservation about calculus is how the limit or the infinitesimal can be equated with zero and in the same breadth not zero. This looks like a mathematical trick, even if a useful one. Under a magnifying lens what appears infinitesimal or tending towards zero can be seen to be physically non-zero.

What is your take on these statements about the infinitesimal, dx

dx ~ 0 (~ = indistinguishable from)

neither dx = 0 nor dx ≠ 0

dx

dx → 0 (→ = becomes vanishingly small)

The statement dx = 0 and dx ≠ 0 cannot both be logically true thus raising suspicion. While not denying the usefulness, suggest to equate dx with the Planck length, 10

Regards,

Akinbo

report post as inappropriate

Akinbo,

Thank you for reading my essay. In truth, I think that all of the FQXI readership is easily able to follow my mathematics but I do feel obliged to give a small warning regarding Linear Algebra and quaternions. Having said that, all that is really needed is to know regarding LA is how to multiply matrices and to understand the meaning of an inverse matrix.

I have read your...

view entire post

Thank you for reading my essay. In truth, I think that all of the FQXI readership is easily able to follow my mathematics but I do feel obliged to give a small warning regarding Linear Algebra and quaternions. Having said that, all that is really needed is to know regarding LA is how to multiply matrices and to understand the meaning of an inverse matrix.

I have read your...

view entire post

Dear Gary,

Can dx have a smallest possible size?

As you pointed out, infinitesimals may not be a difficulty for mathematics, but what of for physics?

Calculus and Cauchy's solution are used to solve Zeno's paradox, what do you think about the last step, which though indeterminate must be taken?

Give these some thought.

Regards,

Akinbo

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Can dx have a smallest possible size?

As you pointed out, infinitesimals may not be a difficulty for mathematics, but what of for physics?

Calculus and Cauchy's solution are used to solve Zeno's paradox, what do you think about the last step, which though indeterminate must be taken?

Give these some thought.

Regards,

Akinbo

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Akinbo,

To me, the infinitesimal can be zero. It is simply a concept. Whether or not it has physical meaning does not alter its usefulness to me as a concept.

For Physics, it does not make any sense to me to think about distances that are smaller than the size of a proton or a neutron since those are the smallest stable particles that have a physical dimension. I'll discuss the electron further in your forum since that is where you pose the question. It is true that both the proton and neutron can be destroyed by high-energy collisions. However, it is not true that either can simply be divided into parts. To do so would simply destroy them.

Regarding Zeno's Paradox for motion ... to me, it is not a paradox. He makes note that you must go 1/2 of the remaining distance each step but he does not mention that at constant velocity it will only take 1/2 of the time of the previous step. To me, Zeno is an example of a missed opportunity. He knows that motion is possible yet his logic gives him doubts. What he needed was a new concept (sum of infinite series) that would connect the two. But he was not able to make the mental step and humans had to wait for 2000 years for Isaac Newton to come along with Calculus. I do not give Zeno a high score ... of course, everything is obvious after someone else shows you the truth.

The discussion between Dr. Klingman and Dr. Maudlin reminds me of Zeno. There is a deeper truth to be had there. I just hope we don't have to wait 2000 years to get a solution.

Best Regards,

Gary Simpson

To me, the infinitesimal can be zero. It is simply a concept. Whether or not it has physical meaning does not alter its usefulness to me as a concept.

For Physics, it does not make any sense to me to think about distances that are smaller than the size of a proton or a neutron since those are the smallest stable particles that have a physical dimension. I'll discuss the electron further in your forum since that is where you pose the question. It is true that both the proton and neutron can be destroyed by high-energy collisions. However, it is not true that either can simply be divided into parts. To do so would simply destroy them.

Regarding Zeno's Paradox for motion ... to me, it is not a paradox. He makes note that you must go 1/2 of the remaining distance each step but he does not mention that at constant velocity it will only take 1/2 of the time of the previous step. To me, Zeno is an example of a missed opportunity. He knows that motion is possible yet his logic gives him doubts. What he needed was a new concept (sum of infinite series) that would connect the two. But he was not able to make the mental step and humans had to wait for 2000 years for Isaac Newton to come along with Calculus. I do not give Zeno a high score ... of course, everything is obvious after someone else shows you the truth.

The discussion between Dr. Klingman and Dr. Maudlin reminds me of Zeno. There is a deeper truth to be had there. I just hope we don't have to wait 2000 years to get a solution.

Best Regards,

Gary Simpson

Gary,

Thanks for your response over at my forum topic. I admit Calculus has achieved a lot but I don't want to let you off so easily :).

You say, to you the infinitesimal can be zero. This runs against the common notion that the infinitesimal is a quantity almost indistinguishable from zero, but not zero, at least as is discussed HERE.

Anyway, the reason for my post is that I have been giving Calculus more thought following your responses. It appears clear that for calculus dx can have no lowest finite value and that being the case, a line would be continuous, having an infinite number of points. Because you have also once considered motion as the creation and destruction of space, and motion as the destruction of the moving object and its creation and reappearance in the next adjacent space, it means all scenarios have at least once featured on your table.

What I want to now know is how the*continuous* line in space can be cut since between any two points there is always a third by definition. Where on this line can you cut, since a point is uncuttable by definition?

Regards,

Akinbo

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Thanks for your response over at my forum topic. I admit Calculus has achieved a lot but I don't want to let you off so easily :).

You say, to you the infinitesimal can be zero. This runs against the common notion that the infinitesimal is a quantity almost indistinguishable from zero, but not zero, at least as is discussed HERE.

Anyway, the reason for my post is that I have been giving Calculus more thought following your responses. It appears clear that for calculus dx can have no lowest finite value and that being the case, a line would be continuous, having an infinite number of points. Because you have also once considered motion as the creation and destruction of space, and motion as the destruction of the moving object and its creation and reappearance in the next adjacent space, it means all scenarios have at least once featured on your table.

What I want to now know is how the

Regards,

Akinbo

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Akinbo,

Thanks for the continued dialog.

I will answer your question concerning making a cut for both a physical structure such as a piece of string and for a mathematical concept such as a continuous line.

Let us say that a piece of string is cut by a knife. The knife is harder than the string. Its atoms are bound to each other more strongly than the atoms in the string. The knife is forced into the space of the string and the bond between some of the atoms in the string is broken. The cut occurs in the space between atoms.

Now let us consider an abstraction such as a continuous geometric line. A cut is to be made between two points but there are an infinite number of such points between any two points ... so where is the cut to be made? Let us be smarter than Zeno and introduce a new concept ....

Let an interval near point x be defined as follows:

(x - delta) < x < (x + delta) where x and delta are both reals

We can now make a cut at point x by taking the limit as delta approaches zero. Essentially, the line is cut into segments by removing point x. Something to remember about real numbers is that they have an infinite number of digits. So, the real number one is 1.000000000 ... ad infinitum. This is equivalent to infinite precision or to taking the limit of the above interval as delta goes to zero.

The more interesting question is what happens to point x? Both line segments approach it but neither segment includes it. I suppose that it could be reconnected to one of the segments but not to both.

Regards,

Gary Simpson

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Thanks for the continued dialog.

I will answer your question concerning making a cut for both a physical structure such as a piece of string and for a mathematical concept such as a continuous line.

Let us say that a piece of string is cut by a knife. The knife is harder than the string. Its atoms are bound to each other more strongly than the atoms in the string. The knife is forced into the space of the string and the bond between some of the atoms in the string is broken. The cut occurs in the space between atoms.

Now let us consider an abstraction such as a continuous geometric line. A cut is to be made between two points but there are an infinite number of such points between any two points ... so where is the cut to be made? Let us be smarter than Zeno and introduce a new concept ....

Let an interval near point x be defined as follows:

(x - delta) < x < (x + delta) where x and delta are both reals

We can now make a cut at point x by taking the limit as delta approaches zero. Essentially, the line is cut into segments by removing point x. Something to remember about real numbers is that they have an infinite number of digits. So, the real number one is 1.000000000 ... ad infinitum. This is equivalent to infinite precision or to taking the limit of the above interval as delta goes to zero.

The more interesting question is what happens to point x? Both line segments approach it but neither segment includes it. I suppose that it could be reconnected to one of the segments but not to both.

Regards,

Gary Simpson

Gary,

Yours is one of the more informed contributions hence my coming back. Your reply resembles the one that Tim Maudlin gave on the question. Mathematically correct, Yes. Physically correct with regards to an extended line? I still have doubts.

For example, what does "interval" mean on a physical line in empty space? Does it not mean some amount of small space? Note that I am not discussing the abstract number line. I am talking of extension, of matter or empty spatial distance.

Then talking of "taking the limit as delta approaches zero". Delta had better not reach zero lest x cease to exist. And for cutting at point x, going by Euclid's definition which I referenced, x can have no parts, so it cannot be cut. Again, x occupies some position, can it be displaced? If so, what does it leave behind on displacement? Wherever, it is displaced to, must also have an x there, can more than one x occupy a point x? I think you get a bit of the dilemma now?

Regards,

Akinbo

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Yours is one of the more informed contributions hence my coming back. Your reply resembles the one that Tim Maudlin gave on the question. Mathematically correct, Yes. Physically correct with regards to an extended line? I still have doubts.

For example, what does "interval" mean on a physical line in empty space? Does it not mean some amount of small space? Note that I am not discussing the abstract number line. I am talking of extension, of matter or empty spatial distance.

Then talking of "taking the limit as delta approaches zero". Delta had better not reach zero lest x cease to exist. And for cutting at point x, going by Euclid's definition which I referenced, x can have no parts, so it cannot be cut. Again, x occupies some position, can it be displaced? If so, what does it leave behind on displacement? Wherever, it is displaced to, must also have an x there, can more than one x occupy a point x? I think you get a bit of the dilemma now?

Regards,

Akinbo

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Akinbo,

Thanks for the continued dialog. I hope that I can give you a fresh perspective on your interesting question.

I will attempt to answer. However, I might need some clarification regarding what you mean by "physical line in empty space". The space cannot be empty if it is occupied by a physical line.

My best interpretation of what you are asking is that you have...

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Thanks for the continued dialog. I hope that I can give you a fresh perspective on your interesting question.

I will attempt to answer. However, I might need some clarification regarding what you mean by "physical line in empty space". The space cannot be empty if it is occupied by a physical line.

My best interpretation of what you are asking is that you have...

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Gary,

Our exchange has been helpful in refining and clarifying my argument (or as it may turn out misconception although I don't think so yet).

- In reply to your request for clarification on what I mean by "physical line in empty space"

To make it have a precedent, let me answer thus. That line is the same line that Newton says in his first law as the path of an object moving...

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Our exchange has been helpful in refining and clarifying my argument (or as it may turn out misconception although I don't think so yet).

- In reply to your request for clarification on what I mean by "physical line in empty space"

To make it have a precedent, let me answer thus. That line is the same line that Newton says in his first law as the path of an object moving...

view entire post

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Gary,

I wrote a form of your derivation in the following that I call quaternion notes.

Cheers LC

attachments: quaternion_notes.pdf

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I wrote a form of your derivation in the following that I call quaternion notes.

Cheers LC

attachments: quaternion_notes.pdf

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Lawrence,

Many thanks ... it looks very similar indeed. I am curious, is this new material or is this old material. Since I am an engineer rather than a mathematician, I do not know what is known ... if that makes sense to you.

I think there is some disagreement however ... you indicate that ij = jk = ki = ijk = -1. This is not what Hamilton states. Hamilton states I^2 = j^2 = k^2 = ijk = -1. Therefore, ij = k; jk = i; and ki = j.

Also, it is not necessary to use the product rule to get the differential with respect to a quaternion although what you present is certainly correct. Part of what I wanted to show was that the quaternion functions should be viewed as a system rather than as four separate problems. The trick then is to solve the system in one step.

If you want to see something really cool, take a look at the quaternion exponential function. It is in the paper at this URL:

http://vixra.org/abs/1412.0257

Jump down to pages 13-15

Thanks for the feedback.

Best Regards and Good Luck,

Gary Simpson

Many thanks ... it looks very similar indeed. I am curious, is this new material or is this old material. Since I am an engineer rather than a mathematician, I do not know what is known ... if that makes sense to you.

I think there is some disagreement however ... you indicate that ij = jk = ki = ijk = -1. This is not what Hamilton states. Hamilton states I^2 = j^2 = k^2 = ijk = -1. Therefore, ij = k; jk = i; and ki = j.

Also, it is not necessary to use the product rule to get the differential with respect to a quaternion although what you present is certainly correct. Part of what I wanted to show was that the quaternion functions should be viewed as a system rather than as four separate problems. The trick then is to solve the system in one step.

If you want to see something really cool, take a look at the quaternion exponential function. It is in the paper at this URL:

http://vixra.org/abs/1412.0257

Jump down to pages 13-15

Thanks for the feedback.

Best Regards and Good Luck,

Gary Simpson

Sorry for being so late, but I have not been on FQXi much the last week. You might want to look at Soiguine's paper in this contest. It is rather complicated, but it works with the geometric algebra of Hestenes and Clifford algebras. The quaternion product is a Clifford algebra.

Cheers LC

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Cheers LC

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You have scaled heights with your subjective work.

Long way to go!

Sincerely,

Miss. Sujatha Jagannathan

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Long way to go!

Sincerely,

Miss. Sujatha Jagannathan

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Comment

This essay is really quite clever, although the math is somewhat difficult. The quaternion nature of 4-space is very nicely and clearly laid out and the device of imagining Newton using Hamilton's quaternion algebra to presage relativity is alluring. But Hamilton was a smart guy too...maybe even smarter than Newton. So why didn't Hamilton come to relativity?

It just seems such a shame that no one has been able to link the quaternion algebra of relativity with the quaternion algebra of quantum spin. We really should have a quantum gravity...

2.0, entertaining

2.0, well written

1.0, understandable

2.5, relevance to theme

7.5

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This essay is really quite clever, although the math is somewhat difficult. The quaternion nature of 4-space is very nicely and clearly laid out and the device of imagining Newton using Hamilton's quaternion algebra to presage relativity is alluring. But Hamilton was a smart guy too...maybe even smarter than Newton. So why didn't Hamilton come to relativity?

It just seems such a shame that no one has been able to link the quaternion algebra of relativity with the quaternion algebra of quantum spin. We really should have a quantum gravity...

2.0, entertaining

2.0, well written

1.0, understandable

2.5, relevance to theme

7.5

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Steve,

Thank you for having a read and giving me some feedback. You are most kind.

You are very correct regarding Hamilton ... he was a smart guy ... a child prodigy from some accounts. Unfortunately, he died prior to the publication of his textbook on quaternions. His son took the responsibility of getting the work published. He worked on the problem until the end of his life. No one can ask more. RIP.

Maxwell originally formulated his work using quaternions but from 1890 - 1895 there was a serious conflict in mathematics between advocates of quaternions and those who favored tensors and four vectors. Quaternions lost and were almost completely forgotten.

I have seen a few things on the internet that indicate that Einstein and Dirac both knew of quaternions. Why they did not attempt to use them, I do not know.

Regarding using them to link QM with gravity ... I have a few ideas but there is much work to do. As I noted in my essay, the inverse of the square of the distance occurs very naturally.

Best Regards and Good Luck,

Gary Simpson

Thank you for having a read and giving me some feedback. You are most kind.

You are very correct regarding Hamilton ... he was a smart guy ... a child prodigy from some accounts. Unfortunately, he died prior to the publication of his textbook on quaternions. His son took the responsibility of getting the work published. He worked on the problem until the end of his life. No one can ask more. RIP.

Maxwell originally formulated his work using quaternions but from 1890 - 1895 there was a serious conflict in mathematics between advocates of quaternions and those who favored tensors and four vectors. Quaternions lost and were almost completely forgotten.

I have seen a few things on the internet that indicate that Einstein and Dirac both knew of quaternions. Why they did not attempt to use them, I do not know.

Regarding using them to link QM with gravity ... I have a few ideas but there is much work to do. As I noted in my essay, the inverse of the square of the distance occurs very naturally.

Best Regards and Good Luck,

Gary Simpson

Hi Gary,

I posted this elsewhere in conversation and I thought I would share this with you to add to our previous conversation.

Here is what Roger Penrose has to say in his book,*The Emperor's New Mind*, p.113… "The system of real numbers has the property for example, that **between any two of them, no matter how close, there lies a third.** It is not at all clear that physical distances or times can realistically be said to have this property. If we continue to **divide up** the physical distance between two points, we should eventually reach scales so small that the very concept of distance, in the ordinary sense, could cease to have meaning. It is anticipated that at the 'quantum gravity' scale (…10^{-35}m), this would indeed be the case.

I think this may help clear up what is meant by dividing a distance. Hence, my asking that assuming, without conceding that the system of real numbers applies to distance, how can a distance be divided if there is always a third element between two elements and going by geometrical considerations these elements are*uncuttable* into parts?

Regards,

Akinbo

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I posted this elsewhere in conversation and I thought I would share this with you to add to our previous conversation.

Here is what Roger Penrose has to say in his book,

I think this may help clear up what is meant by dividing a distance. Hence, my asking that assuming, without conceding that the system of real numbers applies to distance, how can a distance be divided if there is always a third element between two elements and going by geometrical considerations these elements are

Regards,

Akinbo

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Akinbo,

Thanks for the continued dialog.

For someone to say that a distance does not have meaning ... that itself is a meaningless statement to me. The value he gives is 10^-35 meter. I assume that he is referencing the Planck Length. That is shorter than the wavelength of any known radiation. The wavelength of a high energy gamma ray is roughly 10^-12 meter (1 Pico meter). The...

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Thanks for the continued dialog.

For someone to say that a distance does not have meaning ... that itself is a meaningless statement to me. The value he gives is 10^-35 meter. I assume that he is referencing the Planck Length. That is shorter than the wavelength of any known radiation. The wavelength of a high energy gamma ray is roughly 10^-12 meter (1 Pico meter). The...

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attachments: Akinbo_Ojo.pdf

Dear Gary,

You are certainly very good in mathematics and its use. We may not fully agree on some aspects but no matter, as it helps both sides fine-tune their model. In brief, some of the areas of divergence I itemize are:

"The distance that he references is probably closer to the length of a matter wave associated with most or all of the visible universe. So you would need the...

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You are certainly very good in mathematics and its use. We may not fully agree on some aspects but no matter, as it helps both sides fine-tune their model. In brief, some of the areas of divergence I itemize are:

"The distance that he references is probably closer to the length of a matter wave associated with most or all of the visible universe. So you would need the...

view entire post

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Akinbo,

I agree that the dialog is useful. Thank you.

Planck Length ... oops. You are correct. My bad. I was using hyperbole and did not perform a calculation. A wavelength of 10^-35 meter has an energy of 1.986 x 10^11 joules. This is 1.240 x 10^29 eV. The LHC at CERN is 7 x 10^12 eV. So it would require something 15 to 16 orders of magnitude (roughly) more powerful than the LHC to...

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I agree that the dialog is useful. Thank you.

Planck Length ... oops. You are correct. My bad. I was using hyperbole and did not perform a calculation. A wavelength of 10^-35 meter has an energy of 1.986 x 10^11 joules. This is 1.240 x 10^29 eV. The LHC at CERN is 7 x 10^12 eV. So it would require something 15 to 16 orders of magnitude (roughly) more powerful than the LHC to...

view entire post

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

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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

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Gary,

"This text demonstrates that how we think about both Mathematics and Physics can be influenced by the mathematical tools that are available to us." Do math tools come first? Einstein needed clarity for his theory of general relativity, thus utilizing new ventures into Riemannian geometry. Did Math come second here? Or should we say that achievement is built on the foundations of both?

These are questions that you aptly discuss in your essay.

Jim

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"This text demonstrates that how we think about both Mathematics and Physics can be influenced by the mathematical tools that are available to us." Do math tools come first? Einstein needed clarity for his theory of general relativity, thus utilizing new ventures into Riemannian geometry. Did Math come second here? Or should we say that achievement is built on the foundations of both?

These are questions that you aptly discuss in your essay.

Jim

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Jim,

Many thanks for taking the time to read my essay. I think that usually the mathematics comes first. Then when science finds an application for a new concept in mathematics, the mathematicians return to that concept and expand upon it some more. It seems like physics has gotten ahead of math now though.

When Hamilton developed quaternions, I think he was thinking about both math and physics. He knew of developments in electro-magnetism and hoped to incorporate them using anti-commutation.

Best Regards and Good Luck,

Gary Simpson

Many thanks for taking the time to read my essay. I think that usually the mathematics comes first. Then when science finds an application for a new concept in mathematics, the mathematicians return to that concept and expand upon it some more. It seems like physics has gotten ahead of math now though.

When Hamilton developed quaternions, I think he was thinking about both math and physics. He knew of developments in electro-magnetism and hoped to incorporate them using anti-commutation.

Best Regards and Good Luck,

Gary Simpson

Gary,

I also think that it was Euler who reduced Newton's law of motion down to F = ma. Before that no one could understand what Newton was talking about.

You write in a precise and clear manner. I learned something from this essay.

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I also think that it was Euler who reduced Newton's law of motion down to F = ma. Before that no one could understand what Newton was talking about.

You write in a precise and clear manner. I learned something from this essay.

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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

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

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

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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

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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

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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

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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.”

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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.”

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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

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

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

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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

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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.

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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.

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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

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

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.

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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.

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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

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

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

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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

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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

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

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

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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

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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

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.

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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.

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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.

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

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.

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.

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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

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

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

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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

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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

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

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

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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

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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

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

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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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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Dear Gary,

Thank you for your post on my thread.

I posted a reply.

Cheers,

Patrick

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Thank you for your post on my thread.

I posted a reply.

Cheers,

Patrick

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Dear Gary,

My belief is that your excellent essay is worthy of more attention than it has received. I still have not had the free time to work through your derivations to convince myself, but I see nothing to suggest you made any mistakes. I hope that you continue to develop your ideas and I wish to reiterate that I think you will find David Hestenes' Geometric Algebra papers (and books) quite relevant to your interest.

My best regards and appreciation for your comments and kicking the Hornets nest.

Edwin Eugene Klingman

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My belief is that your excellent essay is worthy of more attention than it has received. I still have not had the free time to work through your derivations to convince myself, but I see nothing to suggest you made any mistakes. I hope that you continue to develop your ideas and I wish to reiterate that I think you will find David Hestenes' Geometric Algebra papers (and books) quite relevant to your interest.

My best regards and appreciation for your comments and kicking the Hornets nest.

Edwin Eugene Klingman

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Studying your paper makes me realize just how much dedication you have. You are

You have succeeded...

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John,

Many thanks for taking the time to read and study my essay. I hope it was of benefit to you. If you got a good feel for what I have written, then you picked up most of what I have figured out on the subject over the past few years.

The Lorentz Transform is the cosine term. I am still working through how to apply this to kinematics but it looks very promising.

Regarding the use of quaternions for a 4-D model .... the answer is that quaternions are not applicable to such a model. During the period from 1890-1895, there was a heated debate in the mathematics community regarding the use of quaternions (Hamilton) vs the use of n-vectors (Riemann, Grassmann). The argument for quaternions was that they are uniquely suited to describe 3-D space. The argument against quaternions was that they can not be applied to higher dimensional spaces. Quaternions lost and were essentially abandoned.

It sounds like what you want is a simple 4-vector to apply in Minkowski space-time. That is pretty standard and should not pose a major challenge.

The closest thing that quaternions could offer would be to have one or more of the four terms be a function of time.

Allow me to ask a bit of a snarky question ... Can you point in the direction of time ... or if you prefer, in the direction of i*c*t? If the answer to this is "no", then why do you need or want time as a fourth dimension? In one of the works I have posted to viXra, I show that absolute motion when described using quaternions and Special Relativity produces an effect that is mathematically similar to QM spin and has the bonus of eliminating time as a fourth dimension ... the direction of time becomes linked to the direction of motion.

Best Regards and Good Luck,

Gary Simpson

Many thanks for taking the time to read and study my essay. I hope it was of benefit to you. If you got a good feel for what I have written, then you picked up most of what I have figured out on the subject over the past few years.

The Lorentz Transform is the cosine term. I am still working through how to apply this to kinematics but it looks very promising.

Regarding the use of quaternions for a 4-D model .... the answer is that quaternions are not applicable to such a model. During the period from 1890-1895, there was a heated debate in the mathematics community regarding the use of quaternions (Hamilton) vs the use of n-vectors (Riemann, Grassmann). The argument for quaternions was that they are uniquely suited to describe 3-D space. The argument against quaternions was that they can not be applied to higher dimensional spaces. Quaternions lost and were essentially abandoned.

It sounds like what you want is a simple 4-vector to apply in Minkowski space-time. That is pretty standard and should not pose a major challenge.

The closest thing that quaternions could offer would be to have one or more of the four terms be a function of time.

Allow me to ask a bit of a snarky question ... Can you point in the direction of time ... or if you prefer, in the direction of i*c*t? If the answer to this is "no", then why do you need or want time as a fourth dimension? In one of the works I have posted to viXra, I show that absolute motion when described using quaternions and Special Relativity produces an effect that is mathematically similar to QM spin and has the bonus of eliminating time as a fourth dimension ... the direction of time becomes linked to the direction of motion.

Best Regards and Good Luck,

Gary Simpson

Gary,

Time grows short, so I am revisiting essays I’ve read (3/1/2015) to assure I’ve rated them. I find that I did not rate yours, though I usually do for those I can relate to. I am rectifying that. I hope you get a chance to look at mine: http://fqxi.org/community/forum/topic/2345.

Jim

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Time grows short, so I am revisiting essays I’ve read (3/1/2015) to assure I’ve rated them. I find that I did not rate yours, though I usually do for those I can relate to. I am rectifying that. I hope you get a chance to look at mine: http://fqxi.org/community/forum/topic/2345.

Jim

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Hi Gary,

Thanks for the intellectual exchange during the contest. Hope to engage more on the non-contest blogs after the competition if you are interested. I hadn't rated your essay but now got you within firing range hopefully to make the final list despite the 1-bombings. Hope I make it too but I may not really care that much.

Regards,

Akinbo

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Thanks for the intellectual exchange during the contest. Hope to engage more on the non-contest blogs after the competition if you are interested. I hadn't rated your essay but now got you within firing range hopefully to make the final list despite the 1-bombings. Hope I make it too but I may not really care that much.

Regards,

Akinbo

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It was a pleasure Akinbo. I will periodically visit the forums to see what is up and offer any thoughts that I might have that are useful.

It looks like you will make the finals cut if they accept 40 finalists instead of only 30. As I write this, I am number 43, so no go for me. In any event, good luck.

Best Regards and Good Luck,

Gary Simpson

It looks like you will make the finals cut if they accept 40 finalists instead of only 30. As I write this, I am number 43, so no go for me. In any event, good luck.

Best Regards and Good Luck,

Gary Simpson

Author En Passant replied on Apr. 24, 2015 @ 01:24 GMT unstub

Gary,

I don’t want to insult Sujatha Jagannathan in case she is not an automaton.

Your perception of the “fluency” of her language is right. Her talk seems to me to be “canned” (and I mean that in more ways than one).

But if you are right, then its creators are cheating. They intersperse regular...

view entire post

Gary,

I don’t want to insult Sujatha Jagannathan in case she is not an automaton.

Your perception of the “fluency” of her language is right. Her talk seems to me to be “canned” (and I mean that in more ways than one).

But if you are right, then its creators are cheating. They intersperse regular...

view entire post

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Thanks Gary. I just posted a reply to your well reasoned comment.

All the best,

Akinbo

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All the best,

Akinbo

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Sorry, in addition seeing your interest in the wave equation, what is your assessment/ comment on the correctness of Thomas Erwin Phipps?

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Hello Gary,

Would you mind if I tapped your brain a little? I have a draft of a paper (attachment) and post the abstract below.

Regards and thanks,

Akinbo

*You may reply me here or on my essay blog or better still to: taojo@hotmail.com

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==============================

Abstract: Absurdities arising from Einstein's velocity-addition law have been discussed since the theory's formulation. Most of these have been dismissed as being philosophical arguments and supporters of Special relativity theory are of the opinion that if the math is not faulted they are ready to live with the paradoxes. Here, we now demonstrate a mathematical contradiction internal to the theory itself. We show that when applied to light there is no way to mathematically reconcile the Einstein velocity-addition law with the second postulate of the theory which may have a fatal consequence.

================================================

==========================

attachments: 2__Shorter_version__Application_of_the_velocity-addition_law_to_light_itself.pdf

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Would you mind if I tapped your brain a little? I have a draft of a paper (attachment) and post the abstract below.

Regards and thanks,

Akinbo

*You may reply me here or on my essay blog or better still to: taojo@hotmail.com

===========================================

==============================

Abstract: Absurdities arising from Einstein's velocity-addition law have been discussed since the theory's formulation. Most of these have been dismissed as being philosophical arguments and supporters of Special relativity theory are of the opinion that if the math is not faulted they are ready to live with the paradoxes. Here, we now demonstrate a mathematical contradiction internal to the theory itself. We show that when applied to light there is no way to mathematically reconcile the Einstein velocity-addition law with the second postulate of the theory which may have a fatal consequence.

================================================

==========================

attachments: 2__Shorter_version__Application_of_the_velocity-addition_law_to_light_itself.pdf

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