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RECENT POSTS IN THIS TOPIC

Paul Butler: on 4/26/15 at 4:13am UTC, wrote Dear James, As you requested in your comment to me on my page, I read your...

Neil Bates: on 4/23/15 at 2:39am UTC, wrote Dear James, Your essay is pleasant and readable. It reminds me of the...

KoGuan Leo: on 4/23/15 at 2:34am UTC, wrote Dear James, I looked at your article, I could not find the sentence that I...

Sylvia Wenmackers: on 4/22/15 at 10:28am UTC, wrote Dear James Lee Hoover, Finally, some hints of poetry! :) "The human...

Cristinel Stoica: on 4/21/15 at 12:51pm UTC, wrote Dear James, I enjoyed reading your essay that makes a very good case...

ABDELWAHED BANNOURI: on 4/21/15 at 11:02am UTC, wrote Daer James, The anthropic principle is one of the deepest and most...

Sophia Magnusdottir: on 4/20/15 at 10:30am UTC, wrote Hi Jim, Thank you for drawing my attention to your nice essay :) I think...

Jeffrey Schmitz: on 4/19/15 at 22:49pm UTC, wrote Hi Jim, This work is a poetic juxtaposition of concepts around the theme...


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FQXi FORUM
October 22, 2019

CATEGORY: Trick or Truth Essay Contest (2015) [back]
TOPIC: Connections: Physics, Math, & the Human Brain by James Lee Hoover [refresh]
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Author James Lee Hoover wrote on Feb. 19, 2015 @ 23:26 GMT
Essay Abstract

Euler’s identity, metaphorically, if not in principle, represents an integral connection between physics, math and the human brain. Computational operations, interrelations, combinations, generalizations and abstractions with paths to structure, measurement and transformations – all are represented in one very brief equations. So much math, geometry calculus, and algebra are used in physics to model and theorize physical structures and concepts, that a short representative identity is required. The classical world is what we see and have modeled for centuries. The quantum world requires a speculative, logical, and robust math model to show us how to relate the quantum to the classical. This journey has begun and discoveries have helped to reinforce, actually create, the math tools that continue to be needed.

Author Bio

James Hoover is recently retired from the Boeing Company in Huntington Beach, California, working as a systems engineer. His career in aerospace stretches back over twenty years and involves cost analysis, cost modeling and logistics research. In that span of years he has taught college courses in education, economics, computer science and English. Before his aerospace career, he taught high school. He recently published a science fiction novel called Extraordinary Visitors. His personal interests include studies in particle physics, cosmology, political science, and advanced space flight systems. He has advanced degrees in Economics and English.

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James A Putnam replied on Apr. 8, 2015 @ 00:35 GMT
James Lee Hoover,

A very informative essay. This is not a criticism, it is an admission on my part, I don't yet recognize the central importance assigned to Euler's identity. I will look into it more and think more about it. I do assign central importance to the fine structure constant definition that contains constants that connect major theoretical subdivisions. I feel that Euler's identity must be theoretically important because you point it out. I will work on it. Your essay deserves a high mark.

James Putnam

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Sujatha Jagannathan wrote on Feb. 26, 2015 @ 17:37 GMT
"You've not retired from your learning", I can witness that!

Best Wishes,

Miss. Sujatha Jagannathan

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Author James Lee Hoover replied on Feb. 27, 2015 @ 00:18 GMT
Sujatha,

Thank you for your comment, but having checked out your interesting essay, I am interested in seeing specifics of your perspective on my ideas.

Jim

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Edwin Eugene Klingman wrote on Feb. 26, 2015 @ 23:51 GMT
Dear Jim,

As you note, physics is a science that deals with "matter and energy and their interactions." (I consider the "matter" in this definition to include "anti-matter".) If one, barefooted, kicks a stone, one is left with a very strong feeling that the physical world is real – it matters! But you further note that the union of math and physics "offers more substance"; supplying...

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Gary D. Simpson wrote on Feb. 27, 2015 @ 02:50 GMT
Jim,

Many thanks for an enjoyable read. I would make a revision to your equation for Gt by including $$$$$$ as one of the factors:-) You should read the essay by Shour. You both discuss something similar.

I'm a big fan of Euler but Euler's Equation does not have a zero value. Of course, the real part or the complex part can be zero (one or the other, not both). I tend to view Euler's Equation as a quaternion although it could operate on scalars.

You are completely correct regarding the bigger picture ... the whole is much greater than the sum of the parts ... without any one of the components, we would be limited. I cannot even imagine a level 1 civilization.

Best Regards and Good Luck,

Gary Simpson

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Lloyd Eli Sanchez wrote on Mar. 12, 2015 @ 05:59 GMT
I found your essay very interesting and very well written. Shows a lot

of thought went into preparation of this composition.

While you are correct in stating that "physics depends on the human

brain and accompanying senses to investigate the physical world around

us", the human brain also serves to quantify or describe the

mathematical interactions that describe physical reactions.

The digital computer operates under different conditions than does the

human brain, but this is due to digital computer pre-programming which

makes the computations of known mathematical interactions much more rapid.

The computer can't at this time observe "strange" phenomenon, conclude

that it is a "strange" phenomenon, then classify and perform an

analysis on its own. It can only record strange phenomenon, its limits

being set by programming efficiencies. The human brain however has no

limits to its exploration of the universe and the formation of theories,

although its ability to quantify or describe physical interactions may

suffer from the unknown properties, both mathematical and physical, of

the phenomenon observed.

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Akinbo Ojo wrote on Mar. 17, 2015 @ 13:28 GMT
Hi James,

Well done on your interesting contribution. While I agree with many aspects, I am reluctant to resign to the imaginary number √-1 on the grounds that there aspects of physical reality beyond our comprehension. Unlike Newtonian hypotheses non fingo, I suggest we can be able to figure out hypotheses and test them.

All the best in the competition,

Akinbo

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Peter Jackson wrote on Mar. 24, 2015 @ 22:25 GMT
James,

Thanks for your comments on mine. I now see the basis of your questions. Yes, I agree that entanglement and Godel incompleteness are keys to improving understanding, also that it's the 'way we employ' maths that leads to flaws there, so indeed our minds/brains inadequately evolved, You'll recall my previous essays agree with your identification of the helix as fundamental. Indeed I've now found it even more important than we've realized; See this short video;

Time Dependent Redshift hypothesis.

I found your essay a nice easy read and a quite comprehensive well written eulogy on the merits of mathematics as a tool. You don't identify the abuses, but they are, hopefully, few. I see you seem to agree those I identify, including the great red/green sock switch con trick of QM, confounding logic!

I don't think your score represents the essay and think it should be higher. I'm certainly marking it so. Best of luck in the competition.

Peter

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Peter Jackson wrote on Mar. 25, 2015 @ 09:41 GMT
James,

I saw your note on mine. Someone's trolling. Mine just dived too - I'm sure without even being read! I've just added your (rather higher!) score now, in line with my honest comments (now 4.6).

Best wishes,

Peter

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Thomas Howard Ray wrote on Mar. 25, 2015 @ 16:59 GMT
Jim,

Though it took me a while to get here, it was worth the trip. We find beauty in the same mathematical structures and equations, and largely for the same reasons.

Thanks for commenting in my forum. You get my top vote, and best wishes in the competition.

Best,

Tom

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Michel Planat wrote on Mar. 26, 2015 @ 11:58 GMT
Dear James,

Always glad to detect people of this audience that have faith in mathematics and QM and ability for connecting the two. I like your reference to photosynthesis and entanglement, for QM and biology I also like http://arxiv.org/abs/1310.7761

As for the physical interpretation of Euler's identity, I propose the Bloch sphere representation of the single qubit, in which the south pole |1> and e^(i*pi)|1>=-|1> are the same. In group theoretical terms, this is the two to one homomorphism of SU(2) on SO(3). Yes mathematics is the right language for nature.

What you say about DNA and maths is very challenging and interesting. May be we already have enough maths that can be used in physics and biology to interpret some paradoxes and mysteries.

Best.

Michel

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Michel Planat wrote on Mar. 26, 2015 @ 16:39 GMT
Dear James,

Thank you for your positive feedback. You deserve a similar compliment. You gave me to think about Euler's identity. About chemistry and biology, I have a few ideas based on the same free group (of Grothendieck) that I use for my monstrous topic.

All the best,

Michel

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Torsten Asselmeyer-Maluga wrote on Mar. 27, 2015 @ 13:38 GMT
Jim,

thanks for writing this essay. It contains a lot of ideas and conclusions to agree with. As you know from ym essay, I'm really interesting into the relation between the disciplines like biology, sociology, physics, math etc. Your essay covered all these question.

It reminds me on a discussion with a biophysicist about consciousness and quantum mechanics. New experiments seem to imply that quantum mechanics is needed to get consciousness and higher brain functions. You explained it also at the example of birds finding their route.

Therefore you will also get a high rate from me.

Best

Torsten

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Vladimir F. Tamari wrote on Mar. 29, 2015 @ 01:05 GMT
Dear Jim

I really enjoyed your well thought out and lucidly written essay. Your ideas were presented so well the essay may well have been enjoyed as an after-dinner convention lecture for a group of engineers. As you noted in your comment on my page many of our ideas overlap, for example the highlighting of the Math Brain Physics trinity, or of biological phenomena exhibiting and mediating mathematical functions.

I learned some interesting things from the essay, for example why the Euler formula converts to sine cosine curves when adding a z dimension - or about quantum coherence (entanglement) in photosynthesis - wonderful.

You stress rather too strongly the difference between mathematics and computers while one can argue that digital computation is just a mathematical 'trick' discretizing continuous functions, and utilizing logical operations that are at the base of all mathematics.

As you may know my ideas of physical reality and the theories describing it are rather at odds with what is accepted unconditionally by the physics community, particularly regarding foundational concepts of quantum mechanics. Interpreted from the realism of my Beautiful Universe theory, Bell's Theorem appears as an unnecessary obfuscation of simple transfer of angular momentum across space with the local, causal and realistic interactions of the ether nodes acting like a slippery 'gear train'. Please read Edwin Eugene Klingman's essay debunking Bell.

In a similar vein I also highlight that mathematics, as a product of the mind, can be tricky, supporting multiple scenarios of the same phenomena, whilst physical reality should only have one. Hmmmm.

With best wishes

Vladimir

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Michel Planat wrote on Apr. 1, 2015 @ 08:07 GMT
Dear James,

Can you explain "The cat map reminds me of machinations of Euler's Identity" ? Before I passed to QM I worked a lot on chaos in relation to the understanding of 1/f noise that I finally saw as number theory in experiments. Thanks to you, this year, I realized that Euler's identity has to be kept in mind in relation to the Bloch sphere. As Riemann sphere R is just another representation (a la Felix Klein) of the Bloch sphere, Euler's identity also has a meaning in this context. I just gave a reference on Zivlak's blog. My today favorite objects correspond to three punctures on R or other Riemann surfaces with genus (dessins d'enfants). I spent some time seing them as the molecules of chemistry and biology with moderate success until now (unpublished work).

Cheers,

Michel

ps/ I also rated your work highly a few day ago following our interaction.

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Michel Planat wrote on Apr. 2, 2015 @ 06:51 GMT
Dear James,

Thanks for your note.

There is a blog at the Mathematics Stack Exchange under the title "Euler's identity: why is the e in e^ix? What if it were some other constant like 2^ix?" reminding that a^ix is a logarithmic spiral.

Myself, at the moment, I start from the Riemann sphere R= Complex numbers union infinity as a way to approach the multivaluedness of mathematical knowledge following the giants: Henri Poincaré, Felix Klein and Alexandre Grothendieck. There is the online book by Lando and Zwonkin "Graphs on surfaces and their applications" (2004) where many aspects of this subject are explained. You may be interested to read it.

Best,

Michel

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Joe Fisher wrote on Apr. 2, 2015 @ 14:55 GMT
Dear James Lee Hoover,

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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Janko Kokosar wrote on Apr. 5, 2015 @ 13:08 GMT
Dear James L. Hoover

I see that you deepen in quantum biology. It is necessary to mention that theory of quantum smell of Luca Turin exist also. Besides, a presumption exists also, that number of four base pairs in DNA is a consequence of quantum mechanics. Quantum photosyntesis or quantum smell was already confirmed with a sure result of experiment, I do not know which one. What do you think about Tegmark’s proof, that quantum consciousness does not exist. Do you think that Tegmark’s proof is still valid?

When it will be proven, that consciousness is a quantum phenomenon, the model for it still ever will be necessary. One model is written by me. But it is speculatively, how quantum consciousness changes randomness of quantum measurement. Thus, it is a speculation, that spin 1 of photon can happen more frequently that spin -1. But, Slvain Poirer claims in his essay that he read articles which claims this.

You really gave interestion Euler’s formula.

But, I do not believe that your mentioned telescope will observe antigalaxies. Namely, in Feynman’s ''Lectures On Gravitation'' it is proved that antimatter moves in gravitational field on the same way as the common matter. I hope that experiment in CERN will show, how it is with this.

My essay

Best regards

Janko Kokosar

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Christian Corda wrote on Apr. 8, 2015 @ 10:32 GMT
Hi Jim,

As I promised in my Essay web-page, I have read your nice Essay. Here is a couple of comments:

1) I see that you claim that "With the Santilli telescope, the first detection of anti-matter galaxies occurred, utilizing a new isodual mathematics". I am not sure be this. In fact, on one hand, Santilli's conception of anti-matter is different than the common one as he claims that anti-matter has negative mass (and this violates CPT theorem and Lorentz invariance as consequence). On the other hand, assuming that anti-matter galaxies exist, we should see the night sky completely filled by gamma rays, which we obviously do not see. Assuming the existence of anti-matter galaxies implies indeed also the inter-galactic medium to be filled of anti-particles in addition to particles.

2) I am fascinated by the Eulero identity. I recently found a connection on it in my research on black holes. I was thinking to write my Essay on this issue, but, at the end, I preferred to wrote my Essay on Mossbauer experiment as new proof of general relativity.

In any case, the reading of your pretty Essay was very interesting and enjoyable for me. Thus, I am going to give you an high score.

I wish you best luck in the Contest.

Cheers, Ch.

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Christian Corda wrote on Apr. 9, 2015 @ 12:20 GMT
Dear Jim,

Thanks for the point you raised in my FQXi page. It permits indeed to clarify my position with Santilli also here in FQXi. Santilli is indeed considered a crackpot and a crank by the Scientific Community, see here. Differently from this general judgement, I think that there are parts of Santilli's research which should deserve a better attention, in particular Santilli's research...

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Author James Lee Hoover replied on Apr. 9, 2015 @ 16:56 GMT
Christian,

Not being part of the upper echelons of physics, I appreciate being informed on such anti-matter studies and about reputations in scientific studies. The discovery of antigalaxies seemed questionable, even with my limited knowledge. It is sad that such studies can be elevated in importance in the popular media. It speaks to all the deceptions we see in politics and government. Perhaps they can't be separated because the media is no longer a responsible "Fourth Estate."

Thanks for the scoop.

Jim

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Anonymous wrote on Apr. 10, 2015 @ 06:27 GMT
Dear Jim,

Thank you once again for the comments you left on my essay's page. Yours is certainly a very eclectic essay that touches on many points of contact between mathematics and science. Like you, I find Euler's identity absolutely fascinating, and I wish you good luck in the contest!

Marc

P.S. I am glad Christian Corda (in the thread above) put in perspective Santilli's strange claim of "antimatter galaxies" that you refered to in your essay. It did seem quite peculiar, and indeed it was!

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Member Marc Séguin replied on Apr. 10, 2015 @ 06:29 GMT
That **** auto-log-off issue again! The previous post was mine! :)

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Member Tejinder Pal Singh wrote on Apr. 11, 2015 @ 04:17 GMT
Dear Jim,

Your easy to read essay has a nice collection of thoughts and observations, ranging from Euler's identity to quantum biology. It was nice to see the reference to Khalili's book on quantum biology, which we have enjoyed reading. From the little that we know, the case for quantum entanglement is good in magneto reception and the avian compass; whereas when it comes to photosynthesis, we have of late witnessed skepticism and assertions that the case for entanglement has been overplayed.

You made one statement namely that imaginary numbers appear in quantum theory because there are things in the theory which are hard to imagine. We cannot agree with this Jim! :-) imaginary numbers have a very concrete and well defined role in the theory, without which it would be impossible to formulate the theory. We trace their origin in the theory to the discrete relation between energy and frequency.

You might have already seen this...the book Where mathematics comes from? by Lakoff and Nunez has a very nice discussion of the cognitive origin of the Euler identity.

And Jim we do not seem to find in your essay an explanation for the central question as to why mathematics is so successfully employed in physics. Wonder what your thoughts on this are.

Thank you for an enjoyable read, and with kind regards,

Anshu, Tejinder

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Member Tejinder Pal Singh replied on Apr. 11, 2015 @ 14:09 GMT
Many thanks Jim, for responding on our page, to our post above.

Anshu, Tejinder

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Domenico Oricchio wrote on Apr. 12, 2015 @ 14:26 GMT
Thank you for reading my essay, in a sea of essays.

I read your essay some weeks ago, like everyone else, and I reread it.

I understand the possibility of black hole of antimatter, or antimatter planets, but it seem unlikely the existence of galaxy of antimatter; anyway I don't understand the difference of a photon from antimatter, and a photon from matter, so that it seem that the Santilli's telescope is an extravagance.

The Euler's identity is a bridge between different branches of mathematics, so similar to the Einstein field equations that condense centuries of theory in a simple, and beautiful, equation.

I don't know if the Turing completeness can be applied to the brain-computer equivalence, so that a brain of an elementary organism can be simulated by a usual computer (and in a next time a classical supercomputer could simulate parts of human brain), but it seem possible.

The analogy between mathematics and poetry is perfect; there is pure invention, without restriction except in the composition rules.

I write an essay in a reversing the order of the propositions, so that I try a Wittgenstein's numbering to play with the readers (a citation), and to point out that human knowledge has not a single time evolution, but many possible.

Domenico

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Member Noson S. Yanofsky wrote on Apr. 13, 2015 @ 04:58 GMT
Dear Jim,

Thank you for an interesting essay. I will never look at the Euler Identity in the same way.

All the best,

Noson Yanofsky

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Alma Ionescu wrote on Apr. 13, 2015 @ 17:52 GMT
Dear James,

Thank you for a most interesting read! I enjoyed how you approach the fundamental relations like Euler's identity and yet keep your work informed of the newest research, as the cases from quantum biology you so well describe.

You are making an interesting case, that math today is expanding in purpose and scope from the basic representations of the natural world to the more convoluted ones from biology and genomics. This case makes Winger's question sound a bit dated and a normal consequence of an epoch where the circumstances seemed to favor it only because the advancements of the science were even more limited back then.

Wish you good luck in the contest!

Warm regards,

Alma

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Alma Ionescu replied on Apr. 13, 2015 @ 21:13 GMT
Thank you, Jim! I just rated yours too. Apologies for taking so long to get back to you

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Jonathan J. Dickau wrote on Apr. 14, 2015 @ 04:45 GMT
Enjoyable and fun Jim!

I like what most of what you wove in to your story. The Euler identity connection is a gem. The Santilli telescope and his isodual Math; I am not so sure. I have read some of the papers on those topics, after receiving an e-mail announcement, and I remain unconvinced. Overall, I like your essay; it makes sense. I'll probably have more to say later.

Regards,

Jonathan

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William T. Parsons wrote on Apr. 17, 2015 @ 17:35 GMT
Hi Jim--

Your essay is a great read. Very well written, even though it covers a wide range of fascinating topics. Nicely done. Like everyone else, I especially like your use of Euler's equation. It has always been one of my favorites. Also, thanks for bringing me up-to-speed on the topology of civilizations types.

Quick question: Is the equation involving Gt on page 3 your work? If so, how did you derive it?

On a more personal note, as a pilot, I've always respected Boeing aircraft. Did you ever work on the Triple7? A truly fantastic airplane.

Best regards,

Bill.

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Jeffrey Michael Schmitz wrote on Apr. 19, 2015 @ 22:49 GMT
Hi Jim,

This work is a poetic juxtaposition of concepts around the theme of math and science, which has much to say in-between the lines. The metaphor of Euler’s equation being a poem is carried throughout the work. The style of how ideas are linked, makes it difficult for this reviewer to agree or disagree with the thesis because, as I said, the interesting stuff is like a poem, more in the unvoiced than on the written page.

Hope you do well,

Jeff Schmitz

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Sophia Magnusdottir wrote on Apr. 20, 2015 @ 10:30 GMT
Hi Jim,

Thank you for drawing my attention to your nice essay :) I think you are touching on many good points here, especially for what the role of computers is concerned, and their relation to the human brain. I am not a big fan of the singularity - it seems to me too extreme - but still I think that we will see quite dramatic developments in the near future.

-- Sophia

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ABDELWAHED BANNOURI wrote on Apr. 21, 2015 @ 11:02 GMT
Daer James,

The anthropic principle is one of the deepest and most powerful concepts for understanding our existence. shoud be understood.Indded, I consider my essay an extension to yours.

Sincerly yuors

Bannouri

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Cristinel Stoica wrote on Apr. 21, 2015 @ 12:51 GMT
Dear James,

I enjoyed reading your essay that makes a very good case interweaving many parts of science. Thinking of how complicated biology is from a mathematical standpoint makes it easy to forget that biology is at least partially describable after all. I enjoyed very much the part about Euler’s identity, the apple of so many mathematicians’ eyes. “A wormhole between separate branches of math,” – thank you for reminding me of that quote! The diversity of the topics you approach is admirable and your writing style is great! Wish you the best of luck in your work and in the contest

Cheers,

Cristi

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Member Sylvia Wenmackers wrote on Apr. 22, 2015 @ 10:28 GMT
Dear James Lee Hoover,

Finally, some hints of poetry! :)

"The human brain is analogue; computers are digital" Is the distinction really that clear-cut? Many analogue signals can be used as the basis for digital computing (we just need to distinguish two intervals on the analogue signals) and we can use multiple discrete signals (like single neurotransmitter molecules) to approximate an analogous signal. (I did check the reference you give for this, which mentions the analogue, temporal aspect of neural firing.) What I find curious, somewhat related to this, is that within physics we often switch between discrete and continuous models as well.

Best wishes,

Sylvia Wenmackers - Essay Children of the Cosmos

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KoGuan Leo wrote on Apr. 23, 2015 @ 02:34 GMT
Dear James,

I looked at your article, I could not find the sentence that I wrote above with Peter Jacson on his article on Bell's non-locality that :

"Our Multiverse is infinite, it can not be constrained by limited rule this or that. It must be both this and that simultaneously. It must be both this and that simultaneously."

When I wrote "all this and all that" I had Zhuangzi and Huishi's picture of reality in mind that everything connecting as one. I found this idea is so beautiful and philosophically fullfiling as an earthling living in Sagan's blue dot within an immense Multiverse as the creation of our Ancestor Qbit in his Qbit-centric perspective of creation by itself, for itself and of itself. We and Existence are living in Qbit. Qbit is love with its operating system of Giving first Taking later (love) to create and distribute all that is.

I am glad you invited me to check on your article and I found your article indeed has similar idea as mine and it is well executed and beautiful and fresh. We are indeed connected. We are indeed one with diversity of ideas and myriads of characters. We are also unique in our own way. None is a clone of another. Similarity with a difference.

KQID Zeroth Law is based on KQID-Euler- Einstein combined. I also found Euler is perhaps the greatest mathematician ever.

Best,

Leo KoGuan

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Neil Bates wrote on Apr. 23, 2015 @ 02:39 GMT
Dear James,

Your essay is pleasant and readable. It reminds me of the literate articles written for educated laymen by good scientific communicators, in magazines like "Natural History" or "Harpers". I don't know if there is a specific "innovation" there, but you set the foundational table for readers, and send them looking in various promising directions. You too, appreciate the relevance of "the mind" to these questions. Lately and as noted in your essay (per photosynthesis), I'm heartened to see that quantum mechanics surely does play an important role in biological processes. I am gratified that supposed demonstrations of the inapplicability of QM in cellular systems, such as from decoherence times, don't really block such relevance. I suspect Stuart Hammeroff is right about QM playing an important role in the brain (see my previous FQXi essay for more.)

Here is a minor oddity in the general good sweep of things: your reference to anti-matter galaxies is strange and not convincing, since any such claim is controversial and would require further research and not just faith in a particular unorthodox theory. Also, antimatter should emit ordinary light due to requirements of Maxwell's equations etc. I point out this little wart (;-) as "FYI", your essay is still a good one.

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Paul N Butler wrote on Apr. 26, 2015 @ 04:13 GMT
Dear James,

As you requested in your comment to me on my page, I read your paper and went over it enough to allow me to make a comment on it as you requested. I will post this comment both on your page and mine, so those who read it can easily get the whole picture of how it fits into your comment on my page. To accomplish that completely, I am also including your comments to me on the...

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