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FQXi FORUM
May 25, 2019

CATEGORY: It From Bit or Bit From It? Essay Contest (2013) [back]
TOPIC: Bit from it. Mathematical Clarification by Michael Alexeevich Popov [refresh]
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Author Michael Alexeevich Popov wrote on Jun. 26, 2013 @ 17:30 GMT
Essay Abstract

R.V.L. Hartley in 1928 and C.E. Shannon in 1948 have formulated a fundamental assumption that the most natural choice for any future information theory is the logarithmic function. This assumption became common heuristic in contemporary physics, computer science and quantum information theory. Nevertheless, mathematically speaking, we may suppose that Hartley-Shannon Assumption [HSA] needs taking justification seriously. Our note represents the first attempt of reexamination of such sort of popular assumption and we show how an innocent number-theoretical finding by Shannon became“ unreasonably effective “ abstraction in contemporary theoretical physics, experimental quantum mechanics and cosmology.

Author Bio

Independent researcher and founding director of small private lab.

Download Essay PDF File

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Satyavarapu Naga Parameswara Gupta wrote on Jun. 28, 2013 @ 00:26 GMT
Dear Popov,

Thank you for presenting your nice essay. I saw the abstract and will post my comments soon. I accept you point of view sir,

I am requesting you to go through my essay also. And I take this opportunity to say, to come to reality and base your arguments on experimental results.

I failed mainly because I worked against the main stream. The main stream community...

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Hoang cao Hai wrote on Jun. 28, 2013 @ 04:48 GMT
Send to all of you

THE ADDITIONAL ARTICLES AND A SMALL TEST FOR MUTUAL BENEFIT

To change the atmosphere "abstract" of the competition and to demonstrate for the real preeminent possibility of the Absolute theory as well as to clarify the issues I mentioned in the essay and to avoid duplicate questions after receiving the opinion of you , I will add a reply to you :

1 . THE...

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Vladimir Rogozhin wrote on Jun. 29, 2013 @ 08:57 GMT
Hello, Michael Alexeevich!

You have a very interesting radical ideas on the foundations of mathematics and information theory! Est li u Vas esse na russkom yazyke i drugie raboty? Ne smog nayti vashu stranitsu v seti i vashu pochtu...Moya pochta v moem esse, S uvazheniem, Vladimir

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Author Michael Alexeevich Popov replied on Jul. 6, 2013 @ 14:20 GMT
Vladimir,

Thank You. Vitaly Ginzburg published my article in Uspekhi Physics ( Dec 2003 :173,1382-1384 )I await new website for my lab soon.

Michael

mchlpopov@yahoo.co.uk

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Hoang cao Hai wrote on Jun. 29, 2013 @ 21:45 GMT
Dear Michael

You are still the true fans of Mathematics , so, how is more reasonable?

http://fqxi.org/community/forum/topic/1802

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James Lee Hoover wrote on Jul. 3, 2013 @ 20:10 GMT
Michael,

If given the time and the wits to evaluate over 120 more entries, I have a month to try. My seemingly whimsical title, “It’s good to be the king,” is serious about our subject.

Jim

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Peter Jackson wrote on Jul. 4, 2013 @ 14:47 GMT
Michael,

I'm not a mathematician, but am familiar with geometries. Reading your very concise and interesting essay I started to experience a realization, an apparent hidden likeness with an element of my own work. I'f never considered the Logarithmic qualities of bits, indeed I'm not that familiar with Logarithms.

I find in my essay that between each pair of integers is an inverse Bayesian probability (Quantum uncertainty) distribution equivalent to Godel's n-valued logic, and infer that there are indeed infinitely many smaller cases, which I discuss as in Sample Space, but which may also be considered as Bloch Sphere vectors, which I express as helical. Quantum uncertainty seems to reduce as a fractal or logarithmic function through ever smaller spaces limited only by gamma.

I also suggest the position of the 'Dirac line' to perhaps help clarify mathematics and possibly release both maths and reality from the limitations each impose on the other.

Then I came across your; "Lemma 8. Every real line contains infinitely many pairs of conjugate imaginary bits."

You may have a better picture of what the Logarithmic connection may be and mean for my own conception. See "The Intelligent Bit". I look forward to your views and advice.

Thank you very much for your own essay. it should be rated higher and I'll be very pleased to assist. I didn't anticipate the great value that I think may emerge.

Peter

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Author Michael Alexeevich Popov replied on Jul. 6, 2013 @ 13:58 GMT
Peter,

I agree that your hypothetical quantum computer (HQC) requires a set of coupled quantum bits ( IQbits ) with control gates to manipulate them. You predicted also that such HQC can test some theorems of quantum theory and philosophy.May be.

Today, however, the most promising candidates for the role of quantum bits come from the field of solid state superconducting qubits ( phase, flux and transmon qubits ). 2000-2013 experiments have shown how to entangle and operate these qubits in lab. In fact, my lemmas of mysterious "complex algebra of qubits " attempt to construct new kinds of qubits ( in particular, lemma 11 has exact correspondence with so-called " the flux qubit " ( superconducting qubit) which is based on idea of the cycle in laboratories of Jale,Santa-Barbara,Karlsruhe and Skolkovo.

I believe that mathematical construction of new kinds of qubits and superposition experiments ( in my lab ) may open ways to future quantum computations, having some unexpected applications even in astrophysics.

Respectfully

Michael

mchlpopov@yahoo.co.uk

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Peter Jackson replied on Aug. 4, 2013 @ 20:18 GMT
Michael,

I agree. I hope you'll also comment on commonality with my geometrically based ontology on my blog if you have time to read it. It is about rather more than an IQbit. My last two essays are precursers of the full dynamic unification model.

Very well done, and best wishes for making the top 40.

Peter

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Vladimir Rogozhin wrote on Jul. 10, 2013 @ 14:58 GMT
Hello, Michael Alexeevich!

I found your article in the journal Advances in physics (Успехи физическ
80;х наук 2007/12). It is extremely important conclusions at the end of the article and in the beginning: "One way or another, the evolution of scientific knowledge is the strikingly close to the idealistic picture of nature." Maybe this is consistent with the opinion of Ludwig Faddeev that "mathematics closes physics" (See "The equation of the evil spirit" - Expert № 29-2007 http://expert.ru/expert/2007/29/faddeev/

Sincerely,

Vladimir

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Author Michael Alexeevich Popov replied on Jul. 12, 2013 @ 13:14 GMT
Vladimir,

General.

I consider Ludwig D.Faddeev as an anthropologist of science, investigated cross-cultural differences between physical and mathematical research cultures. Indeed, Physics is not Mathematics, but their dialogue contains metaheuristics and moreover it is very important for human progress. After Einstein’s historical attempts to produce a new kind of “true”...

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Author Michael Alexeevich Popov wrote on Jul. 13, 2013 @ 13:59 GMT
SOME DEFINITIONS.

Dear professor Corda,

Because FQXi contest is not pure scientific forum, I’d like to introduce some common definitions on BHIP ( may be, for readers - poets and philosophers if You agree )

Following Hawking, the black hole (BH) information paradox started in 1967 when Werner Israel showed that the Schwarzschild metric was the only static vacuum black hole...

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Christian Corda replied on Jul. 22, 2013 @ 08:43 GMT
Dear Michael,

Thanks for your particular approach to the BHIP. The metaphor of an encyclopaedia works in my case too. Here, the difference with Hawking's approach is that the emitted radiation is not strictly thermal. Now, the encyclopaedia is not more burned. Instead, one can think as its internal pages have been cut and cut and cut.... an enormous number of times. In other words, the encyclopaedia becomes an enormous puzzle. My mathematical solution permits to reconstruct the puzzle. Thus, it also a final solution of your pseudo-problem.

I have just read your pretty Essay, I am going to comment it at the end of this page.

Cheers,

Ch.

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Author Michael Alexeevich Popov wrote on Jul. 16, 2013 @ 10:58 GMT
Gennady,

Doubters may suggest that in comparison with energy conservation law, Hawking's law of information conservation (The information remains firmly in our universe. Thus, If you jump into a black hole, your mass energy will be returned to our universe but in a mangled form which contains the information about what you were like but in a state where it can not be easily recognized( 2005 )) can be violated ? Moreover, there is no such thing as physical measurement of bits of thermal information in physics .

( copy of comment for Gennady Gorelik )

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Author Michael Alexeevich Popov wrote on Jul. 16, 2013 @ 11:37 GMT
Dear Helmut,

I have some doubts on 'In physics everything is basically bi-polar energy '? As is known, Levi-Strauss, founder of French structuralism ,showed that the whole structure of primitive thought is binary as well. However, tendency to operate with binary counters in all situations in physics could be understood also as taking simplification seriously ?

( copy of comment for H.A.von Schweizer )

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Author Michael Alexeevich Popov wrote on Jul. 16, 2013 @ 13:59 GMT
Dear Jan,

my remark :

Shannon and Other Structuralisms

In comparison with Wheeler Structuralism ( 0-1 structure underpins all of our data and all our sciences ) and Category Structuralism ( by Roman Jakobson, Levi-Strauss, Burbaki, Abramsky and Coecke ) where the universals of human culture exist only at the level of structure,'culinary triangles' or other intuitive ordered things ; Shannon bits mathematics is pure mathematics. Shannon mathematical assumption is based on elementary algebra of common logarithm logax, where the logarithm of x to the base 2 is defined by the equation y = log2 x. This definition is of course applicable only when y is RATIONAL NUMBER. Beyond Shannon rational bits there exists new world of real, irrational, imaginary and complex bits, the world of Weierstrass theorem and the world of real lines contained infinitely many imaginary bits.

Thus, it could be difficult to find real differences between Wheeler binarism and Alternative Gospel of structures , indeed. In fact, similarity between Wheeler and Topos Gospel by Doring & Isham philosophies are obvious.

(Copy of comment for Jan Durham ).

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WANG Xiong wrote on Jul. 18, 2013 @ 13:37 GMT
Dear Michael Alexeevich Popov,

Thanks for your nice essay, well done

Yes!

I agree with you Bit from it.

Enjoy your Mathematical Clarification,

so my essay may interest you, which have less math than yours

Bit: from Breaking symmetry of it

http://fqxi.org/community/forum/topic/1906

Hope you enjoy it

Regards,

Xiong

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Author Michael Alexeevich Popov replied on Aug. 6, 2013 @ 14:17 GMT
Dear Wang,

My later comment, sorry :

When you proposed that 'one day a more complete theory should unify all the three basic concepts Matter-Energy-Information ' , you probably made an assumption that the law of information conservation in future physics is quite possible.

But, sceptics may suggest that in comparison with energy conservation law , Hawking -like law of information conservation (The information remains firmly in our universe. Thus, if you jump into a black hole, your mass energy will be returned to our universe but in a mangled form which contains the information about what you were like but in a state where it can not be easily recognized.( Hawking, 2005 )) can be violated ? Moreover, speaking exactly, there is no such thing as physical measurement of bits of thermal information or generally - universal information in physics.

Best

Michael

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Michel Planat wrote on Jul. 21, 2013 @ 12:07 GMT
Dear Michael,

Your essay deserves more attention, I think.

It seems to me that the bit is a pure (but very convenient) classical convention to manipulate information (after all the electrical current has only two ways). Similarly qubits are conventions, but with a more convenient mathematical structure. In the past years we (me and coauthors) worked a lot on (multiple) qubit and qudit algebraic and geometrical properties and, as you like mathematics, I suggest you have a look at

http://xxx.lanl.gov/abs/1009.3858

In my essay,

http://fqxi.org/community/forum/topic/1789

bits (0 or 1) arise from qubits 'in dessins d'enfants', that are graphs on orientable surfaces such as the Riemann sphere.

May be this is even closer to what you have in mind in your search of a mathematical definition of the bit.

But I agree that Shannon's log function yet does not play a role here.

Best wishes,

Michel

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Author Michael Alexeevich Popov replied on Jul. 23, 2013 @ 12:30 GMT
Michel,

I count myself fortunate to find your recent arXiv articles on Riemann conjecture and its quantum simulations.I try to make something similar but merely in the context of post - quantum cryptology. My initial result ( published in 1999 in France )is connected with introduction of periodic perfect numbers(Bull Sci math 1999,123,29-31),hence, new definitions of prime number theorem, cubic groups and quantum one-way function( Cryptology ePrint Archive, 653/2010 ) are arising. I had found that your attempt to formulate Riemann hypothesis as a property of the low temperature Kubo-Martin-Schwinger states is very original. Your last articles also suggest that beyond very popular Wheeler delusion there exist new world of unknown mathematics and unexpected physics.

best

Michael

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Michel Planat replied on Jul. 23, 2013 @ 14:37 GMT
Dear Michael,

I have not be able to get your 653/2010, may be you can send it to me.

I already checked that the fourth case in your conjecture is not perfect and the fifth case seems out of reach.

There is non-zero interesection between number theory and quantum information processing as you already noticed. May be the perfect numbers are important here, I don't know. Where do you connect your conjecture and RH?

'unknown mathematics and unexpected physics'; yes, a lot of interesting results to appear.

Best wishes,

Michel

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Author Michael Alexeevich Popov replied on Jul. 25, 2013 @ 09:29 GMT
Michel,

P e r f e c t s. It's correct. But it is only a beginning of mystery- Plato ( Euler who made reference to this puzzling place in Republic ) predicted an existence of merely three perfect periodic numbers at all, Ramanujan ( note books ) developed more general technoque using similar assumption and recieved somothing fundamentally different ( he had surprisingly quantum "taste").My new definition of "perfectness" in Bul Sci math is based on elementary proof of impossibility of Euler odd perfect numbers as well.

C u b i c ( already sent ) One - way function in cryptology is another definition of P is not NP solution.

R H . An attempt of Latorre-like superposition of all odds ( 2013 )in quantum computations of pi(x) and the Mangoldt's function.

Best

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Branko L Zivlak wrote on Jul. 21, 2013 @ 15:55 GMT
Dear Popov,

thanks for useful article which you gave us.

regards

Branko

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Christian Corda wrote on Jul. 22, 2013 @ 08:58 GMT
Dear Michael,

As promised in my Essay page, I have read your pretty Essay. It is well written, fascinating and a bit provocative. I appreciated your going beyond Shannon - Hartley Assumption through complex numbers. I encourage you to further develop studies in this direction. As your Essay give a lot of fun to me, I am going to give you an high score.

Cheers,

Ch.

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Sreenath B N wrote on Jul. 22, 2013 @ 13:53 GMT
Dear Michael,

I have down loaded your essay and soon post my comments on it. Meanwhile, please, go through my essay and post your comments.

Regards and good luck in the contest,

Sreenath BN.

http://fqxi.org/community/forum/topic/1827

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Author Michael Alexeevich Popov replied on Aug. 6, 2013 @ 13:47 GMT
Dear Sreenath,

Thank you for your visit.

Because it is physical forum, may I ask you merely one very strange question : How we can make physical measurement of the bits of taking universal information seriously in reality ?

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Akinbo Ojo wrote on Jul. 24, 2013 @ 11:06 GMT
Dear Michael,

I just read your well constructed essay and rated as such. Although a bit technical for me but I will like to know,

1) What is the difference between an imaginary line and a real line?

2) What is the difference between a real bit and an imaginary one?

3) Does line have breadth or zero breadth?

Best regards,

Akinbo

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Author Michael Alexeevich Popov replied on Jul. 25, 2013 @ 10:40 GMT
Dear Akinbo,

Shannon was a mathematician and it is logical to follow his context ( not Wheeler brave speculations ) to understand bit.

In arithmetic we start from the positive integers ( 1,2,3,4,5,6,...) and from the ideas of addition, multiplication,substraction and division. It is easy to test that these operations are not always possible ( 4 - 29, 5 - 7, 2 - 8, 4/29,5/7,... etc )unless we admot new kind of integers ( negative numbers, or more generally, rational numbers )If we include root extraction and the solution of equations, we can find some operations are not possible also unless we admit a new kind of numbers. Mathematicians had found that the extraction of the square root - 1 is not possible unless we go further and admit the complex numbers ( as is known, following mathematicians Einstein, Heisenberg and Schrodinger introduced the square root of - 1 in physics ).Thus,it is practical and productive everywhere ( even philosopher Immanuel Kant made an attempt to introduce negative numbers in philosophy ...)

Complex numbers are sometimes called imaginary.Complex number is not number in the same sense as a rational number ( used by Shannon for bits )It is a pair of numbers (x,y), united symbolically in the form z = x + yi . Hence, it is easy to see, that when y = 0 we say that z is real ( special term for 'post-rational numbers' ), correspondingly, when x = 0 then z is pure imaginary.

Next step.

let ax + by +c = 0 be an equation with complex numbers ( coefficients ). If we give x any concrete complex value , we can find the value of y. Set of pairs of real and complex values of x and y which satisfy the equation is called imaginary straight line, the pairs of them usually are called imaginary points and are said to lie on the line. When x and y are real, the point is called a real point; correspondingly, when a, b, c are all real, the line is defined as real line.etc Hence, we can easy deduce answers for questions 2 and 3.

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Than Tin wrote on Jul. 26, 2013 @ 03:16 GMT
Hello Alex

Richard Feynman in his Nobel Acceptance Speech

(http://www.nobelprize.org/nobel_prizes/physics/laureates/19
65/feynman-lecture.html)

said: “It always seems odd to me that the fundamental laws of physics, when discovered, can appear in so many different forms that are not apparently identical at first, but with a little mathematical fiddling you can show the...

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Author Michael Alexeevich Popov replied on Jul. 29, 2013 @ 11:37 GMT
Dear Than,

I suppose that Physics as a science tries to establish truth but not poetics and political sense of argument.R.Feynman also suggested that physicists have a way of avoiding the politics and subjective tastes in science : if you have an apparatus which is capable of telling how many bits of information given thermal energy ( "termal information" )must contain in the terms of physical measurement, then you can say scientifically about entity information, indeed. Because there is no such thing as physical measurement of the bits of thermal information, then Hawking-like law of information conservation (' The information remains firmly in our universe. Thus, If you jump into a black hole, your mass energy will be returned to our universe but in a mangled form which contains the information about what you were like but in a state where it can not be easily recognized '( Hawking ,2005 )) and its consequences might be considered, unfortunately,as popular illusion.

respectfully

Michael

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Antony Ryan wrote on Aug. 1, 2013 @ 10:19 GMT
Dear Michael,

I think your essay is very good indeed. Top marks from me! Hope it helps. Interesting approach - I'd not thought about investigating logarithms further - seems very logical. You've presented your worked very nicely. Anything around numbers interests me.

I have found a Fibonacci link in my theory which partly unifies the four forces and resolves the three paradoxes of cosmogony. I explored this around Black Holes in my essay, extending to the negative sequence and based on observation and space pathways. Also there's a touch of entropy. I'd be delighted if you had time to look at it.

Best wishes,

Antony

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Author Michael Alexeevich Popov replied on Aug. 1, 2013 @ 13:27 GMT
Dear Antony,

Thank You. Let me imagine a continuation of your n-dimensionality logical game. If rational n-dimensionality ( n = -1,0,1,2,3 and we assume that n is rational number ) is accepted ( i.e. there is a mathematical proof ) we can go further and we may admit a new kind of possible dimensionality, expressed by the square root -1 and complex numbers ( why not ? Einstein and Hawking use the square root-1 as an imaginary time / complex time variable u in physics ).Hence, new unexpected physical generalizations are deduced.

best

Michael

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Antony Ryan replied on Aug. 1, 2013 @ 13:48 GMT
Dear Michael,

I agree that we can indeed use square roots to explore concepts such as this further. I think these could actually apply in experimental results at colliders. Great idea!

Best wishes,

Antony

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Author Michael Alexeevich Popov wrote on Aug. 3, 2013 @ 10:49 GMT
Dear Antony,

Thank you for high rank. Let me imagine a continuation of your n-dimensionality logical game. If rational n-dimensionality ( n = -1,0,1,2,3 and we assume that n is rational number ) is accepted ( i.e. there is a mathematical proof ) we can go further and we may admit a new kind of possible dimensionality, expressed by the square root -1 and complex numbers ( why not ? Einstein and Hawking use the square root - 1 as an imaginary time / complex time variable u in physics ).Hence, new unexpected physical generalizations are deduced.

( copy of my comment for Antony Ryan by 1 Aug 2013 )

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Antony Ryan replied on Aug. 3, 2013 @ 20:05 GMT
Indeed Michael,

I've posted this and it applies to you!

I've lost a lot of comments and replies on my thread and many other threads I have commented on over the last few days. This has been a lot of work and I feel like it has been a waste of time and energy. Seems to have happened to others too - if not all.

I WILL ATTEMPT to revisit all threads to check and re-post something. Your thread was one affected by this.

I can't remember the full extent of what I said, but I have notes so know that I rated it very highly.

Hopefully the posts will be able to be retrieved by FQXi.

Best wishes,

Antony

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Author Michael Alexeevich Popov wrote on Aug. 5, 2013 @ 13:01 GMT
Lost comments by Antony Ryan and my answer ( 1 Aug 2013 ) have been reset.

Thank you who did it.

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Antony Ryan replied on Aug. 7, 2013 @ 22:17 GMT
FQXi fixed the bug. They were transferring servers or something like that.

Regards,

Antony

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Author Michael Alexeevich Popov wrote on Aug. 5, 2013 @ 13:03 GMT
Peter,

My later reading of your essay showed an importance of following passages :

> “ Binary based mathematics relies on the Law of the Excluded Middle between assigned symbols 0,1, A,B, or yes/no for waves”.

“Time itself is a special case. The concept time is a human invention to describe change, often confused with the physical evidence of emissions from...

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Author Michael Alexeevich Popov wrote on Aug. 5, 2013 @ 13:12 GMT
Peter

(technical addition )

' Binary based mathematics relies on the Law of the Excluded Middle between assigned symbols 0,1, A,B, or yes/no for waves'.

Refinement. May be philosophically in the terms of analytical tradition, it is correct, however, mathematically speaking, it is a kind of Wheeler simplification. Indeed, we cannot say that Weirstrass theorem relies on the Law of the Excluded Middle. Technically, in alternative proof of Weirstrass theorem, we divide some interval into two equal parts, when one at least contain infinitely many points. Proceeding in this way we can define a sequence of intervals each of which is a half of its predecessor, and each of which contains infinitely many points. Hence, ideas of points of accumulations, Dedekind theorem, sections of the real numbers into left-hand class / right-hand class, etc. Thus, Wheeler yes/no logic must have limited sense and actual binary mathe-matics is more complicated.

( Copy of my comment for Peter Jackson by 5 Aug 2013 )

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Peter Jackson replied on Aug. 5, 2013 @ 16:06 GMT
Michael,

Many thanks for your post. I think this is very important work and nice to see a recurrent theme in the contest. Yours in particular was excellent and undervalued so I'm very pleased to give it a high score. I do hope you make the final cut.

I re-allocated most of the 'maths' part of my brain to dynamic/kinetics, logic and geometry some time ago! so hope we may stay in touch.

Peter

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eAmazigh M. HANNOU wrote on Aug. 5, 2013 @ 23:06 GMT
Dear Michael,

We are at the end of this essay contest.

In conclusion, at the question to know if Information is more fundamental than Matter, there is a good reason to answer that Matter is made of an amazing mixture of eInfo and eEnergy, at the same time.

Matter is thus eInfo made with eEnergy rather than answer it is made with eEnergy and eInfo ; because eInfo is eEnergy, and the one does not go without the other one.

eEnergy and eInfo are the two basic Principles of the eUniverse. Nothing can exist if it is not eEnergy, and any object is eInfo, and therefore eEnergy.

And consequently our eReality is eInfo made with eEnergy. And the final verdict is : eReality is virtual, and virtuality is our fundamental eReality.

Good luck to the winners,

And see you soon, with good news on this topic, and the Theory of Everything.

Amazigh H.

I rated your essay.

Please visit My essay.

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Author Michael Alexeevich Popov wrote on Aug. 6, 2013 @ 12:42 GMT
Dear Amazigh,

Thank you for your visit.

Some your observations :

'… the duality was obvious to the Egyptians, Greeks, etc.. , And especially the Chinese…

For me the question is quite different : why we do not have yet flushed the irrefutable arguments that prove that the universe is binary, dualistic…

Today, Einstein's relativity is well understood in the...

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Sreenath B N wrote on Aug. 6, 2013 @ 17:24 GMT
Dear Michael,

Thanks for dropping in to my thread and I have answered your question there. Currently your ranking is below 40 and if you want to increase it, immediately contact me at, bnsreenath@yahoo.co.in

Waiting for your response.

Best wishes,

Sreenath

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Author Michael Alexeevich Popov wrote on Aug. 7, 2013 @ 10:54 GMT
Dear Sreenath,

It would be almost impossible to predict another result.Rank is less important but pleasure to participate in global self organized anthropological experiment is beyond any expectations.It is very rare event.Indeed.

Respectfully

Michael

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Paul Borrill wrote on Aug. 7, 2013 @ 21:46 GMT
Dear Michael,

I have now finished reviewing all 180 essays for the contest and appreciate your contribution to this competition.

I have been thoroughly impressed at the breadth, depth and quality of the ideas represented in this contest. In true academic spirit, if you have not yet reviewed my essay, I invite you to do so and leave your comments.

You can find the latest version of my essay here:

http://fqxi.org/data/forum-attachments/Borrill-TimeOne-
V1.1a.pdf

(sorry if the fqxi web site splits this url up, I haven’t figured out a way to not make it do that).

May the best essays win!

Kind regards,

Paul Borrill

paul at borrill dot com

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