Bit from it. Mathematical Clarification by Michael Alexeevich Popov

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

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.

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

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 )

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

Lost comments by Antony Ryan and my answer ( 1 Aug 2013 ) have been reset.

Thank you who did it.

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 'clocks'".

< refinement . When Albert Einstein extended physically accepted list of arithmetical operations so as to include the extraction of the square root of minus 1 in his theory of imaginary time (SR), he admitted the complex numbers and new theory of time. Thus, if we find that some equations of higher degree used in physics are insoluble by aid of complex numbers now, we can go further and admit new math. Hence, new time theory can be connected with new mathematics of non-integer dimensionality spacetime. In 1986 quantum theorists Karl Svozil and Anton Zeilinger proposed that the notion of the dimension of spacetime could be based on"philosophical" measure-theoretic concepts ( admitting the possibility of noninteger dimensions , like 4 - έ ). They introduced in fact an operational dimension which is smaller than the idealized Housdorff dimension. However, they intuitively extended their list of accepted arithmetical operations so as to include the solution of higher equations.Thus,we might to be led to the consideration of numbers of another types in order to introduce complex dimensionality and new time theory, taking technically.

http://www.worldscientific.com/doi/abs/10.1142/S0217751X86000368

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 )

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 physical domain. It explains very well the two points of view of two people moving relative to each other...

Finally, I would like to conjecture that:

-The first principle that governs the universe is Duality. The principle of conservation has come after. Give the same interest to duality that to motion, and things will better.

-We know matter is dual as well as the light is. So if duality is everywhere, how to recognize this fact, as we did for the motion ?

-The Duality is the common language by excellence, the integral link between all the elements of nature, the machine code of the universe '

may suggest some analogies with field findings of structural anthropologists. In particular,French anthropologist Levi-Strauss made reference to 353 duality based different myths in order to prove that the human brain does have a tendency to operate with binary oppositions in all sorts of situations. This result is foundation of structural anthropology and semantic algebra of myths today. However, the sceptics were right as well, because human brain can operate in other ways also. Thus, a fully satisfactory model of the human mind would contain many unknown features which do not occur merely in digital computers. I suppose you can improve your structural linguistic analysis of oppositions by Jakobson's technique based on conjecture that ' in all the languages of the world the complex systems of oppositions between the phonemes are no more than a multi-directional elaboration of a more complex system which is common to all, namely the contrast between consonant and vowel, which through the working of double opposition between compact and diffuse, acute and grave...'

with the best wishes

Michael

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 ?

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