Dear colleagues, at the suggestion of Ed Gerk, we have long and quite fruitfully discussed what the R~Q ratio means for computational physics and for physics in general. And everything would be fine, only here is the final conclusion that sets of capacity greater than Aleph Zero must be completely excluded from the mathematical apparatus of physics - in my opinion, an obvious inflection. The sign ~ , as you know, does not mean equality, especially not identity (unconditional equality), but only equivalence, that is, mutual modeling under certain limiting assumptions. Doing physics, believing that R~Q⇒R≡Q. is about the same as going out into the open ocean not on a 300-meter cruise ship, but on its 3-meter model. Yes, both of these ships float, both have an engine, both are steered. But on a model, you won’t get enough sleep in a comfortable cabin, you won’t eat a delicious lunch in a restaurant, and you’ll sit on a model only on horseback, like on a motorcycle. And what will the first ocean wave do to you ...
To suggest how R can be transformed into Q, see also (just in the given consequence):
https://www.researchgate.net/publication/367542059_NATURAL_BOUNDARIES_OF_e-NEIGHBORHOODS
https://www.researchgate.net/publication/368831616_Rational_numbers_in_the_hyperrreal_world
https://www.researchgate.net/publication/369001394_HYPERREAL_HOMOLOLOGIES_OF_RATIONAL_NUMBERS
Research Proposal CONTINUOUS MODELS IN QUANTUM CONSCIOUSNESS: AMBIGUITY OF DEF...
The following assertion is trivial but useful.
One of the possible ways to model continual n-dimensional structures as discrete ones is to create and describe fractal substructures with rational dimensions (n-1)>D
AO: Thank you for this thread. IMO, a continuum does not exist in nature, cannot be created, and our intuition is too poor to even imagine its details.
All experiments have demonstrated ad absurdum that we live in a quantum world, with at least 4 dimensions (soon to recognize the 5th -- as consciousness without any physical body).
As humans, we all do memory processing in our minds (and mind-2, spiritual), and that is how non-boolean logic and quantum mechanics were discovered, more than 100 years ago (when no computers were available, and one could only see mythically continuing, and smooth, lines, to infinity). One could also see dragons, various djinn, Santa Claus, Baba Iaga, the Tooth Fairy, and other mythology.
Albeit, under a microscope, one could see jagged, interrupted, finite, and dotted lines. Thus, even boolean "certainty" under the LEM (Law of the Excluded Middle) was taken as a Procrustean Bed, in primitive (to today) times.
But, there has been a recent (2022) change, first in mathematics. Functions that are everywhere discontinuous are recognized and exemplified, since the end of 2022 [1], to be able to be differentiate and integrated exactly, and one can use that in a non-boolean setting --- avoiding the LEM.
This also changes physics, and how quantum mechanics solutions can (defensibly) be calculated using rational numbers, that are everywhere discontinuous, and can be compounded as "wheels within wheels" -- to understand pictorially the operation y=mod x in the modular arithmetic of integers -- common today in computers.
On this basis, quantum computing can be considered to have begun in 2022 [1] and use the set Q within a non-boolean set, for decision-making for example. No imaginary numbers are used (think of it: computers do not actually use imaginary numbers -- they simulate if you insist).
Functions that are everywhere discontinuous (i.e., digital) are now common. They are recognized and exemplified, since the end of 2022 [1], to be able to be differentiated and integrated exactly. Non-boolean processing now moves to computers, and becomes ubiquitous, and use FFT (Fast Fourier Transform) -- to improve. Quantum computing will soon become possible ... and in cellphones.
Cryogenic behemoth quantum machines (see figure) consider a "complex qubit" -- two physical impossibilities. They are just poor physics and expensive experiments in these pioneering times. Quantum computing is ... natural. Atoms do it all the time, and the human brain (that is how it was discovered!). Non-boolean processing is done even by lower animals, such as measured in laboratory using a fruit fly, with a limited brain. Fruit flies who don't sleep, not only live less, but cannot remember what is learned.
This changes how anyone can process information in a qualitative and quantitative basis.
Permanent education becomes key to performance, important to students, professors, professionals, the retired, and even to the housewives. The research in RG becomes also education.
CONTINUOUS MODELS IN QUANTUM CONSCIOUSNESS: AMBIGUITY OF DEFINITIONS AS A CONDITION OF CREATION
Let's assume that all material nature is a quantum (ie discrete) system. In this case, how (in a different interpretation - for what purpose) does the quantum brain create continuum and ultra-continuum concepts (models) of physical reality?
To answer this question, let us recall the history of the main definitions of analysis - infinitesimal quantities - first in the schools of Leibniz and Newton, then in the school of Euler. Then - Cantor's "naive" set theory, further discussions around the axiomatics of set theory and the original modifications of set theory and topology generated by this discussion. Finally, Gödel's theorems and modern model theory, starting with hyperreal Robinson numbers, the Levenheim-Skolem theorem and its counterpart in information theory, the Kotelnikov-Shannon theorem. The ambiguity of the main definitions and the resulting ambiguity (bijectivity and even, so to speak, multijectivity) of the corresponding mappings in the space of propositions and in predicates runs like a red thread through this whole story.
Next, let's consider this bijectivity as a "tool" for spontaneous symmetry breaking and, accordingly, multi-scale acts of creation.
https://www.researchgate.net/publication/364926779_UNCERTAINITY_RATIO_SURJECTIVE_MAPPINGS_SPONTANEOUS_SYMMETRY_BREAKING_AND_CREATION-FROM-NOTHING
https://www.researchgate.net/publication/366673118_SPACE-TIME_DUTY_CYCLE_COMPLEMENTARY_FRACTALS_BOSONIC_CONDENSATE_AND_SPACE-TIME_SPLICING
https://www.researchgate.net/publication/366893854_AROMORPHOSIS_IS_A_SOLITON
https://www.researchgate.net/publication/367031975_ON_THE_GENERAL_PRINCIPLE_FOR_THE_QUANTUM_CORRELATION_AND_QUANTUM_ENTANGLEMENT
Comments (so far) are superfluous, but serious developments on this subject are, of course, necessary.
AO: Yes.
That seems like a good question, to clarify.
Knowing (!) that all material nature is quantum (ie discrete) system -- how (in a different interpretation - for what purpose) does the quantum brain create continuum and ultra-continuum concepts (models) of physical reality that it did not experience?
Ego can explain a lot -- forcing one's thoughts, solipsistim. Go to Freud and Jung.
But, there are several other answers possible. One that I personally adopt. God wanted. Therefore, it must be good.
In a more practical level, but running the risk of a human mistake, maybe we experience it in a spiritual world.
Then, we try to apply it, to fail in an impotent reach. The physical world is hopelessly quantum. We are called to adapt.
Plato's Cave metaphor explains it. It is also hopeless to try to guess what is outside the cave. We don't have even the terms to explain it. It is different beyond reason.
Therefore, only by accepting a spiritual reality we may be able to pursue the reason in your question.
Otherwise, we are all the prisoners (for a higher reason) in a Plato's Cave.
All is good. What can the clay (us) say to the vasemaker (God), asked Omar Khayyam. My first answer was that. Postpone understanding, trust first.
AO: To the hypothesis in the question, nothing is lost by setting R=Q. Calculations using a computer are safe and true.
We are able to formulate ANY requirement of conventional field theory, but the reverse is NOT true -- as I published in QM as early as 1978, using differentials of discontinuous functions. The set Q admits solutions that the sets R and C are impotent to reach.
Support comes from:
https://www.researchgate.net/publication/355406882/
https://www.researchgate.net/publication/355406639/
https://www.researchgate.net/publication/286722113/
https://www.researchgate.net/publication/236420748/
https://www.researchgate.net/publication/286640675/
https://www.researchgate.net/publication/286640638/
https://www.researchgate.net/publication/243470610/
https://www.researchgate.net/publication/286625459/
A number is a semiotic concept, and that is why R=Q. Nothing is lost.
A number is not arbitrary in value -- see the prime numbers, they have objective values. Imaginary numbers are not objective. There is nothing arbitrary about this.
Prime numbers do not depend on humans. Any prime number on Earth and in the star Betelgeuse are the same.
Support comes from:
https://www.researchgate.net/publication/290284114/
https://www.researchgate.net/publication/290283924/
https://www.researchgate.net/publication/318661666/
Quantum mechanics is affirmed without any doubt, although it is a story in Plato's Cave.
Mathematical real-numbers, mathematical complex numbers, irrational numbers, Lie numbers, and other man-made concepts must now be rejected, and deprecated after 500 years or more of false use.
The numbers to use in all sciences must be based on natural numbers (set N), integer numbers (set Z), and rational numbers (set Q).
Other sets can be added, for human benefit, such as set Q*, but are not needed by computers.
By the Curry-Howard relationship, this deprecates Gödel's uncertainties.
Dear Dr Gerck, You are absolutely correct in writing here and everywhere that real and other supra-rational numbers are not needed by a computer. Only it would be useful not to forget that even the very idea of a computer - and even more so any of its technical implementation - is not the Absolute, is not God, and therefore it should not be worshipped.
AO: Please correct the spelling of my name, also in the question text. This is an individual error.
OTOH, a computer is not one person. It is the collective result of billions of people and machines -- there cannot be any experimental error (calculate the probability, and you will see).
Comparing with God is IMO useful. Only then can my contribution be properly compared -- as less than nothing. Even if collective. God always holds all the dice (cf. Einstein).
AO: R=Q, and R-Q is not measurable -- with unit probability. Thus, one can consider R-Q=0, and FT=FFT.
The calculus of everywhere discontinuous functions, and step-functions, are now allowed, and absolutely exact [1].
The sets C or G are a mirage -- they mix their parts, "infesting" results. Only calculations with Q can be objective. Therefore, it is a waste of resources (time, cost) to try to calculate with R, C, or G. Only use computer models with B, N, Z, and Q, where B is the set of Booleans.
Computers use only B, and 0^n = 0, 1^n=1, so any multiplication can be effectively avoided. Because division can be exact with modular arithmetic, any division can be avoided as well. The result is that the FFT can be much faster, and no trigonometric result has to be calculated.
Because with R=Q multiplication is repeated addition, and exponentiation is repeated multiplication, both can be calculated by the FFT using B, further accelerating and accelerated. Nothing is lost, as R=Q.
There have been 500 years or more of illusions and mirage. A new mathematics was started in [1] (search reference above or online).
AO, DH, and SPK: Me thinks that you guys and group are just mixing contexts.
You are right when using p-adic numbers, that numbers are arbitrary. There is not even the notion of odd or even numbers. But, that is not true using B, N, Z, or Q.
Thus, whether numbers are arbitrary or not depends on the set that they are defined in, their respective context. For indigenous tribes, some do not know numbers higher than 5. The Mayans used base 20.
In the B set, numbers are either 0 or 1. And 0^n=0, while 1^n=1, so arithmetic is fast and easy. Digital computers only use the B set, and yet can calculate everything. For them, R=Q.
We, humans, can use the Q* set for fast and easy mental calculations. A negative times a negative is a positive.
Quantum computing uses the set Q, to allow calculus with discontinuous functions--as functions must be in the digital world. We see that world in the XXI century.
Your view might or not be useful with p-adic numbers, but it is certainly not universal. So, consider numbers arbitrary in that context.
My preference is what makes physical sense. In my view, C, R, and G, do not. P-adic nunbers; in my view, need to change metric in order to be trustworthy in a physical sense.
All is well in a semiotic sense.
We are correcting a 500-year error. Complex numbers do not exist and are not needed. The Fourier transform and the FFT can use only rational numbers.
The sooner complex numbers are not used, the better. We are suggesting the addition of set Q*.
FFT=FT and can be faster. Galois theory needs to change. Polynomials have coefficients in the set Q. The quintic can be always solved in the set Q. The formula for third-degree polynomials works always in the set Q. The equation for second-degree polynomials change. Discontinuous functions can be differentiated and integrated.
Mathematics evolves.
Responding to a private message, complex numbers were already denied, when we denied mathematical real-numbers [1]. But, how about imaginary numbers, or the Gauss set, using rational numbers to resurrect complex numbers?
No, already denied by Erwin Schrödinger in 1926. And, we deny that today -- using computers, where R=Q and C does not exist.
Soon, we will reveal how the Fourier Transform and the FFT can be a mapping from Q to Q -- and much faster --without passing through C, G, or R. The beginning of that is in the RG preprint "FT=FFT".
We are not saying the last word, but just the first.
If one gets a rational number (which I say happens in any physical measurement), one can get a natural number from it by a known isometric relationship, Q --> N.
One cannot get an irrational number by adding numbers in the set Q -- the set Q is closed for addition, +Q--> Q.
Any number x in the set Q, to add computability, obeys N --> B^n --> B ={0,1}. Computers can calculate anything measurable-- p-adic numbers, and the sets R, C, G, do not appear!
No physical measurement can be in sets R or C (which has two sets R).
No physical measurement can be in set G, either, because a physical path must begin and in the set Q -- and one men can always get two set Q numbers between any two set G numbers.
QED.
The simplicity of a very large number of rational parameters beats hands down the complexity of one parameter in mathematical real-numbers and mathematical complex numbers-- that are not computable or analogly calculable at all.
It is the future. Large numbers of parameters using computable rational numbers-- not unphysical mathematical real-numbers or complex numbers, like the present Standard Model in physics.
Thanks to a private dialogue.
Mathematics now, experimentally, changes. The authority of nature is final. We can only observe and be grateful.
I have been publishing and applying infinite continued fractions since 1976. For example, to solve the Schrödinger equation for bound states. They always result in finite values, physical values, rational number values, the quantum energy levels. That is not, though, my point, important for quantum computing. P-adic numbers do not show up.
To be clear, I did not mean either, what one can explain, with "the argument about irrationality of sqrt(2) has nothing to do with continued fractions, and (provably) only infinite continued fractions lead to irrationals, just like periodic decimals are always rational numbers (and vice versa)".
What I can rationally say is that the connection "no rational expression for sqrt(2) --> "no rationals used for sqrt(2)" is not valid.
A continued fraction expansion using only integers (possibly infinite) exists for every sqrt(x). It is finite and known for sqrt(4), and finite albeit yet of unknown length for sqrt(2).
Now, using quantum computing rules from [1], we can better explore infinite expressions that end in a quantum point, of dimension 0.
This includes all those continued fractions of sqrt(x). They must be rational numbers, because the operation is closed for addition and one only needs addition. They must end in a quantum point, of dimension 0.
One may say: but a continued fraction has division!
Yes, but computers never do division, multiplication, or subtraction.
If one follows the steps in the set B={0,1}, one has only addition and encoding, even to unlimited operations. For a computer, R=Q. All is calculable only with addition and encoding. Humans can, likewise, calculate anything only using the set B, addition and encoding.
Thus, a continued fraction is not a mathematical real-number that cannot be calculated, with probability 1.
A continued fraction expression can always be calculated, and always can use isomorphisms to use only the set B. Computers teach us.
That is why pi must be rational, as one has a very simple summation series for it, using only integers. It does not matter that it is of unlimited length. Computers can reach it.
The infinite opens for us, with quantum computing rules in [1]. Not all infinite expressions are of concern.
All expressions that end in a quantum point, digital, can be calculated with dimension 0, rigorously to unlimited length, using only the set B, addition and encoding.
No real-numbers or mathematical complex numbers. No infinitesimals either. No epsilon-deltas. No p-adic numbers. Mathematics evolves.
Computers teach us, as mathematics becomes an experimental science.
I see it like a Lego, and call it "click-mathematics". Always exact, rigorous. No opposition is possible to [1], or computers. What is right, is now right for all.
REFERENCES
[1] DOI 10.3390/math11010068
The reference of a sentence is its truth value, whereas its sense is the thought that it expresses. Frege justified the distinction in a number of ways.
One can see a number as pointing to its truth value (called a value) and expressing a thought (called its digit in the base used). The digit is the sense, the meaning of the number.
The digit can change at will, subjectively, but the value stays objectively fixed.
Thus one can write F in hexadecimal, while the value is 15 in base 10. The number is this 1:1 mapping between a name ("F") and a value (15).
Or, in reverse, one can use 15 for the name, pointing to the value F.
Or, one can use 1111 in 64-bit signed binary for the name, pointing to the same value (15).
A number is seen as this flexible semiotic concept, very used in computers.
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