It is common to affirm that "One can never perform any measurement whose result is an irrational number."
This is equivalent to say the contrapositive, that anything that can be measured or produced is a rational number.
But the irrational number √2 can be produced to infinite length in finite steps, as 2×sin(45 degrees). It also exists like that in nature, as the diagonal of a unit square.
There is no logical mystery in these two apparently opposing views. Nature is not Boolean, a new logic is needed.
In this new logic, the statements 'p' and 'not p' can coexist. In the US, Pierce already said it. In Russia, Setun used it.
This opens quantum mechanics to be logical, and sheds new light into quantum computation.
One can no longer expect that a mythical quantum "analog computer" will magically solve things by annealing. Nature is also solving problems algebraically, where there is no such limitation.
Gödel’s undecidability is Boolean, and does not apply. The LEM (Law of the Excluded Middle) falls.
What is your qualified opinion?
Basically, you are loosing bits of information when you can't calculate the exact number, it's necessary to cut the computing due to the halting problem. It's not possible to calculate infinite information then you must stop loosing part of the bits. This introduce an uncertainty problem because you haven't all the information, this is an human epistemological issue. The reality has access to all the information, the reality has an oracle, we are limited, not the Nature. I'm not sure that quantum computer will have the power of this oracle, we want to believe that this is true but it may fail due to another limitations not known now. I'm researching the loosing information problem and its connection with the Turing halting problem and Gödel theorems from the point of view of the computing and information theory. These things seem to be deeply connected and they have consequence in physics because physics must calculate to solve problems.
YL: Interesting perspective. But, there are many factors against
"losing information":
1. One can use integers in modular arithmetic. There is no 'approximation" versus using mathematical real-numbers, and one can calculate anything. For example, it is used in cryptography, exactly, with the AES. One cannot use mathematical real-numbers in cryptography, either.
2. Mathematical real-numbers include numbers which nature cannot produce. All measurements, and production, must be based on rational numbers.
3. It is a fiction that 0.999... =1, for example, and nature would halt if that were true. The error would have accumulated in 13.8 billion years.
4. Nature has never halted in 13.8 billion years.
5. Discontinuous functions can be differentiated, although current mathematics thinks otherwise.
6. There is an unbounded class of solutions hidden in current mathematics, due to #5, which exist in physics and mathematics.
7. For example, the second derivative of |x| is not zero, but an impulse function of amplitude 2.
8. #7 works with the second theorem of calculus, and is calculated right by our book and Wolfram Alpha, but not following Courant, Apostol, etc.
9. We give other examples and use in our book "Quickest Calculus" , with PDF free of charge at my RG profile.
Mathematics HAS changed.
As basic resort we are dealing with information, bits, through the computing we are trying to get better information calculating, using empiric data. If you are losing information sometime is not important but sometime explode and you can't know where. Here is the uncertainty and undecidability that we are talking about. Good example is the fluid equations, Euler eq., where for years are trying to understand where they are singulars points where this equation explode. I think that the information about the singular points are outside the equation, you can't find it inside as some people are trying. Algorithms can't create more information than the information you have in your data, this is a proved issue , it can clean your data and give you a better insight about this data but it can't create more info than you have. The algorithms work as Occam razors, minimum expression for maximum coverture. The root of undecidability is in the uncertainty because you can't create more info using algorithms , you must bring more data to get more info using your formulas.
YL: A primary cause of uncertainty is the hypothesis that no matter what you do, the data itself is uncertain. This is Heisenberg uncertainty principle, and Gödel's uncertainty. It is not about using a better method in treating the data.
We do not believe in either,and nature itself would not have lasted 13.8 billion years if both were true.
We see other examples of that, and people take advantage of this view in unethical methods.
So, this is not a question important to just physics and mathematics, but to commerce and life.
More primary than logic, one can look at mereology. There, one can discern 2 factors governing uncertainty: percetual and conceptual.
Then, we can we can resolve four chacteristics of uncertainty -- they can be combined, first, in a Boolean mixture. This can be very illuminating, creating four classifications for any phenomena, thus, a non-Boolean system is born. This allows text and math analysis very well. The four classifications are called: DAOF. See at:
Technical Report What Is Identification, That We Can Identify It? Part I
and
Technical Report What Is Identification, That We Can Identify It? Part II
When trust is added, we get a semiotics system with reference (symbol), sense (meaning), referent (object), and trust. Trust is what makes the DAOF classification possible.
This opens up uncertainty, as a mixture, using two possible systems, from mereology and from semiotics.
Therefore, one can begin to explore uncertainty, and understand the relativity of Gödel's and Heisenberg uncertainties. It is not just dependent on the perceptual, but the conceptual. It is easier to deal first with the conceptual uncertainty.
We do not think and talk about what we see; we see what we are ABLE to talk about, cf. Edgar H. Schein.
Language is more important than we think. Russians can distinguish more tones of purple, because they have words for it. Uncertainty is reduced.
Consider the semiclassical quantization in quantum mechanics. No information is lost, or uncertain. It is exact, that is not an approximation.
It gives results exactly for the free particle and the harmonic oscillator, among other examples.
It is, however, important to realize that the semiclassical quantization has to do with how close F is to the path integral around the classical path.
It is not valid, we all use 3-logic in traffic signals. Support comes from the preprint
https://www.researchgate.net/publication/366124503/ ,
the article published in Mathematica in 2023 at
https://www.mdpi.com/2227-7390/11/1/68/pdf
the free PDF book
https://www.researchgate.net/publication/365130122 ,
the paper-based books at
https://a.co/d/20K6HLx
and other references, including on the new field "quantum circuits".
The comments above show how following the RG ToS is necessary for a scientific conversation.
But the legal responsibility of enforcing the RG ToS falls only on RG, per EU law.
Meawhile, it is suggested to read the question text, with the latest references. Also, private messaging is available.
Can you measure an objects length to be an exact rational number? Say, exactly 1/4 meters? One-fourth meters is also, 0.2500000000000000000 meters, with the zeros repeating forever.
SB: This is the qualitative difference between 1 and 1.000... .
While the second example is a mathematical real-number that never ends, being not measurable numerically according to Gisin (as cited in [1]), the first example is absolutely exact, has zeo variance, and can be readily measurable.
[1] Article Algorithms for Quantum Computation: The Derivatives of Disco...
If algorithmic, the undecidability has to get beyond conventional discontinuity issues in order to become really discrete logics . I look at it from the perspective of type theory, where a program is an example of how a type may be broken down into discrete stages. The answer to this question will depend on how expressive your type system is.
EG You say that all results of a measurement are whole numbers or rational. I can see this for 3 apples. But what if you measure more closely, say by measuring the number of molecules in the apples? You get 3 + x/N where N is the number of molecules in a standard apple. So it's still a rational number. But measurements are taken over time, where apple molecules come and go. We can still assign to this measurement a rational average over time. But if we take an infinite number of measurements, does this average become irrational, in general, and is this irrational average the best, or most accurate measurement, we can do?
SB: If you get a rational number (which I say happens in any physical measurement), you 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 --> Bn --> 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 fact that logic values can be TRUE, FALSE, and UNDEFINED, invalidates Gödel's uncertainties, based on Boolean logic and the Law of the Excluded Middle (LEM). It also allows quantum computing and contradicts Shannon assumption, that information can be modelled by relays, either ON or OFF. The irrationality of irrational numbers can become more evident.
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