■Classification of "Misinterpretation" by stage of occurring

【Confliction and missing of information conveyed】
・Questioner → natural language → programmer
・Programmer → programming language → computer (respondent)
【Confliction of information already have】
・Questioner, programmer, computer (respondent)
Even if the respondent has less information in advance than the questioner, it is not a misinterpretation.
Respondents should be able to answer as much as possible with the information they have.
The two categories can be aggregated if the questioner conveys all the information that the questioner has to the respondent.
If the questioner and the programmer are the same person, there will be no misinterpretation between them.
Only confliction and missing of information conveyed from the programmer to the computer essentially need to be resolved.

■Occurrence Process of "Misinterpretation"

Programmers translate the interpretations they have in their heads into a programming language.
Translation ends when the programmer recognizes that the interpretation before and after translation is the same.
"Misinterpretation" means that the interpretation is different even though the interpretation is recognized as the same.
At that time, there are times when I am convinced that it is correct, and times when I misunderstand due to lack of attention.
If it's just a lack of attention, just be careful.
Even though they are different, they are convinced that they are the same because the object of comparison has been replaced.
If you are sure that "a=b" even though "A≠B", then "A≠a" or "B≠b".
Looking at the written program, if this is what you wanted to do, it will change your perception of what you originally wanted to do.
If the program code is the same, the computer will always do the same operation, so the program code expresses only one interpretation.
On the other hand, the interpretation in my head changes to suit my convenience if multiple interpretations are possible.

■Example of "Misinterpretation"

Problem: What is the probability(P) that the next coin toss will front face?
Information: The result of tossing the coin in the past. heads(H), tails(T), old order.
Interpretation example 1: All averaged, P=30/60=50%
Interpretation example 2: The frequency changes between the first half 30 and the second half 30. Average of the last 30 pieces, P=20/30=67%
Interpretation example 3: The sequence "THH" is repeated in the last 30 numbers, and the next is 0, so P=0%
2 and 3 are false if we assume that the probabilities do not change with each toss of a coin, but there is no such assumption.
1 is also false if there is no assumption that it will not be in a state that is neither heads nor tails.
Even if the probabilities have not changed and we only get heads or tails, there are still insufficient preconditions for the interpretation of 1 to be correct.
Interpretation 1 only answers the frequency of heads in the past. The precondition is that the probability of the next coin toss is equal to its frequency.
Under the preconditions, it seems possible to create a program with correct interpretation, but there is no guarantee that the coin toss performed in the natural world will comply with the preconditions.
In the absence of preconditions, none of interpretations 1, 2, or 3 is correct in modern mathematics.

■Deduction problem, induction problem and misinterpretation

All problems can be divided into deductive problems and inductive problems.
A deduction problem is one in which all the necessary information is available.
The inductive problem is that the information is insufficient, but "I don't know" is not acceptable.
For example, the outcome of a future coin toss is correct with "I don't know", but a better answer is required.
The true answer is "I don't know", but another answer within some preconditions is requied.
A "misinterpretation" is when the preconditions is not what the questioner wanted.
If a person is the questioner, the person's subjectivity determines what is correct.
However, when inferring natural phenomena, it is considered that there is an objective "correctness" because the questioner is the natural world.
The correctness of the interpretation of natural phenomena can be investigated experimentally, but it is not possible to completely distinguish whether a probabilistic phenomenon is accidental or inevitable.
In science, it is common to think that the better interpretation is the one that can explain a wider range uniformly with fewer parameters.
For example, the properties of all atoms can be inferred as combinations of elementary particles without examining them individually.
As for the preconditions of the induction problem, if you examine various examples, you should be able to see the preconditions that are the origin that can explain everything.
Even common sense laws of physics, which have been valid until today, do not guarantee that they will be valid tomorrow, but we have no choice but to think that they are correct in predicting tomorrow.
It is also necessary to determine the criteria for the preconditions of induction problem to be correct as "axioms of induction".

■Exploring the Axiom of Induction

"Axiom" is a premise that it is correct and cannot be proved.
It is easy to point out that an unprovable theory is wrong, but it is difficult to point out that it is correct.
It only makes you think you're probably right when you can't find any more mistakes.
The "Axiom of Induction" can be found by the following procedure.
1. Use the simplest possible inductive reasoning example to determine an inference result that is intuitively clearly correct.
2. Check whether it is still intuitively correct, such as using extreme numbers.
3. Do the same with a slightly more complicated example.
4. Find an inductive inference computation procedure that applies to all the examples considered so far.
5. Assuming more complicated and extreme examples and verifying.
6. Remove the existing axioms from the found calculation procedure and extract the newly assumed "axioms".
7. Intuitively judge whether the new "axiom" is reasonable.