First, we have a question:

Why quantum mechanics has "interpretation"?


in spite that others ( i.e., Newtonian mechanics, the theory of relativity ) do not.

Or, in the first place,

What is the interpretation ( of quantum mechanics)?


This question will be answered below ( or, throughout this book ).

The linguistic Copenhagen interpretation

Axioms 1 (measurement ) and 2 ( causality ) are all of quantum language. Therefore,
$(A_1)$:after learning Axioms 1 and 2 by rote, we nood to brush up our skills to use them through trial and error.
Here, let us recall a wise sayings:
$(A_2)$:experience is the best teacher, or, custom makes all things
and our experience:
(B):A manual helps us to master the rulles quickly.
Thus, we understand: \begin{align*} & \mbox{the linguistic interpretation of quantum mechanics} \\ = & \mbox{the manual on how to use Axioms 1 and 2} \end{align*} Although the linguistic interpretation (= the linguistic Copenhagen interpretation) is composed of many statements, the simplest and best representation may be as follows.
The linguistic interpretation (This will be explained in $\S$3.1)
The most important statement in the linguistic interpretation is
"Only one measurement is permitted"


To put it strongly, we say the following opposite statements concerning the linguistic interpretation:
$(E_1)$:through trial and error, we can progress without the linguistic interpretation.
$(E_2)$: all that are written in this note are a part of the linguistic interpretation.

which are the same assertions from the opposite standing points. In this sense, there is a reason to consider that this lecture note is something like a cookbook. Of course, these (i.e., (E$_1$) and (E$_2$)) are extreme representations.



$\fbox{Note 1.5}$ Kolmogorov's probability theory starts from the following spell:
$(\sharp_1):$Let $(X, {\mathcal F}, P)$ be a probability space. Then, the probability that an event $\Xi ( \in {\mathcal F})$ happens is given by $P(\Xi)$
Through trial and error, Kolmogorov found his extension theorem, whose spirit says that
$(\sharp_2):$ Only one probability space is permitted.
This surely corresponds to the linguistic interpretation "Only one measurement is permitted." That is, \begin{align*} \overset{\mbox{(the most fundamental theorem)}}{\underset{\mbox{ (Only one probability space is permitted)}} {\fbox{Probability theory}}} \overset{\mbox{ (correspondence)}}{\longleftrightarrow} \overset{\mbox{(the linguistic interpretation)}}{ \underset{\mbox{(Only one measurement is permitted)}} {\fbox{Quantum language}}} \end{align*} In this sense, I want to assert that
$(\sharp_3):$Kolmogorov is one of the main discoverers of the linguistic interpretation.

Therefore, I am optimistic to believe that the linguistic interpretation "Only one measurement is permitted" can be acquired, after trial and error, if we start from Axioms 1 and 2. In fact, I myself acquired skill of the linguistic interpretation with this method.



So, I consider, as mentioned in (E$_1$), that we can theoretically do well without the linguistic interpretation.Also, one of our purposes may be to assert

the superiority of Axiom 1 the above spell ($\sharp_1$).
which is equivalent to
Assertion: quantum language is the theoretical statistics of the future