10.1: The most important unsolved problem---what is causality?
The importance of "measurement" and "causality" should be reconfirmed in the following famous maxims:

$(C_1):$ $\qquad$There is no science without measurement.
$(C_2):$$\qquad$Science is the knowledge about causal relationship.
which should be also regarded as one of the linguistic interpretation in the wide sense.

10.1.1: Modern science started from the discovery of "causality."
When a certain thing happens, the cause always exists. This is called causality. You should just remember the proverb of \begin{align} \mbox{ "smoke is not located on the place which does not have fire" } \end{align} It is not so simple although you may think that it is natural. For example, if you consider
$\qquad$ This morning I feel good. Is it because that I slept sound yesterday? $\;\;$ or is it because I go to favorite golf from now on?
you may be able to understand the difficulty of how to use the word "causality". In daily conversation, it is used in many cases, mixing up "a cause (past)", "a reason (connotation)", and "the purpose and a motive (future)."
It may be supposed that the pioneers of research of movement and change are \begin{align} \left\{\begin{array}{ll} \mbox{Heraclitus(BC.540 -BC.480): "Everything changes."} \\ \\ \mbox{ Parmenides (born around BC. 515): "Movement does not exist." } \\ \mbox{(Zeno's teacher)} \end{array}\right. \end{align} though their assertions are not clear. However, these two pioneers (i.e., Heraclitus and Parmenides ) noticed first that "movement and change" were the primary importance keywords in science(= "world description") , i.e., it is

\begin{align} & \color{magenta}{\mbox{ [The beginning of World description ]} } \\ \\ = & \mbox{ [The discovery of movement and change ] } \\ = & \left\{\begin{array}{ll} \mbox{ Heraclitus (BC.540 -BC.480) } \\ \\ \mbox{ Parmenides (born around BC. 515) } \end{array}\right. \end{align}

However, Aristotle(BC384--BC322) furthermore investigated about the essence of movement and change, and he thought that

all the movements had the "purpose."

For example, supposing a stone falls, that is because the stone has the purpose that the stone tries to go downward. Supposing smoke rises, that is because smoke has the purpose that smoke rises upwards. Under the influence of Aristotle, "Purpose" continued remaining as a mainstream idea of "Movement" for a long time of 1500 years or more.

Although "the further investigation" of Aristotle was what should be praised, it was not able to be said that "the purpose was to the point." In order to free ourselves from Purpose and for human beings to discover that the essence of movement and change is "causal relationship", we had to wait for the appearance of Galileo, Bacon, Descartes, Newton, etc. \begin{align} \mbox{ Revolution to "Causality" from "Purpose" } \end{align} is the greatest history-of-science top paradigm shift. It is not an overstatement even if we call it "birth of modern science". \begin{align} \overset{\scriptsize{\mbox{the birth of world description}}}{ \underset{\scriptsize{\mbox{ (Heraclitus, Parmenides, Zeno)}}} {\fbox{Movement}} } \xrightarrow[\scriptsize{\mbox{Aristotle :( About 1500 years) }}]{\scriptsize{\mbox{ "purpose"}}} \overset{\scriptsize{\mbox{the birth of modern science}}}{ \underset{\scriptsize{\mbox{ ( Galileo, Bacon, Descartes, Newton)}}} {\fbox{Causality}} } \end{align}

$\fbox{Note 10.1}$ I cannot emphasize too much the importance of the discovery of the term: "causality". That is,
$(\sharp):$ Science is the desipline about phenomena can be represented by the term "causality" (i.e., "No smoke, no fire" )
Thus, I consider that the discovery of "causality" is equal to that of science.

10.1.2: Four answers (a)--(d) to "what is causality?"

As mentioned above, about "what is an essence of movement and change?", it was once settled with the word "causality." However, not all were solved now. We do not yet understand "causality" fully. In fact,
Problem 10.1 "What is causality?" is the most important outstanding problems in modern science.
Answer this problem!

There may be some readers who are surprised with saying like this, although it is the outstanding problems in the present. Below, I arrange the history of the answer to this problem.

$(a):$ [Realistic causality]: Newton advocated the realistic describing method of Newtonian mechanics as a final settlement of accounts of ideas, such as Galileo, Bacon, and Descartes, and he thought as follows. :
$\quad$ "Causality" actually exists in the world. Newtonian equation described faithfully this "causality". That is, Newtonian equation is the equation of a causal chain.
This realistic causality may be a very natural idea, and you may think that you cannot think in addition to this. In fact, probably, we may say that the current of the realistic causal relationship which continues like
"Newtonian mechanics$\longrightarrow$ Electricity and magnetism$\longrightarrow$ Theory of relativity$\longrightarrow$ $\; \cdots \;$"

is a scientific flower.

However, there are also other ideas, i.e., three "non-realistic causalities" as follows.
$(b):$ [Cognitive causality]: David Hume, Immanuel Kant, etc. who are philosophers thought as follows.
$\quad$ We can not say that "Causality" actually exists in the world, or that it does not exist in the world. And when we think that "something" in the world is "causality", we should just believe that the it has "causality".
Most readers may regard this as "a kind of rhetoric", however, several readers may be convinced in "Now that you say that, it may be so." Surely, since you are looking through the prejudice "causality", you may look such. This is Kant's famous "Copernican revolution", that is, \begin{align} \Large{\color{magenta}{ \mbox{ "recognition constitutes the world." }} } \end{align} which is considered that the recognition circuit of causality is installed in the brain, and when it is stimulated by "something" and reacts, "there is causal relationship." Probably, many readers doubt about the substantial influence which this (b) had on the science after it. However, in this book, I adopted the friendly story to the utmost to Kant.

$(c):$ [Mathematical causality(Dynamical system theory)]: Since dynamical system theory has developed as the mathematical technique in engineering, they have not investigated "What is causality?" thoroughly. However,
$\quad$ In dynamical system theory, we start from the state equation (i.e., simultaneous ordinary differential equation of the first order) such that \begin{align} & \left\{\begin{array}{ll} \frac{d\omega_1}{dt}{} (t)=v_1(\omega_1(t),\omega_2(t),\ldots,\omega_n(t), t) \\ \frac{d\omega_2}{dt}{} (t)=v_2(\omega_1(t),\omega_2(t),\ldots,\omega_n(t), t) \\ \cdots \cdots \\ \frac{d\omega_n}{dt}{} (t)=v_n (\omega_1(t),\omega_2(t),\ldots,\omega_n(t), t) \end{array}\right. \tag{10.1} \end{align} and, we think that
$(\sharp):$ the phenomenon described by the state equation has "causality.
This is the spirit of dynamical system theory (= statistics ). Although this is proposed under the confusion of mathematics and world description, it is quite useful. In this sense, I think that (c) should be evaluated more. Also, we recall Pythagoras' words:
The origin of everything is the mathematics

$(d):$ [Linguistic causal relationship (Measurement Theory)]:

The causal relationship of measurement theory is decided by the Axiom 2 (causality; $\S$10.3) of this chapter. If I say in detail,:

$\quad$ Although measurement theory consists of the two Axioms 1 and 2, it is the Axiom 2 that is concerned with causal relationship. When describing a certain phenomenon in quantum language (i.e., a language called measurement theory) and using Axiom 2 (causality; $\S$10.3), we think that the phenomenon has causality.

Summary 10.2
The above is summarized as follows.
$\quad$ (a) World is first$\;\;$
(b) Recognition is first$\;\;$
(c) Mathematics(buried into ordinary language) is first $\;\;$
(d) Language (= quantum language) is first

Now, in measurement theory, we assert the next as said repeatedly:
$\quad$ $\quad$ Quantum language is a basic language which describes various sciences.
Supposing this is recognized, we can assert the next. Namely,
$\quad$ $\color{magenta}{\mbox{ In science, causality is just as mentioned in the above (d).}}$
This (d) is my answer to "What is causality?", and I explain these details after the following paragraph.

$\fbox{Note 10.2}$ Consider the following problems:
$(\sharp_1):$ What is time (space, causality, probability, etc. ) ?
Two kinds of answers are presented. That is,
$(\sharp_2):$ The answer of "What is XX?" $\left\{\begin{array}{ll} \mbox{(a): To show the definition of XX } \\ \\ \mbox{(b): To show how to use the term "XX" } \end{array}\right. $
The answer (b) is due to the linguistic world view. In this book, the answer to the quaestion ($\sharp_1$) is presented from the linguistic point of view.