8.4: Cogito--- I think, therefore I am---

Recall the following figure.

The following example may be rather unnatural, but this is indispensable for the well-understanding of dualism.

Example 8.8 [Brain death ] Consider the classical basic structure

\begin{align} [C_0(\Omega ) \subseteq L^\infty ( \Omega, \nu ) \subseteq B(L^2 ( \Omega, \nu ))] \end{align} Let $\omega_n$ $(\in \Omega=\{\omega_1,\omega_2,\ldots, \omega_N \}$) be the state of Peter. Let ${\mathsf O}_{12}$ $=$ $(X_1 \times X_2 ,$ $2^{ X_1 \times X_2 } ,$ $F_{12}{}{{=}} F_1 {\mathop{\overset{qp}{\times}}} F_2)$ be the brain death observable in ${L^\infty (\Omega)}$ such that $X_1=\{ T, {\overline T}\}$ $X_2=\{ L, {\overline L}\}$, where $T$ $=$ $\mbox{"think"}$, ${\overline{T}}$ $=$ "not think", $L$ $=$ $\mbox{"live"}$, ${\overline{L}}$ $=$ "not live". For each $\omega_n$ $(n=1,2,\ldots,N)$, ${\mathsf O}_{12}$ satisfies the condition in Table 8.1.
Since $[F_{12}( \{ {\mbox{ T}}\} \times \{ \overline{\mbox{ L}}\} )](\omega_n) =0$, the following formula holds: \begin{align} [{\mathsf O}_{12}^{(1)};{ \{ {\mbox{ T}} \} }] \underset{ {\mathsf M}_{L^\infty (\Omega)} ({\mathsf O}_{12} , S_{ [\omega_n] }) }{ \Longrightarrow} [{\mathsf O}_{12}^{(2)};{\{ {{\mbox{ L}}} \}}] \end{align} Of course, this implies that
 $(D_1):$ $\qquad$ Peter thinks, therefore, Peter lives.
This is the same as the statement concerning brain death. In the above example, we see that
 $\quad$ $\qquad$ observer$\longleftrightarrow$doctor, $\qquad$ system$\longleftrightarrow$Peter,
The above (D$_1$) should not be confused with the following famous Descartes' saying (= cogito proposition):
 $(D_2):$ $\qquad$ "I think, therefore I am".
in which the following identification may be assumed:
 $\quad$ $\qquad$ observer$\longleftrightarrow$I, $\qquad$ system$\longleftrightarrow$I
And thus, the above is not a statement in dualism (=quantum language). In order to propose Figure 8.2 (i.e., dualism) ( that is, in order to establish the concept "I" in science), he started from the ambiguous statement "I think, therefore I am". Summing up, we want to say the following irony:
 $(B):$ Descartes proposed the dualism (i.e., Figure 8.2) by the cogito proposition (D$_2$) which is not understandable in dualism.
 $\fbox{Note 8.1}$ It is not true to consider that every phenomena can be describe in terns of quantum language. Although readers may think that the following can be described in measurement theory, but we believe that it is impossible. For example, the followings can not be written by quantum language: $$\left\{\begin{array}{ll} \mbox{ ① tense---past, present, future ---} \quad \quad \\ \mbox{② Heidegger's saying"In-der-Welt-sein"} \\ \mbox{③ the measurement of a measurement (Wigner's friend),} \quad \\ \mbox{④ Bergson's subjective time} \\ \mbox{⑤ observer's space-time}, \quad \\ \mbox{⑥ Only the present exists ( due to Augustinus(354-430))} \\ \mbox{⑦ principle of constancy of light velocity} \end{array}\right.$$ We have to recall Wittgenstein's sayings \begin{align} \mbox{ The limits of my language mean the limits of my world} \end{align} or $$\mbox{ What we cannot speak about we must pass over in silence. }$$ If we want to understand the above words ①--⑦, we have to propose the other scientific languages ( except quantum language). In fact, in order to describe ⑦, Einstein proposed the theory of relativity.