12.4: Two kinds of absurdness ---idealism and dualism

This section is extracted from

 $(\sharp):$ S. Ishikawa, "Measurement Theory in the Philosophy of Science" $\quad$ arXiv:1209.3483 [physics.hist-ph] 2012

Measurement theory (= quantum language ) has two kinds of absurdness. That is,
 $(\sharp):$ ${\mbox{ Two kinds of absurdness }} \left\{\begin{array}{ll} \mbox{idealism} {\cdots} \mbox{linguistic world-view} \\ {\mbox{ The limits of my language mean the limits of my world} } \\ \\ \mbox{dualism}\cdots {\mbox{Descartes=Kant philosophy}} \\ {\mbox{ The dualistic description for monistic phenomenon } } \end{array}\right.$
In what follows, we explain these.

12.4.1: The linguistic interpretation --- A spectator does not go up to the stage

Problem 12.13 [A spectator does not go up to the stage] Consider the elementary problem with two steps (a) and (b):

$(a):$ Consider an urn, in which 3 white balls and 2 black balls are. And consider the following trial:
 $\bullet$ Pick out one ball from the urn. If it is black, you return it in the urn If it is white, you do not return it and have it. Assume that you take three trials.
(b):

Then, calculate the probability that you have 2 white ball after (a)(i.e., three trials).

Answer Put ${\mathbb N}_0$ $=\{0,1,2,\ldots\}$ with the counting measure. Assume that there are $m$ white balls and $n$ black balls in the urn. This situation is represented by a state $(m,n) \in {\mathbb N}_0^2$. We can define the dual causal operator ${\Phi^*}: {\cal M}_{+1}({\mathbb N}_0^2)$ $\to {\cal M}_{+1}({\mathbb N}_0^2)$ such that

\begin{align} {\Phi^*}(\delta_{(m,n)}) = \left\{\begin{array}{ll} \frac{m}{m+n} \delta_{(m-1,n)}+\frac{n}{m+n} \delta_{(m,n)} & \quad ( { \mbox{when} \;\; m \not= 0 \; )} \\ \delta_{(0,n)} & \quad {(\mbox{when } m = 0 \; )}. \end{array}\right. \tag{12.17} \end{align}

where $\delta_{(\cdot)}$ is the point measure. Let $T=\{0,1,2,3\}$ be discrete time. For each $t$ $\in T$, put $\Omega_t = {\mathbb N}_0^2$. Thus, we see:

\begin{align} & {[\Phi^*]}^3 (\delta_{(3,2)}) = {[\Phi^*]}^2 \left( \frac{3}{5}\delta_{(2,2)} + \frac{2}{5}\delta_{(3,2)} \right) \nonumber \\ = & {\Phi^*} \left( (\frac{3}{5} (\frac{2}{4} \delta_{(1,2)} +\frac{2}{4} \delta_{(2,2)} ) + \frac{2}{5}( \frac{3}{5} \delta_{(2,2)}+\frac{2}{5} \delta_{(3,2)} ) \right) \nonumber \\ = & {\Phi^*} \left( \frac{3}{10} \delta_{(1,2)} +\frac{27}{50} \delta_{(2,2)} + \frac{4}{25} \delta_{(3,2)} \right) \nonumber \\ = & \frac{3}{10} ( \frac{1}{3} \delta_{(0,2)}+\frac{2}{3} \delta_{(1,2)} ) +\frac{27}{50} ( \frac{2}{4} \delta_{(1,2)}+\frac{2}{4} \delta_{(2,2)} ) + \frac{4}{25} ( \frac{3}{5} \delta_{(2,2)}+\frac{2}{5} \delta_{(3,2)} ) \nonumber \\ = & \frac{1}{10} \delta_{(0,2)}+\frac{47}{100} \delta_{(1,2)} +\frac{183}{500} \delta_{(2,2)} + \frac{8}{125} \delta_{(3,2)} \tag{12.18} \end{align}

Define the observable ${\mathsf O} =({\mathbb N}_0,2^{{\mathbb N}_0}, F^)$ in $L^\infty (\Omega_3)$ such that

\begin{align} [F^{}(\Xi)](m,n) = \left\{\begin{array}{ll} 1 & \qquad (m,n ) \in \Xi \times {\mathbb N}_0 \subseteq \Omega_3 \\ 0 & \qquad (m,n ) \notin \Xi \times {\mathbb N}_0 \subseteq \Omega_3 \end{array}\right. \end{align}

Therefore, the probability that a measured value "$2$" is obtained by the {{measurement}} ${\mathsf M}_{L^\infty ({\mathbb N}_0^2)}(\Phi^3{\mathsf O},$ $S_{[(3,2)]})$ is given by

\begin{align} [\Phi^3 (F (\{2\}))](3,2) = \int_{\Omega_3} [F(\{2\})](\omega) ({[\Phi^*]}^3 (\delta_{(3,2)}) )(d \omega) = \frac{183}{500} \tag{12.19} \end{align}
$\square \quad$

The above may be easy, but we should note that
 $(c):$ the part (a) is related to causality, and the part (b) is related to measurement.
Thus, the observer is not in the (a). Figuratively speaking, we say: \begin{align} \mbox{ A spectator does not go up to the stage } \end{align} Thus, someone in the (a) should be regard as "robot".

$\fbox{Note 12.4}$ The part (a) is not related to "probability". That is because The spirit of measurement theory says that
 $\quad$ there is no probability without measurements.
although something like "probability" in the (a) is called "Markov probability".

12.4.2:In the beginning was the words---Fit feet to shoes

Remark 12.14 [The confusion between measurement and causality ( Continued from Example 2.31)] Recall Example 2.31 [The measurement of "cold or hot" for water]. Consider the measurement ${\mathsf M}_{L^\infty ( \Omega )} ( {\mathsf O}_{{{{{c}}}}{{{{h}}}}},$ $S_{[\omega]} )$ where $\omega=5 \mbox{°C}$. Then we say that

 $(a):$ By the { {{measurement}}} ${\mathsf M}_{L^\infty ( \Omega )} ( {\mathsf O}_{{{{{c}}}}{{{{h}}}}}, S_{[ \omega(=5)]} )$, the probability that a measured value $x(\in X =\{{{{{c}}}}, {{{{h}}}}\})$ belongs to a set $\left[\begin{array}{cc} {} \emptyset (={\text {empty set}}) \\ \{ \mbox{c}\} {} \\ \{ \mbox{h} \} \\ \{ \mbox{c} ,\mbox{h}\} \end{array}\right]$ is equal to $\left[\begin{array}{cc} {} 0 \\ {} [F(\{ c \})] (5)=1 \\ {} [F(\{ h \})](5) =0 \\ {} 1 \end{array}\right]$
Here, we should not think:
 $\quad$ $\qquad$ "5°C" is the cause and "cold" is a result.
That is, we never consider that
 $(b):$ $\qquad \qquad$ $\underset{\mbox{(cause)}}{\fbox{5 °C}} \longrightarrow \underset{\mbox{(result)}}{\fbox{cold}}$
That is because Axiom 2 (causality; $\S$10.3) is not used in (a), though the (a) may be sometimes regarded as the causality (b) in ordinary language.

$\fbox{Note 12.5}$ However, from the different point of view, the above (b) can be justified as follows. Define the dual causal operator ${\Phi^*} : {\cal M}([0, 100]) \to {\cal M}(\{{{{{c}}}}, {{{{h}}}}\})$ by \begin{align} & [{\Phi^*} \delta_\omega ](D) = f_{ {{{{c}}}} }(\omega) \cdot \delta_{\mbox{ C}} (D) + f_{{{{{h}}}} }(\omega) \cdot \delta_{\mbox{ H}}(D) \qquad (\forall \omega \in [0,100],\;\; \forall D \subseteq \{{{{{c}}}}, {{{{h}}}}\}) \end{align} Then, the (b) can be regarded as "causality". That is,
 $(\sharp):$ $\mbox{ "measurement or causality" depends on how to describe a phenomenon. }$
This is the linguistic world-description method.

Remark 12.15 [Mixed measurement and causality] Reconsider Problem 9.5 (urn problem:mixed measurement). That is, consider a state space $\Omega=\{\omega_1, \omega_2 \}$, and define the observable ${\mathsf O} = ( \{ {{w}}, {{b}} \}, 2^{\{ {{w}}, {{b}} \} } , F)$ in $L^\infty (\Omega)$ in Problem 9.5. Define the mixed state by $\rho^m =p \delta_{\omega_1} +(1-p) \delta_{\omega_2}$. Then the probability that a measured value $x$ $(\in \{ {{w}} , {{b}} \})$ is obtained by the mixed measurement ${\mathsf M}_{L^\infty(\Omega)}({\mathsf O}, S_{[{}\ast{}] }(\rho^m) )$ is given by

\begin{align} P(\{ x \}) &= \int_\Omega [F(\{ x \})]( \omega) \rho^m(d \omega) = p [F(\{ x \})](\omega_1) + (1-p) [F(\{ x \})](\omega_2) \nonumber \\ &= \left\{\begin{array}{ll} 0.8 p + 0.4 (1-p) \quad & (\mbox{when }x={{w}}{}\; ) \\ 0.2 p + 0.6 (1-p)) \quad & (\mbox{when }x={{b}}{}\; ) \end{array}\right. \tag{12.20} \end{align}

Now, define a new state space $\Omega_0$ by $\Omega_0=\{\omega_0\}$. And define the dual (non-deterministic) causal operator ${\Phi^*}: {\cal M}_{+1}(\Omega_0)$ $\to {\cal M}_{+1}(\Omega)$ by ${\Phi^*}(\delta_{\omega_0})$ $=p \delta_{\omega_1} +(1-p) \delta_{\omega_2}$. Thus, we have the (non-deterministic) causal operator ${\Phi}: L^\infty (\Omega)$ $\to L^\infty (\Omega_0)$. Here, consider a pure measurement ${\mathsf M}_{L^\infty (\Omega_0)}(\Phi{\mathsf O}, S_{[\omega_0]})$. Then, the probability that a measured value $x$ $(\in \{ {{w}} , {{b}} \})$ is obtained by the measurement is given by

\begin{align} P(\{ x \}) &= [\Phi (F (\{ x \}))](\omega_0) = \int_\Omega [F(\{ x \})]( \omega) \rho^m (d \omega) \\ &= \left\{\begin{array}{ll} 0.8 p + 0.4 (1-p) \quad & (\mbox{when }x={{w}}{}\; ) \\ 0.2 p + 0.6 (1-p)) \quad & (\mbox{when }x={{b}}{}\; ) \end{array}\right. \end{align}

which is equal to the (12.20). Therefore, the mixed measurement ${\mathsf M}_{L^\infty (\Omega)}({\mathsf O}, S_{[{}\ast{}] }(\nu_0) )$ can be regarded as the pure measurement ${\mathsf M}_{L^\infty (\Omega_0)}(\Phi{\mathsf O}, S_{[\omega_0]})$.

$\fbox{Note 12.6}$
In the above arguments, we see that
 $(\sharp):$ $\qquad \qquad \mbox{ Concept depends on the description }$
This is the linguistic world-description method. As mentioned frequently, we are not concerned with the question "what is $\bigcirc \bigcirc$?". The reason is due to this $(\sharp)$. "Measurement or Causality" depends on the description. Some may recall Nietzsche's famous saying:
 $\bullet$ $\qquad$ There are no facts, only interpretations.
This is just the linguistic world-description method with the spirit: "Fit feet (=world) to shoes (language)"
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$\fbox{Note 12.7}$ In the book "The astonishing hypothesis" by F. Click (the most noted for being a co-discoverer of the structure of the DNA molecule in 1953 with James Watson)), Dr. Click said that
 $(a):$ You, your joys and your sorrows, your memories and your ambitions,your sense of personal identity and free will,are in fact no more than the behavior of a vast assembly of nerve cells and their associated molecules.

It should be note that this (a) and the dualism do not contradict. That is because quantum language says:
 $(b):$ Describe any monistic phenomenon by the dualistic language (= quantum language )!
Also, if the above (a) is due to David Hume, he was a scientist rather than a philosopher.
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