12.4: Two kinds of absurdness ---idealism and dualism
This section is extracted from
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)"
|
}
| $\fbox{Note 12.7}$ |
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.
|
}
