14.4 : Zeno's paradoxesAchilles and a tortoise
 14.4.1:What is Zeno's paradox?
 14.4.2:Dynamical system theoretical approach to Zeno's paradox
 14.4.2.1:The formulation od dynamical system theory
 14.4.2.2:Dynamical system theoretical answer to Zeno's paradox
 14.4.2.3:Why isn't dynamical system theoretical answer authorized
 14.4.3:Is quantum linguistic anser authorized?
In this section, we explain our opinion for Zeno's paradox ( the oldest paradox in science ): that is, \begin{align} \mbox{ What is the meaning of Zeno's paradox? } \end{align} 14.4.1: What is Zeno's paradox?
Although Zeno's paradox has some types (i.e., "flying arrow", "Achilles and a tortoise", "dichotomy", "stadium", etc.), I think that { these are essentially the same problem}. And I think that the flying arrow expresses the essence of the problem exactly and is the first masterpiece in Zeno's paradoxes. However, since "Achilles and the tortoise" may be more famous, I will also describe this as follows.
$\bullet$  Consider a flying arrow. In any one instant of time, the arrow is at rest. Therefore, If the arrow is motionless at every instant, and time is entirely composed of instants, then motion is impossible. 
[Achilles and a tortoise]
$\bullet$  I consider competition of Achilles and a tortoise. Let the start point of a tortoise (a late runner) be the front from the starting point of Achilles (a quick runner). Suppose that both started simultaneously. If Achilles tries to pass a tortoise, Achilles has to go to the place in which a tortoise is present now. However, then, the tortoise should have gone ahead more. Achilles has to go to the place in which a tortoise is present now further. Even Achilles continues this infinite, he can never catch up with a tortoise. 
In order to explain \begin{align} \mbox{ "What is Zeno's paradox?" } \end{align} we have to start from the following Figure. That is, we assert that \begin{align} \mbox{Zeno's paradox can not be understood without the following figure:} \end{align}
$(A):$  Descartes=Kant philosophy and the philosophy of language have no power to describe Zeno's paradox (Paradox 14.9). 
$(B_1):$  How do we describe Zeno's paradox (Paradox 14.9) in terms of Newtonian mechanics? 
$(B_2):$  How do we describe Zeno's paradox (Paradox 14.9) in terms of quantum mechanics? 
$(B_3):$  How do we describe Zeno's paradox (Paradox 14.9) in terms of the theory of relativity? 
$(B_4):$  How do we describe Zeno's paradox (Paradox 14.9) in terms of statistics (i.e., the dynamical system theory) ? 
$(B_5):$  How do we describe Zeno's paradox (Paradox 14.9) in terms of quantum language? 
$(C):$  What is the most proper world description for Zeno's paradox (Paradox 14.9)? 
$(D):$  "to solve Zeno's paradox (Paradox 14.9)" $\Longleftrightarrow$ "to answer the above $(B_5)$" 
$(E):$  The answer of the above (C) is just quantum language 
$\quad$  Describe "flying arrow" and "Achilles an a tortoise" in (classical) quantum language! 
Before the answer of Problem 14.11, we give the answer to the Problem (B$_4$), i.e., the dynamical system theoretical answer. However, in order to do it, we have to start from the formulation of dynamical system theory in what follows
14.4.2: The answer to $(B_4)$: the dynamical system theoretical answer to Zeno's paradox
14.4.2.1: The formulation of dynamical system theory
Although statistics and dynamical system theory have no clear formulations, as mentioned in Chapter 13, we have the opinion that statistics and dynamical system theory are the same things. At least, the following formulation (i.e., the formulation of dynamical system theory in the narrow sense) should belong to statistics.
Dynamical system theory is formulated as follows. \begin{align} \underset{\mbox{}}{\fbox{Dynamical system theory}} = \underset{\mbox{}}{\fbox{①:State equation}} + \underset{\mbox{}}{\fbox{②:Measurement equation}} \tag{14.9} \end{align}
①: $ \underset{\mbox{}}{\fbox{State equation}} $ is as follows. Let $T={\mathbb R}$ be the time axis. For each $t ( \in T)$, consider the state space $\Omega_t = {\mathbb R }^n$ ($n$dimensional real space). The state equation (Chap. 13 (13.2)) is defined by the following simultaneous ordinary differential equation of the first order
\begin{align} & \underset{\mbox{}}{\fbox{State equation}} = \left\{\begin{array}{ll} \frac{d\omega_1}{dt}{} (t)=v_1(\omega_1(t),\omega_2(t),\ldots,\omega_n(t),\epsilon_1(t), t) \\ \frac{d\omega_2}{dt}{} (t)=v_2(\omega_1(t),\omega_2(t),\ldots,\omega_n(t),\epsilon_2(t), t) \\ \cdots \cdots \\ \frac{d\omega_n}{dt}{} (t)=v_n (\omega_1(t),\omega_2(t),\ldots,\omega_n(t), \epsilon_n(t),t) \end{array}\right. \tag{14.10} \end{align} where $\epsilon_k(t)$ is a noise ($k=1,2, \cdots, n $).②: $ \underset{\mbox{}}{\fbox{Measurement equation}} $ is as follows. Consider the measured value space $X = {\mathbb R }^m$ ($m$dimensional real space). The measurement equation (Chap.13 (13.2)) is defined by
\begin{align} & \underset{\mbox{}}{\fbox{Measurement equation}} = \left\{\begin{array}{ll} x_1(t)=g_1(\omega_1(t),\omega_2(t),\ldots,\omega_n(t),\eta_1(t), t) \\ x_2(t)=g_2(\omega_1(t),\omega_2(t),\ldots,\eta_n(t),\eta_2(t), t) \\ \cdots \cdots \\ x_m(t) =g_m (\omega_1(t),\omega_2(t),\ldots,\eta_n(t), \eta_n(t),t) \end{array}\right. \tag{14.11} \end{align}where $g(=(g_1, g_2, \cdots, g_n)): \Omega \times {\mathbb R}^2 \to X$ is the system quantity and $\eta_k(t)$ is a noise ($k=1,2, \cdots, m $). Here, $x(t)(=(x_1(t), x_2(t), \cdots, x_n(t)))$ is called a motion function.
14.4.2.2: The dynamical system theoretical answer to Zeno's paradox
Let $q(t)$ be the position of the flying arrow at time $t$. That is, consider the motion function $q(t)$.
$\bullet$ 
Note that the following logic (i.e., Zeno's logic ) is wrong:

[The dynamical system theoretical answer to "Achilles and a tortoise (in Paradox 14.9)"]
For example, assume that the velocity $v_q$ [resp. $v_s$] of the quickest [resp. slowest] runner is equal to $v(>0)$ [resp. ${\gamma}v \; ( 0<{\gamma}<1)$]. And further, assume that the position of the quickest [resp. slowest] runner at time $t=0$ is equal to $0$ [resp. $a \; (>0)$]. Thus, we can assume that the position ${\xi (t)}$ of the quickest runner and the position $\eta (t)$ of the slowest runner at time $t$ $( \ge 0)$ is respectively represented by
\begin{eqnarray} \left\{\begin{array}{ll} \xi (t) =vt \\ \eta(t) = {\gamma}vt + a \end{array}\right. \tag{14.12} \end{eqnarray}$\bullet$  Calculations 
The formula (14.8) can be calculated as follows (i.e., (i) or (ii)):
[(i): Algebraic calculation of (14.8)]:
Solving $\xi(s_0)=\eta(s_0)$, that is, \begin{align} vs_0 = \gamma v s_0 + a \end{align} we get $s_0= \frac{a}{{(1\gamma)} v}$. That is, at time $s_0= \frac{a}{{(1\gamma)} v}$, the fast runner catches up with the slow runner.
[(ii): Iterative calculation of (14.8)]:
Define $t_k$ $(k=0,1,...)$ such that, $t_0=0$ and
\begin{align} t_{k+1}= \gamma v t_k + a \;\;\;\; (k=0,1,2,...) \end{align}
Thus,
we see that
$t_k=\frac{(1{\gamma}^k)a}{(1{\gamma})v}$
$(k=0,1,...)$.
Then,
we have that
as $k \to \infty $.
Therefore, the quickest runner catches up with the slowest
at time $s_0 =\frac{a}{(1{\gamma})v}$.
[(iii): Conclusion]:
After all, by the above (i) or (ii), we can conclude that
$(\sharp):$  the quickest runner can overtake the slowest at time $s_0 =\frac{a}{(1{\gamma})v}$. 
14.4.2.3: Why isn't the Answer 14.13 authorized?
We believe that Answer 14.13 is not the wrong answer of Zeno's paradox. If so, we have to answer the following question:
$(F):$  Why isn't the Answer 14.13 accepted as the final answer of Zeno's paradox? 
We of course believe that
$(G_1):$  the reason is due to the fact that statistics (=dynamical system theory) is not accepted as the worldview in Figure 14.10 
Or equivalently,
$(G_1):$  the linguistic worldview is not accepted as the worldview in Figure 14.10 
If so, the readers note that
$(H):$  the purpose of this note is to assert that the linguistic world view should be authorized in Figure 14.10. 
14.4.3: Quantum linguistic answer to Zeno's paradoxes
Before reading Answer 14.14 ( Zeno's paradox(flying arrow) ), confirm our spirit:
$(I):$  The theory described in ordinary language should be described in a certain world description. That is because almost ambiguous problems are due to the lack of "the worlddescription method". 
$(J):$  it suffices to describe "motion function $q(t)$ in Answer 14.13 (flying arrow)" in terms of quantum language. Here, the motion function should be a measured value, in which the causality is concealed. 
In Corollary 14.7, putting \begin{align} q(t)=y_t( = g_t(\phi_{{t_0},t } (\omega_{t_0} ))) \end{align} we get the timeposition function $q(t)$.
$\qquad \qquad $Fig.1.1:The history of worlddescriptions 
Although there may be several opinions, we consider that the followings (i.e., (K$_1$) and (K$_2$)) are equivalent:
$(K_1):$  to accept Figure 14.10(=Fig.1.1):[The history of the worldview] 
$(K_2):$  to believe in Answer 14.14 as the final answer of Zeno's paradox 
$\fbox{Note 14.2}$ 
I think that
"the flying arrow" is Zeno's best work.
If readers agree to the above answer,
they can easily answer the other Zeno's paradoxes.
Also,
it should be noted that
Zeno of Elea (BC. 490430)
was a Greek philosopher
(about 2500 years ago).
Hence, we are not concerned with the historical aspect of Zeno's paradoxes.
Therefore,
we think that
Also, for the quantum linguistic spacetime, see $\S$10.7 ( Leibniz=Clarke correspondence). I doubt great philosophers' opinions concerning Zeno's paradoxes without Figure 14.10. 