14.4 : Zeno's paradoxes---Achilles and a tortoise

Contents:

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}

Figure 14.10 [=Figure 1.1 in $\S$1.1: The location of quantum language in the history of world-description]
Fig.1.1: the location of "quantum language" in the world-views
It is clear that
 $(A):$ Descartes=Kant philosophy and the philosophy of language have no power to describe Zeno's paradox (Paradox 14.9).
However, we have the following problems:
 $(B_1):$ 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?
And, finally, we have
 $(C):$ What is the most proper world description for Zeno's paradox (Paradox 14.9)?
We assert that
 $(D):$ "to solve Zeno's paradox (Paradox 14.9)" $\Longleftrightarrow$ "to answer the above $(B_5)$"
and conclude that
 $(E):$ The answer of the above (C) is just quantum language
Therefore, it suffices to answer the above (B$_5$) , that is,

Problem 14.11 [The meaning of Zeno's paradox]

 $\quad$ 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.

Formulation 14.12 [The formulation of dynamical system theory in the narrow sense]
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.

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:

 $\bullet$ for each time $t$, the position $q(t)$ of the flying arrow is determined. $\Longrightarrow$ the motion function $q$ is a constant function
Thus, 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

\begin{align} \big( \xi (t_k), \eta (t_k) \big) & = \big( \frac{(1-{\gamma}^k)a}{1-{\gamma}}, \frac{(1-{\gamma}^{k+1})a}{1-{\gamma}} \big) \nonumber \\ & \to \big( \frac{a}{1-{\gamma}},\frac{a}{1-{\gamma}} \big) \tag{14.13} \end{align}

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?
 $(F):$ 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 world-view in Figure 14.10

Or equivalently,

 $(G_1):$ the linguistic world-view is not accepted as the world-view 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.

 $(I):$ 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 world-description method".
Therefore,
 $(J):$ 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.
This will be done as follows.

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 time-position function $q(t)$.

 $\qquad \qquad$Fig.1.1:The history of world-descriptions

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 world-view] $(K_2):$ 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. 490-430) 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
 $(\sharp):$ "How did Zeno think Zeno's paradoxes?" is not important from the scientific point of view.
and
 $(\sharp):$ What is important is "How do we think Zeno's paradoxes?"

Also, for the quantum linguistic space-time, see $\S$10.7 ( Leibniz=Clarke correspondence). I doubt great philosophers' opinions concerning Zeno's paradoxes without Figure 14.10.