Section outline

  • The philosopher Zeno of Elea proposed many paradoxes—statements that sound logical but lead to contradictory conclusions—to explore motion. His paradoxes led  some philosophers to use infinite geometric series to argue that motion—and, in fact, any change—was impossible.
    ::埃莱亚的哲学家泽诺(Zeno of Elea)提出了许多自相矛盾的说法,这些说法听起来很合乎逻辑,但却导致自相矛盾的结论 — — 来探索运动。 他的自相矛盾导致一些哲学家使用无限几何序列来论证这一动议 — — 事实上,任何变化都是不可能的。

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    One of his most well-known motion paradoxes is the Dichotomy paradox: 
    ::他最著名的运动自相矛盾之处之一是切除术自相矛盾:

    That which is in locomotion must arrive at the half-way stage before it arrives at the goal. – as recounted by Aristotle , Physics 1
    ::运动中的东西必须在到达目标之前到达中途阶段。 亚里士多德所描述的物理1

    Here we will use the idea in the Dichotomy paradox to argue that an arrow could never hit its target. (The arrow itself is at the center of its own paradox; see below.) Zeno's argument goes like this: Imagine you have shot an arrow, and it has to fly 64 feet to reach its target. First it has to fly half the distance, or 32 feet. Then it has to fly half the remaining distance, or 16 feet. Then it flies half of that distance, so it flies 8 feet.
    ::在这里,我们将使用小切口悖论中的概念来论证箭头不可能击中目标。 (箭头本身是其悖论的中心,请看下面)。 Zeno的论点是这样的:想象你射了箭,它必须飞64英尺才能达到目标。首先,它必须飞行一半距离,或32英尺。然后,它必须飞行剩下的一半距离,或16英尺。然后,它飞了一半距离,所以飞了8英尺。

    1. Find the 1st 10 terms in this sequence. 
    ::1. 在这一顺序中找出第10个术语。

    2. Find a formula for the nth term of this sequence. 
    ::2. 为这个序列的 nth 术语寻找公式。

    Since any distance can be divided in half, the arrow has to take an infinite number of steps to reach its goal. Since you are adding an infinite number of steps together, it would take an infinite amount of time to reach the goal. This means the arrow will never reach the target.
    ::由于任何距离可以一分为二, 箭头必须采取无限多的步骤才能达到它的目标。 由于您正在一起添加无限多的步骤, 达到目标需要无限的时间。 这意味着箭头永远不会达到目标 。

    3. Find the 1st five partial sums of this sequence. 
    ::3. 找出这一顺序的第一和第二部分部分总和。

    4. Find the infinite sum of this sequence, if possible. 
    ::4. 如有可能,找出这一序列的无限总和。

    When people argued that everyone could see that the arrow hit the target, Zeno explained that it only appeared to move. In reality, he said, nothing was moving, and the appearance was an illusion. This is the Arrow paradox. 
    ::当人们认为每个人都能看到箭射中目标时,泽诺解释说,箭只看似在移动。 事实上,他说,没有动静,外观是一种幻觉。 这就是箭的悖论。

    If everything when it occupies an equal space is at rest, and if that which is in locomotion is always occupying such a space at any moment, the flying arrow is therefore motionless.
    ::如果它占据平等空间时的一切都处于休息状态,如果运动中的物体在任何时刻总是占据这种空间,则飞行箭是无动于衷的。

    – as recounted by Aristotle, Physics 1
    ::- 正如亚里士多德所描述的,物理1

    5. Explain what Zeno failed to understand about the motion of the arrow. 
    ::5. 解释Zeno为何不理解箭的动向。

    by TED-Ed describes the Dichotomy paradox and how to resolve it.
    ::TED-Ed描述了小切口悖论和如何解决它。

    References
    ::参考参考资料

    1. "Zeno's Paradoxes," last edited May 6, 2017, 
    ::1. 2017年5月6日编辑的《泽诺的悖论》