89. Zeno's Paradoxes and the Problem of Motion

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Lecture Notes

Main Topics #

• Parmenides and the Principle of Non-Contradiction: Parmenides insisted on the impossibility of something both being and not being. This principle cannot be rejected without contradiction, making it foundational to all thought.
• The Problem of Change: Change appears to require something both to be and not be, creating an apparent contradiction. Yet change is more evident to the senses than the abstract principle. Aristotle resolves this by properly understanding how change occurs without contradiction.
• Zeno’s Defense of Parmenides: Zeno developed six arguments against motion to defend his teacher’s thesis that change is impossible.
• The Continuous and Divisibility: The key to resolving Zeno’s paradoxes lies in understanding that the continuous is divisible in potency (infinitely) but not in actuality (all at once).
• Potency vs. Actuality: The human mind tends to make what is merely potential into something actual in imagination, creating false contradictions.
• Time as Continuous: Time, like any continuous magnitude, is not composed of indivisible nows but is infinitely divisible.

Zeno’s Six Arguments #

1. The Dichotomy (Not Getting Out the Door) #

• To traverse a distance, one must first reach the halfway point
• Before completing that, one must reach the half of the remaining distance
• This creates an infinite series of required destinations
• Conclusion: one can never complete a finite distance
• Aristotle’s Solution: Time is divisible proportionally with distance. One has infinitely many nows available to traverse the infinitely many points. The divisions are potential, not actual.

2. Achilles and the Tortoise #

• The swiftest runner cannot overtake the slowest if given a head start
• By the time Achilles reaches where the tortoise was, the tortoise has moved further
• This process repeats infinitely
• Berquist’s Analysis: Mathematically, if Achilles is twice as fast and the tortoise has a one-unit head start, they meet at a determinate point (two units). The error is treating infinite potential divisions as actual divisions that must be counted.
• Aristotle’s Solution: Same as Dichotomy—divisions are potential, not actual.

3. The Arrow (Standing Still in Flight) #

• In each indivisible now, the arrow occupies a space equal to itself
• Something occupying a space equal to itself is at rest
• Therefore, in each now the arrow is at rest
• Since time is composed of nows, the arrow is always at rest
• Aristotle’s Solution: Time is not composed of indivisible nows. The continuous is not made up of indivisibles. Without this false assumption, there is no paradox.

4. The Stadium (Relative Motion) #

• Three rows of equal magnitudes (A’s, B’s, C’s) move in such a way that B’s move toward C’s
• In the time a B passes one A, it passes two C’s
• This suggests equally fast bodies travel at different speeds depending on the reference frame
• Aristotle’s Solution: Speed must be measured relative to something. Compared to the A’s (at rest), B moves distance x. Compared to the C’s (in opposite motion), B covers distance 2x in the same time. The fallacy is equivocating on what the body is moving relative to.

5. Change Between Contradictories (Being/Not-Being) #

• When changing from being a man to not being a man, you must be both man and not-man
• This contradicts the principle of non-contradiction
• Aristotle’s Solution: Change involves divisible bodies. You can be partially white and partially not-white during change. For instantaneous changes (e.g., death), the change occurs at an instant (a now) that is the endpoint of a process. You never experience your own death while alive—it occurs in the last now of life, when you are no longer alive.

6. Circular Motion #

• A revolving sphere remains in the same place
• Yet motion requires change of place
• Aristotle’s Solution: One part of the sphere succeeds to where another part was. This is a genuine form of local motion, though different from whole-body translation.

Key Philosophical Principles #

The Danger of Actualizing the Potential #

• The human mind tends to imagine potential divisions as if they were actual
• Locke’s problem with “triangle in general” exemplifies this: we imagine a triangle with determinate sides, but a universal triangle is all triangles in potency, none in act
• When we imagine infinite points on a line segment, we make them actual in imagination and create an insoluble problem

Distinction in Ability vs. In Act #

• Something can be infinitely divisible in ability (potency) without being actually divided
• Similarly, a triangle in general contains all possible triangles in potency (equal sides, unequal sides, right angles, etc.) but none of these actualized simultaneously
• Adding a specific determination (equilateral, scalene, etc.) actualizes one of the potential forms

The Role of Reference Frames #

• The Stadium argument reveals the fundamental principle of relativity: motion and speed are relative to a frame of reference
• Equally fast bodies appear to move at different speeds when measured against different reference frames (one at rest, one in opposite motion)
• This ancient paradox anticipates Einstein’s principle of relativity

Important Definitions #

• Continuous (continuum): A magnitude divisible infinitely in potency but not composed of indivisible parts
• Potency/Actuality: Potency is what can be or can be done; actuality is what is or is being done. The continuous is infinitely divisible in potency but limited in its actual divisions at any moment.
• Now (nunc): An indivisible instant of time; time is not composed of nows but is continuous
• Local Motion: Change of place; the most evident form of motion to the senses

Examples & Illustrations #

The Door Example #

• Simplified version of Dichotomy paradox: you must reach halfway to the door, then half of what remains, then half of what remains, etc. The infinite sequence should prevent exit, yet we clearly can walk through doors.

Alphabet Recitation #

• If there were infinitely many letters, a child reciting the alphabet would never finish, even if reciting each letter instantaneously.

Geometric Proof Chains #

• Euclid’s proofs sometimes use A to prove B, B to prove C, C to prove D, etc. If this chain were infinite, one would never reach a conclusion.

Definition Chains #

• Square is defined as “quadrilateral,” which is defined as “rectilineal plane figure,” which is defined as “plane figure,” which is defined as “figure.” If this regress were infinite, nothing could be defined.

Two Rectangles with Same Perimeter #

• Theorem II.5 from Euclid’s Elements: two rectangles can have identical perimeter but different areas
• This restated as a question (“which rectangle with the same perimeter has more area?”) arouses wonder more than the formal theorem
• Concrete application: ancient geometers defrauded land buyers by trading property of greater area for less perimeter (the buyers thought they were gaining because perimeter decreased)

Questions Addressed #

• How can something traverse an infinite number of points in finite time? The points are infinite only in potency (divisibility); one has correspondingly infinite nows available. The error is treating potential infinity as actual infinity.
• Why did Zeno’s arguments seem compelling even to ancient philosophers? They reveal genuine difficulties in understanding continuity, infinite divisibility, and the nature of time that require careful philosophical analysis to untangle.
• What does the Stadium argument reveal about motion? It shows that speed and motion are always relative to a chosen reference frame; calling something “equally fast” requires specifying what it is moving relative to.
• How does death occur if change requires being and not-being? Death is an instantaneous change (generation ceases). The final moment of life is the instant when the change from alive to dead occurs; you never experience being dead while alive, so there is no contradiction.
• Is Zeno’s reasoning valid or fallacious? The arguments are formally sound but rest on false premises (especially that time is composed of indivisible nows, or that infinite potential divisions must be actually traversed). Zeno’s error is not in logic but in assumptions about the nature of the continuous.

Connection to Modern Physics #

• Louis de Broglie and Quantum Theory: The French physicist noted connections between Zeno’s paradoxes and problems in quantum theory, particularly regarding instantaneous velocity (motion in the indivisible).
• Schrödinger’s Observation: Questioned whether instantaneous velocity might be “fictional” rather than physically real.
• Instantaneous Velocity in Newtonian Physics: Uses the concept of straight-line motion tangent to circular motion at a single point—a limiting case that echoes Zeno’s concerns about motion in the indivisible.
• Relativity Theory: The Stadium paradox anticipates Einstein’s principle that motion is relative and requires specification of a reference frame.