Posted: May 31st, 2023

Philosophical Paradoxes: Unraveling Zeno’s Paradox of Motion

Philosophical paradoxes.

ESSAY I: On philosophical paradoxes Write an essay of about 3 pages, or 750 words. (Printed in double-spaced format.) Structure your essay in response to the specific questions below. Your TA will instruct you on how to turn in your essay (in electronic or hard copy, etc.). First: What is a paradox, i.e., a strict (logical) paradox? What is a “paradoxical” proposition, or philosophical “puzzle”? How does a strict paradox differ from such a puzzle? Second: By way of example, what is Zeno’s paradox of motion? Explain how this particular paradox is formulated. Third: By way of philosophical critique: What do you make of Zeno’s paradox of motion? Is there a true paradox in the phenomenon of motion itself? Or in the nature of space, or in the nature of time: given that motion is defined as change in spatial location over time? Is Zeno’s paradox arguably solved (or resolved or dissolved) by drawing on more recent ideas from mathematics or philosophy? Our main text is Sorensen’s A Brief History of the Paradox: Philosophy and the Labyrinths of the Mind. We draw ideas from Sorensen’s observations and extend our analysis of paradoxes as in Lecture and Discussion. We have extended our analysis of Zeno’s paradox in terms drawn from Cantor’s theory of “transfinite” numbers (formulated in terms of his mathematical development of set theory). And, subsequently, we consider McTaggart’s theory of time (as McTaggart argues that time itself is unreal). You may consider how Cantor’s mathematical theory of measures of “infinity” may impact Zeno’s argument. Or you may consider how the nature of action, in “hypertasks”, impacts Zeno’s argument. Or you may consider how McTaggart’s theory of time itself may impact Zeno’s argument. Or you may consider how exactly these more idealized technical models — of motion or space or time — apply to the concrete phenomenon of motion. After clearly expounding Zeno’s paradox of motion itself, your task in the third part of your essay is to focus on a particular aspect of Zeno’s paradox that you think revealing, and to analyze Zeno’s paradox further in terms of that aspect. In the Sorensen text: Cantor’s ideas are addressed along the way, in relation to different problems, on pages 54 (hypertasks), 57 (infinities), 317 (continuity, the calculus), 322 (set theory, cardinality of sets), 345 (rule-following as in counting). McTaggart’s ideas are addressed on pp. 173-176 and again 184-196.

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Philosophical Paradoxes: Unraveling Zeno’s Paradox of Motion

Introduction:
Philosophical paradoxes have long captivated the human mind, challenging our understanding of logic, reality, and existence. These paradoxes often arise from seemingly contradictory propositions or thought experiments that provoke profound philosophical puzzles. In this essay, we will delve into Zeno’s paradox of motion, a classic paradox that examines the nature of movement and the concept of infinity. We will explore the formulation of Zeno’s paradox, critically analyze its implications, and consider how modern ideas from mathematics and philosophy shed light on its resolution.

Understanding Paradoxes:
A strict or logical paradox is a statement or argument that leads to a contradiction when analyzed logically. These paradoxes challenge the foundational principles of logic and reason, often defying conventional understanding. On the other hand, a paradoxical proposition, or philosophical puzzle, refers to a statement or situation that appears self-contradictory, counterintuitive, or conceptually challenging but does not necessarily violate logical principles.

Zeno’s Paradox of Motion:
Zeno’s paradox of motion is a well-known philosophical puzzle dating back to ancient Greece. Zeno of Elea formulated this paradox to question the possibility of motion by highlighting the apparent contradictions within it. The paradox consists of a series of arguments and thought experiments, with the most famous being the dichotomy paradox and the Achilles and the Tortoise paradox.

In the dichotomy paradox, Zeno argues that for motion to occur, one must first cover half the distance, then half of the remaining distance, and so on, resulting in an infinite number of steps. As a result, Zeno claims that motion is impossible since one can never reach the end of an infinite series of steps.

The Achilles and the Tortoise paradox involves a race between Achilles, the swift Greek hero, and a slow tortoise. Zeno asserts that if the tortoise is given a head start, Achilles can never overtake it. According to Zeno, for Achilles to reach the tortoise, he must first reach the point where the tortoise started, but by the time Achilles reaches that point, the tortoise has moved slightly ahead, and so on indefinitely.

Analysis of Zeno’s Paradox of Motion:
Zeno’s paradox of motion challenges our intuitive understanding of movement and the nature of space and time. However, various philosophical and mathematical perspectives shed light on its resolution.
a) Mathematics and Cantor’s Theory: Cantor’s mathematical development of set theory and the concept of transfinite numbers provide a framework to address Zeno’s paradox. Cantor’s theory of infinities and continuum enables us to understand that even an infinite series of steps can have a definite end or limit. By employing mathematical concepts like limits and converging series, Zeno’s paradox can be resolved.

b) Hypertasks and Action: Hypertasks, conceptual tasks that involve infinitely many steps, offer insights into resolving Zeno’s paradox. By understanding that an infinite number of steps can still be completed within a finite time, we challenge Zeno’s assumption that motion cannot occur due to the infinite subdivision of space and time.

c) McTaggart’s Theory of Time: McTaggart argues that time itself is unreal, presenting an alternative perspective on Zeno’s paradox. If time is an illusory construct, Zeno’s argument based on temporal progression loses its foundation. By examining the nature of time and its relation to motion, we can reinterpret Zeno’s paradox in light of McTaggart’s theory.

Further Analysis: Cardinality of Sets and Rule-Following:
One aspect of Zeno’s paradox worth exploring is the relationship between cardinality of sets and the notion of rule-following. Cantor’s

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