Zeno Of Elea And His Paradoxes, Part 1

Studying under the great Parmenides, Zeno of Elea lived from 490 to 430 BCE, writing on topics ranging from mathematics, science, and philosophy. From all academic perspectives, Zeno's significance to intellectual history lies in his contribution to and development of the concept of infinity. In fact, most consider Zeno to be the first thinker in the West to demonstrate the problems with infinity in practical application.

Although we lack almost all of Zeno's work, we learn most about him through Plato, Aristotle, Proclus, and Simplicius. The majority of our sources, however, derive from Aristotle's writing on Zeno. In fact, because we have almost no primary sources, many scholars have filled in the missing gaps of Zeno's arguments with educated and very researched guesses.

It is important that I also highlight some interpretive issues. Zeno spent a majority of his time on what is currently referred to as his "Paradoxes." Most philosophers traditionally interpret Zeno's paradoxes as supporting arguments to the monistic metaphysics of his teacher, Parmenides. Others interpreters say he meant to oppose Parmenides, while some contend he merely meant to contest the ideas of motion that were commonly held in his time. Still yet, recent researchers claim his paradoxes responded to Pythagorean philosophy.

Since scholarship finds Zeno's philosophy very problematic to interpret, and thorough contemplation of Zeno's work requires more mathematics than I am willing to write about, I will espouse here the nine paradoxes that scholars have extrapolated from Zeno's philosophy by means of the traditional interpretation when applicable.

The Achilles Paradox. Imagine Achilles and another -- obviously slower -- runner. When the slower man starts running, Achilles then chases after him. However, by the time Achilles reaches the point where the other man presently is, the runner will have moved on to a new point. Then Achilles must run to a new point, from which the runner, again, has already moved, ad infinitum. From the traditional interpretation, Zeno wishes to discredit motion, or change, as a mere illusion in accordance with Parmenides' philosophy.

The Racetrack Paradox. Scholars also refer to this as the progressive dichotomy. The paradox supposes a runner that begins a race at a fixed point, the starting line, and quickly moves to another fixed point, the finish line. However, according to Zeno, by the time he traverses half the distance of the track, the distance between start and finish, he must again traverse half the distance of the remainder, then half of the next remainder, ad infinitum. We see in yet another way how Zeno suggests motion and change is an illusion, or better yet, an impossible goal.

The Arrow Paradox. We see here another paradox on the illusion of movement. If we assume that time exists as a succession of "timeless" moments, Zeno suggests the idea of an archer. The archer will shoot an arrow, but this arrow can only take up a distance in space that equals the length of the bow. Since in every "moment" the arrow cannot move in or out of this space, because that would require time, or a new moment, then the arrow stays perpetually in some place. Since a place cannot move, the arrow itself never moves but stays stuck in a particular place. Motion is only illusory.

The Stadium or Moving Rows Paradox. Zeno here proposes a very weak paradox, at least in its assumption, but highlights a very important concept in Physics. However, this paradox will take several sentences to explain. In this paradox, he wishes to refute a commonly held belief of the time. The belief held said that a body of fixed length that traverses the fixed distance of another body will do so in the same amount of time if the former body were to traverse the second distance (or body) again.

Zeno contests this theory, proposing another paradox. Imagine a stadium where there are three equal, parallel, horizontal, and linear tracks. On track A, there is a stationary vehicle A, that rests in the center of the track; on track B, there is a vehicle B that starts from the very left of the track and moves at a constant speed, X, toward the right of the track; and on track C, there is a vehicle C that starts from the very right of the track and moves at a constant speed, X, toward the left of the track. It turns out that vehicles B and C pass one another in half the time that it takes for either vehicle B or C to pass A. He merely points out what we now consider relative velocity, but in this scenario, he stretches the analogy in attempt to state the following point that Aristotle rephrases in his Physica: "it turns out that half the time is equal to its double."

For diagrams and a similar, yet longer explanation, read this article on Zeno's Moving Rows in the Stanford Encyclopedia of Philosophy.

Limited and Unlimited Paradox. Should there exist many things in the world but only in a limited amount, in contrast to a world in which only one thing exists, we may suppose at first two things existing. Zeno would state, that for these two objects to exist, they must have distinctive characteristics that separate them, but for the objects to be separated, there must also exist a third thing, whether it be a generic thing, a space of separation, or a quality of separation. For three things to exist, then there must be a fourth... ad infinitum. In order that many things could exist in a limited amount, they must actually be unlimited as well, and this is an obvious contradiction. Zeno, as a result, concludes with Parmenides' thesis that the world is One.

In the second and final segment, I shall continue with the final four paradoxes of Zeno and consider their importance to the intellectual history of philosophy, mathematics, and science.

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Edmund Husserl on the Essential Laws of Consciousness

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"If all consciousness is subject to essential laws in a manner similar to that in which spatial reality is subject to mathematical laws, then these essential laws will be of most fertile significance in investigating facts of the conscious life of human and brute animals."

-Edmund Husserl

A video on Husserl, Heidegger and Phenomenology:

Primary Source: The Essential Husserl: Basic Writings in Transcendental Phenomenology (Studies in Continental Thought) by Edmund Husserl

Secondary Source: Husserl (The Routledge Philosophers) by David Woodruff Smith

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Thomas Hobbes on Facts as the 'Brute Beasts' of the Intellectual Domain

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"All generous minds have a horror of what are commonly called "Facts". They are the brute beasts of the intellectual domain."

-Thomas Hobbes

A video of a class lecture at Yale on the sovereign state and Thomas Hobbes' Leviathan:

Primary Source: The Leviathan by Thomas Hobbes

Secondary Source: Hobbes: A Very Short Introduction by Richard Tuck

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Gottfried Wilhelm Freiherr von Leibniz on Simple Ideas and Primary Principles

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"Finally there are simple ideas of which no definition can be given; there are also axioms or postulates, or in a word primary principles, which cannot be proved and have no need of proof."

-Gottfried Wilhelm Freiherr von Leibniz

 

A video about Leibniz, Modality and Possible Worlds:

Primary Sources: Discourse on Metaphysics and Other Essays by Gottfried Wilhelm Freiherr von Leibniz

Secondary Sources: Cambridge Companion to Leibniz by Nicholas Jolley

Leibniz: An Intellectual Biography by Maria Rosa Antognazza

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Gilles Deleuze on Power, Impotence, and Maliciousness

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"It is always from the depths of its impotence that each power center draws its power, hence their extreme maliciousness, and vanity."

-Gilles Deleuze

Again... another video from the European Graduate School. These guys have some really good stuff, you should check out their page on YouTube. In any event, here is a 2004 lecture that Jacques Derrida gave on Gilles Deleuze.

Primary Source: Anti-Oedipus: Capitalism and Schizophrenia by Gilles Deleuze and Felix Guattari

Secondary Source: Gilles Deleuze (Routledge Critical Thinkers) by Claire Colebrook

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Georg Wilhelm Friedrich Hegel on Thinking and Generalization

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"An idea is always a generalization, and generalization is a property of thinking. To generalize means to think."

-Georg Wilhelm Friedrich Hegel

Another video from the European Graduate School. Slavoj Zizek speaks on "A Return to Hegel" : 

Primary Source: The Science of Logic by Georg Wilhelm Friedrich Hegel

Secondary Source: The Accessible Hegel by Michael Allen Fox

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Hannah Arendt on Freedom and Necessity

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"Man cannot be free if he does not know that he is subject to necessity, because his freedom is always won in his never wholly successful attempts to liberate himself from necessity."

-Hannah Arendt

This is a lecture put on by the European Graduate School by Judith Butler, in which she discusses Hannah Arendt, Ethics, and Responsibility:

Primary Source: The Portable Hannah Arendt by Hannah Arendt edited by Peter Baehr

Secondary Source: Hannah Arendt: For Love of the World, Second Edition by Elisabeth Young-Bruehl

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John Dewey on the Gravity of Thinking

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"Anyone who has begun to think, places some portion of the world in jeopardy."

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A short video by Davidson Films on the life and works of John Dewey:

Primary Source: Liberalism and Social Action by John Dewey

Secondary Source: John Dewey and American Democracy by Robert B. Westbrook

 

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Plato on Philosophy as Music

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"Philosophy is the highest music." 

-Plato

A version of Plato's Allegory of the Cave in clay:

Primary Source: Five Dialogues of Plato edited by G.M.A. Grube

Secondary Source: Introduction to Philosophy and Writings of Plato by Thomas Taylor

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Friedrich Nietzsche on Friends and Writing

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"A good writer possesses not only his own spirit but also the spirit of his friends."

-Friedrich Nietzsche

A short exposition on Nietzsche's idea of Nihilism and the Death of God:

Primary Source: Basic Writings of Nietzsche (Modern Library Classics) by Friedrich Nietzsche 

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Secondary Source: Nietzsche: Philosopher, Psychologist, Antichrist by Walter Kaufmann

 

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Michel Foucault on Archaeology and Mankind

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"As the archaeology of our thought easily shows, man is an invention of recent date. And one perhaps nearing its end."

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John Frow of the University of Melbourne gives a lecture on Michel Foucault:

Primary Source: Discipline and Punish by Michel Foucault

Secondary Source: Cambridge Introduction to Foucault (Cambridge Introductions to Literature) by Lisa Downing

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Parmenides Of Elea

Parmenides of Elea, often consider the Grandfather or Father of Western Philosophy, lived and wrote during the early to middle part of the fifth century BCE. His didactic poem On Nature survives only in fragments, while the "Proem," the introductory verses to On Nature, have been fully preserved.

The story begins with Parmenides traveling to the home, or underworld, in which the goddesses Day and Night reside. Two parts make up the work, the first being "The Way of Truth" and the second "The Way of Appearance." Parmenides writes in an apocalyptic style which begs a number of hermeneutical questions; however, Parmenides seeks to critique the metaphysical theories proposed before him.

Truth, to Parmenides, was necessarily eternal and immutable. He lays out a thought experiment. First, two methods of inquiry exist, to seek that which is and that which is not. Something either is or is not. To him, the latter is impossible because to speak of something that is not is unintelligible. Something cannot be known if it does not exist. Therefore, inquiry seeks that which is.

Thinkers prior to his era proposed metaphysical first principles that fundamentally required change or motion, a necessity Parmenides found absurd. For change and motion exist, something must transition from an is not to an is, and that which is notis an impossibility. Therefore, his first principle(s) started with the purest form of that which is: eternity and immutability.

Parmenides' conclusions leave us with quite a few reservations. From his perspective, our empirical view of the world must change and/or re-explain notions such as change, generation, motion, and asymmetry. On the other hand, this narrative may caution us against the fallible reasons of man, as explained in a lengthy cosmology put forth by the goddess.

Parmenides certainly questioned the thinkers of his day, but thinkers today are still up in the air about his work. The most common interpretation of Parmenides reads him as a material monist, where the prime substance of the world is simply eternity and immutability. This is, however, very short-sighted. A better interpretation may suggest that Parmenides wishes to juxtapose two ways of thinking: Truth versus appearance, theory versus practice, reason versus experience, so on and so forth. This interpretation allows for Parmenides critique of earlier philosophers and suggests that metaphysics must not only consider change and motion as fundamental principles but also eternity and immutability.

An additional interpretation explains that Parmenides is developing a "meta" physics, an eternal and immutable substance/world, from which the changing and moving physical world, known as physics, comes. Likewise, scholars have built on this interpretation with modal and perspectival adjustments, in which "appearances" are different modes or perspectives of some monistic substance.

In any event, Parmenides opens up new questions to the dialogue of philosophy that proceeded him. Plato, in particular, based hisParmenides on his work, about which he probably learned a great deal from Socrates, who Parmenides supposedly met in Socrates' younger days on a trip to Athens. Likewise, Plato's Republic certainly shares many themes with Parmenides' work, most obviously through Plato's "Allegory of the Cave."

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