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WyCnet's avatar

Is Zeno's paradox of Achilles and the tortoise really a follower's trap?

Asked by WyCnet (184 points ) October 13th, 2012

In a metaphoric sense Achilles is the student of the master Tortoise. To learn from the master the student has to take every step the master took, and pass through every point where the master has been. The master had set the dimensions of the stride and once the student who reaching the last place where the master has been, does not break stride with the master, going one up on the master, the student will never surpass the master. In other words the follower is always one step behind.

The rhetorical point is the trailblazing Master slowly making progress and it does not matter how quickly Masters can raise students to their level, once students cannot trailblaze themselves then they remain within the Master’s step, forever trapped in the follower’s trap.

The initial conditions of the paradox give the tortoise as being ahead. Achilles must pass through all the points where the tortoise has been. However Achilles is passing through those points quicker than the tortoise can. Eventually Achilles gets to the last place where the tortoise had covered, the tortoise had just left. At this juncture the dialog turns its attention to mathematical constructs and their validity in Reality.

The paradox makes reference to the slower tortoise and a quicker Achilles. However the mathematics has a hidden variable which is the unit of distance that makes slower commensurate with faster. If time were infinitely divisible, which it is conceptually, it is easy to claim the tortoise is at another place as Achilles gets to the tortoise’s last place, however the unit of distance that takes the rhetoric away from the adjectives, slower amd faster, will catch the tortoise with his foot in the air as Achilles reduces the distance between him and the tortoise.

Thus to break out of the paradox it has to be accepted as (1) a follower’s trap, or alternatively, (2) the rules of the competition have to be interpreted for a proper application in Reality thereby avoiding the rhetoric.

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15 Answers

Thammuz's avatar

You’re overanalizing the issue here. Zeno also suggested the arrow paradox, which is the same as the tortoise but without a moving target.

It’s a completely different issue made to try and refute the concept of multiplicity and change, because if there are infinite fractions of x you need to traverse an infinite to get from point A to point B, which is impossible, therefore nothing actually is moving, it’s all an illusion.

Nobody paid much attention to zeno, not even his master Parmenides, supposedly, because he was clearly double bonkers on toast.

WyCnet's avatar

@Thammuz::The crux of this illusion is that the real number line in mathematics is not always suitable to describe a real life situation. In the macro world those infinity of points are smoothed over by cognitive landmarks of Society, but still buried in the sequence that leads to the turnpike.

zenvelo's avatar

To answer the question, No. It is a mathematical parable.

gailcalled's avatar

For endless self-referential discussions of this topic, see

Godel, Escher, and Bachödel,_Escher,_Bach by Douglas R. Hofstadter

SavoirFaire's avatar

No. Zeno’s paradoxes were constructed to defend Parmenides’ view that there was no such thing as change, which is itself best understood as an early statement of eternalism about time. According to Parmenides, whatever change we might think we see is an illusion based on our limited way of viewing the universe. In fact, all things and all times exist eternally. Though an object may only exist for the span of three seconds, those three seconds exist eternally. Thus the object exists eternally. We think it’s gone, but it’s really just out of sight.

The paradox itself is easily resolved by calculus, which reveals Zeno’s math to be faulty. This same is true of several of Zeno’s paradoxes, though others require philosophical responses (such as the paradox of the arrow). In any case, it is important to note that the failure of the paradoxes does not itself show that Parmenides was incorrect in his conclusions. What Zeno was attempting to do was show his fellow Greeks, who at first considered Parmenides’ views to be absurd, that commonsense had worse troubles. It worked. Parmenides and Zeno were extremely influential on ancient Greek philosophy.

LostInParadise's avatar

Here is what I take away from the debate between the followers of those two most influential and opposing pre-Socratic philosophers, Parmenides and Heraclitis.

How can we describe a world that is constantly in flux? There is a duality that you can see in our very language: noun and verb, being and becoming, matter and energy, variable and subroutine.

In the constantly changing universe, there are islands of stability within a specified section of space-time. We can point to planets and stars, ecosystems, species and individual plants and animals. The philosophical insight from relativity is that matter and energy are convertible into one another and the insight from quantum mechanics is that there is a trade -off between knowing where something is and where it is headed.

In the end, entropy overpowers everything and all dissipates into nothingness. Maybe Plato had a way around this. Perhaps ideas, in either the form of abstract mathematics or scientific law, is permanent. In the multiverse view of things, through the infinite formation of individual universes, we have constant change but finite possibilities, so that at any moment there is infinite repetition of all that is possible.

ETpro's avatar

Welcome to Fluther, @WyCnet, even though you are obviously an illusion since no change is possible in this ever changing Universe. Darned illusions… They’re everywhere you look, aren’t they.

I think, since spacetime is one thing, that the Planck length is its smallest possible division. Neither space nor time is infinitely divisible.

Thammuz's avatar

@WyCnet I know, hence why my “Zeno was batshit” (or, rather, willfully ignorant of the obvious) comment.

WyCnet's avatar

@SavoirFaire :: ”... Zeno’s paradoxes were constructed to defend Parmenides’ view that there was no such thing as change,”...

In the paradox as I view it, the ,mathematics is the competition, between master and student, when the student wishes to be recognized as equal to the master or has surpassed the master. If students fail in the competition they have fallen into the follower’s trap. The competition begins when the student is just about caught up with the master.
If you notice the competition is worded to benefit the master, so at first blush the student has to demand a fair hearing.

WyCnet's avatar

@ETpro :: The Planck length is a limatation where the maths has to be applied, but rhetoric of the Mind has no boundary. In the competition you gave yourself a chance to succeed, by placing a unit of equality, simething good for both of you. I would measure the big foot, dang.

SavoirFaire's avatar

@WyCnet I understand how you are viewing it. I’m just letting you know that you’re wrong. Zeno created the paradoxes for very specific purposes that are well attested to in the surviving literature.

WyCnet's avatar

@SavoirFaire :: I agree that there has been much talk about the purposes of the paradox since Zeno, but from my perspective if what lay in Zeno’s mind was similar to the master-student paradox, and then laid out to fool the world as a service for his master, Zeno had surpassed his master and was now serving himself, and all those mathematicans who sought to sum up to infinity. The only real sum to infinity is progressing through life.

SavoirFaire's avatar

@WyCnet I’m sorry, but I’m going to have to ask you to repeat that in a language other than utter nonsense. There is no evidence whatsoever that your reading has even the slightest credibility.

Thammuz's avatar

@WyCnet As far as i know:

a) Zeno came way later than the prominent mathematician philosophers like Thales and Pitagoras, and by his time most of the maths related issues were over and done with, at least from a philosophical standpoint.
b) Even if he did have anything to do with mathematics, he was a disicple of Parmenides and never openly refuted his master’s ideas, so he wouldn’t be argiung for multiplicity and/or trying to argue about summing up to infinity. Rather, he would be arguing about how summing two numbers means one of the two numbers already was the sum of the two or some shit like that, because there is no multiplicity yada yada.
c) I never heard anything about debates on the purpose of Zeno’s paradox.

So in short: The fuck are you talking about?

Also, i re-read your comment: The crux of this illusion is that the real number line in mathematics is not always suitable to describe a real life situation.

Wrong. The crux of the illusion is deliberately misinterpreting the concept of fraction as adding quantity instead of just being a smaller part of the same overall quantity. Something the greek understood, by the way.

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