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Objectivist Insight Needed for Achilles vs. Tortoise

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I'm curious to see if Objectivists on here can offer a practical explanation for Zeno's paradox of Achilles vs. the Tortoise. I realize that Aristotle attempted to tackle it, but his explanation doesn't seem complete to me since he resorts to "potential halves" instead of "actual halves," thereby not actually resolving (or refuting) Zeno's reasoning; or perhaps I'm not understanding him correctly and we may agree more than I think. I have posted my resolution to the "paradox" beneath the selection.

"t is impossible for [Achilles] to overtake the tortoise when pursuing it. For in fact it is necessary that what is to overtake [something], before overtaking [it], first reach the limit from which what is fleeing set forth. In [the time in] which what is pursuing arrives at this, what is fleeing will advance a certain interval, even if it is less than that which what is pursuing advanced … .And in the time again in which what is pursuing will traverse this [interval] which what is fleeing advanced, in this time again what is fleeing will traverse some amount … . And thus in every time in which what is pursuing will traverse the [interval] which what is fleeing, being slower, has already advanced, what is fleeing will also advance some amount." http://plato.stanfor...ox-zeno/#ParPla

Here is my take on it:

Zeno's critique of motion as illusion does make sense in the realm of mathematics, and this is where I see Zeno's flaw. Mathematics doesn't exist in reality (similar, if not identical, to mathematical formalism [which happens to be contra mathematical Platonism], which I indirectly learned I agree with while researching solutions to this paradox). Mathematics is inherently a mental construction. Laws of physics, biology, properties of matter . . . none of these actually exist in physical reality.

We can take a system of any kind, superimpose it onto reality, and while it might line up as a perfect stencil (for instance, I'm not going to walk off a cliff anytime soon even though a system tells me I will fall to my death and I'm saying the system is fake), the system itself is purely subjective. It is the same with infinity. There is no way to even conceptualize infinity much less see it in nature. Thus there is no way for us to manipulate it and use it or view it outside of a subjective construct that doesn't actually exist.

Zeno's example of Achilles does not work in reality. Achilles would blow by the turtle in all accounts, and we know this from experience. Can we mathematically explain it? No, and that's why his paradox frustrates so many people. But the truth of the matter lies in the distinction between the mental system of mathematics or physics and what is real. Zeno maintained that his paradox showed that movement and change is an illusion, but it really doesn't. Mathematics and all subjectively constructed systems are the illusion.

Between any two independent objects, there is a gap. However, a kilometer is not a yard, and a yard is not a foot. It depends on how you look at it. The fact, though, is that no matter how you look at it, measurements don't actually exist. Basic dimensions of length, width and depth certainly exist at different degrees, but the fact remains: the middle of anything is purely subjective, just as the length of a distance between two objects is purely subjective; not only that, the systems we're using to measure and find the middle of anything don't actually exist! That's it, that's the answer: mathematics and other systems of measurements are mental constructs, therefore there is no paradox of any kind. Sense-experience gives us all the information we need. In other words, I'd put my money on Achilles; I don't know many who wouldn't.

Edited by wisdomlover

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ARI had a presentation by Pat Corvini entitled "Achilles, the Tortoise, and Objectivity in Mathematics".In this presentation, she related the story by Zeno to a dirt track with two starting block a set distance apart. Selecting two different speeds, one for Achilles and one for the Tortoise, she began marking the distances as they progressed by placing pins in the dirt to represent the progress. Eventually, the pins become separated by a grain of sand. You can no longer proceed because the grain of sand prohibits you from sticking a pin into the track.

Repeating the scenario with high speed digital photography, even the resolution of the digital photography eventually resolves down to two pictures which are indistinguishable from one another due to the size of the pixel representation..

Mathematics, a humanly contrived system for measurement, allows you to solve for all intents and purposes without regard for resolution. Math can "stick the pins in the grain of sand" or "resolve the equation beyond the pixel resolution.

Another interesting experiment you can mathematically do for yourself is to start with a millimeter of length. Divide it by two. Divide the result by two. Repeat until you reach the size of a Planck unit. Take that same millimeter and multiply it by two. Multiply the result by two. Repeat until you reach the alleged size of the universe (Keep in mind, we calculate galaxies to be ~14.5 billion l.y. away. If that is a radial estimate from earth or our Milky Way Galaxy, that would establish a diameter of 29 l.y.) What you will discover is that within 100 repetitions either way, the mathematics is providing you with results to which there is no known referents to represent. The power of mathematics is the ability to resolve without regard to resolution. The power of the mind allows you to grasp this fact when presented with something like Zeno's paradox.

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I just noticed one more point you made about their being "no way to even conceptualize infinity". ITOE identifies many different conceptual constructs. Concepts of entities. Concepts of motion, which presumes a concept of entity. Concepts of materials, which also presupposes a concept of entity. It also provides one that encompasses infinity, concept of method. Infinity, as a method, provides you with the instructions of what to do in a particular situation. In the case of the number system, infinity amounts to nothing more than to what ever number you have reached, add one more. If you try to conceive infinity as a number or an amount, it fails, it is the method of "adding one more". The problem I had initially with the concept of eternity, I kept trying to envision it as 'an infinite amount of time'. How ever much time I envision, I could always "add more time". Resolving infinity in this regard helped me to understand that eternal actually deals with the fact that in fact, time is inapplicable to eternal.

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Zeno's critique of motion as illusion does make sense in the realm of mathematics, and this is where I see Zeno's flaw. Mathematics doesn't exist in reality (similar, if not identical, to mathematical formalism [which happens to be contra mathematical Platonism], which I indirectly learned I agree with while researching solutions to this paradox). Mathematics is inherently a mental construction. Laws of physics, biology, properties of matter . . . none of these actually exist in physical reality.

This is good. The Laws of Physics do not instruct matter what it must do, they instruct men what they can think about nature without contradiction. That said. there is such a thing as matter and it does have properties, discoverable intrinsic attributes.

Zeno's example of Achilles does not work in reality. Achilles would blow by the turtle in all accounts, and we know this from experience. Can we mathematically explain it? No, and that's why his paradox frustrates so many people. But the truth of the matter lies in the distinction between the mental system of mathematics or physics and what is real. Zeno maintained that his paradox showed that movement and change is an illusion, but it really doesn't. Mathematics and all subjectively constructed systems are the illusion.

This is less good. There is a critique to be applied to the mathematics, which is that the times required to traverse the ever smaller increments of distance are proportionally smaller such that the infinity of infinitesimal distances does not add up to an infinity of time to cross the original distance. There are actually two infinities of method at work here ( infinitesimals of distance and time) and they cancel each other out.

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I would go about Zeno's paradox in a somewhat different way. While Achilles never reaches the tortoise in the scenario, neither does time ever get past the limit. And in fact if you wanted to put in the work, you can calculate how long it takes Achilles to catch the tortoise (in reality) by the limiting series of distances. So I would put it as Zeno's paradox is true in the relative context of finite time defined by the limits. In terms of physics, the wave nature of quanta means you can't keep subdividing distance down to infinitesimals.

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The Zeno paradox is merely an artifact resulting from of an unfortunate parameterization (mathematization) of the problem. With a different parameterization, that is with a different division of the time or space intervals, the „problem“simply disappears. In fact, Achilles does not move by performing steps to successively reach the anterior positions of the tortoise, he just walks one 3-foot step at a time.

The incorrect/unfortunate parameterization of a physical problem, for example an unfortunate choice of the origin of coordinates, is well known to lead to false singularities and other absurdities. In cosmology, in some coordinate systems the Schwarzschild radius (sphere) of a non-collapsed star is a singularity, but not a true one, and can be eliminated by various coordinate transformations.

Sasha

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I have to say, I posted this to the Mises.org forum and you guys have done a much better job of answering it. Objectivism is so much better than idealism. Check out some of the responses here, and feel free to create accounts on there and bring some Objectivist insight to them; many of them need it!

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Is the world continuous ? I mean, is there say a distance of length 2*Pi actually ? I'm thinking here of the circle with radius of 1 meter, whose perimeter mathematically is 2*Pi. If the smallest unit of length is Plank's distance 'h', then are we to assume that the circle as is actually "pixelated" ? Could the smallest amount of matter be 'h', but we could still think of half of 'h' ?

Suppose that you answer "yes", distance is discreet, in steps of 'h'. What about 'time' ? Does this also imply that we can not think of continuous time, because time is only a record of movement of objects?

Zeno's paradoxes in fact criticize both views of continuity and discreetness of time. Quote from http://cerebro.xu.edu/math/math147/02f/zeno/zenonotes.html

Let us now step back and reconsider the four paradoxes and try to filter out Aristotle's perspective on the matter. In the Dichotomy and Achilles, Zeno argues that motion is impossible from the hypothesis that time is a continuous phenomenon. In the Arrow and the Stadium, he argues that motion is impossible from the hypothesis that time is a discrete phenomenon. Cobbling these together into one meta-argument gives us an even more powerful conclusion: regardless what your stand is on the nature of time, continuous or discrete, the conclusion is that motion is impossible. Has Zeno convinced you of this?

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How about just looking at speed in terms of miles per hour/ kilometres per hour.

for example, lets say it is a five hundred meter race. the tortoise has a five meter head start. lets say that the runner runs at ten meters per second, the tortoise at one.

the runner will reach the finish line, 500m away, in 50 seconds (500/10=50)

the tortoise will reach it in 495 seconds (500-5(head start)=495)

seeing as the runner reaches the finish line much faster then the tortoise, but starts further back, we have to logically conclude that the runner overtook the tortoise.

Edited by joojie

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Mises is so gross. It is sad that the Stirnerite of all people came to the best solution

Stirnerite hatred of abstractions and rules though isn't really helpful for solving problems though.

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