The Instigator
m93samman
Con (against)
Winning
20 Points
The Contender
Kiwi13cubed
Pro (for)
Losing
14 Points

Achilles and the Tortoise Paradox

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Post Voting Period
The voting period for this debate has ended.
after 10 votes the winner is...
m93samman
Voting Style: Open Point System: 7 Point
Started: 11/1/2010 Category: Miscellaneous
Updated: 6 years ago Status: Post Voting Period
Viewed: 3,712 times Debate No: 13526
Debate Rounds (3)
Comments (44)
Votes (10)

 

m93samman

Con

Shown here http://hubpages.com... about a third of the way down the page.

_____

In the paradox of Achilles and the tortoise, Achilles is in a footrace with the tortoise. Achilles allows the tortoise a head start of 100 meters. If we suppose that each racer starts running at some constant speed ( very fast and very slow), then after some finite time, Achilles will have run 100 meters, bringing him to the tortoise's starting point. During this time, the tortoise has run a much shorter distance, say, 10 meters. It will then take Achilles some further time to run that distance, by which time the tortoise will have advanced farther; and then more time still to reach this third point, while the tortoise moves ahead. Thus, whenever Achilles reaches somewhere the tortoise has been, he still has farther to go. Therefore, because there are an infinite number of points Achilles must reach where the tortoise has already been, he can never overtake the tortoise.
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I am going to attempt to prove that this paradox does not stand under scrutiny and analysis, which is why I am Con. My opponent's burden is to defend this paradox, as Pro. I look forward to an interesting exchange of ideas.

If my opponent begins, he should pass on the last round to make the number of rounds of debate even. Otherwise, I will begin in round 2.
Kiwi13cubed

Pro

Very well; I shall accept and make the first argument, thus intending to pass on the last round as you have requested.
I would like to thank you for this interesting debate which is sure to reveal some interesting thoughts and ideas, as well as to wish you the best of luck.

My understanding of this debate is that I am to defend the paradox of Achilles and the Tortoise.
According to dictionary.com, there are four definitions of "paradox" (1):
1. a statement or proposition that seems self-contradictory or absurd but in reality expresses a possible truth.
2. a self-contradictory and false proposition.
3. any person, thing, or situation exhibiting an apparently contradictory nature.
4. an opinion or statement contrary to commonly accepted opinion.

Definitions three and four merely state that something is apparently contradictory or contrary to something, and make no statements about their actual truth. Thus, I do not think this is what my honorable opponent wishes to debate.
This leaves definitions one and two which, rather ironically, are apparent opposites. However, based on what I have inferred by Con's argument, I assume definition one is the one intended for me to defend.

In order to eschew confusion, I shall henceforth refer to something meeting the conditions of definition one as a Paradox, and the example of Achilles and the Tortoise as a paradox, as we have not yet ascertained that it is a Paradox.

As such, I have the burden of proving:
[1] that the paradox of Achilles and the Tortoise seems self-contradictory or absurd, and
[2] that the paradox of Achilles and the Tortoise expresses a possible truth.

I think that there is little dispute over [1], but I shall nonetheless make my case.
Achilles can run faster than the tortoise and, as I'm sure nearly anyone will agree, a faster person (or creature) will always manage to overtake a slower one, provided that we assume the rates of each are constant, neither one tiring or requiring a break of any sort. I think that these assumptions are fairly valid, as they seem to have been implicitly assumed in the original stating of the paradox. This paradox states that Achilles will not catch the tortoise, and since Achilles can run faster, this in contradictory. Thus, this paradox meets the first condition of being a Paradox.

I shall now consider [2].
If we consider in turn each iteration of Achilles running to the tortoise's previous position, Achilles gradually will get closer to the tortoise.

My contention is that there are two ways of looking at this.
First, we could consider that the amount of time each successful iteration takes also approaches zero, and so Achilles will reach the tortoise after a finite amount of time. We could go on to prove this, with one method or another, but I do not feel that is necessary. However, there appears to be little conflict over whether or not this is paradoxical and, indeed, this interpretation is not my main point.

The other interpretation, however, is. Regardless of the number of iterations considered, Achilles will not catch the tortoise. One could argue that he would get close enough for most practical purposes, but this is not the debate. The most relevant definition of overtake that I find is "to catch up with and pass, as in a race; move by" (2); by this definition, even if Achilles reaches the tortoise after an infinite number of iterations, he will still not be ahead and thus not have overtaken the tortoise. Either way, this interpretation clearly shows some "possible truth" that, after an infinite number of iterations, the tortoise will still not have been overtaken.

Thus, my contention is that Zeno's paradox of Achilles and the Tortoise is, indeed, a Paradox.

(1)http://dictionary.reference.com...
(2)http://dictionary.reference.com...
Debate Round No. 1
m93samman

Con

I thank my opponent for his response.

*DISCLAIMER- ALL NUMBERS ARE ARBITRARY*

My opponent's argument, in brief, is just a rewording of the paradox, that goes something like this: iteration 1 of Achilles getting to the previous position of the tortoise takes 'x' amount of time. Iteration 2, takes x - a1, iteration 3 takes x - a2, ad infinitum to x - an (the numbers and the 'n' following the 'a' go as sub-scripts). X is a constant, defined by the distance between the tortoise and Achilles all divided by Achilles' speed; which is why distance divided by velocity equals time.

===Beginning of Refutation===

X will continue to decrease, according to logic, because Achilles' velocity is greater than that of the tortoise. Let's say he is 5 times as fast. 100 meters will become a difference of 20 meters; then, the time covered will reduce from 5 seconds to 1 second. Then, to 1/5th of a second, ad infinitum. If you take any number divided by (5 multiplied by infinity) it will reach zero. Of course, we can't take zero seconds to do something. Thus, the point at which the time will become something of 0.0000000~~~~(infinite zeros)1, is the point at which the tortoise will be overtaken.

===Con Argument(s)===

My case is simple. The problem with analyzing the paradox by the terms of the paradox is that it looks at the scenario in micro-terms. We have to zoom out; let's look at it this way. The finish line is 1000 meters from the start, and the tortoise is 100 meters ahead of Achilles, who is at the finish line. At 100 meters per minute, Achilles will reach the finish line in 10 minutes. At 20 meters per minute, the tortoise will reach the finish line in 45 minutes. Somewhere during Achilles' 20 minutes, he must have overtaken the tortoise.

Moreover, it seems logically narrow-minded to consider the paradox prima facie. To claim that a head start means you will always be in first place is simply ridiculous; head starts are what I give to little kids to motivate them to try and run faster. As they get more athletic, the head start becomes less and less, but I still overtake them; it just takes a longer time.

Finally, note what my opponent concludes with. "after an infinite number of iterations, the tortoise will still not have been overtaken." My opponent is wrong here; because the time is decreasing by a constant multiplier every time, we are adding what is an infinite series that CONVERGES; so, the sum of the infinite addition of this series will be a whole number, not infinity. Information on converging series: http://en.wikipedia.org...

(Note differences between divergent and convergent)

I look forward to my opponent's response.
Kiwi13cubed

Pro

First off, I would like to thank my opponent for his quick reply.

I would like to remind the audience of the definition of paradox established earlier, "a statement or proposition that seems self-contradictory or absurd but in reality expresses a possible truth."
My opponent himself has asserted that the paradox is absurd by declaring that "claim[ing] that a head start means you will always be in first place is simply ridiculous;" this paradox claims such rationale and, by my opponent's own statement, this paradox is ridiculous or, to use a synonym, absurd.

However, he has not addressed that to be a Paradox something only need express a "possible truth." This means that, in some way, it has some truth, but not necessarily from every point of view. Compare this to a fable of some sort; it expresses a truth, generally in the form of a moral, but not every piece of it makes sense. Generally they involve animals talking; this obviously is not a reality, but rather a device for expressing the truth.
Similarly, Zeno stated his paradox to challenge the conventional way of thinking, not that it would be flawless in every manner.

Again, I am not trying to claim that the tortoise will win the race. My opponent has clearly proved that he will not, unless the head start is enough to overcome slower speed of the tortoise. This, however, is not the point.

I instead contend that, given an infinite number of iterations, the tortoise will not be overtaken. Yes, the infinite number of iterations will take a finite amount of time. Again, this is not my point. This paradox is instead trying to point out an interesting truth: that no matter how many iterations are taken, Achilles will not overtake the tortoise. This is the "possible truth" that makes it a Paradox.

Perhaps this same point could have been illustrated in a similar example:
Achilles and the tortoise are taking place in a similar race, but with a slight modification. Every arbitrary unit of time (say, every hour perhaps), Achilles will run to where the tortoise originally was and stop. The tortoise will also run as Achilles does and then stop at the same time as Achilles. By assigning the same amount of time to each iteration, it should be clear that regardless of the number of iterations that Achilles will not catch the tortoise.
Now, this was not the paradox stated, but the kernel of insight (the "possible truth") granted is the same.

My opponent claims that the "problem with analyzing the paradox . . . is that it looks at the scenario in micro-terms." I contend that this is not a problem at all, but rather the entire point. Clearly, Achilles will win, but by analyzing it step-by-step it appears that the tortoise will. This makes it a Paradox.

I contend that the paradox of Achilles and the Tortoise is, indeed, a Paradox and that you, the voter, should vote "Pro" because it fits both parts of the definition of a Paradox.
Debate Round No. 2
m93samman

Con

I thank my opponent for his reply.

It seems my opponent is doing two things that are disadvantaging him. First; he has a very evident contradiction. Namely, he says:

1) " I am not trying to claim that the tortoise will win the race. My opponent has clearly proved that he will not"
2) "I instead contend that, given an infinite number of iterations, the tortoise will not be overtaken."

Clearly, these are incompatible statements.

Second, he says:

"However, he has not addressed that to be a Paradox something only need express a "possible truth." This means that, in some way, it has some truth, but not necessarily from every point of view." He compares this to a fable with talking animals. This isn't the case here; yes, the tortoise can talk and discuss the terms and conditions of the race, but no, the situation doesn't express a possible truth of any sort. In no way could a tortoise ever beat a man unless the man literally fell asleep, or was paralyzed, or had some other situation that rendered a tortoise superior to him physically, but that is besides the point.

Basically, what it all comes down to is a resolute understanding of the paradox, which my opponent has not arrived at. He understood clearly, the MEANING of a paradox, and demonstrated this by clearly stating "you, the voter, should vote "Pro" because it fits both parts of the definition of a Paradox.". That is beside the point; actually it proves my point. His final analysis to arrive at this conclusion is

"My opponent claims that the "problem with analyzing the paradox . . . is that it looks at the scenario in micro-terms." I contend that this is not a problem at all, but rather the entire point. Clearly, Achilles will win, but by analyzing it step-by-step it appears that the tortoise will. This makes it a Paradox."

Notice how he says, "Clearly, Achilles will win". He agrees, this paradox is simply ridiculous when looked at. Even step by step, the analysis is false because Zeno, who created this paradox, assumed that distance is continuous while time is discrete (hence the step by step analysis which leads to the fallacy). Had he done the rational and realized that time is in fact a continuous entity, we would never have come across this problem.

The resolution is negated.
Kiwi13cubed

Pro

As agreed in the first round, I make no statements in this round.
Debate Round No. 3
44 comments have been posted on this debate. Showing 1 through 10 records.
Posted by m93samman 6 years ago
m93samman
"even though we know it isn't true, it seems to make sense"

He made that statement, yea, but I felt I pointed out well enough that the assumptions made by kiwi assumed that we only analyze at a microscopic level. If you step back, none of what is said is even remotely sensible.

Btw I'm not trying to affect your vote, I just like discussing stuff.
Posted by Ore_Ele 6 years ago
Ore_Ele
That is true, but I felt that putting the calculus aside, kiwi did a better job defending the paradox, that even though we know it isn't true, it seems to make sense (thus making it a paradox).
Posted by m93samman 6 years ago
m93samman
@OreEle: What you said makes sense, but the paradox isn't strictly limited to calculus.
Posted by Ore_Ele 6 years ago
Ore_Ele
I have to go with Kiwi on this, because of the semantics of the word "overtaken" which implies to actually pass, while the calculus proof only goes to the point of catching, not overtaking. Therefore, the calculus on its own cannot support overtaking.

But truly, I think that both debators did very well, and if possible, would love to give both the win (since whoever ends up losing has done better then many other people that have "winning" debates).
Posted by Kiwi13cubed 6 years ago
Kiwi13cubed
lovelife - what a meaningful vote!
Posted by m93samman 6 years ago
m93samman
Gotcha. Although, I digress.
Posted by Naustics 6 years ago
Naustics
Sometimes you must limit your self and sometime you should not, just be moderate don't let any one call you crazy or let anymore call you dumb 0_o
Posted by lovelife 6 years ago
lovelife
Lol
Posted by m93samman 6 years ago
m93samman
Oh you just think you're SOOOO cute now don't you? :P
Posted by lovelife 6 years ago
lovelife
There I voted
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Vote Placed by gavin.ogden 6 years ago
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