Are Zeno's paradoxes logical fallacies?

[Alan McDougall]

This could also be a topic in the maths forum, but I would like to approach the philosopher Zeno and his famous paradoxes in this sub-forum

 

Are Zeno's paradoxes logical fallacies?

 

One paradox is about a race between Achilles and a tortoise, with the tortoise given a half way start to the finish line.

 

According to Zeno's ,logic, Achilles will never catch up and pass the slower tortoise , because the distances between them are just subdivided smaller and smaller infinitely.

 

There have been solutions to this paradox, but my point are his paradoxes really fallacies?

 

[swansont]

What fallacy do you propose is present?

 

AFAIK the paradoxes simply represent the absence of knowledge of infinitesimals. The premise is incorrect, so the conclusion is invalid. There need not be any incorrect logic involved.

[tar]

Alan,

 

I remember my disbelief in the impossibility of the logic being correct in determining that the runner could never logically reach the tortoise because I knew, he absolutely would and could, so the logic must be wrong. Although the logic has no fault, the answer comes to me, along the lines of swansont in our inability to handle infinitesimals logically in our minds, in the same way that they are actually handled in the real world. That is, the theory, occuring in our minds does not have to follow the actual rules of time and distance that the real world imposes upon itself. Our 10th grade math class intuition of how the race would turn out, in this regard, is superior to the most intelligent and well thought out, logical proof.

 

From wiki article on Zeno's paradox:

 

"Pat Corvini offers a solution to the paradox of Achilles and the tortoise by first distinguishing the physical world from the abstract mathematics used to describe it.^[39] She claims the paradox arises from a subtle but fatal switch between the physical and abstract. Zeno's syllogism is as follows: P1: Achilles must first traverse an infinite number of divisions in order to reach the tortoise; P2: it is impossible for Achilles to traverse an infinite number of divisions; C: therefore, Achilles can never surpass the tortoise. Corvini shows that P1 is a mathematical abstraction which cannot be applied directly to P2 which is a statement regarding the physical world. The physical world requires a resolution amount used to distinguish distance while mathematics can use any resolution"

 

Thus math is an abstraction of what is already physical and real, and as humans we already have a solid intuition of space and time, whose rules and interactions fit together and operate, quite seemlessly, regardless of any rent in such a fitting and consequential fabric, that we might hypothetically impose.

 

We cannot therefore bypass the actual, with the hypothetical, and the logic of reality trumps the logic of any model of it that we can formulate.

 

If a paradox between the physical and the mental exists, I would first fault the mental as potentially working off a false premise, or an incorrect model, or attempting a transference or grain size shift or perspective change that failed to take everything into account...rather than fault reality, which already fits together in a quite error free fashion.

 

Regards, TAR2

[Alan McDougall]

Alan,

 

I remember my disbelief in the impossibility of the logic being correct in determining that the runner could never logically reach the tortoise because I knew, he absolutely would and could, so the logic must be wrong. Although the logic has no fault, the answer comes to me, along the lines of swansont in our inability to handle infinitesimals logically in our minds, in the same way that they are actually handled in the real world. That is, the theory, occuring in our minds does not have to follow the actual rules of time and distance that the real world imposes upon itself. Our 10th grade math class intuition of how the race would turn out, in this regard, is superior to the most intelligent and well thought out, logical proof.

 

From wiki article on Zeno's paradox:

 

"Pat Corvini offers a solution to the paradox of Achilles and the tortoise by first distinguishing the physical world from the abstract mathematics used to describe it.^[39] She claims the paradox arises from a subtle but fatal switch between the physical and abstract. Zeno's syllogism is as follows: P1: Achilles must first traverse an infinite number of divisions in order to reach the tortoise; P2: it is impossible for Achilles to traverse an infinite number of divisions; C: therefore, Achilles can never surpass the tortoise. Corvini shows that P1 is a mathematical abstraction which cannot be applied directly to P2 which is a statement regarding the physical world. The physical world requires a resolution amount used to distinguish distance while mathematics can use any resolution"

 

Thus math is an abstraction of what is already physical and real, and as humans we already have a solid intuition of space and time, whose rules and interactions fit together and operate, quite seemlessly, regardless of any rent in such a fitting and consequential fabric, that we might hypothetically impose.

 

We cannot therefore bypass the actual, with the hypothetical, and the logic of reality trumps the logic of any model of it that we can formulate.

 

If a paradox between the physical and the mental exists, I would first fault the mental as potentially working off a false premise, or an incorrect model, or attempting a transference or grain size shift or perspective change that failed to take everything into account...rather than fault reality, which already fits together in a quite error free fashion.

 

Regards, TAR2

 

Thanks for your informative post! A non- mathematical solution might be that Achilles finally catches up with the tortoise where there is less space between them than the smallest infinitesimal fundamental particle. Achilles in his very long effort to catch the tortoise has also reduced to the smallest possible size of a fundamental particle, thus; Achilles cant jump over himself and reaches the tortoise in his final jump.

 

I know this is not a solution to the paradox , Zeno had no knowledge of modern physics, however, I still want to muse around this and other fascinating paradoxes for a while.

[tar]

Alan,

 

Then as additional helpers in you muses understand convergence and limits.

 

My favorite helper is the abstract ball that bounces exactly up half the distance it drops and down again, bouncing a logically infinite amount of times, before coming to rest in a finite amount of time.

 

Of course the real world takes care of bringing the ball to rest, regardless of the math, because at some point, perhaps the planck length, the ball no longer could be considered leaving the ground on the bounce, eliminating any subsequent fall.

 

Regards, TAR

[Alan McDougall]

Alan,

 

Then as additional helpers in you muses understand convergence and limits.

 

My favorite helper is the abstract ball that bounces exactly up half the distance it drops and down again, bouncing a logically infinite amount of times, before coming to rest in a finite amount of time.

 

Of course the real world takes care of bringing the ball to rest, regardless of the math, because at some point, perhaps the planck length, the ball no longer could be considered leaving the ground on the bounce, eliminating any subsequent fall.

 

Regards, TAR

 

Yes even clapping your hands together, is a type of Zeno paradox! Planck length is the point at which our reality begins.

Edited January 5, 2014 by Alan McDougall

[Wso]

Technically moving at all should be impossible using that logic, because you can not traverse an infinite amount of small subdivided distances. Of corse if you can not move then the whole argument is invalid anyway, if you can never move 1 foot how would you get halfway to 20 feet? When using infinities, it's best just to stop right there, infinity has a pot of strange properties. Instead of using infinity to describe something, try to substitute it with a variable (this May or may not help in all, if any cases). The idea that we can never move is invalidated by the fact that as I type this, I am moving. You can use infinity and make a lot of logical fallacies, all of them invalid due simply to the use of infinity.

[swansont]

The idea that we can never move is invalidated by the fact that as I type this, I am moving.

 

Hence the use of the word "paradox"

[Alan McDougall]

Technically moving at all should be impossible using that logic, because you can not traverse an infinite amount of small subdivided distances. Of corse if you can not move then the whole argument is invalid anyway, if you can never move 1 foot how would you get halfway to 20 feet? When using infinities, it's best just to stop right there, infinity has a pot of strange properties. Instead of using infinity to describe something, try to substitute it with a variable (this May or may not help in all, if any cases). The idea that we can never move is invalidated by the fact that as I type this, I am moving. You can use infinity and make a lot of logical fallacies, all of them invalid due simply to the use of infinity.

 

I am not sure exactly what you are getting at, " in Zeno's paradox Achilles could can move/run at first" and then "move/run half that distance" and so on, the next time etc, only later was he faced with ever smaller divisions into the infinitesimal.

 

Hence the use of the word "paradox"

 

I once wrote a short document on the apparent enigma or paradox of movement, I will try and locate it.

Edited January 6, 2014 by Alan McDougall

[Dekan]

I think Wso is right, - Zeno's argument is invalid.

 

Doesn't it mean that you can't get shot by a gun, if you're running away - the shell/bullet can never catch up with you?

[imatfaal]

I think Wso is right, - Zeno's argument is invalid.

 

Doesn't it mean that you can't get shot by a gun, if you're running away - the shell/bullet can never catch up with you?

 

That is why it is a paradox - Zeno uses an arrow in one of his argument but to the same effect. He seems to show two contradictory arguments must both be true - thus it is called Zeno's paradox not Zeno's patented arrow stopper. From the beginning everyone knew something was awry - people got shot, tortoises were overtaken, snails climbed out of wells etc; the problem was to show where in the ineffable logic the mistake was.

[Alan McDougall]

I think Wso is right, - Zeno's argument is invalid.

 

Doesn't it mean that you can't get shot by a gun, if you're running away - the shell/bullet can never catch up with you?

 

In the real world Zeno is obviously wrong, but strangely his logic is hard to fault.

[pears]

Swansont is right. The premise (that an infinite number of points cannot be traversed) is incorrect. Therefore the logic of the argument is irrelevant. If the premise were correct and the conclusion false there would be a deductive fallacy. As it is it's just a false premise.

[overtone]

Zeno's logical arguments are proofs by contradiction that the assumptions they are based on are incorrect - as in any valid logical argument that leads to false conclusion.

 

All logical arguments begin with the word "if", or to quote the principle: "A proof shows you where to concentrate your doubts". The difficulty in Zeno's case is in explicitly stating the assumptions he exploited - comprehending the situation. The assumptions are common habits of human thought hidden from careless or naive consideration, and it takes a bit of work to ferret them out. The difficulty we often have in figuring out what a magician has done is another example of this fact of human thinking and perception. People are easily and habitually led to take things for granted - make assumptions - that do not in fact obtain.

 

One of the assumptions Zeno exploits is that an infinity of intervals of time takes an infinity of time to traverse - we are used to seeing an infinity of spatial intervals packed into a finite space, but it's harder to "see" an infinity of time intervals packed into a finite amount of time. Once you recognize that situation, the invalidity of the word "never" pops into view - you discard the implicit assumption that an infinity of time intervals can "never" be traversed, and the paradox based on it vanishes.

[Alan McDougall]

Swansont is right. The premise (that an infinite number of points cannot be traversed) is incorrect. Therefore the logic of the argument is irrelevant. If the premise were correct and the conclusion false there would be a deductive fallacy. As it is it's just a false premise.

 

Thus speaking is the Zeno's paradox, under discussion a fallacy?

[imatfaal]

 

Thus speaking is the Zeno's paradox, under discussion a fallacy?

 

 

No - per overtone and others - the problem is his assumptions. A fallacy is a flaw in logic - an argument with a logical formal fallacy can still be true or false but that is luck. A valid argument ie one without logical fallacy must be true iff the assumptions/axiomata are true. If an axiom is not true then the truth-value of your conclusion again is anyone guess. Zeno's logic is valid - however he makes an assumption (we are not sure exactly what) which is untrue and therefore his conclusion cannot stand; we can also tell empirically that this conclusion is false.

 

The logic is fine - but the he makes is that an assumption that any infinite set of actions will not be able to be completed in a finite amount of time. Now Zeno did not talk about summing infinite series and modern philosophers have argued that the fact that we can now sum infinite series does not impact on Zeno's paradoxes; however this does strike one as a little too nice. Those who maintain the paradox still remains have taken a very esoteric and recondite position - the resolution of the paradox as normally phrased is the use of the sum of a convergent infinite series. I am pretty sure we have no direct writing by Zeno - only reports by later Philosophers - so quibbling on the details seems a little pedantic.

[Alan McDougall]

 

 

No - per overtone and others - the problem is his assumptions. A fallacy is a flaw in logic - an argument with a logical formal fallacy can still be true or false but that is luck. A valid argument ie one without logical fallacy must be true iff the assumptions/axiomata are true. If an axiom is not true then the truth-value of your conclusion again is anyone guess. Zeno's logic is valid - however he makes an assumption (we are not sure exactly what) which is untrue and therefore his conclusion cannot stand; we can also tell empirically that this conclusion is false.

 

The logic is fine - but the he makes is that an assumption that any infinite set of actions will not be able to be completed in a finite amount of time. Now Zeno did not talk about summing infinite series and modern philosophers have argued that the fact that we can now sum infinite series does not impact on Zeno's paradoxes; however this does strike one as a little too nice. Those who maintain the paradox still remains have taken a very esoteric and recondite position - the resolution of the paradox as normally phrased is the use of the sum of a convergent infinite series. I am pretty sure we have no direct writing by Zeno - only reports by later Philosophers - so quibbling on the details seems a little pedantic.

 

Agreed like I stated in post 12

 

In the real world Zeno is obviously wrong, but strangely his logic is hard to fault.

 

[hoola]

zeno's paradox shows the fallacy of any infinities within the universe...If it takes an infinite period of time to progress any finite distance, what you are describing is the singularity, pre-big bang.......only a potential of described of rabbits and hares moving within the singularity...luckily, reality has the quantum to keep things from freezing up, thus allowing movement....edd

[tar]

Hoola,

 

Huh?

 

It does NOT take an infinite amount of time to travel a finite distance. The infinite amount of divisions of time that you suggest are within a finite amount of time, are not really. Well they are in thought, but you cannot freeze time, while you are thinking about it, because it marches on, without you. Thus by the time you have imaginarily divided a second in half and then in half again, its already the next second, and you are dividing a past second, in retrospect.

 

If you want to talk about a future second, you can speculate that it will unfold in an infinite amount of infinitely tiny parcels, but when the designated second comes, the whole thing will unfold in exactly a finite second. So the infinite divisions were either an inappropriate, imaginary prediction, or an infinite amount of infinitely small divisions of a second will add up to exactly a second, provided the sum of the infinite series you have constructed, adds up to 1.

 

Regards, TAR

[hoola]

tar, you have misread my thread...I said "if" it takes a infinite period...... I certainly agree with it no doing so, poking fun of the idea of infinites, by declaring their logical existence to cause a lack of motion, hence the singularity reference.......after all, the paradox, if valid would have allowed no motion of the inflation of the big bang....so no physical universe....edd

[tar]

Hoola,

 

Oh.

 

Poking fun at Zeno? Perhaps he was poking fun at logic, and we all thought he was serious.

 

Perhaps its another thread, but I don't know what it would be entitled. I do have a bantering deficiency myself, and often take literarly what is meant to be an obvious exageration or false statement, but "I thought you were joking", turns out to be quite an insult, if the other is serious.

 

This is a tough board, especially the speculation section, where individuals often take an insight, or idea or logical assessment very seriously, and nobody else pays any attention, or just laughs it off.

 

I "laughed" at a grammar mistake back in the fall, that the president of my company made. Thought his staff must have been remiss in letting it through. Turns out it was intentional and part of a carefully thought out marketing strategy. A "twist" that could be either considered "clever" and meaningful if you are on the "inside"...or just plain wrong and stupid if you are "not in the know."

 

I thought it was wrong and stupid for my 5th grade teacher to tell me to "look it up" in the dictionary, when I did not know how to spell it, in the first place. (being that you had to know how to spell it, to find it in the dictionary). She evidently was serious and did not see the humor in her instruction.

 

Regards, TAR

Knowing the difference between humor and seriousness is as easy as won, to, three.

What sensible individual would ever open the dictionary to the O section and start scanning for a word that starts with a W sound.

 

I think us English speaking folk were handicapped from the very beginning with one, two, four. They stuck the w, that should have been at the start of the first number, in the middle of the second number where it has no place, and added a letter in 4 just to throw u off, in case you where inappropriately looking for a pattern to follow.

 

They must have been joking around.

To...for...sics...ate..who do we appreciate? Number spellers! Number spellers! Number spellers! YEAH!

or is it ait, or aight, I can never remember.

[Alan McDougall]

Humor can make a friend out of an enemy, but it is not always a good thing and has its place just like all other forms of emotional expression. Some people take debating much too seriously, try to show how much more they are able to grasp concepts than "normal people", they want to win the argument at all costs, no matter what another person brings up, even if it is valid and true, they try to counter any argument with their supposed, subjective superior logic.

Edited January 13, 2014 by Alan McDougall