Can the Zeno Paradox be Resolved Through Infinite Divisions?

[Zenoman]

I may have discovered an explanation to the Zeno Paradox. However, in this explanation, the concept of a race between to objects at different speeds is simplified into one object traveling a given distance. The distance traveled in this example can be represented by the variable D. If the distance traveled were divided into an infinite number of segments, then each segment could be represented by D/(infinite). Therefore, if each of these D/(infinite) segments were added together an infinite amount of times since there is an infinite amount of divisions then if would be D/(infinite) * infinite which would equal D.

The same is true with time, and as the number of divisions goes to infinite the time to cross those divisions goes to 1/(infinite) then it is easy to resolve the paradox.

Questions...comments?

 

[Hurkyl]

I may have discovered an explanation to the Zeno Paradox.

Then first, you need to precisely state Zeno's paradox. There are many different ways to interpret it. Some interpretations are extremely easy to refute, due to an obvious error in the argument. I've seen others interpret Zeno's paradox as one of the first arguments (by reductio ad absurdum) that that error really is an error. (It probably wasn't so obvious back then)

If the distance traveled were divided into an infinite number of segments,

What segments?

then each segment could be represented by D/(infinite).

What do you mean by (infinite)? And why do you think you can divide by it? And how would the result represent a segment?

Therefore, if each of these D/(infinite) segments were added together an infinite amount of times

You can't add an infinite number of times. What operation are you really intending to use? (e.g. the basic definition of infinite sum from calculus?)

since there is an infinite amount of divisions

What is a division? And why would there be infinitely many of them?

then if would be D/(infinite) * infinite

What does 'infinite' mean here? And why do you think you can multiply D/(infinite) by it?

which would equal D.

And why would you think it gives this result?

 

[DaveC426913]

I may have discovered an explanation to the Zeno Paradox. However, in this explanation, the concept of a race between to objects at different speeds is simplified into one object traveling a given distance. The distance traveled in this example can be represented by the variable D. If the distance traveled were divided into an infinite number of segments, then each segment could be represented by D/(infinite). Therefore, if each of these D/(infinite) segments were added together an infinite amount of times since there is an infinite amount of divisions then if would be D/(infinite) * infinite which would equal D.

The same is true with time, and as the number of divisions goes to infinite the time to cross those divisions goes to 1/(infinite) then it is easy to resolve the paradox.

Questions...comments?

D/infinity*infinity does NOT result in D; it results in undefined - which means 'this has no unique answer'.

 

[Zenoman]

What segments?

The segments that the line is divided into. If we have a line of length D and we divide it into an infinite amount of sections then each section would be the width of D/(infinite)

To answer another of your questions right off the bat, (infinite) is equal to infinite just put the parenthesis to be helpful.

Would you like me to define every word of my post to you?

What is a division? And why would there be infinitely many of them?

A division is: 1. the act or process of dividing; state of being divided.
Dividing is: 1. to separate into parts, groups, sections, etc.

There is an infinite amount of divisions because that is the paradox.

What does 'infinite' mean here? And why do you think you can multiply D/(infinite) by it?

2. indefinitely or exceedingly great: infinite sums of money.

I can multiply D/(infinite) by infinite because its the same as adding D/(infinite) and infinite amount of times.


Do me the great honor of learning basic words such as sections and dividing.

 

[Zenoman]

If infinity/infinity does not equal one then I have found the other solution to the paradox. This would be that infinity is not well enough defined to solve a paradox involving infinite.

 

[pallidin]

Perhaps this can help:

When dealing with points A and B of finite length, infinite segmentation is classically impossible.
To go even further, one can not segment infinity, as that would be inherently contradictory. What is one-half of infinity as opposed to infinity itself? (for example)

Anyway just some thoughts...

 

[pallidin]

Another point I'd like to offer is that you can keep dividing mathematically forever, but there are practical limits in the real world... even the quantum level.

When one keeps mathematically dividing below certain levels it simply has no real meaning or effect on reality.

 

[HallsofIvy]

If infinity/infinity does not equal one then I have found the other solution to the paradox. This would be that infinity is not well enough defined to solve a paradox involving infinite.

All you've really said here is that you do not understand the "paradox". I suspected that when I saw that this had been posted under "General Physics".

 

[DaveC426913]

Would you like me to define every word of my post to you?

No need to be snooty. And yes, you may need to define terms. That's what Zeno's Paradox is about - our nebulous grasp on meanings versus reality. We find that our words make a lot of assumptions, as you are finding you've done in your argument.

 

[DaveC426913]

If infinity/infinity does not equal one then I have found the other solution to the paradox. This would be that infinity is not well enough defined to solve a paradox involving infinite.

There is nothing in the paradox about infinity. Infinity only shows up in flawed attempts at a solution.

The solution to the paradox is to eliminate the infinite. The reason it took so long to be solved is because every other philosopher fell into the same infinity trap that you did.

 

[rcgldr]

Isn't infinity a part of Calculus? There are methods used to calculate infinite sums. Integration to find area under a curve is the limit of the area as the number of segments approaches infinity.

In the case of two objects traveling at different but constant speeds, the ratio of the change in position versus the change in time is constant for each "interval", so the rate of closure and eventually passing by remains constant, even if the limit of the number of intervals approaches infinity while the size of the intervals approaches zero at the point where the faster object passes the slower one.

Another example, is a ball bouncing with a fixed percentage loss in energy on each bounce. The total time the ball bounces is fixed, but the number of times an abstract ball bounces is infinite. (In real life, eventually the bounces become smaller than the deformation of the ball, so it stops bouncing a bit sooner).

 

[Hurkyl]

If infinity/infinity does not equal one then I have found the other solution to the paradox. This would be that infinity is not well enough defined to solve a paradox involving infinite.

I would agree that your concept of 'infinite' does not seem to be well-defined. However, mathematics gives it a more than adequate treatment (even to the satisfaction of philosophers!) -- so if you're serious about reflecting about 'infinite' things, you really need to study mathematics.

 

[DaveC426913]

Isn't infinity a part of Calculus? There are methods used to calculate infinite sums.

Absolutely. And therein will be an answer. But mere arithemetic - dividing and multiplying -i.e. mixing natural numbers and infinities - isn't it.
That's where the philosphers and the OP went awry.

 

[rcgldr]

Absolutely. And therein will be an answer. But mere arithemetic - dividing and multiplying -i.e. mixing natural numbers and infinities - isn't it.
That's where the philosphers and the OP went awry.

True, but I'm not sure of the OP's math background. If you replace "infinite" with "n", and take the limit as n approaches infinite, then the OP had the right idea.

 

[RandallB]

True, but I'm not sure of the OP's math background. If you replace "infinite" with "n", and take the limit as n approaches infinite, then the OP had the right idea.

The right idea to prove what? Was Zeno right or wrong in what he was trying to prove?

The OP seems to think Zeno was claiming any length is equal to an infinite length and any time duration is equal to an infinite duration of time. Or that that is true and thus negates what Zeno was trying to claim.
I cannot tell because the OP never says what Zeno was claiming to prove.

But neither does anyone else here which makes me think no one in this argument really knows what Zeno was trying to prove.

 

[rcgldr]

But neither does anyone else here which makes me think no one in this argument really knows what Zeno was trying to prove.

This might help:

http://en.wikipedia.org/wiki/Zeno's_paradoxes

It's a paradox, which could mean that the author already knew the solution, but still considered it an interesting exercise for a potential reader.

 

[Zenoman]

Infinity/Infinity does not equal one in some cases... true. Such as 5x/x as x-> Infinity. However in this case, the n/n as n-> infinity does equal one.

Zeno claimed that movement wasn't possible because there was an infinite number of divisions to overcome. What I am saying is that because the number of devisions gets infinitely large, the time to cross those would be infinitely small. So these infinities would cancel.

If you can't figure out if I am proving or disproving the paradox perhaps it would be helpful to read the Paradox in full from the link Jeff provided. I was mistaken not have added that in my OP.

 

[DaveC426913]

Infinity/Infinity does not equal one in some cases... true. Such as 5x/x as x-> Infinity. However in this case, the n/n as n-> infinity does equal one..

Yes.

Zeno claimed that movement wasn't possible because there was an infinite number of divisions to overcome. What I am saying is that because the number of devisions gets infinitely large, the time to cross those would be infinitely small. So these infinities would cancel.

No.

 

[pallidin]

Your assuming, and thus calculating, that infinity resides between two finite points.

 

[pallidin]

This is not possible.

 

[rcgldr]

Your assuming, and thus calculating, that infinity resides between two finite points.

Or that an infinite number of points exist between two finite points, similar to the fact that an infinite number of real numbers exist between any two real numbers.

 

[epkid08]

Or that an infinite number of points exist between two finite points, similar to the fact that an infinite number of real numbers exist between any two real numbers.

Your comment seems out of the blue. How do you get "...an infinite number of points exist between two finite points," from $$\frac{\infty}{\infty}$$?

I think what you're trying to say is that as x approaches 0, t also approaches 0.
Zeno's paradoxes are nonsense anyways...

 

[rcgldr]

Your assuming, and thus calculating, that infinity resides between two finite points.

Or that an infinite number of points exist between two finite points, similar to the fact that an infinite number of real numbers exist between any two real numbers.

Your comment seems out of the blue. How do you get "...an infinite number of points exist between two finite points," from $$\frac{\infty}{\infty}$$? I think what you're trying to say is that as x approaches 0, t also approaches 0.
Zeno's paradoxes are nonsense anyways...

No, just a response to the comment that infinity resides between two "finite" points. A better analogy to the OP's paradox is that an infinite number of events (the location where the faster object reaches a point where the slower object was at the the previous event), can exist within a finite distance and/or time.

 

[RandallB]

Zenoman;
You and others are continuing to argue as if Zeno thought things like motion, finite points, distance, or things at distances were real things.
Did any of you look at the Wiki link Jeff gave? With other links there it does a reasonable job of explaining Zeno did not believe those were real.
And creating an infinity from something that does not exist proves nothing to Zeno.

The reduction to the absurd – was not the method used to falsify Zeno’s conclusions as some seem to indicate in this thread.
It was the method Zeno used to prove his point that the idea that any “thing” real moves at all was false.

Zeno’s argument was with the “pluralists” (pluralism at the time believed things really do move within a plenum similar to what Newton centuries later called absolute space and absolute time)

The point of Zeno’s arguments is he was used the logic of those that believe things do move “pluralists” to construct a paradox based on their own pluralist rules of movement that reduces to an absurd result; therefore refuting as absurd the idea that anything moves at all;
Thus supporting Zeno’s monism (what today we would call “Absolute Monism”) which believe that all is just one thing within which we and everything only exist as essentially an idea with nothing real actually moving.

As the Wiki info states in pure logic no one since then though the 20th Century has yet to falsify the Zeno Dichotomy Logic without first assuming in principle that things move; And that in logic terms incorrectly “begs the question” posed in the first place.

Nothing in this thread certainly not the OP brings something new to resolve the issue as raised by Zeno.

Science seems to do just fine though by ignoring the Dichotomy and like paradoxes as meaningless sophist arguments. They quite literally were put forward by sophists, the problem is justifying with logic that sophist ideas are meaningless.

If you wish to resolve Zeno’s Dichotomy, it needs to be done in the context it was presented without “begging the question”.

Extra note:
Also since “Science”, (General Physics included) can be said to include as a principle foundation that Sophist agreements are not useful;
it is only logical that science cannot resolve a sophist problem.
That is this is more of a Logic and Philosophical issue; it might be better placed in the Philosophy Forum.

 

[DaveC426913]

The point of Zeno’s arguments is he was used the logic of those that believe things do move “pluralists” to construct a paradox based on their own pluralist rules of movement that reduces to an absurd result; therefore refuting as absurd the idea that anything moves at all;
Thus supporting Zeno’s monism (what today we would call “Absolute Monism”) which believe that all is just one thing within which we and everything only exist as essentially an idea with nothing real actually moving.

While I don't doubt that you're making a valid point, I cannot understand what you are saying, especially in the above. It is grammatically so poorly-formed as to be unintelligible.

 

[Zenoman]

I think even Zenoman must agree with you there Dave.

 

[RandallB]

The point of Zeno’s arguments is he was used the logic of those that believe things do move “pluralists” to construct a paradox based on their own pluralist rules of movement that reduces to an absurd result; therefore refuting as absurd the idea that anything moves at all;
Thus supporting Zeno’s monism (what today we would call “Absolute Monism”) which believe that all is just one thing within which we and everything only exist as essentially an idea with nothing real actually moving.

While I don't doubt that you're making a valid point, I cannot understand what you are saying, especially in the above. It is grammatically so poorly-formed as to be unintelligible.

Sorry the word “was” should have been edited out –
however I suspect your trouble is not with the language so much as following the thought processes involved in formal logic.
I’ll attempt to translate phrase by phrase:

[The point of Zeno’s arguments is he used the logic of those that believe things do move “pluralists”]
pluralists Zeno's opponents “believe things do move” - most of use do too.
Zeno did not believe ANYTHING “moved”.
Zeno’s favored methods - reducing a argument to the absurd - requires taking your opponents position to explain some problem with the ultimate objective to assert something so ridiculous as to make you opponents look foolish. (I suspect he was rather “rude” about it)

[ to construct a paradox based on their own pluralist rules]
The problem of course must be stated and rigorously worked out in the rules of his opponent.
In this case it is clear rules of movement require any thing must first cross half the distance to a line before it can EVER cross the line.

[that reduces to an absurd result]
One of the rules of “pluralists” is that space or distance can always be further divided. (kind of gets into can space be quantized or not).
With that for Zeno it was clear that there would always be a halfway point the must first be traveled before the Line could EVER be reached – and logically EVER becomes NEVER.
AS the line can NEVER be reached since there will always be a halfway point that must be reached first!

[therefore refuting as absurd the idea that anything moves at all]
Zeno was not saying an object could move but not reach the goal, He was saying that since the very rules of movement could not justify the goal being reached was an absurd result based on the idea of motion - Therefore the rules and very idea of movement at all was absurd.

[Thus supporting Zeno’s monism (what today we would call “Absolute Monism”)]
He was supporting his own “religious science” of monism.

[believe that all is just one thing]
His version was not that elements divided down to smaller things eventually reaching just one element single identical form that made up Earth wind and fire (or quarks electrons etc.) from which all other things are made. When he said "one thing" he meant just one single non-moving thing.

[within which we and everything only exist as essentially an idea]
Meaning that everything we see and even ourselves do not exist, Reality is within that one thing where we are just thoughts or ideas with thoughts and ideas of our own.

[ with nothing real actually moving. ]
Not even within that one thing is any movement needed; only ideas with ideas unknown to other ideas with its own ideas; all existing without need of any physical movement; certainly not the kind of motion we think we are seeing in our misguided view of reality.

The problem is most do not approach Zeno on the terms he established for the paradox. Which makes the debate pointless. And under Zeno's terms most true philosophers recognize that these paradoxes have not been resolved at least not in favor of rational science.


It is absurd to debate Zeno or attack his logic if you don’t understand or know what he was arguing or under what terms he made his points. Maybe it can be translated into scientific terms but it has not been here.
I have my own way of kicking Zeno’s arguments to the curb (which is where I and most think they belong) but it is too long for here and IMO it is a logic debate not a science forum issue.
And the closest we have to a logic forum is the Philosophy Forum.

 

[Hurkyl]

The problem is most do not approach Zeno on the terms he established for the paradox.

On the contrary; most peoples' approach to this paradox is to demonstrate that the step

With that for Zeno it was clear that there would always be a halfway point the must first be traveled before the Line could EVER be reached – and logically EVER becomes NEVER.
AS the line can NEVER be reached since there will always be a halfway point that must be reached first!​

is not justified. In particular, they take their best guess as to what Zeno might have been thinking -- usually something like "each individual subgoal takes time, and so infinitely many subgoals must take infinitely much time" -- and actively demonstrate that the reasoning is invalid.

(To reject Zeno's argument, it is of course sufficient to simply point out the lack of justification for this step. But it seems to be the popular practice to go beyond simply pointing out the omission and attempt to actually demonstrate the flaw)


Since (to the best of my knowledge) Zeno does not actually state his rationale for this step, we cannot be sure what he really thinking here. However, what Zeno really meant is a mostly irrelevant question, since we have the more practical matter of considering what people today are thinking when they consider this argument.

 

[rcgldr]

Going back to the bouncing ball with a fixed loss on each bounce. The ball bounces an infinite number of times in a finite amount of time. This is close to a real world example, except it there's a point when the bounce height is less than the deformation, but if the center of mass movement is used, then it's a close approximation to the idealized case where the number of bounces or vertical movements is infinite in a finite period of time.

 

[RandallB]

Facing Zeno

On the contrary; most peoples' approach to this paradox is to demonstrate that the step

With that for Zeno it was clear that there would always be a halfway point the must first be traveled before the Line could EVER be reached – and logically EVER becomes NEVER.
AS the line can NEVER be reached since there will always be a halfway point that must be reached first!​

is not justified. In particular, they take their best guess as to what Zeno might have been thinking -- usually something like "each individual subgoal takes time, and so infinitely many subgoals must take infinitely much time" -- and actively demonstrate that the reasoning is invalid.

Nothing contrary in that demonstration at all, Zeno’s point was that using the reasoning based on motion is invalid – your agreeing with Zeno here.

(To reject Zeno's argument, it is of course sufficient to simply point out the lack of justification for this step. But it seems to be the popular practice to go beyond simply pointing out the omission and attempt to actually demonstrate the flaw)

But that is exactly what Zeno was hoping for – to show that when he uses the logic and rules of motion to build and argument, only a flawed absurd result is obtained – additional points about that flaw are points in Zeno’s favor – he appreciates the help. [/QUOTE] Since (to the best of my knowledge) Zeno does not actually state his rationale for this step, we cannot be sure what he really thinking here. [/QUOTE] Again, it is not his rational – it is our rational in believing in the rules for motion that he is applying against us! The modern knowledge of Zeno’s thinking is known - how he logically argues this point is included in advance modern philosophical logic, but is still based on the basic principles of logic (reducing to the absurd & begging the question) taught in Logic 101. Often scientist don’t included the more advanced training of Formal Logic in their training and knowledge. That is why many miss the point that Zeno wants his argument to be shown as absurd, because it is a argument based on motion – and that in the end was his objective – a logical debate demonstrate our ideas of motion were simply absurd.
Yes we are sure what he was thinking – that is want the Wiki information & links Jeff provided pointed out – Modern Experts in Formal Logic and Philosophical Debate know and understand very well what Zeno did and they tell use that a rebuttal to the Zeno proof against our believe in motion has never been made using sound formal logic. At a minimum those modern experts should be considered if you want to argue this issue.

However, what Zeno really meant is a mostly irrelevant question, since we have the more practical matter of considering what people today are thinking when they consider this argument.

And there it it is - the statement Zeno was waiting for as you debate him on the marble steps in Athens.

Zeno will say: Of course Hurkyl you are absolutely correct all that matters is “what people are thinking” just has I have defined Absolute Monism, not that when the imagine things like a bouncing ball that anything actually in fact moves – I glad to see you finally agree with my position.​

As he walks of laughing at both of us for believing things are real and really move; He has yet another debate victory. You are not the first his logic has defeated – you will not be the last.

Zeno is not an easy debate to face. Unless you only put up a strawman version of him.

 

[pallidin]

Zeno appears to have been resurrected more times than Lazarus.

 

[Hurkyl]

Zeno’s point was that using the reasoning based on motion is invalid – your agreeing with Zeno here.

You seem to have completely misinterpreted me. Hopefully, the following is clear:

Zeno's argument is invalid.​

And just to make sure, I will point out that that is not the same thing as

Zeno presented a valid reductio ad absurdum.​

In fact, assuming the consistency of Peano arithmetic, Zeno's argument doesn't merely contain a gap: it is provably a non sequitur.


(In the above, by "Zeno's argument" I refer to what you have presented as his argument)

 

[RandallB]

Zeno appears to have been resurrected more times than Lazarus.

I’m not aware of the sophism of Zeno have every haven been seriously “resurrected” - I’d need to see some credible reference for that. AFAIK that has been dead for centuries displaced by the accepted assumption that things do move.
Is that assumption supported by a legitimate refutation of Zeno’s Paradoxes? NO!
It should be clear that no such formal falsification has ever been provided, but reminding folks of that does not resurrect his science.

And when I say no formal falsification of the Zeno paradoxes has ever been provided that includes incomplete declarations and assumptions such as:

…. the following is clear:

Zeno's argument is invalid.​

In fact, assuming the consistency of Peano arithmetic, Zeno's argument doesn't merely contain a gap: it is provably a non sequitur. …

These assumptions here stated as facts directly contradict the claims made in the Wiki links about what real experts in formal logic say about them. If there is evidence for these claims then the Wiki information should be corrected.
I’m no expert but if they are “provably a non sequitur” or have been legitimately falsified that should be a big deal in the world of formal logic & reasoning.
Just show us the proof, peer reviewed not in a physics journal willing to accept assumptions, but a mathematics journal reviewed by real experts in formal logic.
To the best of my knowledge no such paper or proof has been accepted as complete – and the Wiki information is not a misrepresentation in need of correction.

 

[DaveC426913]

I thought that Zeno's paradox was resolved with the intro of the concept of "instantaneous velocity"? i.e how one can calculate a velocity as the duration approaches zero. That's the resolution is it not?

 

[Hurkyl]

Just show us the proof,

I interpret "space" to mean Q³.
I interpret "time" to mean Q.
Let f be the function f(t) = (t, 0, 0).
I interpret "the line" to mean the line given by the equations x = 1, z = 0.
I interpret "Achilles reaches a point P" to mean that there exists a time t satisfying f(t) = P.
I interpret "Achilles reaches the line" to mean reaching any point satisfying the equations of the line.

Then, we observe that, in this model Achilles reaches each of the half-way points as well as the line. The existence of this model presupposes only Peano arithmetic, and thus proves the theorem

Peano arithmetic is consistent ==> Your presentation of Zeno's argument is an invalid argument.​

Did I not formalize it how you intended? Too bad, your fault for being imprecise. Does this not resemble the semantics you intended? Doesn't matter, I'm asserting you've made a formal fallacy. However, I don't believe I've made use of either of these freedoms in the above argument.


Just for fun...

ithat there would always be a halfway point the must first be traveled before the Line could EVER be reached – and logically EVER becomes NEVER.​

would you care to take a stab at justifying this passage?