Can the Zeno Paradox be Resolved Through Infinite Divisions?

[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?

 

[RandallB]

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?

what is so hard about that?
Zeno uses our logic of motion and space to define a halfway point that will always be between our object and the finish line – if there is always something between it and the finish how can the finish ever be reached. As Zeno says this shows it cannot and will never be reached – he uses our rules of space and motion to produces absurd results, his paradox still stands.

These forums expect a statement like “Zeno's argument is an invalid argument” to be supported by more than a “Straw man” debate - you have an obligation to present obvious points Zeno would raise not just pretend he would stand mute.

Peano arithmetic is 19th century math on number theory – where is your justification to extending that to real distances that you assume can be traversed by things in motion. If you have only assumed that to be true then your just begging the question.
Do you really think Zeno would stand mute as if made of straw if we use such flawed logic?

If you don’t know what "begging the question" is, Too Bad open a book on logic. If you want to claim Zeno’s paradox’s as false arguments you need to use formal logic to do so.
If you really think the experts already motioned are wrong, name the peer reviewed papers that show that and use them to correct the Wiki Information to agree with your opinions. Keep us updated on your progress in that effort.

Otherwise just because I agree with the science of motion is no reason to accept your logic or assumptions like "instantaneous velocity" over Wikipedia information that real experts say these paradoxes have not been formally rejected.

 

[CRGreathouse]

Otherwise just because I agree with the science of motion is no reason to accept your logic or assumptions like "instantaneous velocity" over Wikipedia information that real experts say these paradoxes have not been formally rejected.

I'm surprised you put such faith in Wikipedia.

 

[Hurkyl]

what is so hard about that?
Zeno uses our logic of motion and space to define a halfway point that will always be between our object and the finish line – if there is always something between it and the finish how can the finish ever be reached. As Zeno says this shows it cannot and will never be reached – he uses our rules of space and motion to produces absurd results, his paradox still stands.

That is not a valid argument. Instead of using a chain of deductive reasoning to justify your conclusion, you are appealing to your ignorance about how the conclusion could fail.

This error is particularly egregious because you are not ignorant about how the conclusion could fail -- you are perfectly aware of the traditional calculus-based "sum of series" argument. (Which, according to Wikipedia, was already known as far back as Aristotle).

These forums expect a statement like “Zeno's argument is an invalid argument” to be supported by more than a “Straw man” debate - you have an obligation to present obvious points Zeno would raise not just pretend he would stand mute.

An "obvious" point Zeno would raise is

each individual subgoal takes time, and so infinitely many subgoals must take infinitely much time​

which I've already presented and invalidated in this thread. The only other "obvious" point I can imagine Zeno bringing up is

Each event in a ordered chain of events must either be the first event, or have an event immediately preceding it. Similarly for last and succeeding.​

which, incidentally, is tantamount to assuming a priori that infinite divisibility is impossible. But that's easily invalidated by pointing out there's no justification given for such an assumption. In fact, physics allows the order type chain of events can be any suborder of R.


But -- you've already rejected this method to refuting Zeno through a bit of sophistry¹: you've pretended that such an argument is merely providing the other half of Zeno's contradiction rather than demonstrating an error in his reasoning. I pointed this out earlier, but you didn't bother to defend or retract your assertion. I didn't press the issue because I thought it interesting to take up another line of attack. But since you are pressing the issue, discussion cannot proceed until you address my objection.

1: Pun intended: I mean this in the pejorative sense.




As for the other line of attack -- I had assumed you were aware of some of the basic relationships between syntax and semantics: suppose we are considering whether or not a proposition P can be proven by deductive logic. One necessary condition is that P must be true in every semantic interpretation of our language -- no matter how strange or convoluted it might be. Contrapositively, if any interpretation can be constructed where P is false, then P cannot be proven deductively.

On the assumption that Peano arithmetic is consistent, I have provided an interpretation that serves as a counterexample. Furthermore, I assert that it's a 'reasonable' counterexample, on the grounds it's essentially the same as (a particular example of) the way we normally interpret the notions of 'space', 'time', and 'motion', except I have replaced the continuum R with the rational numbers.

Since your argument is informal, it can never be truly refuted, because you can always surprise us at the last minute and say "Haha, this whole time I was really making the a priori assumption that motion is impossible, I just never bothered to explicitly state it!" (of course, at this point we'd all dismiss your argument as vacuous) But I can (and have) refuted the particular argument you have put forth.

 

[RandallB]

Since your argument is informal, it can never be truly refuted, because you can always surprise us at the last minute and say "Haha, this whole time I was really making the a priori assumption that motion is impossible, I just never bothered to explicitly state it!" (of course, at this point we'd all dismiss your argument as vacuous) But I can (and have) refuted the particular argument you have put forth.

That is impossible – I was clear that Zeno was presenting a argument based on the a priori assumption of motion to build an absurd result. You cannot possible be so dense as to miss that point.

If you seriously think that all philosophers agree with you, then site an authority in the field and have them see to that that Wikipedia is revised based on that authority to remove the statement:

However, some philosophers insist that the deeper metaphysical questions, as raised by Zeno's paradoxes, are not addressed by the calculus. That is, while calculus tells us where and when Achilles will overtake the Tortoise, philosophers do not see how calculus takes anything away from Zeno's reasoning that concludes that this event cannot take place in the first place. Most importantly, many philosophers do not see where, according to the calculus, Zeno's reasoning goes wrong …​

Absent that I’ll only be able to logically assume you are not really serious.
Your illogical logic resorting to insults disqualify you as useful as a mentor on this topic – so see you in another thread on another subject someday, but you and I are done in this one.

 

[Crosson]

That is impossible – I was clear that Zeno was presenting a argument based on the a priori assumption of motion to build an absurd result. You cannot possible be so dense as to miss that point.

If you seriously think that all philosophers agree with you, then site an authority in the field and have them see to that that Wikipedia is revised based on that authority to remove the statement:

However, some philosophers insist that the deeper metaphysical questions, as raised by Zeno's paradoxes, are not addressed by the calculus. That is, while calculus tells us where and when Achilles will overtake the Tortoise, philosophers do not see how calculus takes anything away from Zeno's reasoning that concludes that this event cannot take place in the first place. Most importantly, many philosophers do not see where, according to the calculus, Zeno's reasoning goes wrong …​

Absent that I’ll only be able to logically assume you are not really serious.
Your illogical logic resorting to insults disqualify you as useful as a mentor on this topic – so see you in another thread on another subject someday, but you and I are done in this one.

Randall, you place relatively too much confidence in your authorities, and relatively too little confidence in your own ability to reason.

As someone who holds degrees in mathematics, physics, and philosophy, I hope you would consider that I agree with Hurkyl, for what it's worth. I believe that, for example, the rebuttal in post #35 is definitive.

If you truly wish to argue that Zeno's paradox is not resolved, then let us examine the article that is linked in the wikipedia web page:

http://philsci-archive.pitt.edu/archive/00002304/

I hope that anyone who has a moderate training in mathematics, logic, and philosophy will see that this paper only succeeds in supporting its premise in as much as it also trivializes Zeno's arguments. The author has a pompous attitude, and he callously makes claims of impossibility without any hint of a rigorous proof; it is a crackpot paper.

 

[Hurkyl]

Your illogical logic resorting to insults

If you believe I have insulted you, then you should use the 'report post' feature to make the other mentors (and the administration) aware of my behavior. Assuming you're serious, I'm not really sure what prompted your opinion -- was it the phrase 'appeal to ignorance'? That wasn't an insult: I am using it in its technical meaning describing a class of formal fallacies resembling that of justifying a position on the basis that one is unaware of evidence to the contrary.

Incidentally, there was another error I failed to point out:

Zeno uses our logic of motion and space to define a halfway point that will always be between our object and the finish line

He only demonstrates such a halfway point exists when the object is not yet at the finish line.

 

[Diffy]

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)

Zenoman,

Here is something to think about from your original arguement:

I'd argue that since there are an infinite amount of points in a line segment, if you could divide a line segment into an infinite amount you wouldn't end up with points! And as Euclid said, a point is that which has no width. But you are saying it has a width of D/(infinite). How is this any different from (D+10)/(infinite)?

The problem really is you are trying to treat infinity as a number, and it is a concept, not a number.

 

[Diffy]

Also I have feel I have a much deeper understanding of the paradox due to Hurkyls and RandallB's lively debate! Keep it up guys!