Are Zeno's paradoxes solved by modern science?

[Jeff]

1 minute ago, Song of the Dao said:

 

"The Method of Exhaustion", in other words, I give up! Let's just say it is this much!

Calculus is an approximation, a statistic. It turns the infinite into the finite. It is a conceptualization machine, making something out of nothing! Hmmm, something coming from nothing...where have I heard that before?

You can see it here. They take "lengths" and divide them up smaller and smaller.
http://tutorial.math.lamar.edu/Classes/CalcII/ArcLength.aspx
 

"We’ll do this by dividing the interval up into n equal subintervals"! What? how long is n? It is imaginary, infinite!

 

That is why you can solve it with the constant watching of the tortoise (infinite looks).  That is how calculus works, infinite looks trumps the infinite individual lengths, so you can catch and pass that tortoise. Look at the tortoise itself, not where he was (old method of exhaustion).

[wandelaar]

Calculus doesn't need infinite or infinitesimal numbers for a rigorous foundation, limits defined by means of the (ε, δ)-definition suffice. See:

 

https://en.wikipedia.org/wiki/(ε,_δ)-definition_of_limit

 

However there are also some rigorous approaches to calculus that use infinite and infinitesimal numbers. See for instance:

 

https://en.wikipedia.org/wiki/Non-standard_analysis

[wandelaar]

@ Jeff

 

In my opinion the logical fallacy in Zeno's reasoning is that he supposes the impossibility of accomplishing infinitely many acts within finite time.

[wandelaar]

@ Song of the Dao

 

I agree that even measurement systems introduce a subjective element. But they are tested against reality by doing experiments at the same time that the more theoretical elements of our scientific world view are tested. Measurement systems that don't somehow do justice to the world as it is will automatically prove insufficient sooner or later.

 

Edited January 2, 2019 by wandelaar

[Jeff]

33 minutes ago, wandelaar said:

@ Jeff

 

In my opinion the logical fallacy in Zeno's reasoning is that he supposes the impossibility of accomplishing infinitely many acts within finite time.

 

That is just the point, as it is constantly watching the tortoise, not the location of where he was when you looked. Watch the tortoise and it is easy to catch him.  

[wandelaar]

4 minutes ago, Jeff said:

That is just the point, as it is constantly watching the tortoise, not the location of where he was when you looked. Watch the tortoise and it is easy to catch him.  

 

I don't see what watching the tortoise has to do with it, as long as they move with same velocity as before they may look at whatever they like...

[Jeff]

2 minutes ago, wandelaar said:

 

I don't see what watching the tortoise has to do with it, as long as they move with same velocity as before they may look at whatever they like...

 

That is how calculus works (logically), and hence not subject to the "method of exhaustion" problem.

[wandelaar]

15 minutes ago, Song of the Dao said:

How is that accomplished? If I have a finite life how can I count to infinity?

 

Good! The fact that it seems impossible to accomplish infinitely many acts within finite time is exactly what makes Zeno's paradoxes hard to solve ... until you see that there actually are infinite collections of acts that can be completed in finite time.

[wandelaar]

3 minutes ago, Jeff said:

That is how calculus works (logically), and hence not subject to the "method of exhaustion" problem.

 

The method of exhaustion isn't problematic as far as I know. But there is a logical error in Zeno's reasoning and as soon as one sees it, the paradox is solved. Or mathematically speaking: an interval of finite length can indeed contain infinitely many intervals of finite lengths without thereby becoming infinitely long itself. That is all one needs to know, and then the reasoning of Zeno falls apart.

[wandelaar]

10 minutes ago, Song of the Dao said:

Explain?

 

Walk from one side to the other side of your room. Now to do that you first have to traverse half of the distance, then half of the remaining distance, then half of the still remaining distance, etc. Nevertheless you will doubtlessly succeed because the time necessary for the subsequent acts of traversing the subsequent half's of the remaining distance are continually getting shorter sufficiently fast. And in fact all of those times fit into the time that it takes you to walk from one side of your room to the other.

[wandelaar]

If you don't like to walk you may also look at this picture:

 

https://www.britannica.com/science/geometric-series/media/229825/19629

 

Quote

If one can do an infinite amount of things in a finite amount of time that should hold true in all instances.

 

Not at all! No reason to suppose that.

[wandelaar]

You are drifting off into empty speculation. That's a very old joke. And indeed - I cannot proof that all kind of exotic ideas like we are living in a simulation, the world is a dream, we are continually fooled by devils, etc. are not true. But the only reasonable conclusion one can draw from that is that we cannot be absolutely sure about the world or even about ourselves. The old sceptics hundreds of years before Christ already knew that. But your life goes on, and apparently you keep using your computer as if there actually exists people who appear as Bums on this forum, so in practice you don't even take your own illusion-speculation seriously yourself. And that's OK because there is no reasonable way to take those speculations serious, except that is as warnings against pretensions of absolute knowledge.

 

Edited January 2, 2019 by wandelaar

[Rocky Lionmouth]

 

6 hours ago, Taoist Texts said:

err, u all just admitted that you dont even know why paradoxes were created or what they even mean. Is not that a reason to suspect that y'all do not comprehend them?

 

I admitted nothing and if i did it’s probably true.

[Rocky Lionmouth]

So no takers on the ”Zeno had a sharp wit to test their reasoning amd a good sense of humor to boot”-theory? We’re still talking about paradoxes devised by one of the stoics, of Socratic descent by way of the cynic lineage yes?

 

[Lost in Translation]

If Achilles' desire were to reach the tortoise then the paradox would hold. But Achilles' desire is not to reach the tortoise but surpass him. Thus Achilles has as his benchmark not the distance that the tortoise has travelled but the length of the racecourse.

[Rocky Lionmouth]

Just now, Lost in Translation said:

If Achilles' desire were to reach the tortoise then the paradox would hold. But Achilles' desire is not to reach the tortoise but surpass him. Thus Achilles has as his benchmark not the distance that the tortoise has travelled but the length of the racecourse.

 

Describe the lenght of a racecourse please.

[wandelaar]

I never said that science gives us an absolutely accurate picture of reality, and no modern scientist would dare to claim that either.

[Lost in Translation]

9 minutes ago, Rocky Lionmouth said:

 

Describe the lenght of a racecourse please.

 

It does not matter. All that matters is that the locations of the tortoise and of Achilles are functions on the length of the race, not of each other.

[Rocky Lionmouth]

1 minute ago, Lost in Translation said:

 

It does not matter. All that matters is that the locations of the tortoise and of Achilles are functions on the length of the race, not of each other.

 

Oh come on, you left me hanging on a ”How long is a piece of string?” joke.

[Lost in Translation]

1 minute ago, Rocky Lionmouth said:

 

Oh come on, you left me hanging on a ”How long is a piece of string?” joke.

 

My apologies. The course is 1.6 x 10^3 meters long.

[wandelaar]

Is it so bad? Take a rest then...

[OldDog]

8 hours ago, Taoist Texts said:

Zeno took on the project of creating these paradoxes because other philosophers had created paradoxes against Parmenides' view.

 

Yes, I read that also. But why a paradox? Why not simply refute the opposition with straight forward factual logical argument? Why go to the trouble of creating a mind boggling argument? Is there something about a paradox that ...?

 

 

[wandelaar]

5 minutes ago, OldDog said:

Yes, I read that also. But why a paradox? Why not simply refute the opposition with straight forward factual logical argument? Why go to the trouble of creating a mind boggling argument? Is there something about a paradox that ...?

 

He did refute the opposition with his paradoxes, or at least he thought he did. The opposition were those philosophers who thought that motion is real and that there are more than one things in the world. Zeno's teacher Parmenides claimed that there is actually only one changeless being, and that all else is illusion. See: https://en.wikipedia.org/wiki/Parmenides#Thought

[Taoist Texts]

5 hours ago, OldDog said:

 

Yes, I read that also. But why a paradox? Why not simply refute the opposition with straight forward factual logical argument? Why go to the trouble of creating a mind boggling argument? Is there something about a paradox that ...?

Good questions. Yes there is something special about the paradox besides it being harder to understand.

A parable was simply the way they presented their arguments back then, the arguments which added to the general knowledge 

But a paradox format of a parable was used to mark the end of all possible knowledge.

Paradox means that we do not know and will never ever know.

 

With his 'paradoxes' Zeno did 3 things:

1. Demonstrated that the very fabric of our world consists of nothingness.

2. Marked the end of all knowledge on that particular issue

3. Refuted Parmenides' argument against nothingness. (Refuting the atomists too.)

 

Quote

Parmenides made the ontological argument against nothingness, essentially denying the possible existence of a void. According to Aristotle, this led Democritus and Leucippus, and many other physicists,^[27] to propose the atomic theory, which supposes that everything in the universe is either atoms or voids

 

Zeno's paradoxes, when rephrased, are very simple and very irrefutable:

Splitting a unit of space or a unit of matter  into progressively smaller bits will eventually end in bits that are ungraspable to our senses or to our measuring tools, i.e this process will end in nothingness. 

Which in turn means that the spacematter consists of nothingness.

Also, it means that the spacematter is fundamentally unknowable. 

 

That's all there is to it. Simple, no?

[OldDog]

Seems that there are two kinds of paradox. True paradox and pseudo- or quasi- paradox.

 

The quasi-paradox being simply tricks of definitions, semantics or reasoning where the intent is to produce an argument that cannot be rationally resolved, thus bringing further inquiry to an end and apparently establishing a truth or untruth. In such quasi-paradoxes, if one takes the time to analyze the premise, suppositions and reasoning of the paradox, the deception may be discovered. 

 

In true paradox, no deception can be found and the only conclusion that can be attained is that the reasoning mind is imperfect and not reliable, thus pointing to the need for some other means of understanding.

 

Seems that Daoism recognizes that true paradox exists and, realizing the inability of the rational mind to deal with it, poses that suspending the reasoning mind offers a directions for continued inquiry. 

 

Just musing about paradox.