Zeno's paradoxes

[Cumannach]

What do you think about Zeno's paradoxes?

Leaving aside their relation to dialectics, is it possible to solve the 'motion is impossible' paradox?

Some mathematicians I hear think they created a formal solution to it- but is a formal symbolic system actually valid as a philosophical solution, a solution to a problem of actual space, and actual time as we experience it?

[ÑóẊîöʼn]

Isn't it only a "paradox" if one doesn't have calculus?

Edited to add: Which therefore means it is only a "paradox" if one doesn't have the appropriate tools to analyse the situation.

[Rosa Lichtenstein]

Zeno's 'paradoxes' only work because of his sloppy use of language.

I have de-fused the alleged 'paradox of motion' here:

http://homepage.ntlworld.com/rosa.l/page%2005.htm

Motion is not the least bit 'contradictory', so Hegel's use of this paradox was a big mistake (as was Engels's attempt to appropriate Hegel's blunder).

Here is a summary of one part of my argument:

It is not easy to reconstruct the rationale behind Engels's (and thus Hegel's) conclusion that motion is contradictory, but it seems to depend on this line of argument (beginning with a rejection of an apparent contradiction, in E1a):

E1: An object cannot be in motion and at rest at one and the same time.

E2: If an object is located at a point it must be at rest at that point.

E3: Hence, a moving body cannot be located at a point, otherwise it would not be moving, it would be at rest.

E4: Consequently, given E1, a moving body must both occupy and not occupy a point at one and the same instant.

[E1a: An object can be in motion and at rest at one and the same time.]

But, if this is Engels's (or even Hegel's) rationale, then he/they give no reason why we should prefer one contradiction (E4) over another (E1a). And yet, E1a is a familiar truth, for it is surely possible for an object to be at rest with respect to one frame of reference and yet be in motion with respect to another.

On this, Robert Mills had this comment to make:

"Another way of stating the principle of equivalence, a way that better reflects its name, is to say that all reference frames, including accelerated reference frames, are equivalent, that the laws of Physics take the same form in any reference frame…. And it is also correct to say that the Copernican view (with the sun at the centre) and the Ptolemaic view (with the earth at the centre) are equally valid and equally consistent!" [Mills (1994), pp.182-83. Spelling altered to conform to UK English.]

[It is worth recalling that the late Professor Mills was co-inventor of Yang-Mills Theory in Gauge Quantum Mechanics, and thus no scientific novice.]

Hence, in one frame, the Earth is stationary, in another is it moving. But, in that case, if E1a is true, E4 cannot follow, and the imputed rationale behind Engels's 'contradiction' evaporates.

Furthermore, Engels's conclusion clearly depends on an object moving between locations with time having advanced not one instant, that is, his conclusion implies that the supposed change of place must occur outside of time -- or, worse, that it happens independently of the passage of time --, which is incomprehensible (as even Trotsky would have admitted):

"How should we really conceive the word 'moment'? If it is an infinitesimal interval of time, then a pound of sugar is subjected during the course of that 'moment' to inevitable changes. Or is the 'moment' a purely mathematical abstraction, that is, a zero of time? But everything exists in time; and existence itself is an uninterrupted process of transformation; time is consequently a fundamental element of existence. Thus the axiom 'A' is equal to 'A' signifies that a thing is equal to itself if it does not change, that is if it does not exist." [Trotsky (1971), pp.63-64.]

And yet, how else are we to understand Engels's claim that a moving body is actually in two places at once? In that case, a moving object would be in one place at one instant, and it would move to another place with no lapse of time; such motion would thus take place outside of time. But, according to Trotsky, that sort of motion would not exist, for it would not have taken place in time.

This would further mean that while we may divide position as finely as we wished -- so that no matter to what extent the spatial aspects of a body's location were partitioned, we would always be able to distinguish two contiguous points, allowing us to say that a moving body was in those two places at once --, while we can do that with location, we cannot do the same with respect to time.

Engels's 'argument' thus depends on the claim that while the location of a particular body is subject to infinite divisibility (an assumption which, one presumes, is necessary to support the claim that moving bodies must be in two places at the same time, no matter how microscopically close together the latter are -- which implies that spatial locations can be given in endlessly finer-grained detail), the time interval during which the said body occupies this or that or any other location is not subject to similar division. Now, this is an a priori and non-symmetric restriction -- that is, this restriction is applied to time, but not to space --, which is impossible to justify on either empirical or logical grounds.

[Not one single DM-fan, as far as I am aware, has ever even tried to justify this one-sided stipulation. In fact, it is equally clear that not one DM-fan is even aware of it!]

If this constraint is waved (as surely it should be!), it would mean that no matter how close together the two locations are that a body is supposedly in, we can always specify a time interval during which locomotion occurs -- or, perhaps map two moments isomorphic to them. That done, the alleged 'contradiction' vanishes.

Again, the only way to neutralise this response is to counter-claim that a body must be motionless if it is in a certain place at a certain time (as we saw in E2). In that case, if it is moving, a body must be in two places at the same time.

But, that just repeats the non-symmetrical restriction noted above (and the suspect derivation upon which doubt was cast earlier). If we can divide up places more finely so that it is possible to say an object is in two of the latter while the 'instant' during which this occurs stays the same, then we can surely do likewise with respect to time, specifying two times for each of these two places (or, at least, a time interval in which such a change of place occurs).

Once more, none of this is the least bit surprising since Engels's claims about motion and change date back to the a priori speculations of that ancient mystic Heraclitus -- a thinker who did not even bother to base his wild ideas on anything remotely like evidence (having derived his 'profound' conclusions about all of reality for all of time from what he thought was true about the possibility of stepping into a certain river!) --, and to an Idealist conundrum invented by Zeno.

[Of course, these observations dispose of the DM-claim that contradictions between space and time are only to be expected since reality is fundamentally contradictory. This is because this 'contradiction' plainly results from a lop-sided convention that interprets one of these as continuous (place) and the other as discrete (time). But, if they are both treated in the same way, there is no contradiction.]

Moreover, Engels also failed to note that several other paradoxical consequences follow from his ideas. One of these is that if a moving body is anywhere, it must be everywhere, all at once. This is because his argument depends on the idea that a moving body must be in two places at the same time -- i.e., in, say, P1 and P2 --, otherwise it would be stationary. This allows him to derive a 'contradiction': a moving body must be in two places at once, and it must both be in and not in at least one of these at the same moment.

But, clearly, if the said body is in P2 it must also be in P3 in the same instant. If this is denied, then the conclusion that a moving body must be in one place and not in it at the same instant, and in another place at the same time, will have to be dropped.

However, if it is still held true that at one and the same instant a moving body is in one place and not in it, and that it is in another place at that time (otherwise it would be stationary), then it must be in P3 at the same instant that it is in P2, or it would not be moving while at P2, but would be stationary at P2.

In that case, such a body must be in at least three places at once. Unfortunately, the same argument now applies to P3, and to P4, and so on...

Hence, assuming that the said body is still moving while at P2, by the application of a sufficiently powerful induction, it can be shown that any moving body must be everywhere if it is anywhere, all at the same instant!

Now that is even more absurd than Zeno's ridiculous conclusion!

But that's Diabolical Logic for you!

More details and references can be found here:

http://homepage.ntlworld.com/rosa.l/...Essay_Five.htm

[Even more details at the link I posted above.]

[Rosa Lichtenstein]

Noxion:

Isn't it only a "paradox" if one doesn't have calculus?

Edited to add: Which therefore means it is only a "paradox" if one doesn't have the appropriate tools to analyse the situation.

In fact, it's not a paradox to begin with.

http://homepage.ntlworld.com/rosa.l/page%2005.htm

[ÑóẊîöʼn]

In fact, it's not a paradox to begin with.

That's why I was using quotation marks. It is manifestly apparent that movement is possible - the so-called "paradox" only arose due to the limitations of the thinkers at the time.

[Cumannach]

Zeno's paradox of motion was in a single inert frame of reference.

In inert frame R, there are two points A and B. A tortoise starts at A and is traveling to B, but to reach B he must get halfway there, and reach halfway....

Now why should a symbolic logical system that allows for infinite series to sum to a finite number be a valid solution for this real world philosophical problem?

And it's not a solution to say, I observe motion therefore it can't be paradox.

[Rosa Lichtenstein]

Cummanach

Zeno's paradox of motion was in a single inert frame of reference.

In inert frame R, there are two points A and B. A tortoise starts at A and is traveling to B, but to reach B he must get halfway there, and reach halfway....

Now why should a symbolic logical system that allows for infinite series to sum to a finite number be a valid solution for this real world philosophical problem?

And it's not a solution to say, I observe motion therefore it can't be paradox.

Your mistake is to treat physical space as if it were isomorphic to to R3 (the Real manifold). Since it is impossible to show there is an isomprhism here without this assumption being taken for granted (in other words, without this being a circular argument), there is no paradox.

Zeno's problems arose, too, when he thought he could treat physical space as if it were a mathematical space.

Indeed, it is, but only for Pythagorean/Platonists.

His other 'paradoxes' (including that of the arrow, material division, and the Stade) also fall at the same fence.

And this is quite apart from the other objections I raised above, which you simply ignored (no surprise there...).

[Rosa Lichtenstein]

Noxion:

That's why I was using quotation marks. It is manifestly apparent that movement is possible - the so-called "paradox" only arose due to the limitations of the thinkers at the time.

Fair enough, but it is possible to re-pharse Zeno's 'paradoxes' in ways that make them immune to your objections (i.e., those based on post-Weierstrassian Real Analysis -- the modern calculus).

My approach cuts his 'paradoxes' of motion off at the knees (by showing that they arise from a series of fundamental confusions based on his sloppy use of language), and thus it does not require the (in the end) ineffectual input of modern mathematics.

[Cumannach]

I ignored the rest because as I said in the OP I'm leaving DM aside, something I know you're generally incapable of doing.

But what about the issue Wesley Salmon raises in his introduction to a book about the paradoxes. on page 16

http://books.google.ie/books?id=0AzP...efox-a#PPP1,M1

the whole introduction is available in the preview.

[Rosa Lichtenstein]

Cummanach:

I ignored the rest because as I said in the OP I'm leaving DM aside, something I know you're generally incapable of doing.

Ah, bottled it again, I see...

But what about the issue Wesley Salmon raises in his introduction to a book about the paradoxes. on page 16

http://books.google.ie/books?id=0AzP...efox-a#PPP1,M1

the whole introduction is available in the preview.

Not only have I seen it, I studied this book (and many others as part of my degree), but he, and the others in that book (apart, perhaps, from Max Black) make all the usual mistakes -- some of which I outlined in my earlier post -- you know, the one to which you could not respond.

By the way, Salmon's book is nearly 40 years old, and many of the articles it contains are even older.

There has been much work done on these alleged 'pardoxes' since.

You can find references to some of that work in my Essay Five:

http://homepage.ntlworld.com/rosa.l/page%2005.htm

My solution (which is to dissolve these 'paradoxes') breaks entirely new ground.

[Nulono]

Zeno's paradoxes ignore mathemeatics. If you have an infinite number of events, you can have them occur in finite time if the durations decrease exponentially.

If event 1 takes a half a second, event 2 takes a quarter, 3 an eight, 4 a sixteenth, and so on, an infinite number of events can occur in only one second!

[Rosa Lichtenstein]

Nulono:

Zeno's paradoxes ignore mathemeatics. If you have an infinite number of events, you can have them occur in finite time if the durations decrease exponentially.

If event 1 takes a half a second, event 2 takes a quarter, 3 an eight, 4 a sixteenth, and so on, an infinite number of events can occur in only one second!

In fact, Zeno's 'paradoxes' aren't paradoxes to begin with, but are, like so much traditional philosophy, based on a sloppy use of language. In which case, there is no need to look to mathematics to help us out here.

[Invariance]

Hmm, my calculus teacher said that this could be 'solved' using the concept of a limit, as ∆x approaches zero but doesn't = 0, to avoid the problem of a 0/0 which is what happens when you try to find instantaneous velocity, i.e the velocity of something occurring without any change in duration of time, yet since nothing can move when time is paused, instantaneous motion seems absurd...I'm not so convinced by any mathematical explanation, since I think that, as Rosa says, Zeno is treating mathematical space as the same as physical space, which is not infinitely divisible. I think that not only are philosophers confused in the answers they give, they are confused in the questions they ask.

[Rosa Lichtenstein]

Vinnie:

Hmm, my calculus teacher said that this could be 'solved' using the concept of a limit, as ∆x approaches zero but doesn't = 0, to avoid the problem of a 0/0 which is what happens when you try to find instantaneous velocity, i.e the velocity of something occurring without any change in duration of time, yet since nothing can move when time is paused, instantaneous motion seems absurd...I'm not so convinced by any mathematical explanation, since I think that, as Rosa says, Zeno is treating mathematical space as the same as physical space, which is not infinitely divisible. I think that not only are philosophers confused in the answers they give, they are confused in the questions they ask.

Check out the superior 'delta-epsilon' approach to the calculus developed 150 years ago by Weierstrass:

http://www.karlscalculus.org/x2_1.html

which many think has solved this problem.

But, if you have a look at the literature, you will see that philosophers who accept Zeno's argument (albeit in an updated form), and that includes many dialectical materialists, have gotten around this 'difficulty'. For example, check out the articles published in Marx's Mathematical Manuscripts:

http://www.marxists.org/archive/marx...ipts/index.htm

There, Russian theorists openly question the validity of Weierstrass's method.

So, my approach, which shows just where Zeno (and Hegel, etc.) went wrong is much to be preferred since it does not depend on such obscure technicalities.

You can find the above in Sections (3) to (10) here (use the Quick Links at the top):

http://homepage.ntlworld.com/rosa.l/page%2005.htm

There, I develop the very first Wittgensteinian demolition of the rationale 'underpinning' this obscure backwater of ancient Idealism --, which, if my argument is correct, provides the first non-technical solution to this 'problem' in 2500 years -- by showing it isn't a problem to begin with!

[gilhyle]

All of which would be fine except for the rather silly idea that two thousand years of philosophy has been simply 'sloppy' - no one in all that time had the discipline of our Rose.....I think not; rather you have missed the point that there are reasons, drivers for people to speak in ways which limit the conceptualised divisibility of time.

[Rosa Lichtenstein]

Gilhyle:

All of which would be fine except for the rather silly idea that two thousand years of philosophy has been simply 'sloppy' - no one in all that time had the discipline of our Rose.....I think not; rather you have missed the point that there are reasons, drivers for people to speak in ways which limit the conceptualised divisibility of time.

Unfortunately for you, it is quite easy to show that this is indeed the case.

Anyway, you are an excellent example for the prosecution; the more you post, the stronger the case becomes...

[synthesis]

Rosa's right here, although I don't think it's as cut-and-dry as she does. They are paradoxes, but they're paradoxes of language, not physics.

[Rosa Lichtenstein]

KF:

Rosa's right here, although I don't think it's as cut-and-dry as she does. They are paradoxes, but they're paradoxes of language, not physics.

These are only 'paradoxes' if one is determined to use language in rather odd ways.

[synthesis]

These are only 'paradoxes' if one is determined to use language in rather odd ways.

Well, that's what Zeno did. I just think the mountain of effort undertaken by mathematicians to "solve" the paradoxes is rather silly, because we know they don't accurately describe reality. I mean, isn't a lot of philosophy just "using language in odd ways"?

[Rosa Lichtenstein]

KF:

Well, that's what Zeno did.

Indeed, and that's why his 'paradoxes' aren't paradoxes.

I mean, isn't a lot of philosophy just "using language in odd ways"?

Indeed, all of traditional philosophy is as you describe.