Wednesday 4 January 2012

Explaining Zeno's paradoxes


I was aware from mid-teens of Zeno's paradox about Achilles and the Tortoise* (for example as a Tortoise and Rabbit in Tom Stoppard's play Jumpers) - but I never properly got around to understanding just precisely what led to the paradox, why the mode of reasoning was wrong.

I see now this was an oversight and an error. 


The penny dropped when reading pages 137-9 of The Master and his Emissary, by Iain McGilchrist:

Zeno's paradoxes ... rest on the adoption of the left hemisphere's view that every flowing motion in space or time can be resolved into a series of static moments or points that can then be summed to give back the living whole. The 'seamless' fluidity of motion in space or time is 'reduced' to a series, akin to the series of still frames in a cine film.


In essence, the paradox arises from the inbuilt assumption that every dynamic phenomenon can be resolved into a series of static units that can then be summed to recreate the dynamic whole.

There is absolutely no grounds for this assumption - this metaphysical assumption concerning the nature of reality, this basic assumption which frames all further analysis - indeed the assumption is obviously refuted by experience as demonstrated by the paradox.


(A further assumption is that metaphysics ought not be contradicted by plain, commonsense experience - since the best Ancient Greek philosophy was meant to enrich normal life and plain commonsense experience - not usurp and replace it. This is a major constraint on philosophizing, and if it is abandoned - as modern culture has abandoned it - then philosophy becomes an open-ended and unbounded activity with no grounds for choice between philosophical ideas, only subjective preferences or varieties of instrumental expediency.)


But with the paradox we (somehow) fail to notice we have made this metaphysical assumption and built it into the analysis - hence the paradox.

And it is suddenly obvious how deep an insight it was to understand this paradox, and what trap it is for humans to fail to recognise it.


All my life in science, medicine and education I have observed the progressive destruction of these professions by the failure to recognise Zeno's paradox; the failure to recognise that dynamic skills cannot be broken down into static units then rebuilt. They cannot. They cannot.

It does not matter how long you make the forms to be completed, how elaborate the checklists, how comprehensive the managerial oversight.

The problem remains: in making movement static you have frozen the movement - movement is not movement any longer if it has been frozen.


Achilles never catches-up with the tortoise

Because both have been killed.

(To freeze a living entity is to kill it.) 


What complexity of static summaries does is to make things worse not better - because few, crude and simple static pictures of a complex activity cannot be mistaken for the real thing, whereas humans are bewildered by complexity into assuming it is completeness.


*Achilles races the tortoise. He allows that tortoise a 'start' at point A, a little ahead of Achilles. They both begin to run. Achilles runs to the tortoises starting point A; but by this time the tortoise has moved a little ahead from A to point B. Then Achilles runs to point B - but by this time the tortoise has reached point C. So Achilles runs to point C, but by now the tortoise is at point D. And so on. Achilles never catches-up with the tortoise.

I sometimes think the Ancient Greeks did all the philosophy that was worth doing, that needed doing (especially metaphysics); and everything else since has been worse than nothing at all, since it made us lose sight of their insights. All we need to do is retain and re-express.


Daniel said...

I remember a great feeling of perplexity, followed by a great feeling of relief, when I was first taught Zeno's paradox. I must have been about 15 or 16.

It's an all-time great paradox because, given the rules of the game, it's absolutely insoluble. That's why I was so perplexed.

But, of course, the whole point of the paradox is to point out how "frozen-time" reasoning doesn't match up with common sense. So my second feeling, after perplexity, was relief. It's actually a rather joyous thing to realize: that measuring things with a stopwatch or a ruler is indeed very useful, but also hopelessly incomplete (if we want to make our measurements join up with reality).

I went on to all kinds of errors after my Zeno realization, of course. But I don't think I ever again (or at least not seriously, not for longer than a few confused moments) fell into the trap of taking measured "reality" to mean real reality.

And yet, as you have pointed out on this blog before, scientists and scientistic thinkers (i.e. all of us?) operate under this same, obviously false delusion all the time!

Achilles never gets there, not because the measurements are wrong, but because they are incomplete! They don't measure reality.

Bruce Charlton said...


Indeed. But why are we so powerless to resist this being imposed?

McG has it that this is precisely because the Left brain cannot make sense of the Right (whereas the Right comprehends the Left) - the very inadequacy of the static snapshots means that they are - from their own perspective - un-refutable!

How many times have I been told about the latest managerailly imposed Left brain stuff, in effect: "Yes we know it is nonsense, indeed dangerous nonsense - but we have to do it anyway"

Why do we have to do nonsense? To *have to* do nonsense is evil and corrupting, as the Communists knew and exploited; as PC and the mass media know full well.

It will not stop until the whole thing collapses, or people refuse to do nonsense - I used to hope for the latter, indeed I expected people to snap out of their suicidal insanity as its effects became obvious - but I see no sign of it, no sign at all - presumably corruption begets corruption, insanity begets insanity - the up-front cost of honesty and sanity gets greater and greater...

Proph said...

All this points to an interesting question which I've been struggling with for a while. We seem to be coming around to the perception that the modern mode of being is radically defective, and that these defects are probably largely beyond his control. I've likened it to autism, Eric Voegelin likened it to insanity or disease.

What does this mean for the modern who rejects God and revels in sin? It seems to me this is a very real impediment to grace, yet one not largely freely enacted. It's enough to make one wonder if the Calvinists are right.

Wurmbrand said...

I am a tenured professor at a small state university, but if I refused to have anything to do with "measurable learning outcomes" and so on I would probably be regarded as insubordinate (NB "insubordination" is not defined at my university -- I have checked!) and disciplined, and, if I continued to refuse, would be fired. So I try to do good things even if "subversively."

Bruce Charlton said...

@Dale - the ratchet of corruption - make people do evil, make them say evil is good - corrupt them.

It has been very effective!

senexada said...

A weaker claim is that the static model may require infinite frames to reflect reality. The left brain struggles with infinity, but it can still do meaningful computations. For instance, it can compute the sum of infinite series such as (1/2) + (1/4) + (1/8) + ... And Zeno's paradox can be somewhat understood in a left-brain fashion using the mathematics of infinite series.

I agree with your greater theme against reductionism. And I'm intrigued by the mystery of whether time and space, at its core, is continuous or discrete. But I'm unsure that Zeno is the best example against reductionism. The mathematics for understanding it is fairly mature & capable, even if hindered by unintuitive elements.

The Continental Op said...

Zeno's "paradox" of Achilles and the tortoise requires exponentially diminishing time slices, until they become infinitesimally small. That is the trick that Zeno uses to make his "paradox." If the time slice is 0 seconds, they're both standing still (as you said, they're now frozen, dead). If they're both standing still, with the tortoise in the lead, Achilles will never pass the tortoise. Duh!

As soon as you take constant time slices, say 0.1 seconds, then the paradox falls apart like a house of cards in a hurricane.

Mathematicians will say that "infinitesimally close to 0" is not zero, but I'm an engineer, and by golly, it's zero.

It's basic trickery.

The Continental Op said...

Another way of seeing Zeno's paradox is that you are watching the race in replay. The closer Achilles gets to the tortoise, the faster the replay slows down. The replay slows down until it stops, with the tortoise just in the lead. Of course Achilles then will not pass the tortoise, because nothing is moving.

Stopping time: now that is some trick. Reality does not have a 'pause' button.

HenryOrientJnr said...

I believe according to quantum physics there is actually a discrete "smallest unit of distance" - the Planck length. Hence, the error in assumptions is actually backwards. The flowing motion of the right brain is an illusion and the digital series of the left brain is correct.

The paradox occurs because when you get down to the quantum level you are either at i or i-1 Planck units from the goal but never at i-1/2.

The Continental Op said...

The flowing motion of the right brain is an illusion and the digital series of the left brain is correct.

At this point science is weird and useless, unless it drives some scientists mad to where they break down and finally cry out to God.

Anonymous said...

I agree with sexenada. Reductionism is wrong and I tend to agree with McGilChrist, but Zeno's paradoxes are not the best example to prove that.

There a lot of experiments that suggest that reductionism is wrong, but materialist scientist want to hide under the rug. It would be good to make a list.

I think good candidates for this list are Bell's theorem or the experiments of Sheldrake with rats.


I am not sure if Planck distance means that the movement is discrete or that we cannot say anything about movement between the Planck distance. But I accept I am not very good in this quantum things. If someone corrects me, I will be glad.

Having said that, I'm still with McGilChrist, because when we get to the quantum level, we have a wave function that evolves in a continuous way. A wave function where are not only involved the Achilles and the tortoise's particles but the surrounding particles as well.

The left hemisphere conceptualizes this as a simple linear sequence of two bodies approaching. It's a Disneyesque portrait of a very complex interaction, as far as I know.


Bruce Charlton said...

Thanks to commenters for such excellent discussion.

What I take from it is an increased awareness of the ground of philosophy.

At the first level of analysis, Zeno's paradox is as I stated it; but of course we moderns have (for a few hundred years) mathematical ways of dealing with infinite numbers of static states.

The big question is whether or not this affects the first interpretation; does calculus 'solve' the paradox?

My gut feeling is that mathematics is orthogonal to the problem, does not solve it because it is a different kind of thing.

And I have another gut feeling that there is a hazard in trying to push further back beyond the primary paradox, in continuing to keep doing philosophy when philosophy has done its work...

So, I think the proper attitude is to say - yes I get it; rather than - can I find a solution to it?

Gabe Ruth said...

I'm with Continental Op (and I'm also an engineer). For the longest time I really thought I was an idiot because I failed to see what was so paradoxical about this particular paradox. Sometimes I still think that.

Cantillonblog said...

This is spot on.

See the following quote from Alfred Marshall, a key name in economics at Cambridge and indeed in all of Britain.

"The element of time is a chief cause of those difficulties in economic investigations which make it necessary for man with his limited powers to go step by step; breaking up a complex question, studying one bit at a time, and at last combining his partial solutions into a more or less complete solution of the whole riddle. In breaking it up, he segregates those disturbing causes, whose wanderings happen to be inconvenient, for the time in a pound called Ceteris Paribus. The study of some group of tendencies is isolated by the assumption other things being equal: the existence of other tendencies is not denied, but their disturbing effect is neglected for a time. The more the issue is thus narrowed, the more exactly can it be handled: but also the less closely does it correspond to real life. Each exact and firm handling of a narrow issue, however, helps towards treating broader issues, in which that narrow issue is contained, more exactly than would otherwise have been possible. With each step more things can be let out of the pound; exact discussions can be made less abstract, realistic discussions can be made less inexact than was possible at an earlier stage. (Alfred Marshall, Principles of Economics, Bk.V,Ch.V in paragraph V.V.10)."