Monday, November 29, 2010

Rough Jottings on Zeno's Argument and Paradoxes Generally

Graham Priest has an article on dialetheism and paraconsistency at the NYT's "Stone" blog. I was struck by what he says about Zeno's paradox:

Here we can’t just accept the conclusion: we know that the car can get to point B. So something must be wrong with the argument. In fact, there is now a general consensus about what is wrong with it (based on other developments in 19th-century mathematics concerning infinite series). You can do an infinite number of things in a finite time — at least provided that these things can be done faster and faster.

The consensus, which is sometimes called the Standard Solution, can only get "You can do an infinite number of things in a finite time - at least provided that these things can be done faster and faster," if it provides a solution to the paradox, not vice versa. And it's not so clear that the Standard Solution does everything that would be required to resolve/solve/dissolve the Paradox. And, indeed, when you look at elaborations of how the Standard Solution is supposed to solve the paradoxes, such as the IEP article on the subject, all the responses very clearly end up being, "Wherever there's a problem, we assume that you can do what Zeno denies can be done, and make a distinction based on that assumption." Which is fine, but one should be quite clear that this is not what solving a paradox is: to make this response work you need not only to show that you avoid the paradox and have provided a consistent answer; you need also to show that there is independent good reason to reject the assumptions of the paradox you are rejecting. People have given the conclusion Priest gives for literally ages -- Aristotle explicitly mentions it in response to the Dichotomy. What has always been in question is what can support this conclusion without begging the question. This is why Aristotle makes his famous distinction between actual and potential infinites here: actuality and potentiality are more fundamental concepts than anything in the Paradox itself, and are presupposed by the Paradox itself, and so can be used simultaneously to argue that its assumptions are wrong and that an alternative is right. Whether one agrees with Aristotle as to the actual answer or not, this is the way to dissolve a paradox. The Standard Solution doesn't do this, and the usefulness of the mathematics that yield it is neither here nor there: it always remains open to accept both the Standard Solution and the Paradox by being an anti-realist about the former and a realist about the latter. This is a regular problem with mathematical solutions to non-mathematical problems: we know for a fact that extremely useful mathematical solutions can fail to correspond to reality, and therefore it's always an option to be anti-realist about any of them. The Standard Solution at most shows you that, if certain things are true about change, then a consistent system can be had in which Zeno's Paradox can't be formulated. It doesn't show that those things are true about change, but at most that if we at least pretend that they are we get right answers without having to worry about the Paradox. The Paradox is evaded for practical purposes, which is good, and the evasion is shown not to be inconsistent with itself, which is very good, but if you leave it at that, the Paradox has not been resolved. Indeed, if you leave it at that, you clearly show that you have no idea what resolving/solving/dissolving a paradox is. And showing that every feature of the (extraordinarily complex) Standard Solution has real, independently establishable, physical counterparts is a massive challenge that no one has ever undertaken.

It reminds me a little, actually, of common responses to the Preface Paradox, which was talked about at length in the blogosphere some time back. The usual response was to posit as true whatever was required to make there no longer be a paradox. This is an evasion of the paradox, and a perfectly reasonable thing to do for practical purposes, but it doesn't deal with the paradox at all: what you need to do is prove that what you are positing really is true and doesn't just save the appearances. And when you set out to do that you find that independent proof of the assumptions they make is extraordinarily difficult (and that many of the proposals for handling the paradox that are taken by their proponents as just obvious are inconsistent with each other).

It's also not really correct to say that the Standard Solution is a general consensus: there is no general consensus. The Standard Solution is easily the dominant single position, but it has quite a few robust rivals, and when you take those into account it's not even always clear that the Standard Solution is accepted by a majority of people. Lots of people still prefer Aristotelian approaches of various kinds, at most thinking that they need to be refined; lots of people are constructivists; lots of people prefer appeal to infinitesimals; and so forth. No one of these groups is even close to being as dominant as the Standard Solution group, but all together make a very sizable bunch. What we have is a dominant proposal; this is very different from a general consensus.

Further, Zeno's Paradox wasn't an argument that there could be no consistent mathematical system without the Paradox; it was an argument that actual motion through actual space and time was something opponents of Parmenides and Zeno (it is not, and never has been, clear whether Zeno was attempting to defend Parmenides or merely to criticize the arguments proposed against Parmenides, because we don't know how closely Zeno himself actually followed Parmenides) could not coherently account for. This is as certainly known as anything else about Zeno's Paradox; it's one of the few things that is agreed on by the three major interpretations of the Paradox (that Zeno was defending Parmenidean monism against common-sense pluralism; that Zeno was a nihilist attacking both monism and pluralism; that Zeno was not defending any particular position but only raising problems). The genuinely important question is not, "Is there some set of assumptions that, if true, would dissolve the paradox?" because the answer to that is "Of course," regardless of the paradox. The question, "What proves these dissolving assumptions actually true?" is more important, but is still not the most important question. The most important question is: "What are the common features of the accounts of motion Zeno's Paradox on its own makes it impossible to accept?" That is, people often look at paradoxes upside down: they focus on finding things that at least evade the paradox. There are always a great many of those, and sometimes the answer that dissolves the paradox is actually not interesting, beyond dissolving the paradox. What is really and consistently valuable in a paradox -- and it always remains even when the paradox is dissolved -- is what it tells us about what we can't accept.


  1. John Perry11:46 AM

    Admittedly, the philosophy of mathematics isn't my particular specialty, but I studied this very problem once, and I don't think it's correct to say that the standard solution essentailly says that you can do an infinite number of things in a finite amount of time.

    Rather, the standard solution assumes that the system is a continuum as opposed to a discrete space. This seems to be what confuses people: we look at a continuum as if it were discrete. The SS then goes on to say that there are certain properties of a continuum that one can approximate as finite sums, and the more one refines the sum -- not necessarily according to time ("faster"), in fact I would prefer to refine it according to size ("smaller"), which applies to time but to other things as well -- the smaller the error in the approximation. But this is only subject to assumptions similar to this for the language of limits: if the size of the allowable error decreases arbitrarily, we can refine the measurement and keep the actual error within the allowable range.

    But I don't think it's accurate to say that you are ever doing infinitely many things in finite time: if you think you are, then you're in the discrete mindset when dealing with a continuum.

    Probably someone who knows more about this than I can correct me, but it's also possible that both positions can be held.

  2. branemrys12:09 PM

    The 'faster' has been bothering me since I read it, for reasons I haven't been able to place, but getting into the finer details of the Standard Solution often makes my head spin, so I just gave it to Priest. Now that you put it that way, I think you are probably right that the description makes the mistake of talking about the continuum as if it were discrete.

  3. Ocham2:54 PM

    I dont' follow the argument.  I can see that 1-4 are true.  But why is 5 true?  How are 1-4 supposed to imply 5?

    1.  The tortoise has a head start, so if Achilles hopes to overtake it, he must run at least to the place where the tortoise presently is

    2. but by the time he arrives there, it will have crawled to a new place

    3. so then Achilles must run to this new place,

    4. but the tortoise meanwhile will have crawled on, and so forth.

    5. Achilles will never catch the tortoise, says Zeno.

  4. branemrys4:30 PM

    I'm not sure why they'd have to imply 5; Zeno is certainly using additional assumptions (e.g., about divisibility of space and time). But if we take the 'and so forth' seriously, the tortoise is always ahead of Achilles for any finite number of iterations (every time Achilles reaches where the Tortoise was the Tortoise will have crawled ahead of that point at least a tiny fraction of the distance), and if we aren't talking about finite iterations we have to explain how Achilles catches the Tortoise.

  5. Ocham3:28 AM

    <p><span>>> I'm not sure why they'd have to imply 5</span>
    </p><p><span> </span>
    </p><p><span>I was quoting almost exactly the version of the argument given in the IEP.<span>  </span>On the assumption the argument is valid, the premisses would ‘have to’ imply 5.</span>
    </p><p><span> </span>
    </p><p><span>Clearly there are other assumptions that have to be made.<span>  </span>On your point about the ‘and so on’.<span>  </span>I can’t see how that works.<span>  </span>Suppose Achilles aims at the exact spot Y where he is going to overtake the tortoise.<span>  </span>Clearly when he has reached that spot, he will have reached the tortoise.<span>  </span>If he reaches any spot X before that, he will have not reached the tortoise.<span>  </span>So all the ‘argument’ appears to be saying, it seems, is that if we take any point X before Y, then there is some distance to go.<span>  </span>And if we take any spot X’ between Y and X, there is still some distance to be ‘and so on’.<span>  </span>But this ‘and so on’ doesn’t prove anything.<span>  </span>It proves simply (or rather, it assumes that) we can take any distance whatever, and cut it somewhere.<span>  </span>How does that prove ‘Achilles will never reach the tortoise’?</span>
    </p><p><span> </span>
    </p><p><span>Priest (in the article you cite) , has a slightly different version of the argument.<span>  </span>He says (I paraphrase)<span>  </span>In order to get from a to b, you must first get to each point between a and b. But there are infinitely many points between a and b.<span>  </span>Hidden premiss: to get to something = to do something.<span>  </span>Therefore to get from a to b in a finite time, you must do infinitely many things.<span>  </span>But you can’t do infinitely many things in a finite time.<span>  </span>Therefore etc.<span>  </span>But there is much to challenge there.<span>  </span>Is the hidden premiss correct?<span>  </span>Is ‘getting to’ a point the same as ‘doing something’?<span>  </span>Can we actually ‘get to’ a mathematical point?<span>  </span>How? We can cross such a point, of course.<span>  </span>But then the argument amounts to this: there are infinitely many collections of finite distances between a and b, and we can traverse any such collection in a finite time. Indeed, clearly we can, for the total length of any such collection will be the length between a and b.</span>
    </p><p><span> </span>
    <span>Aristotle mentions the argument several times in the Physics, arguing that we must [...]

  6. branemrys7:18 AM

    I'm not understanding your argument. Yes, everyone knows that if Achilles aims to overtake the tortoise he will. That's part of the point. Likewise, it is precisely part of the point of the argument that "<span><span>if we take any point X before Y, then there is some distance to go"; </span></span>thus the conclusion that every time Achilles reaches where the tortoise was, the tortoise has moved beyond that point and therefore Achilles always has some distance to go.

    You seem to be operating on the assumption that a paradox requires a real inconsistency; but as far as I can see this is manifestly not true -- most paradoxes consist of apparent, not real, inconsistencies. Likewise, the IEP article did not make the claim that the argument precisely as given there was logically valid, and indeed makes the point that it's difficult to know precisely how the steps of any of Zeno's arguments went (and that the version we usually use is actually a commentator's adaptation of Aristotle's version of Zeno's argument); so I don't know why we need to have the assumption that the argument should be valid precisely as stated.

    But I must be missing something in your argument.

  7. <p><span><span>>>I'm not understanding your argument. Yes, everyone knows that if Achilles aims to overtake the tortoise he will. That's part of the point. Likewise, it is precisely part of the point of the argument that "if we take any point X before Y, then there is some distance to go"; thus the conclusion that every time Achilles reaches where the tortoise was, the tortoise has moved beyond that point and therefore Achilles always has some distance to go.<span>  </span></span></span>
    </p><p><span><span> </span></span>
    </p><p><span><span>It’s not an argument, it’s a critique of an argument.<span>  </span>The argument deliberately selects a point where Achilles will not have caught up with the tortoise.<span>  </span>So it states a triviality: if you select any point before Achilles reaches the tortoise, then Achilles will not have reached the tortoise.<span>  </span>Your final statement “Achilles always has some distance to go” does not follow.<span>  </span>I.e. the statement ‘at any point before Achilles reaches the tortoise, Achilles will not have reached the tortoise’ does not imply ‘Achilles always has some distance to go [from the tortoise]’.<span>  </span>Clearly the antecedent is true, but the consequent is false, and the antecedent is perfectly consistent with ‘Achilles is no distance from the tortoise’.</span></span>
    </p><p><span><span> </span></span>
    </p><p><span><span>I didn’t follow your next point about ‘apparent inconsistency’.<span>  </span>I’m not seeing any apparent inconsistency at all. Even an apparent consistency needs to be made apparent.</span></span>
    </p><p><span><span> </span></span>
    </p><p><span><span>I comment on a slightly tidier version of the argument here.</span></span>
    </p><p><span><span> </span></span>

  8. branemrys1:14 PM

    Critiques of arguments are arguments, so your comment on that point makes me wonder  if there is some specific thing you are attempting to do that I'm just not getting.

    In one sense the argument deliberately identifies a point at which Achilles will not have caught up with the tortoise, because that is what Zeno is doing; but that's not what the argument is an argument to -- the argument is that every point is such a point, because for any point at which Achilles has caught up to where the tortoise was, there are more points from that to where the tortoise has by that time moved to. But Achilles always has to catch up to where the tortoise was before he can overtake the tortoise. And if at every point Achilles reaches there are points that still have to be traversed to reach the tortoise, Achilles always has some distance to go to catch up with the tortoise.

    Perhaps the problem is that you keep reformulating it in terms that require the assumption that there is a point at which Achilles reaches the tortoise. We all know that there is such a point, for reasons that have nothing whatsoever to do with the argument, but the argument itself does not appeal to such an assumption or presuppose that any such point exists. Perhaps the way to see the apparent inconsistency (which most people see) is to bracket that assumption for the moment, and only then formulate the argument.

  9. Ocham4:38 PM

    My second version above does not require the assumption that Achilles will have reached the tortoise.  I am simply pointing out the triviality of saying that, at any point where Achilles has not reached the Tortoise, (and identifyintg a point where the tortoise *was*, but is no longer, is such a point), is a point where Achilles has not reached the Tortoise.

    >>your comment on that point makes me wonder  if there is some specific thing you are attempting to do that I'm just not getting.

    I am simply stating that the inconsistency is not apparent to me. 

    >>And if at every point Achilles reaches there are points that still have to be traversed to reach the tortoise, Achilles always has some distance to go to catch up with the tortoise.  

    So the argument is now:

    (1) If at every point Achilles reaches there are points that still have to be traversed to reach the tortoise, Achilles always has some distance to go to catch up with the tortoise

    (2) At every point Achilles reaches there are points that still have to be traversed to reach the tortoise

    (3) Therefore Achilles always has some distance to go to catch up with the tortoise (i.e. will never catch the tortoise).

    Justification for premiss (2) please.

  10. Ocham3:03 AM

    PS thinking about your point on my 'assumption that there is a point at which Achilles reaches the tortoise'.  I'm not assuming this.  I'm simply saying that this assumption is not in any way inconsistent with the premisses of Zeno's argument as stated in the IEP.  That's all I'm saying.  A time-honoured way of questioning the validity of an argument is to point out that the premisses can be true and the conclusion false.  (Where conclusion = 'Achilles won't reach the tortoise').

    I think you agree with this (since you agreed earlier that the argument as stated doesn't logically imply the conclusion).   But you also mention the idea of an 'apparent inconsistency' is an intriguing one.  We can turn this into the idea of 'apparent validity' - if p is apparently inconsistent with q, that suggests the argument 'p, therefore not-q' is 'apparently valid'.  And I tend to agree with you that the Zeno argument does have this quality.  It always seems to work 'at first sight'.  But then as soon as you write it down and make the premiss and conclusion transparent, it fades into thin air.  Why is that?  Is there some hidden premiss that we failed to supply when we detailed the argument?

    More generally, isn't this true of many philosophical arguments?  As soon as you analyse them the initial puzzle seems to vaporise.  Yet at the same time, you feel the analysis has missed something out. A bit like when you look closely at the brush-strokes of the painting to see how the artist achieved a particular effect, but as soon as you do so, the effect vanishes, and all you see is brush-strokes. 

    Just some random thoughts.

  11. branemrys8:39 AM

    Yes, but your 'triviality' ends up just being the paradox if there is no point at which Achilles catches the tortoise. So calling it a triviality and criticizing the argument for it just seems odd.

    Likewise, (1) - (3) aren't a restatement of the paradox, as you seem to be suggesting with the phrase "the argument is now"; they're a description of why the conclusion of the paradox is different from your triviality. You had said that your criticism was that the argument states a triviality. You can't mean that it has a triviality as a premise, because that's not a criticism of an argument; every kind of argument can be formulated with trivialities as premises -- that's what a triviality logically is, something that can be introduced at any stage of an argument and used as a premise at any point . So the only way saying that the paradox states a triviality is a criticism of the argument is if the triviality is supposed to be the conclusion. (1)-(3) were my explanation of why the triviality is not the conclusion of the argument.

  12. branemrys8:58 AM

    I agree that 'apparent validity' is an important concept (it's essentially what's involved in medieval discussions of sophistics, or at least a good portion of it), but I don't think it has any straightforward link with apparent inconsistency. For one thing, the latter is a much broader concept, ideas we have of one thing can be apparently inconsistent or apparently consistent, but that has no connection to validity. Even in the case of arguments I don't think it works the way you are suggesting: if p is apparently inconsistent with q, it doesn't follow that 'p therefore not-q' is apparently valid, because there might be other things that make 'p therefore not-q' obviously invalid. I'm not sure that the paradoxicality of the Zeno paradox has any connection to apparent validity: for one thing, people who have no idea what validity is can still regard the paradox as paradoxical, which shouldn't be the case if its paradoxicality required assessment of apparent validity.

    I do think that what you say is true of many philosophical arguments, although it's also true that sometimes it's an artifact of defects in the analysis rather than anything to do with the argument itself  -- logical positivists slapping 'pseudo-problem' on things, for instance, rather than anything genuinely illuminating.

  13. Ocham3:40 AM

    Add appreciate some thoughts on the argument given here - it seems to me the argument is so obvious, or superficially so, that there must be a literature on it. But I'm not aware of any.  Interested in your thoughts.

  14. Ocham3:41 AM

    Sorry, wrong link.  The argument is here (Cantor's Angel).


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