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What do Zeno's paradoxes tell us about motion?


To: Pat F.
From: Geoffrey Klempner
Subject: What do Zeno's paradoxes tell us about motion?
Date: 25 February 2005 11:02

Dear Pat,

Thank you for your email of 15 February, with your University of London essay in response to the question, 'What, if anything do Zeno's paradoxes tell us about motion?'

There are good things about this essay which would impress an examiner. You show that you are aware of the difficulty of resolving Zeno's paradoxes of motion, and you demonstrate that you knowledgeable about contemporary physics and maths and the light that these cast on Zeno.

However, there is a lack of clarity in the argument, in particular, concerning the relation between the logical analysis of motion, and empirical discoveries which suggest that motion in the actual world is different from what it might have been in some other possible worlds.

Let me first deal with the point about Aristotle. Here is a quote from unit 9 of The First Philosophers (attached):

172. If Zeno's argument is not meant to be valid, where is the fallacy? As Aristotle comments there is no special difficulty with the idea of covering an infinitely divisible distance in a finite time, since the time itself may be thought of as infinitely divisible. The question is rather how we are to conceive of the infinite number of separate steps that need to be taken in order for Achilles to catch the tortoise; or, more generally, as Zeno shows in the paradox of The Stadium, the infinite number of steps that need to be taken if any object is to move any distance. Aristotle thought it sufficient to distinguish between 'actual' and 'potential' infinity: it is impossible for an infinite number of separate steps to actually take place in a finite time, but we may legitimately talk of a 'potentially' infinite number of steps; steps which do not actually exist as such, but still each have the 'potential' to exist if one were to direct one's attention to that particular step. Thus, the infinite series of ever tinier distances that Achilles has to cover in order to catch the tortoise exists only as the result of our being able to continue mentally dividing the total distance he has to run into ever tinier parts; the members of that series are not separate steps he has to take, one after the other.

172. Now is this an adequate response to the paradox? Let us agree that Achilles does not have to perform an infinite number of physical actions: all he does is take three strides, and he's there. The question is what would count, in principle, as a complete description of everything that actually happens in the process of his catching the tortoise. One might argue that the reality of the actual situation does go beyond any finite description that anyone could give of it. An infinite number of events do actually take place, an uninterrupted sequence of happenings corresponding to the infinite number of ever tinier distances that Achilles has to cover. If that is correct, then Aristotle's explanation is not adequate as it stands. Achilles does take an infinite number of distinct steps, in the sense that there actually occur an infinite number of events of his moving just that bit closer to the tortoise; even though we could never be aware of them as such. We must then say that the number of steps that can take place in a finite time is actually infinite, in Aristotle's sense. But it is one thing to know what one 'has to say', and quite another to know what one would mean in saying it. As in the case of Zeno's argument against the existence of a plurality of objects, the initial reaction that 'there must be a fallacy' gives way to the realization that a lot of philosophical work needs to be done in order to meet fully the challenge of the paradox.

If I was writing an answer to this question, the first thing I would do is attempt state what we perceive or think motion to be. The upshot of your essay is that Zeno has successfully shown that motion is different from what we thought; but this conclusion would be clearer if the reader was in a position to compare the two conceptions, the pre-reflective concept of motion and the reflective.

What exactly is the relevance of the quantization of time and space? In his paradoxes concerning divisibility, Zeno shows that he was aware of the alternatives, and designed his argument to cover both possibilities: that physical reality is ultimately continuous and that physical reality is ultimately discrete.

Take the paradox of the arrow. I couldn't quite make this out, but you say, 'This is a paradox and it in fact holds [in the sense that the] because it is true of the universe that the universe is inherently paradoxical because of its quantum nature.' Well, how exactly? If time and space are discrete then there is no such thing as 'motion' as pre-reflectively understood, full stop. What we pre-reflectively term 'motion' is merely a sequence of static states, like the frames of a movie which follow on from one another so rapidly that we are not aware of the discontinuity. How clever of Zeno, someone might think, to have anticipated quantum theory! But there is no necessity that things should have been this way. The conclusion to draw for a possible universe where time and space are continuous is that motion is more than just the occupation of different positions at different times. An object X could occupy different positions at different times and not be 'moving'; for example, if it occupied point A for an instant, then point B and so on. (Imagine a Star Trek set up, but where the arrow disappears and reappears continuously.)

I don't know what to make of the mathematical paradox which you cite (actually this is new to me). However, your statement, 'This is a fundamental paradox within mathematics as Zeno's paradox of the arrow etc. are fundamental philosophical paradoxes' would not, in my view, convince an examiner that this example was relevant to the discussion of Zeno. You can hardly claim that Zeno anticipated this mathematical paradox.

Again, I don't quite see from what you say in your essay how you are justified in drawing the conclusion, 'To embrace Zeno is not to say that there is no change but rather to say that we cannot fully express change as an objective quality that is separate and distinct from the way that we choose to measure it.'

If this statement means anything to me, it suggests what I said earlier, that there are 'pre-reflective' and 'reflective' concepts of motion and change. As in other areas where precise analysis and formalization replace pre-reflective notions, it is possible that we may be faced with a choice of alternative formalizations, alternative 're-definitions' of motion. In that sense, I can understand that there might be room for a 'choice'.

All the best,