## Was Zeno unfair toward Achilles in his paradox?

Was Zeno unfair toward Achilles in his paradox? Last week I was reading the Croatian edition of Bryan Magee’s “The Story of Philosophy” and he reminded me of Zeno’s famous “Achilles and the tortoise” paradox. Here is how the paradox goes (taken from Wikipedia): “In the paradox of Achilles and the Tortoise, Achilles is in a footrace with the tortoise. Achilles allows the tortoise a head start of 100 meters. If we suppose that each racer starts running at some constant speed (here instead of ‘one very fast and one very slow’ I would stick to the original: Achilles is twice time faster than the Tortoise), then after some finite time, Achilles will have run 100 meters, bringing him to the tortoise's starting point. During this time, the tortoise has run a much shorter distance, 50 meters. It will then take Achilles some further time to run that distance, by which time the tortoise will have advanced farther; and then more time still to reach this third point, while the tortoise moves ahead. Thus, whenever...

Dear Robert,

You are right. The key to understanding the paradox is that although Achilles must complete an infinite number of tasks in order to catch up to the Tortoise, he can do so in a finite amount of time, since each successive task takes much less time than its predecessor (as you noted). Of course, today we understand how to add an infinite sequence of terms that converge to a finite quantity. But this wasn't well understood until millenia after Zeno -- and the logical foundations for doing so required Cauchy and Weierstrass in the nineteenth century. So we shouldn't be too hard on old Zeno.

By the way, you might find it amusing to consider some more recent Zeno-like puzzles, such as the "New Zeno" discussed by Stephen Yablo in the journal ANALYSIS, vol 60 (April 2000).

## Is it possible to divide something into an infinite amount of parts?

Is it possible to divide something into an infinite amount of parts?

I've nothing against Sean Greenberg's answer, but I figured I'd just add a word or two on a further relevant distinction here. Infinite divisibility is not the same as the possibility of dividing something into infinitely many parts. At least, it doesn't need to be understood in that way. There's a distinction that goes back at least as far as Aristotle, between the actual infinite and potential infinite, and the notion of infinite divisibility can be interpreted in either way. If we interpret infinite divisibility in the sense of the potential infinite (which, for what it's worth, is how Aristotle himself understood -- and endorsed -- the concept), this will mean that, no matter how small something might be, it can still be divided into still smaller parts. You can cut something into two halves, divide each of those to yield four quarters, divide each of these to yield eight eighths, and carry on going without ever needing to stop dividing. Mathematically, there is no greatest power of two: so, no matter how many pieces this process of subdivision has already yielded, there is always the potential for that number to be doubled by further subdivision. But the thing to appreciate is that the number of pieces will always remain finite. Mathematically, there are infinitely many positive integers, but each one of them individually is finite. The whole point about an infinite process is that it can never be completed. The very word suggests this: the prefix 'in' is a negation while the term 'finite' (from the same etymological root as 'finished') indicates a terminus. As Aristotle put it: 'The infinite turns out to be the contrary of what it is said to be. It is not what has nothing outside it that is infinite, but what always has something outside it.' (Physics, 206b34-207a1).

So much for the potential infinite. One might alternatively maintain that it should in principle be possible for an infinite process -- of division, or whatever else it might be -- to be completed. For instance, if one believes that God is actually omnipotent, then He at least ought to be able to divide something infinitely many times. After all, what good is infinite power if it can't be exercised infinitely? Perhaps He makes the second cut half a minute after the first one, makes the third cut a quarter of a minute after that, makes the fourth cut an eighth of a minute later, and so on. Then, after the whole minute has elapsed, His work will be complete. And how many parts will He then have produced? Why, infinitely many of them.

Historically, I think it's fair to say that the notion of infinite divisibility has more frequently been embraced in the potential sense, but there have been a few who've pressed for actual infinite divisibility too. But note that there's a problem arising here, one that doesn't arise under the potential interpretation. I don't say an insurmountable problem, necessarily, but it is one that will certainly need to be addressed. Suppose we do allow a process of infinite division to be actually completed, so as to yield infinitely many parts. Are these parts extended or aren't they? If they have no extension, no size whatsoever, then where did the extension of the original object go? It seems that the bulk of the original object has not merely been dispersed but has actually been annihilated. And that doesn't seem right, that we should be able to annihilate the bulk of something just by moving its parts around. On the other hand, if each of these parts has some extension individually, then it seems that infinitely many of them together must have infinitely much. So, simply by moving the parts of our original object around, we'll have generated an infinite bulk out of a finite one. And that doesn't seem right either. Consequently, many of those who favoured the 'actual' interpretation of the notion of infinite divisibility rejected that notion in favour of a theory of necessarily indivisible atoms. That, I think, is the reason why infinite divisibility has found more support when understood in the 'potential' sense, for there the same problem doesn't arise. Even if there's no limit to the number of parts you can get through division, that number will still be some finite number n, and the size of each one can unproblematically be an nth of the size of the original object.

## I have a series of questions about Time, motion, and space. Or maybe they are

I have a series of questions about Time, motion, and space. Or maybe they are the same question expressed different ways in an attempt at clarity. Is the concept of "Time" possible apart from the concept of "Change"? In what ways are the two concepts different? Is it possible for "Time" to exist apart from "Change"? Can anything truthful be said about "Time" that does not also apply in an identical way to "Change"? Is "Change" possible without "Motion" of some kind? If even at an atomic level? Could time exist if nothing moved? How is the concept of time possible without the concept of motion? How is it in anyway different? How can space be conceived apart from the relation we refer to as 'distance' between two or more objects? If there was only one object in the universe how would space be conceived or possible? The same question applies to motion, how is it conceivable unless there is movement in relation to some other body? I am not a philosopher. I'm a high school drop out and...

You may be a high school drop out, but you have a genius for asking great questions! Let me try to break up the questions a bit. There is a difference between motion and change insofar as motion appears to involve physical objects and events. If there is motion, there is change, but some philosophers have either denied the existence of physical objects or events (some idealists) or they are theists who believe that there was a time when God (an immaterial / non-physical reality) existed and there were no physical objects. These philosphers would allow that change could exist, but without motion. In any case, once you have change, you have time, for change presumably involves there being one time when X occurs and then another time when X is not in the same state. If motion ceased, would time cease? Not necessarily, if there could be a nonphysical reality (God or souls or...) that change. But what if all change ceased? Would time then cease? Well, if by 'all change' we include 'temporal change' then I suppose the answer would have to be 'yes', but let's refine the question. Imagine all physical and non-physical (if there are any) realities ceased to involve or undergo any changing states; imagine everthing (as it were) freezes and there is no change in thinking, feeling, breathing etc. Can we imagine this happening for, say, 10 minutes and then everything starting back up again? Well, no one would know there had been a 10 gap, and indeed the very idea of there being a gap of 10 minutes as opposed to 9 suggests we can make sense of clock time when there are no changes among any clocks anywhere. Even so, I think the thought experiment makes some sense, and insofar as it does, then there is some reason to think that time is more basic than non-temporal changes.

An analogy with space may be useful. One reason for thinking that space is more than the spatial objects that make up the spatial world is as follows: Can you imagine everything spatial doubling in size in an instant? I think one can, though this would be utterly undetected in our experience. People would still be the same heighth, the moon would still be the same distance from earth according to all our systems of measurement. Nonetheless, there could be a fact of the matter that every spatial thing doubled.

Space and time, I suggest may be more fundamental than motion or change. You may need space and time for there to be motion, as well as change.

## It seems obvious that a line of length 4 is longer than a line of length 2; but

It seems obvious that a line of length 4 is longer than a line of length 2; but couldn't we just as easily say that the two lines are equally made up of an infinite number of points?

You are right that the points in a 4 inch line segment can be put into one-to-one correspondence with the points in a 2 inch line segment. Think of a line swinging through both line segments, the way a door swings through a shorter path nearer its hinge and a longer path further from the hinge. The swinging line matches any point in one with a point in the other. Therefore, they have the same number of points--an infinite number. However, that is not a strike against the claim that the line segments have different lengths. The points are dimensionless, and the length of a line segment is not a function of the number of its dimensionless points. So the 4 inch line segment is still twice the length of the other.

## Doesn't time travel involve space travel too? If I travel back in time one year,

Doesn't time travel involve space travel too? If I travel back in time one year, say, in order to be in the same 'place' as I started, I'd need to travel across countless millions of miles of space, since the planet has moved during the last year. Since such instant space travel contradicts Einstein, how come so many philosophers seem to think it's possible? Martin, Wales, UK

Nice conundrum. Here is a stab at it. If, in the example, time travel is traveling back one year of time in an instant of another time dimension--call it metatime--then Einstein has not been contradicted. He is silent about how much space can be covered in an instant of metatime. So time travel, conceived this way could be possible even given our actual laws of nature, if there is metatime. If, however, there is no metatime, then traveling back in time would be a case in which what would normally be a later stage of one's life occurs before what would normally be an earlier stage (see David Lewis, "The Paradoxes of Time Travel"). For this to be possible, the laws of nature would already have to be different than ours in such a way as to also allow that what would normally be the very next stage in ones' life occur far away from the current stage. If it is conceivable that the laws of nature be different than what they actually are then time travel would be conceptually possible. And this is the sense of 'possible' most philosophers appeal to in saying that time travel is possible. In other words, it is conceptually possible that something happen that contradicts Einstein.

## Someone asked [http://www.askphilosophers.org/question/1411], "How do we know

Someone asked [http://www.askphilosophers.org/question/1411], "How do we know our right hand from our left hand when there is literally nothing that can be said about one which cannot be said about the other"? Mark Lange posed this question in turn, "Suppose there were a universe that was utterly empty throughout its history except for a hand (unattached to any body) floating in it. (Pretty gruesome, but let's not think too hard about that!) Would that hand be a right hand or a left hand? Now we cannot appeal to the hand's relations to other things to give it its handedness, since there are no other things." The thumb is on different sides of each hand. Put the palm down and you can tell which hand it is by looking to see if the thumb is on the inside or outside. What am I missing? Gloves come in left and right, you know? You could even tell this in a void.

When you imagine a space that is empty of everything except one hand, you are still imagining the appearance of that hand from a particular point of view (or, perhaps, from several different points of view). That point of view is what tells you it is a right hand versus a left hand, for it is from that point of view that the thumb extends to the left of the the palm rather than to the right.

Some philosophers (e.g. Berkeley) have claimed that imagining anything requires you to imagine the existence of a viewing subject. There is a difference, however, between imagining how a hand looks from a particular point of view and imagining that someone is occupying that point of view. (This distinction is nicely clarified in an important article by Bernard Williams, entitled "Imagination and the Self" .) You can imagine what a particular hand looks like from a particular point of view without imagining that there is anyone occupying that point of view.

If you agree with Berkeley that imagining something in space requires you also to imagine an observer of that thing,then it is not possible to imagine a space that contains a hand and nothing else. At best, you could conceive of a hand in an otherwise empty space -- supposing or stipulating the existence of such a thing without actually visualizing it. To further stipulate that it is a right rather than a left hand would only be meaningful, though, insofar as one supposed that there is a possible point of view onth hand -- a point of view that could be taken if there were also a subject in that space.

## I'm thinking about relative position (left, right, up, down, ahead, behind). My

I'm thinking about relative position (left, right, up, down, ahead, behind). My general question is whether you think that these three oppositions (left/right, up/down, ahead/behind) have the same "status". For instance, for every point moving on a straight line, there is a meaningful and precise difference between ahead and behind, but not necessarily between left and right or up and down. Another example: for any (physical) object on the surface of a planet, the difference between up and down is clear, but not the remaining two oppositions. Another one: if it is settled, in a given 3D situation, what is left and right, then it is also necessarily settled what is ahead and behind, and what is up and down, but this does not (always) work the other ways around. What do you say? And do you think that the opposition between inside and outside has some relation to the other ones?

Offhand, it's not clear why we'd think there's a difference in status among these oppositions. Once we fix a point on a line as the "origin," it's still up to us which direction counts as ahead and behind. What's up where I am on earth is down from the point of view of folks across the center from me. And so on. Space is isotropic; any direction is as good as any other. (And just a side note: if we fix left and right, we haven't fixed up and down. Imagine holding your arms out and rotating 180 degrees around the axis they define. You'd flip up and down, and also ahead and behind.)

Still, there are some interesting points in the neighborhood. In our space, there's such a thing as "handedness": you can't turn a left hand into a right hand by sending it along some path in space. Our space is "orientable." But some possible spaces are non-orientable as the surface of a Möbius strip demonstrates. Likewise, in our space, there's an absolute distinction between inside and out, but that's a fact about our space, as the concept of a Klein bottle illustrates.

## Is Zeno's paradox really refuted by the fact of someone's walking? ("Solvitur

Is Zeno's paradox really refuted by the fact of someone's walking? ("Solvitur ambulando" - L. Carroll)

Zeno most famous paradoxical argument seems to show that Achilles can never overtake the tortoise.

Plainly, the conclusion of Zeno's argument is false: that can be shown by Achilles just walking along, overtaking the tortoise! That's why the argument -- which seems to go from true premisses via plausible reasoning to the palpably false conclusion -- is a paradox.

But of course, just re-iterating that the conclusion is false doesn't solve the paradox, if that means explaining just where Zeno's reasoning goes wrong. It's perhaps not helpful, then, to talk about "refuting" the paradox, for that's ambiguous. It could mean showing the conclusion is false, or it could mean explaining where the bug is in Zeno's reasoning. Doing a bit of walking (Achilles overtaking the tortoise yet again) suffices for the first, but not for the second!

## I cannot understand how things move. Consider the leading point of a pool ball:

I cannot understand how things move. Consider the leading point of a pool ball: for the ball to move, that leading point has to dematerialise from Point A and materialise at Point B. When I attempt to explain this to others, they invariably respond with something along the lines of 'But it just moves a small distance'. This is what causes me a problem because, regardless of the distance moved, small or large, the leading edge of the pool ball must be in one place at one moment, and the next moment, it is in a different place. What else can this be other than dematerialisation / materialisation. Which, as I understand, is not possible. So how do things move?

I shall begin with a 'philosophical' kind of answer, the kind of answer that philosophers ever since Aristotle's time might have given. (Indeed, it is closely related to the answers that Aristotle himself gave to Zeno's paradoxes of motion. Perhaps you're already familiar with those paradoxes: but, if not, then I'd invite you to look them up, for you might enjoy pondering them). I think the flaw in your question lies in that phrase "the next moment". In the case of space, you seem to be treating it as continuous in the sense that, between any two points, no matter how close they might be, there will still be further spatial points between them -- so that to jump straight from one to the other would have to involve some sort of teleportation, bypassing all those intervening points. And yet (as a philosopher might tell you) time itself is equally continuous, and in exactly the same way. At any given moment of time, there is simply no such thing as the next moment. The continuous nature of time means that, between any two moments, let's call them t0 and t1, there must be an intervening moment, call it t0.5. And, between t0 and t0.5, a further moment, t0.25. And then also t0.125, t0.0625, t0.03125, etc., all standing between you and the moment you initially took to be the 'next' one. In a certain sense (and I don't intend this as an account of how motion works physically; just how it could work, logically), the mistake is to try to build up a big motion out of lots of little ones. The big motion ought to be the starting point. (It is said that Diogenes' response, when he heard Zeno spouting off about his 'proof' that motion was impossible, was simply to walk across the room!). Once you have the entire motion, between A and B, only then should you start to break it down and contemplate its component parts: getting half way between them by t0.5, getting a quarter of the way by t0.25, etc. The fact that there is no mathematical end to this process of breaking the motion down -- as opposed to trying to build it up from its 'least' parts -- means that there is no moment at which the object has to cross any real distance at all.

That, as I say, is the kind of answer that a 'philosopher' might give: but, particularly when it comes down to the kinds of topics that are nowadays studied by physicists, we philosophers ought to accept that we can't do everything on our own. (I've mentioned Aristotle already in this reply. Of course, in his day, there was no distinction to be drawn between a philosopher and a physicist -- but that's no longer the case). Now, I am not a physicist, and so here I cannot even pretend to approach the full story. But, for a start, quantum mechanists seem quite comfortable with the notion that an object might indeed just dematerialise from one place and materialise in another. Indeed, according to quantum mechanics, it's not at all clear that an object is ever in any fully determinate place at all. And then the string theorists will go on to tell you that, when you get down to the level of something called the "Planck length" (of the order of 10-35 metres, about a trillion trillion times smaller than something already as tiny as an atom -- a shorter distance than I suspect your friends could ever even have approached imagining!), alongside something called the "Planck time" (of the order of 10-44 seconds -- if anything, even more mind-bogglingly tiny!), then everything to do with space and time starts to go a bit haywire. For a start, there are ten dimensions down there! Now, it's not yet clear where all this cutting-edge physical research is going: but, who knows, maybe space and time will turn out not to be quite as continuous as Aristotle suggested after all. Although space and time certainly do still remain fascinating topics for philosophers, and philosophers surely do still have something to offer in this area, Einstein and his ilk taught us that we're not really competent to lay down the law about them on the basis of pure a priori speculation alone.

But, rather like Diogenes, I'm tempted just to get up and walk across the room. No one seriously believes that motion doesn't exist: the philosophers will explain how it's possible that there should be such a thing at all, and the physicists will endeavour to find the laws of nature that explain how it actually works in the real world.

## Space and time are measured in hours and metres, value is measured in utility.

Space and time are measured in hours and metres, value is measured in utility. In these three fundamental scales, I have read that zero and the unit are arbitrary. I can see that there is no beginning of time, and no bottom to the universe and no absolutely valueless state of affairs, but it seems perfectly sensible to talk of two states of affairs being of equal value, in which case the difference in value would be zero. Two durations could be of equal length, as could two bodies. So is there a non-arbitrary zero in space, time and value that corresponds to the difference in length, duration or utility between the equally long, enduring or valuable?

It may be that there are two questions hidden here. You're right: if we can compare things in terms of length or duration or utility, then we'll sometimes be able to say that they're the same on this scale -- that if we subtract one value from the other, we get zero. But there's another question: is there such a thing as a thing's having zero length, taking zero time or possessing zero utility?

Length and duration are not quite the same sorts of scales as utility. Length and duration are ratio scales. It makes sense to say that this stick of wood is twice as long as that one. Turns out that this goes with the fact that there is such a thing as having no length or lasting for no time. In these cases, we have a natural zero. However, it may not make sense to say that one thing has twice as much utility as another. Utility scales are interval scales. All that matters are the ratios of the differences.

Let's make this a bit more concrete. I might rate the utility of a cup of coffee at 1, the utility of a cup of tea at 3 and the utility of a glass of beer at 6. That makes it look as though the utility of a cup of tea is three times the utility of a cup of coffee, and that the utility of a glass of beer is twice that of a cup of tea. But for purposes of decision theory, what matters is that the difference between the utility of the tea and the coffee is two-thirds of the difference between beer and tea. As far as decision theory is concerned, we preserve all the relevant information if we re-write the utilities this way:

coffee: 5; tea: 9; beer: 15

Notice that the utility of tea no longer appears to be three times the utility of coffee. Likewise, the utility of beer no longer appears to be twice the utility of tea. But the difference between 9 and 5 -- i.e, 4 -- is 2/3 of the difference between 15 and 9 -- i.e., 6.

For that matter, we could even represent the same utilities as

coffee: 0; tea: 2; beer: 5

or even as

coffee: -20; tea: -14; beer: -5

When we start mixing our utilities and our probabilities together in the way that decision theory says we should if we want to figure out what to do, all that matters are the ratios of the intervals.

It could still be that there's a natural zero point for utilities -- a sort of neutral point, as it were. But decision theory can get along without assuming that.

So yes: if we can say that two things are equal on some scale, that automatically means that we can say that the difference between them on that scale is zero. But whether the scale has a natural zero point, as in "having zero length" or "having zero utility" is another question.