Is it possible for a physical object to be four-dimensional? This depends a bit on what you mean by a "physical object". But it seems plausible to say that the could be a four-dimensional world with four-dimensional objects in it, and why should we not in talking about such a possible world call such objects physical objects? I would address your question in the same way. There could be a one-dimensional world, and there could be objects in it. Again, it seems plausible to say of these one-dimensional objects that they are the physical objects of that one-dimensional world. But then again, if someone where to feel very strongly that the expression "physical object" should be reserved for three-dimensional objects, I for one would not want to get into a long argument over this point.

Now another question you may have in mind is whether there can be one-dimensional objects in our three-dimensional space. We can certainly describe such objects geometrically, so they are possible in this sense (and, again, the question whether we should call such an object a physical object seems to me to be of little interest). Whether the existence of such objects is consistent with the natural laws of our universe and, if so, whether we could detect them, are different questions which, I think, you are not asking.

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).

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.

It depends on what one means by the 'thing' in question. According to the principles of geometry, a line is infinitely divisible, although of course it has basic components, points. But what of ordinary, middle-sized objects, such as tables and chairs? Certain early modern philosophers, such as Leibniz, believed that material things were not only infinitely divisible, but that they were also actually divided. (Hence, Leibniz concluded, material things weren't real things, because they lacked what he called a 'true principle of unity', and the only 'real' things were souls, which, being immaterial, could not be infinitely divided.) In a recent article, "Van Inwagen and the Possibility of Gunk," the metaphysician Ted Sider argues that it is possible that objects in the world are infinitely divisible and even lack basic constituents, like the points of a line. According to Sider, it is logically possible that the world consists only of 'atomless gunk', that is, "that it divides further into smaller and smaller parts," and, unlike a line, "does not even have atomic parts..._all_ parts of such an object have proper parts. Now one might think that current physical theory is incompatible with the possibility of what Sider calls 'a gunk world'. Sider anticipates this objection: "Scientists discovered that hydrogen ‘atoms’ have proper parts. Then they discovered that protons have proper parts. At one point, at least, it was a legitimate scientific hypothesis that this process could go on forever, that there is no end to the world’s complexity. A metaphysical theory should not have the consequence that a legitimate scientific hypothesis is metaphysically impossible. So we ought to accept the possibility of material objects made of gunk." I agree with Sider that "a metaphysical theory should not have the consequence that a legitimate scientific hypothesis is metaphysically impossible"; however, it's not clear to me that it is indeed a live scientific hypothesis at this point that everything is subdivisible. Since I, at least am inclined to take current physical theory as the best guide to understanding the nature of reality--although, of course, this is a controversial assumption--if current scientific theory does not allow for the actual divisibility of things into an infinite number of parts, then I think that the answer to your question is 'no'. (Of course, to be sure, even if current scientific theory does not allow for a 'gunk world', future discoveries might reveal that it is indeed possible. In that case, I would of course revise my answer to your question.)

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.

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.

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.

You make a very interesting point. If time travel takes a second, then since a later Earth - say Earth in a year - might be zillions of miles away (i.e. more than 186,000), I must travel faster than light, which is impossible. But how long does my time travel take? How do we know that it takes a second? After all, if on the new or later Earth it is a year later, presumably it took me a year to get "there", the same amount of time as it took the Earth itself, even if it felt instantaneous. So there is a difficulty about the meaning of "How long does my time travel take?" If we move in time, or times moves past us, there is a difficulty about the concept of the speed of the movement. Movement in space is distance divided by time, so movement in time, or the movement of time, it seems, is time divided by time; and it is hard (as D.C. Williams pointed out ages ago in "The Myth of Passage") to attach any sense to this idea. This is the interest of your point for me; how do we attach sense to the speed of time travel? In what reference frame does it take place?