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?

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.

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.

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 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 t_0.5. And, between t₀ and t_0.5, a further moment, t_0.25. And then also t_0.125, t_0.0625, t_0.03125, 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 t_0.5, getting a quarter of the way by t_0.25, 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 Zeno, 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.

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.

Right, so it seems you think the argument is self-undermining. It assumes that you can get to the midpoint, C, and then it goes on to prove that motion from C to the endpoint B is impossible. Maybe we need to rethink our assumption that we could get to C! And indeed, other versions of this paradox of Zeno's work in that way. In order to get from A to B, this version runs, we need to get to the midpoint C. But in order to get from A to C, we need to that interval's midpoint, C1. And in order to get from A to C1, we need to get to its midpoint C2, ad infinitum.

The strategy is always the same: to find a way of taking something finite (in this case, the racetrack) and dividing it into infinitely many parts; then arguing that a related task (here, running to the finish line) that looked to be finite really involves an actual infinite number of subtasks (here, reaching all the midpoints); and then concluding that, because one cannot complete an infinite number of tasks, the original task is impossible.

All these steps have captured the imagination -- of mathematicians, philosophers, poets. Blake wrote about the first:

To see a World in a Grain of Sand
And a Heaven in a Wild Flower,
Hold Infinity in the palm of your hand
And Eternity in an hour.

The in/at variation is a convention of the English language and has no equivalent in many other languages. It seems to mark no significant underlying distinction, and your question is then one about proper English.

Understood in this spirit, I would say that you are both right. With some places we use "at", with others "in". Consider two buildings, for example, my school and my house. One could say that I am in the first building or at school. And one could say that I am in the second building, or in my house, or in my home, or at home, or at my place.

I assume a grammarian could give you a general rule about when we use "in" and when "at". But, as my example shows, this rule cannot draw on the type of location alone.