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 to be 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.

A number of paradoxes have been attributed to Zeno. One of them is the Paradox of the Runner: in order for a runner to get to the finish line, she needs to cross the first half of the track. Once she's done that, she needs to cross half the distance from the halfway mark to the finish line. Once she's done that, she needs to cross half the distance from that point to the finish line; etc. It seems that there are infinitely many finite intervals that she needs to traverse before she makes it to the finish line. But it's impossible to accomplish in a finite amount of time infinitely many tasks, each of which takes a finite amount of time. Therefore, the racer cannot make it to the finish line.

It's common to hear that the solution is to appreciate that the sum of infinitely many finite quantities can be finite. Mathematicians have taught us, we're told, that the infinite sum:

1/2 + 1/4 + 1/8 + 1/16 + ...

actually sums to 1. So, if we view the racer as traversing the first half of the racetrack in half a minute, the next quarter of it in 15 seconds, etc., then we can see that she'll reach the finish line in exactly one minute. This result actually requires the subtleties of the calculus, a branch of mathematics that was placed on a firm footing only in the 19th Century. So it's no shame on Zeno if he didn't appreciate this solution.

Is that the end of the matter? Perhaps it is unless there remain disputes about the mathematical result. How could there be such disputes in mathematics!? Does anyone actually think that this infinite sum doesn't sum to 1? No. It's rather that not all mathematicians and philosophers would agree on how to understand the claim that this infinite sum sums to 1. When the claim is spelled out, it involves quantification over infinite totalities. And there has been substantial and difficult disagreement about how precisely to understand such quantification. For a bit more discussion, see Question 139.

With regard to both questions, I understand you as imagining that objects are longer or shorter in all dimensions (not merely in one dimension). So spheres would still be spheres, except larger or smaller ones. Right?

On your first question, this is not possible if we hold fixed the laws of nature holding in this universe. To illustrate: In your Twin Solar System, scaled up by a factor of 2, Twin Earth would have eight times as much mass, and gravity near its surface would be roughly twice as great (surface gravity is proportional to the planet's mass divided by the square of it radius). Like the Earth, a scaled-up object would have eight times as much mass, so the gravitational force acting on it (its weight) would be 16 times greater. Now imagine this object suspended by a string. This string would be thicker in two dimensions, hence four times stronger. But the object's weight would be 16 times greater! So, on Earth, the string may be sufficiently strong to support the object even while on Twin Earth the counterpart string would not be strong enough to support the counterpart object. Examples could be multiplied. A Twin-Earth parachutist would also have 16 times as much weight as her counterpart on Earth while her parachute's surface would be only four times as large. Likewise for scaled-up planes, where the wing surface area (creating lift) would not keep up with the increased weight; scaled-up replicas of planes we use here would not be able to fly on Twin Earth. (The examples ignore more subtle differences: Because Twin Earth has higher surface gravity, it would have higher atmospheric pressure and density than our Earth, and it would also be a bit more compressed which would lead to a further increase in its surface gravity.) You see the general point: In many ways, other things would not be the same in a scaled-up (or scaled-down) solar system, because some of its parameters (e.g., forces acting) would vary with the scaling factor, others with the square of the scaling factor, and so on. Interestingly, however, one thing could be the same: Twin Earth could be circling Twin Sun in a stable orbit once a year. Gravitational and centrifugal forces acting on Twin Earth would balance out, both being 16 times what they are in our solar system.

On your second question, yes, I think this would be meaningless for lack of a common benchmark in terms of which lengths could be measured in both universes.

The thesis that time is more fundamental than space is not uncommon among philosophers -- although the significance attached to this, and the meaning of 'fundamental' varies widely. At least arguably, Aristotle, Leibniz, Kant and Heidegger, are committed to some variety of this claim.

Kant's argument has some similarities to yours. All propositions about things and events must, when fully analysed, include a subordinate proposition about time (if only the location in time of the act of thought itself). But not all propositions about things and events must include a subordinate proposition about space. Kant then uses this analysis to argue further that the basic categories of all thought must be understood to be rules for the determination of time relations.

Our experience of objects (including ourselves) in space and time seems vital to our human existence, and I'm not sure what it would mean to say that either spatiality or temporality is more important than the other.

Since thinking about events that may have happened in the past or events is not literally time travel, so spatiality seems to "beat" temporality with respect to ease of travel, which your question refers to. The difficulty of self-directed travel through time doesn't mean that temporality is unimportant, however.

One can think about different times, but one can also think about different places; and although one can not choose when to be, one can choose where to go.

Space is not expanding "in" anything else. The distances between points in space are increasing, but not because they are moving through some "superspace" that contains space.

Mathematicians distinguish between two different approaches to defining geometric properties of a space: the extrinsic approach and the intrinsic approach. The extrinsic approach involves relating the space to some larger space that it sits inside; the intrinsic approach makes use of only the space itself, and not some larger space that it sits inside.

For example, suppose we want to study the curvature of the surface of the earth. One way to see that the surface of the earth is curved is to image a flat plane tangent to the surface of the earth at some point. We can detect and measure the curvature of the surface of the earth by noting that the surface deviates from the tangent plane, and measuring the size of this deviation. But this deviation takes place within the 3-dimensional space that the surface of the earth is embedded in, so this is an extrinsic measure of the curvature. The curvature can also be detected by making measurements that take place entirely on the surface of the earth. For example, if you lay out a large triangle on the surface of the earth and measure the angles of the triangle, you will find that they add up to more than 180 degrees. This measurement makes no reference to a larger space containing the earth's surface, so it is an intrinsic measure of the curvature of the surface.

Cosmologists use only the intrinsic approach when discussing the geometry of spacetime. Thus, none of this discussion involves any reference to a larger space that spacetime sits inside. Although they may use words that seem to suggest such a larger space, such as "expansion" or "curvature", those words are always being used to refer to some intrinsic property of spacetime itself, and not some relationship between spacetime and a larger space.