## I was thinking about Zeno's paradox of motion today and decided on an

I was thinking about Zeno's paradox of motion today and decided on an explanation that I'd like to check. As I've heard the paradox stated, one premise is that in order to get from A to B you have to first get to the midway point, call it C. Then there are other premises resulting in the conclusion that motion is impossible. But doesn't the above premise already allow for the possibility of motion, making you agree that motion to C is possible before going on to claim that motion to B is not? Perhaps there is another way to state the paradox, then? Thanks much.

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.

## A friend and I were discussing our philosophy class a while ago, and somehow we

A friend and I were discussing our philosophy class a while ago, and somehow we got onto the subject of the properties of things and the definition of a place. We began to argue about whether you can be in an object or in a place. I said that you can only be in an object and to be in a place is impossible. But you can be at a place. Example: you are in the building, but you are at the DMV. She said the opposite. That it is possible to be in a place. Who is correct?

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.

## How is Zeno's paradox solved? Thanks, andrea

How is Zeno's paradox solved? Thanks, andrea

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.

## Is the physical world proportional? What I mean is: is it possible, for instance

Is the physical world proportional? What I mean is: is it possible, for instance, that we find a solar system exactly like ours except for the fact that every object (planets, stones, animals, trees, etc.) is one thousand times longer or less long? What if only twice longer? And what about a different universe where even atoms (and elementary particles, if they have any length at all) were one thousand times "longer"? Is this meaningless?

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.

## Is it sensible to think that time is more fundamental than space, because one

Is it sensible to think that time is more fundamental than space, because one can just close one's eyes and relive memories, going back in time or prospectively go forward in time to predict something, without actually changing your position in space?

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.

## Science states that space is endless, and ever expanding. But, if we are inside

Science states that space is endless, and ever expanding. But, if we are inside the planet earth, the planet earth is inside the galaxy, the galaxy is inside space, then what is space inside? What is it expanding in? And if space is endless, how can it expand?

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.

## Hello,

Hello, I submitted the following question a few days ago, but it has not been posted as far as I can tell. Perhaps the submission did not go through, but it is also possible that it was not posted because someone thought that the question had already been asked. Just in case, I post it again. Please notice that my question is quite different from questions like "Is the universe infinite?" or "Does the universe have an end?". So here it goes: Are there two points in the universe such that, if you take the straight line through these two points and lay out yard sticks along that line to measure the distance between those two points, no finite number of yard sticks is sufficient to do so. In other words, are there infinite distances in the universe? Again, please notice that this is NOT the same question as "Is the universe infinite?" The universe could be infinite without there being an infinite distance between any two points. Many thanks for responding.

I believe that the answer to this question is "No". But it's a question for a physicist, really, not one for a philosopher nor even for a mathematician. One can certainly describe metrics on spaces that behave in the kind of way you suggest. But whether the universe is such a space is an empirical question.

## I understand points as entities with zero extension. (Is this correct?) Yet

I understand points as entities with zero extension. (Is this correct?) Yet infinitely many points are said to compose space. It seems like even infinitely many zeros could never add up to a finite non-zero value. So, what's up with points? If they don't have any extension, what are they? As a follow up, does it make sense to think about points in space in a different way from how we think about points in time?

Yes, a point has length, depth, and height zero. So do two points, three points, and even as many points as there are natural numbers. But if you have as many points as there are real numbers (of which there are more than there are natural numbers), then that set of points may have some positive length, depth, or height, though it may not. (In that case, they will not have zero length, depth, and height but may have no assignable length, depth, or height.) The branch of mathematics in which such things are studied is called "measure theory".

Exactly what a point is is another question. In mathematics, points may be regarded in a wide variety of ways, as is convenient. Are there any points in space itself? That's a disputed question, and an empirical one, not one on which philosophers can pronounce.

## Everything has to take up space, so what is outer space taking up?

Everything has to take up space, so what is outer space taking up?

More space: someone more versed in astronomy can correct me if I'm wrong, but I believe "outer space" just refers to spatially-extended bits of the universe beyond our galaxy. I'm not sure I agree, though, that everything has to take up space. What about numbers or colors or God(s)...?