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What makes Xeno's paradox paradoxical?

What makes Xeno's paradox paradoxical? It sounds more like a trick question than a bona fide paradox. Achilles and the tortoise are going to have a half-mile race, and Achilles gives the tortoise a 1/4 mile head start. Suppose Achilles runs as fast as a decent male high school track athlete, and he can cover 1/2 mile in 2-1/2 minutes. He gives the tortoise a head start of 1/4 mile. According to a quick internet search, the average turtle moves at 3 to 4 mph. Let's say our tortoise is particularly fast, and moves at 5 mph. It thereby takes the tortoise 3 minutes to cover 1/4 mile. Achilles finishes 30 seconds ahead of the tortoise. Where's the paradox?

The reasoning you gave illustrates why Zeno's example has a chance of counting as a paradox at all. As you show, of course Achilles will overtake the tortoise. But Zeno claimed to have equally good reasoning showing that Achilles never overtakes the tortoise. That's the paradox: apparently good reasoning in favor of each of two incompatible claims.

For Zeno's reasoning and a critique thereof, see sections 3.1 and 3.2 of this SEP entry.

I am a bit bewildered when I try to think about empty space. Does it make sense

I am a bit bewildered when I try to think about empty space. Does it make sense to think about space insofar as it is space? What sort of existence, if any, does it have? Is it nothing? Thank you!

There are two major views about space, and they give different answers to your question.

One view is "substantivalism." On this view, space really is a thing of a certain sort—a substance. Space would exist even if nothing else did. Needless to say, space it not like things as we usually think of them, but it has its own sort of reality. For Newton space was, among other things, a system of absolute positions. Newton believed that there was an absolute distinction between rest and motion, and that called for a corresponding system of positions. However, the points of space were otherwise indistinguishable; one point was intrinsically like any other.

In contemporary physics space and time are deeply intertwined, and we talk about space-time. Space time in general relativity is mathematically like a field (think of the electromagnetic field), and unlike Newtonian space, the points of space-time aren't all alike. This goes with the idea that space-time itself is curved. Roughly, the curvature at two different points of space-time (represented by a mathematical object called a tensor) can be different.

The alternative to the substantival view of space (or space-time) is relationalism. According to relationalists, space(-time) doesn't have any independent existence. To talk about space is really to talk about relationships among physical objects. For example: the fact that one object "takes up more space" than another" might simply amount to the fact that the second object would fit inside the first. The fact that A is further from B than from C might mean that if we had a large collection of rods whose ends would meet up if we brought them together (intuitively, are of the same length), we would have to lay more of those rods end-to-end to reach from A to B than to reach from A to C. More complicated variations on this idea lead to a relational account of the space-time of general relativity. On the relational view, there is no such thing as space or space-time, but there are spatial/geometric facts about the physical things that make up the world.

I know that there have been numerous contributions in philosophy discussing the

I know that there have been numerous contributions in philosophy discussing the divisibility of matter, e.g. Zeno's paradoxes. Are there contemporary debates regarding this topic still? Do you think it's plausible that matter can be divided infinitely? When we hear of experiments in modern physics where particles are collided and break into smaller pieces, does this constitute a division of matter? I understand I've asked a lot here. I hope the questions are related to each other enough that they can be addressed in a single response. Thank you!

Are there contemporary debates regarding this topic still?

To judge from the SEP article on mereology, the infinite divisibility of matter is indeed a topic of contemporary debate. See especially section 3.4 here:

Do you think it's plausible that matter can be divided infinitely?

I'd distinguish between (1) the claim that every bit of matter is composed of smaller bits of matter and (2) the claim that, as a matter of physical law, those smaller bits of matter can always be pulled apart. (1) is a logically weaker claim than (2), so (1) can be plausible even if (2) isn't. I myself find (1) to be plausible. I take no stand on (2).

When we hear of experiments in modern physics where particles are collided and break into smaller pieces, does this constitute a division of matter?

Yes. Or at least I can't see why it wouldn't.

Have Zeno's paradoxes of motion actually been satisfactorily solved? Physicists

Have Zeno's paradoxes of motion actually been satisfactorily solved? Physicists and mathematicians I've read on the matter seem to regard them as no longer important, but never explain convincingly (for my money) why they're not still important. Have philosophers said anything interesting about them recently? Could you either succinctly explain how they've been solved or point me in the direction of good recent discussions?

I recommend starting with the SEP entry on the topic, available here.

There's an article not cited by the entry that may be relevant because it takes a skeptical view of the standardly accepted solution to one of the paradoxes: "Zeno's Metrical Paradox Revisited," by David M. Sherry, Philosophy of Science 55 (1988), 58-73. Sherry argues that the standardly accepted solution "defuses" the paradox but is too ad hoc to count as a "refutation" of Zeno's reasoning.

I am looking for resources on a seemingly simple issue. I believe the seeming

I am looking for resources on a seemingly simple issue. I believe the seeming simplicity of this issue is quite deceptive: What is a "surface?" What allows anything to "touch?" Where does philosophy stand on this issue? Thank you for your time.

You should consult:

Stroll, A., 1979, ‘Two Concepts of Surfaces’, Midwest Studies in Philosophy 4: 277-291.

Stroll, A., 1988, Surfaces, Minneapolis: University of Minnesota Press.

The two concepts of a surface are the physical one, in which a surface can be pockmarked or scored, and the geometrical one, in which it is an ideal or geometrical object.

Can space be cognized by only verbal means or does it require experience to be

Can space be cognized by only verbal means or does it require experience to be understood? Let me show you what I am getting at. You could never imagine what the color red is from a description of it and I think most people see that as an intrinsic limitation on language. No matter how sophisticated the person listening/describing or how sophisticated the language used you would never know what red is without an experience of it. Is space equally ineffable when it comes to descriptions of it?

Imagine there's a pure, disembodied intellect, and you somehow have the ability to communicate with it. It's a very clever intellect, so it's perfectly receptive to abstract, a priori mathematics: but it has never had any experience of spatial things, and it wants you to explain space to it. How might you go about this?

Well, first you might explain the number line. You invite it to consider an infinite set of objects (we'll call them 'real numbers'), all different from one another, but continuously ordered in two directions from a particular element that we'll call 'zero' (or 'the origin'): ever greater to a positive infinity, and ever less to a negative infinity. And now, with the number line in place, you invite the intellect to take three such lines. That is to say, you invite it to consider an infinity of ordered triples of the form <x, y, z>, where x, y and z are all real numbers from this same set, but are capable of varying independently of one another. Let's call each of these triples a 'point', and call its three individual elements 'Cartesian coordinates'. And then you add some more specific definitions, framed in these terms. For instance, you define a 'sphere' as a set of points such that x2+y2+z2=r2. (Where r is yet another number: we can call that one the 'radius'). And then, once you feel that you've got enough definitions in place, for different kinds of 'shapes' (for that's what we'll call such sets of points), you proceed to offer the intellect a bunch of axioms, still framed in these same terms. Let's opt for Euclid's axioms. (At this point, perhaps the mind asks you: "So am I to take it that these axioms define the only way that space could be?" You might reply: "No, other systems do exist, the so-called 'non-Euclidean' geometries: but, for present purposes, let's just keep things nice and simple"). And then, once the intellect has had a while to mull over all this, and to digest all the mathematical consequences of the axioms you've given it, you can proudly declare that you have succeeded in teaching it geometry. It now knows all that there is to know about shapes and the relations between them.

Once you're satisfied that it's comfortable with that, you then carry on. You proceed to explain that certain shapes instantiate things that we call 'sensible qualities'. The intellect is puzzled: "What are those like?" But here, you concede, your powers of communication do unfortunately run out. You might say: "There's one that we call 'red', but I'm afraid I can't convey to you what red is actually like, because you could never imagine what it is from a description of it. Sensual experience is necessary, before you can get the idea, and that's precisely what you're lacking." But at least you can tell the intellect a few things about these qualities. For a start, there are lots of them. And they're all different... only some are more different than others. They're connected amongst themselves by higher-order relations, and in particular a relation that we call 'resemblance'. The quality we call 'red', for instance, resembles the quality we call 'orange' more than it resembles the one we call 'green', and much more than it resembles the one we call 'rough'. But the one we call 'rough' kind of resembles the one we call 'prickly'; more so, at any rate, than it resembles the one we call 'smooth', and a lot more than it resembles 'quiet'. And then some pairs among these qualities can combine together, while others can't. For instance, the qualities 'red' and 'smooth' can both be spread over the same set of points, while the qualities 'red' and 'green' cannot be. And so on.

At least, the same shape can't be red and green at the same time: but then, we haven't introduced time into the story at all yet. Okay then, let's do that. We'll just add a fourth coordinate, so that our points will now be of the form <x, y, z, t>. And then let's add some more axioms. Perhaps we might make them Newton's laws. Or just pick your own favourite system of mechanics: you can make it Einstein's if you prefer. For, where geometry deals with shapes and their abstract spatial relations at an instant, what mechanics tells you is how one configuration of shapes will evolve into another through time. So we'll plug in some mechanical laws and... we now seem to have a world.

And we've managed to convey all of this to our friendly disembodied intellect, apart from the sensible qualities that are supposed to be giving some real character to these shapes, as they interact and move around through time. But those sensible qualities don't constitute space. They do characterise the objects that are in space (at least in relation to our own sensory capacities): but the space itself is something distinct from them all. It's an eternal and universal structure that's fit to give a home to them. And that structure, it seems, has been adequately communicated to the disembodied intellect: for the structure is a purely mathematical one. Fundamentally, all that it really is is a bunch of numbers, ordered into certain sets, plus a handful of axioms and laws. And, once the intellect has grasped that, what's missing? It still doesn't know what it feels like to experience objects in space: but that's just a matter of how things make us feel, which is independent of the space itself. As far as genuinely spatial properties and relations are concerned, you can take any you like: 'round', 'square', 'thin', 'thinner than', 'half as big as', 'further away from', 'three times as far as', 'in between', or even something as precise as '4.7 metres long' (given that this is really just expressing a comparison between a given object and an agreed standard). Indeed, once we factor in time, we can even add 'growing', 'moving', 'accelerating at 9.8 m/s2', etc. All of these properties and relations can be adequately understood just by taking some Cartesian coordinates and doing a bit of elementary arithmetic on the numbers themselves.

Since nothing could change without some kind of movement, and time would not be

Since nothing could change without some kind of movement, and time would not be perceivable without some kind of change, why isn't time fundamentally motion. Likewise, since space would not be perceivable without some sort of motion, why isn't space fundamentally motion as well? In other words, what part of space or time is conceivable without bringing motion into the explanation?

The reasons you gave for thinking that time is fundamentally motion and that space is fundamentally motion seem to depend on this principle: If A isn't perceivable (or isn't explicable) without some kind of B, then A is fundamentally B. But that principle looks false. Motion isn't perceivable without some kind of perceptual apparatus, but that doesn't imply that motion is fundamentally perceptual apparatus. Motion isn't explicable without some kind of explanation, but that doesn't imply that motion is fundamentally explanation. Furthermore, if time and space are both fundamentally motion, are time and space identical to each other? Even physicists who talk in terms of "spacetime" nevertheless talk about time as a separate dimension of spacetime; I don't think they regard time and space as one and the same.

One might also question whether space, or the perception of space, requires motion. When I stare at my index fingers held one inch apart, I perceive them as occupying different spaces, and I judge there to be "empty space" between them, but I don't think I'm relying on the perception of motion in that case.

For an argument that time can pass without any change, have a look at Sydney Shoemaker's classic article "Time Without Change" (1969). I also found some lecture notes about the article at this link.

Can the theory that everything that exists exists in time and space, which is

Can the theory that everything that exists exists in time and space, which is materialism as I understand it, explain how things have motion as well? Motion is not itself a thing that can be located within time and space it is only the word that we apply to the effect of something changing position in a continuous manner. But if the only things which exists exists in time and space what is there to move the things that is in motion? Certainly not something else which is in time and space since that demands as well an explanation for it's movement.

This is a deep question or set of questions! The history of the philosophy of motion is fascinating as is the general philosophy of space and time. There are historically significant arguments to the effect that to account for motion in the cosmos, one needs to posit an unmoved mover --God (as developed in the work of Thomas Aquinas in the 13th century). If you are interested in this line of reasoning, you may wish to take a look at more recent articulations of the cosmological argument: you can find these in the Stanford Encyclopedia of Philosophy under 'Cosmological Arguments' and in the entry 'Philosophy of Religion' --entries are free and available online (as this website makes clear). These entries will speak to your sense that something more is needed to account for space-time as currently conceived.

There may be two things to keep in mind as you reflect on the philosophy of motion. First, while motion is not a thing in the sense that it is not a concrete individual object (a rock) it is not is not (necessarily) immaterial (or incompatible with materialism). Motion seems, rather, closely tied in with time; without time, there would be no motion. Also, while some do define materialism as the view that everything that exists is in time and space, this is not universally accepted. Significant philosophers (from the 17th century Cambridge Platonists to G.E. Moore in the 20th century) thought that certain things are non-physical (sensations, and for the Cambridge Platonists the soul) was spatial but not material.

You are on to a vital, historically fascinating issue. Richard Sorabji has published a number of important books that address the rich and creative ways in which motion have been conceived of since Zeno. A close look at his work will prove (I wager) to be very rewarding! You may still have questions un-answered but not un-addressed, and Sorabji is brilliant at bringing to light ancient sources that often go overlooked these days.

Would the idea of 3 dimensional space be possible without vision?

Would the idea of 3 dimensional space be possible without vision?

The answer seems pretty clearly to be yes. Touch and hearing both convey information about dimension. Think, for example, about the fact that a sound can be above you, or in front, or two the side. Or think of how you could tell that object A is taller than object B, but object B is wider than object A just by using your sense of touch.

If you're interested, here's a link to a video about a remarkable Turkish painter, blind from birth but able to convey subtle information about perspective.

Is it possible for a physical object to be 1-dimensional?

Is it possible for a physical object to be 1-dimensional?

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.