# My friends and I have gotten into an argument over whether or not there is/are opposites to a circle. Both sides have some valid points, but the main idea is whether or not there are opposite shapes.

### I can't think of any ordinary

I can't think of any ordinary sense of "opposite" that allows for the existence of opposite shapes (i.e., closed plane figures). But you and your friends could invent a technical sense of "opposite" that allows for opposite shapes. Maybe the opposite of a shape is the mirror image of the shape along the vertical axis, or along the horizontal axis, or along some oblique axis, provided that opposite shapes never look the same. On that definition, a circle wouldn't have an opposite shape, but a triangle could. In any case, there's nothing to disagree about until you have a suitably precise definition of "opposite shape," which again I think you'll have to stipulate, because ordinary language doesn't supply one.

# 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

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 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

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: http://plato.stanford.edu/entries/mereology. 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 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 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.

Excellent questions. I'm glad to hear you're looking into this issue. I think philosophers and scientists often throw around talk of "surfaces" much too glibly. I recommend starting your search with the SEP entry on the concept of a boundary, available here . It contains a lot of information relevant to your questions and a bibliography with several useful references.

# 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...