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.

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 .

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.

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.

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.

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...