4 ms·
I have never understood why Zeno's paradoxes are considered so profound. Consider what an instant of time is. Is it a non-zero chunk of time smaller than any o
by zaptheimpaler 13y ago
I have never understood why Zeno's paradoxes are considered so profound.
Consider what an instant of time is. Is it a non-zero chunk of time smaller than any other finite chunk of time? In this case, we are defining an instant as an infinitesimal chunk of time. If we posit that such a thing exists, then why can't an infinitesimal distance (which the arrow travels in that time) exist?
The other option is to define an instant of time as no time at all. Not the smallest, simply zero. Saying that since you cannot move a non-zero distance in zero time is valid. However, a chunk of time is not composed of an infinite series of zero time chunks, just like 0.00000...1 is not equal to 0 + 0 + 0... There is a gap between zero and infinitesimal that cannot be bridged by zeros alone.
- jamesrcole 13y ago> Consider what an instant of time is. Is it a non-zero chunk of time smaller than any other finite chunk of time? Here's how I think of Zeno's paradox: it is saying that if time/space are continuous then there are no such things as finite chunks of them, therefore they must be discrete.
- zaptheimpaler 13y agoThats a nice way to look at it - Zeno's paradoxes are proofs by contradiction that time/space are discontinuous!
- monjaro 13y agoThat's not true though. Zeno's paradoxes are all pretty easy to resolve using first-year calculus. They certainly don't prove that time or space is discontinuous.
- jamesrcole 13y agoCalculus doesn't resolve the paradox. It's just a tool, and it doesn't involve any literal division of anything into infinities. You're making a map vs. territory mistake. [edit] i can't reply to monjaro's comment, so I'll put my reply here: does calculus prove that there are actually an infinite number of positions between any two positions?
- monjaro 13y agoCalculus is a tool for reasoning about exactly this kind of thing. If you think there is an actual paradox present, please explain what it is.
- ajuc 13y agoIf you don't belive you can apply calculus to solve the problem what makes you believe you can use division or addition to define the problem? These are tools too.
- monjaro 13y agoAs a response to your edit, no it doesn't. So what? If you can infinitely divide space in the manner of Zeno's paradox, then the sum is well defined and easy to do. If you can't, the paradox doesn't apply to reality. What's the problem?
- Jach 13y agoThe map vs. territory issue at hand depends on whether you take Zeno's paradoxes to be statements about physics or statements about logic. If it's the latter, calculus resolves those with ease. If it's the former, well, then you have to look at the physics. And the physics are described in our maps by math which includes differential equations of complex numbers, and hence the paradoxes are again resolved as best as we're capable unless some new representation of physics comes along that makes the paradoxes manifest.
- ajuc 13y agoIt's actually the other way around - if space AND time is discrete Zeno Paradoxes are paradoxes, because you cannot have finite sum of infinite number of elements, if the elements can't get arbitrarily small. So if anything Zeno proved (by contradiction) that space and time behave like they are continuous, at least for events in human scale (because arrow does move so our assumptions were wrong). Now we see that this breaks at quantum level.
- deleted 13y ago[deleted]
- taejo 13y agoNot everything that is obvious today was obvious in the fifth century BC; for example, it took until the third century for Archimedes to point out that the number of grains of sand on the earth is finite. One could say that continuity and related concepts was not properly understood until the 19th century. In some ways, the knowledge of the average high school student today far exceeds that of the greatest geniuses of history.
- zaptheimpaler 13y agoI thought that Zeno's paradoxes are still considered unresolved as of now though.
- modarts 13y agoMost basic calculus courses clear them up by around the time they introduce students to the concepts of limits.