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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.
Not 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.
I thought that Zeno's paradoxes are still considered unresolved as of now though.
Most basic calculus courses clear them up by around the time they introduce students to the concepts of limits.
> 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.
Thats a nice way to look at it - Zeno's paradoxes are proofs by contradiction that time/space are discontinuous!
That'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.
This article is very low quality. The account of Zeno's paradox is wrong on pretty much every count, which makes it hard to trust anything else the article claims.
The Ask A Mathematician page linked is quite a bit better.
http://www.askamathematician.com/2012/03/q-is-the-quantum-ze...
You're right, that's a much better article.
Tabloid level at best, indeed. I dropped after the second paragraph. Its obvious the guy understands shit about Science and about Greek science in particular. Take one theory out f context and loosely tie it with quantum physics and you have enough paper for your next toilet trip.
Seems okay to me - what precisely do you think they got wrong?
"He did this by setting up a series of paradoxes that showed, among other things, that half a given span of time is equal to twice that given span of time, that time and space are neither continuous nor discrete, and that nothing ever moves. Ever."
None of those things are true. One minute isn't equal to four minutes, Zeno didn't prove anything about time and space being neither continuous nor discrete, and things clearly do move.
The only correction I'd make to their sentence is change "showed" to "appeared to show". And his paradoxes did appear to show all those things. Of course empirically you (and the ancient Greeks) could see that they must be wrong - the challenge was to find the logical flaw in his arguments. Finding that took a very long time.
Perhaps I'm being overly pedantic, but in my mind "showed" and "appeared to show" mean very different things.
A low quality article on the Gawker network? Surely not!
More info here for those interested
http://physics.stackexchange.com/questions/47252/simple-expl...
The picture in the article fits perfectly, not only because of Neo stopping time, but because, more importantly, at least to me, this looks like if the Universe has its own Garbage Collector. Which could mean that we are living a simulated reality [1], something like the Matrix!
I think so, too.
We have gaps in reality, the planck length, planck time etc.
Also, we have Gödel's incompleteness theorems, which tells us about the gaps in theories we can't fill.
You are assuming more about the Planck units than is currently known. Directly from wikipedia:
"There is currently no directly proven physical significance of the Planck length; it is, however, a topic of research."
> Gödel's incompleteness theorems, which tells us about the gaps in theories we can't fill
That's mathematical, not physical, so that would mean that in the parent universe somehow _something else constitutes a formal language_, which requires changing what it means to be a subset, which requires changing set theory. So if GITs are invalid in the parent universe, it means, roughly either that you can compare something to itself, and find that it contains different things that itself (absurd), or that list comprehensions are logically impossible (not as obviously absurd, but still "whaaa?").
Planck length and time are curious; but incompleteness is a mathematical construct, so it's true for all universes, simulated or not.
The problem with this explanation is that checking more often the state of the quantum state makes it decay more "slowly" it doesn't stop it totally.
> ... Check on it after three seconds, and it probably will have decayed. But, Misra and Sudarshan argue, check on it three times in one second intervals, and it will most likely not have decayed. ...
Let's invent numbers for an example, with 3->5. It you check a system after 5 seconds the probability that it has not yet decayed "is" 1-0.05 . It you check a system five times in the 5 seconds interval the probability that it has not yet decayed "is" (1-0.002)^5=~1-0.01 . The measurements makes the decay probability smaller, but it's not reduced to 0 as the article try to induce, to make it similar to the Zeno paradox.
The important point is that at the quantum level every measurement changes the system [1]. So the result of the experiment is affected by the measurements. (At the classical level, some measurements are negligible, and it's safe to ignore them.)
[1] of an operator that doesn't commute with the Hamiltonian of the system
> In any given instant, the arrow has to appear motionless. If it wasn't motionless, there would be two instants, one in which the arrow was at one position and one in which the arrow was in another position.
I don't follow. Why can't the arrow be in both places at the same time? With the probability of one vs the other dependent on the velocity and the size of the time slice.
Remember, the arrow is one of Zeno's paradoxes. He's not describing reality; he's trying to impose a conceptual-only understanding of reality on the thing, itself, and ending up in a logically impossible situation.
His description of the arrow's motion, or the runner's traveling half the distance to the finish line, and then half the remaining distance, and then half the distance left after that, and so on, and thus never actually finishing, are exactly that: descriptive, not prescriptive.
That sounds nice. Unfortunately, knowing nearly nothing about any of this, I'm having a hard time imagining what it means to "check on" an atom and how that could influence its state.
Sometimes it seems to me that every day words become mere metaphors when applied to physics.
The problem with Quantum Physics (well, physics in general, but in classical physics it's mostly ignored) is that in order to make a measurement, you need to interact with the system thus changing it's state. You might for example measure perturbations in an electric field, but by applying the electric field you influenced the state of the system. This is also an important part of say the uncertainty principle. One way to think about why you can't measure position and momentum simultaneously to infinite accuracy is that perfect measurement of one would require shooting it with photons of zero momentum, while perfect measurement of the other would require shooting it with photons of infinite momentum. The real fun comes in when you leave a system alone for a while because then everything becomes probabilistic.
They use the term 'observe', which typically means to shoot photons or electrons at it. A laser is a common option.
This is a pretty reasonable use of the term, as when most people observe anything it's typically by capturing photons that previously bounced off it.
I don't know if I understand it correctly.
If space and time is discrete, and movement is deterministic - Zeno paradox is a paradox - object moving at speed 1 minimal division of space (MDS) per 2 minimal divisions of time (MDT) isn't at any valid point of space after 1 minimal division of time which is paradoxical.
Calculus solved this, but it has assumption that space and time is continuous.
Now we have experiments that shows us it's not true, so probably the assumption about determinism isn't true. So for example object moving 1 MDS per 2 MDT "really" moves 1 MDT per 1 MDT with probability 0.5.
Am I getting this right?
> Every time you check on it, it will revert to its "original" measured state, and the clock will start over.
This makes me uneasy - I feel like there is a violation of thermodynamic laws in there somewhere. Decay is not the only thing that is probabilistic, tunneling is too, and if you can manipulate things so it's more like to go in one direction vs the other, you can reverse entropy.
Nothing wrong in reversing entropy, if you do it by adding energy into the system. I guess shooting photons counts.
Ah, like Maxwell's demon.
What I don't get is wouldn't you see the same effects in a classical mechanical wave: http://en.wikipedia.org/wiki/File:Polarizacio.jpg ?
Saying that this effect "stops the world" doesn't seem right to me. It just "appears" to have stopped only for the observer, who observes the world at very very small intervals.
A mathematician and an engineer agreed to take part in an experiment. They were both placed in a room and at the other end was a beautiful naked woman on a bed. The experimenter said every 30 seconds they would be allowed to travel half the distance between themselves and the woman. The mathematician said "this is pointless" and stormed off". The engineer agreed to go ahead with the experiment anyway. The mathematician exclaimed on his way out "don't you see, you'll never actually reach her?". To which the engineer replied, "so what? Pretty soon I'll be close enough for all practical purposes!".
It's a lame enough joke that I wonder whether it was really necessary to drag it out again, considering how it is likely to make many female visitors to the site uncomfortable.
I'm all for avoiding sexist jokes and comments, but I can't really see what's wrong with this.
Are we becoming so puritanical we can't even admit that a male person is attracted to a "naked beautiful woman"?
Yes. Why just yesterday I read that male time travellers are pigs for abusing their information advantage to start relationships with women. That's sexist and also no woman would ever do anything like that even if she were allowed to time travel as the principle protagonist.
I like the joke, but to play the devil's advocate it's slightly sexist to assume the mathematician and the engineer are both male.
Perhaps I am corrupted with recent news coverage, but I'd have liked a simple statement that the woman was actually willing. Perhaps cruelly tempting them or something. But you could replace the woman with $500, promise of tenureship or any other desirable thing and the story would still be funny without sexually objectifying women.
He may be attracted, but what does she think? What is she doing there naked, to begin with?
That's pretty funny, but the mathematician must be one who lived before the invention of calculus, which resolves such paradoxes. (With either infinitesimals or limits.)
Not in this case. In Zeno's paradox, as the distance halves, so too does the time. In this case, the time for each halving is constant at 30 seconds. This means the mathematician is right and theoretically they'll never reach the woman (except practically, as pointed out).
I think mathemathician considered getting there after inf * 30 seconds to be "never".
The important difference between this and Zeno paradox is - in Zeno paradox time is also divided.
They could ask the woman to come to them.
Good Sir! That's the fuckin' funniest thing I've read in a while.