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!
> 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?").
> 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.
> 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.
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.
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.
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.
> 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.
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.
Calculus 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?
If 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.
As 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?
The 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.
It'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).
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.
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!".
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).
Good point, I guess I can see in this case the mathematician might realize that while he can cross the full distance easily enough with infinite time, he might not want to wait around for infinite time...
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.
If one were male and the other were female, the joke wouldn't work.
If both were female it wouldn't work, as (if you're male like me) you're aware of our inherent and unfortunate sexism (our rational brain shuts down upon viewing an attractive female).
The difference between sexism and acceptable behaviour is knowing this limitation of our selves and acting respectfully with the other sex.
I chose my words poorly; I didn't mean "assume" in the interpretation of the joke, I meant the joke writer was presumptive in using gender roles like that.
The original assumption that the joke is sexist relies on presuming that the joke makes women into objects used only for sex, but not vice-versa for men. So by claiming that the joke is sexist I think we can reasonably assume the person making the claim believes the engineer and the mathematician to be male.
This version includes a male pronoun to be explicit for one of the two, other versions include male pronouns to be explicit for both of them. One could also use a 'her' instead somewhere in the joke since lesbians exist but that would needlessly complicate it and possibly cause confusion.
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.
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.
I don't think the joke would work as well with just any other desirable thing. The joke sets up the scene with two geeks and a sexy woman and there's some amount of expectation that the joke is going to be about their relative sex drives or the mathematician only knowing how to have sex in theory or something similarly boring and familiar. It's then somewhat surprising when it turns into a joke about philosophical paradoxes and is therefore funnier with the surprise.
The attraction is fine; the issue is that the woman is treated as an inert object in the story, rather than a person with her own free agency equal to the men.
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.
"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.
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.
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.
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
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[ 4.3 ms ] story [ 174 ms ] threadhttp://physics.stackexchange.com/questions/47252/simple-expl...
[1] http://en.wikipedia.org/wiki/Simulated_reality
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.
"There is currently no directly proven physical significance of the Planck length; it is, however, a topic of research."
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?").
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.
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.
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.
Sometimes it seems to me that every day words become mere metaphors when applied to physics.
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.
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.
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.
[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?
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.
The important difference between this and Zeno paradox is - in Zeno paradox time is also divided.
Are we becoming so puritanical we can't even admit that a male person is attracted to a "naked beautiful woman"?
If one were male and the other were female, the joke wouldn't work.
If both were female it wouldn't work, as (if you're male like me) you're aware of our inherent and unfortunate sexism (our rational brain shuts down upon viewing an attractive female).
The difference between sexism and acceptable behaviour is knowing this limitation of our selves and acting respectfully with the other sex.
Edit: typo
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.
http://www.askamathematician.com/2012/03/q-is-the-quantum-ze...
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?
> ... 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