Consider Movement is possibly a strong illusion which works, constructed by consciousness adapted to time/space. Hence one can solve a paradoxical movement riddle by using consciousness to create an alternate weak illusion of movement, by not moving.

Who says that?

In hopes of considering Zeno in an evolutionary/historic sense and as noted in the referral by Zeno (our Zeno) Zeno was a compatriot of Parmenides who saw the world as a "unity", or "one" and therefore denies the discrete to which measurement is primary.

"Parmenides sees the Untrue, the Deceiving, in sense-perceived, external nature. He sees what alone is true in the Unity, the Imperishable that is seized by thought. Zeno tries to come to terms with, and do justice to, the thought experience by pointing out the contradictions that result from a world view that sees truth in the change of things, in the process of becoming, in the multiplicity that is shown by the external world. One of the contradictions pointed out by Zeno is that the fastest runner (Achilles), etc...." from " the Riddles of Philosophy", chapter 2:

http://wn.rsarchive.org/Books/GA018/...18_p01c02.html

Prehistorical man which continued up into early Greece and even into our current era didn't bring "thought" into action in his perception of the world but had a 'soul experience" or "feeling experience' of the world. He felt himself to be a "part of" not "separate from" the cosmos, inclusive of the earth. His consciousness consisted of "imaginative pictures" which, included in this "feeling experience" gave him a direct knowledge of his surroundings. the senses that we have today were not as powerful but his external world was more nebulous and not nearly as clear as our present day. Yes, he had senses, but the total power of these senses were not as strong as we possess during our times.

He felt himself to be a part of the cosmos, inclusive, with the separation, in a sense, in movement and fructifying to our present existence. The "separation", one can say, began with the ancient Greek thinkers in which "thought" came to the fore and mankind , in graduation, worked through the power of "thought" as a personal attainment.

It appears that Paramenides , who denied externals, living only in "thought" carried forth this ancient state of immersion in the cosmos along with the new found ability of thoughts and thinking. Therefore, "all is a unity". the thought world is unity and cannot countenance particularity. He felt it and "thought it" in his passion of understanding.

The salient point is that "thoughts and thinking" was "new' and the birth of the intellect began within the Greek mind.

As an important aside one cannot "measure" in the thought world for there has to be something to measure which is our materiality or ponderability to which thoughts and thinking are not a part thereof. You can't weigh a thought nor measure its distance nor can our perspective of "time" be appreciated in the thought world but there is "time" within that world and for this we can get a glimmer of the cosmic time within the "memories " of man. Time tied to materiality is the time of our sciences and is relatively "new" as is the "intellect" and its considerations.

Echos of Paremenides can be seen in Hegel who only saw the world as "spirit" which is exactly the world of thoughts and thinking. Of course Madnak is here and I wish he were here.

There were other thinkers including Democritus , Plato and Aristotle and I'll refer you to my reference of Greek Thinkers. It is natural that Aristotle should deny Zeno as it is the entire whoop and wharf of Aristotle to bring thinking and thoughts to the external world, as precursor to modern science.

The intellect "cuts" or is "discursive " and therefore our concepts of species, genera, periodic table of the elements,etc.... all of which are mandated by the intellect, a methodology of thinking and thought. Parmenides demurs.

The number 1 surpasses infinitely many numbers less than the number 1. It does not have to "do" anything for that to be true. It only has to "be". Zeno claims it does have to "do" and what it has to "do" is impossible. Yet there the number 1 sits, doing nothing, oblivious to Zeno's confusion.


PairTheBoard

Good thoughts are boring.

I just picked up from the table and put on my wrist my analog watch. Disproving, I believe, all of Zeno's paradoxes. Very boring.

On the "no one ever dies" thing - I just learned that I have a 40% chance* to live to age 85 (I'm currently almost 62). I almost certainly want to be dead before I reach 85**. But as you say, according to Zeno - I will never get there anyway. So no worries.



*Calculations are based on mortality data from the Society of Actuaries Retirement Participant 2000 Table (Vanguard website).

** I have only a 1% chance of making it to 100. I find that very comforting.

Paradox solved.

https://www.google.ie/search?q=infin...cl-wa5TvM6AmAs

Something you found on the internet told you this so it must be true, got it.

Errr....he said that "some say" it drove Cantor insane and you asked who said it. Linking people on the internet saying it caused his insanity is a fine proof that what he said is true.

There are better arguments for why no one ever really dies and you just watch a bunch of people who you think have died.....still no proof though

http://en.wikipedia.org/wiki/Quantum...nd_immortality

There is no such thing as continuous space time and therefore continuous motion in a trajectory. Those are classic concepts not realized in the quantum world. Essentially consider this; The notion of a particle that exists out there traveling in a well defined trajectory x=f(t), with nobody there to observe it but truly generating an objective reality in that idealization of a path that is a well known as function of time, doesnt exist.

To exist you must be observed/interacted with.

The uncertainty principle secures that you cant know the location and speed of a particle (momentum) with infinite precision which is exactly what is required by claiming you know x=f(t) (momentum is related to the perfectly known derivative etc) .

Big systems are collection of small particles that are quantum systems and for which such trajectories dont exist but only as classical limits. The big systems are quantum systems as well but they are examined from a macroscopic perspective that ignores the fine structure eg of their edges.

To me there is absolutely no problem at all. There is nothing there to worry about. As you split distances and observation segments you will at some point arrive at the inconsistency of the classical picture.


For example you can imagine both particles as wavepackets with some different speeds (here v1 and v2 separated by a distace of d units) starting as gaussian wavepackets.


Again i suggest seeing the link on wavepackets that involves such a solution for an original gaussian wavepacket at t=0 that enjoys the ideal uncertainty relation (ie the equality of the general inequality Δx*Δpx >=ħ/2)

http://en.wikipedia.org/wiki/Wave_packet

Such wavepacket is the solution of the Schrodinger equation and the square of its amplitude, when you solve the equation in time with the initial condition that it starts as a Gaussian wavepacket corresponding to the initial positions (eg at 0 with speed v) , is the probability density to find the particle in some location x,t.

It comes out as (in some simple units selected) ;



(which is not properly normalized here yet, also they use k0 for the speed v )

Notice that its still more likely in that picture to find the particle in the classic trajectory x=v*t. But it is not a given that you will find it exactly there. Its experimental position is a random variable obeying a normal distribution now.

An electron for example that has speed 0.1c has momentum uncertainty if you demand to localize it within one micron that corresponds to a speed uncertainty of 50m/s. You cant know both with as much accuracy as you like.

If you consider 2 wavepackets now, 2 ahead of 1 by d and with different speeds v2, v1 the probability densities of their locations as function of time will be; (it remains a Normal curve but the width and average evolve with time and the width in particular spreads with time now)

f1(x,t)=(2/Pi)^(1/2)*1/(1+4t^2)^(1/2)*Exp(-2*(x-v1*t)^2/(1+4*t^2))

f2(x,t)=(2/Pi)^(1/2)*1/(1+4t^2)^(1/2)*Exp(-2*(x-d-v2*t)^2/(1+4t^2))

For the above time evolved wavepacket assumption.


One can always ask what is the chance 1 (the faster) is ahead of 2 if we measured them and that will be given by a combined probability density P(x1>x2) same as P(x1-x2>0). Since both f1,f2 are normal and while spreading traveling remain normal with widening widths (standard deviations) the difference is also normal with the average the difference of averages ie the value d-(v1-v2)*t (their classical distance in time) and a new sd since both have common sd just 2^(1/2)*the sd of each above which was 1/2*(1+4t^2)^(1/2) for a general time t.



As they approach it will start becoming obvious as the 2 densities start to overlap that well in advance of the classical trajectories v1*t and v2*t+d meeting that the chance that 1 is ahead of 2 is nonzero and significant now (which was close to 0 initially in that picture of wave-packets for all practical purposes).

As a result you never have to get to any infinitesimal small distance between the 2 with the conviction than 2 is still ahead. Well in advanced of some extraordinarily small divisions of distances you will have arrived in finite time at situations where which one is ahead is a cloudy statement already answered by the new normal;

f21(x=x2-x1,t)=1/(Pi)^(1/2)*1/(1+4*t^2)^(1/2)*Exp(-(x-(d-(v1-v2)*t))^2/(1+4*t^2))



The following charts show the 2 wavepackets at t=0 and the distribution of their difference of locations by any experiment (clearly a bell curve around 10 as expected for an initial distance of 10 units).





And now the chart when 1 sec has passed and classically object 2 is still ahead (at 11 and the fast at 10 where it was before ) but any experiment now can go either way with significant probability to claim 1 is ahead already.



Notice the area on the left of x=0 below the curve is the chance now that 2 is already ahead in measurements even if classically 1 is still ahead but they are close enough now for the ambiguity of the quantum nature of the particles to come in play.


As expected at 1.11111... =d/(v1-v2)=10/(10-1) (the classical meeting point in time) the distribution is 50-50 for who is ahead and the 2 wavepackets are perfectly identical.




(In this example i assumed both to be traveling in 2 parallel lines to avoid any interaction between the 2 particles that will happen in real life if collinear)

I'm posting this because Masque just said space and time isn't continuous.
This guy's reasons for saying so are different though, since he's saying we're living in a simulation.

I'm not sure it's been conclusively shown that space and time are not continuous.

Is it a logical problem or is it a problem with the logic?


PairTheBoard

I think the bolded below is the key to the argument and what Zeno presents as the logical problem when in fact it's the problem in Zeno's logic.


Zeno: In order to travel from 0 to 1 on a continuum you must traverse infinitely many points before getting to 1. e.g. the points (1-1/2), (1-1/4), (1-1/8), (1-1/16), ... ; However, logic tells us that the job of traversing all the points in the list requires the traversing of a final point in the list for the completion of the job. Since there is no possible final point in the above list it is not possible to complete the job. Therefore it's impossible to travel from 0 to 1 on a continuum.

PTB: What rule of logic tells us that "the job of traversing all the points in the list requires the traversing of a final point in the list for the completion of the job"? Is it because Zeno says so? Or because it seems like it should be the case? Or is it just some so called "logic" Zeno has pulled out of thin air? Isn't it really just faulty intuition based on our experience with finite lists?

In my view, there's nothing wrong with the logic that says the job is completed when you get to the number 1 and you do so having traversed all the points in the infinite list without ever traversing any "final point" in the infinite list


PairTheBoard

The rabbit has infinite pace when passing the turtle. It passes all those points faster and faster, until reaching infinity.

It has eternal life at that point.

Both the rabbit and the tortoise are passing infinitely many points every nanosecond of the trip. There's nothing special about the nanosecond in which the rabbit overtakes the tortoise nor the particular infinite list of points Zeno identifies there. In fact it's a rather paltry list being countable. They both pass uncountably many points every nanosecond. They are passing infinitely many infinite sets of points every nanosecond.

We should have a thread on the Cantor Set. That's down the rabbit hole.


PairTheBoard

By this logic, not only do we never die but we are never born to begin with.


PairTheBoard

An infinitely small amount of distance is just that, infinitely small. Just because we can give it a name and then also claim there are an infinite amount of those infinitely small increments of distance within any finite space, say one meter, does not mean it takes an infinite amount of time to travel one meter. It still just takes a finite amount of time, say one second, that we can also break down to an infinite number of infinitely small fractions of one second that all sum up to a finite amount of time. It's still just a finite distance and finite period of time we're describing. Stop trying to confuse everyone, maths.

It seems clear that the explanation for this "paradox" was given in statement #13.

Zeno proposed a construction which creates a series whose limit is the time T when Hercules overtakes the turtle. Since any finite number of terms of that series produces a time t<T, if one considers only times included in the series then Hercules does not overtake the turtle.

Thus Zeno's paradox reduces to the observation "at all times before overtaking the turtle, Hercules does not overtake the turtle".

Deep.

Of course, Zeno can be forgiven since the mathematical basis of the limit of an infinite series was not available to him. Modern thinkers have less excuse.

One can see the error in the usual statement of the "paradox".

The keyword is "never". Never means "at no point in time, past or future". But that is clearly false.

The other comments about the nature of time and space and its nature as continuous or discrete are all important questions. But they are not captured, anticipated or clarified by Zeno's observation.

In the case of the majority of P/PU posters then this would be a great benefit. Not to mention the other areas of usefulness in time and place. But this is a digression.

In fact, according to Zeno, nothing ever happens al all. The universe is static and at a complete standstill. Everything that you think happens really doesn't. Solipsism?

Its amusing that one of the words you've used here is capable of eliciting an emotional reaction from an otherwise stone-cold and methodical individual, who places very little weight on emotion and bias therein.

Can we guess who? or will it be revealed to us?

At the classical level that you accept exact trajectories and a crystal clear unambiguous spacetime that all those observations/considerations are described within, all that Zeno establishes is that the slow object will remain ahead for a finite amount of time.

He is unable to understand how you can have motion because he wants to see motion as transition in discrete steps from one location to another. He is treating traveling 1 meter and 1/2 or (1/2)^100 meter as equivalent things in the transition sense. All these involve having to go through an infinite amount of points within an infinite amount of time points that both form however finite intervals however small we make them. We can have a correspondence between the infinite elements of 2 continuous sets. Eg x=v*t and time is real. No explanation is provided of how one goes from one point to another. Nobody describes i mean what is the process of motion at an elementary level, because there is no such level, there are no steps to be performed that evolve one state to another in some sequence of actions. A discrete simulation of course is such step process. And we do not have this here either. The world is not a grid.

Velocity is simply the rate of change of this function of location in time. A concept such as velocity is a limit or the result of averaging (sampling) that assumes the real thing is a limit as well only approximating it.

Of course the real world doesnt work that way.

The real wold is not continuous. The real world is not discrete either. Haha. The real world is an experience that is described to a degree, in a satisfactory sense, by the assumption of a continuum. And then Quantum Mechanics comes to remind us the unfinished business and the illusion we built our world around, based on intuition derived by experience as its perceived by a macroscopic system/observer/brain.


The process Zeno wants to perform cannot occur in the world we live. It cannot because notice what it requires from you. To be able to talk about the movement of a particle from say point A to point B in the kind of accurate terms required when you start talking about very small distances and time intervals, the knowledge of both location and speed is necessary to a high accuracy.

What does Zeno say? After n sequences of examining what each particle does one by one you will be forced to argue that in the time the fast object covers the last segment, the slow one has to advance a little bit further. How much is it that it advances?
If the time of that step n was t(n) the displacement of the slow object will be v2*t(n). You then need a time t(n+1)=v2*t(n)/v1 to take the fast object there too and so on the eternal cycle goes. This is a geometric process that eventually since v2<v1 takes the time interval and the corresponding v2*t(n) successive displacements of the slow object both to as small sizes as you want.

In order for the process to maintain its integrity, accuracy in the knowledge of the speed v2 and the last location of the particle will be required. If the original uncertainties were DV and DX this is not a significant problem for the first few steps in general. But as the distance of the last segment gets smaller than DX we get in trouble. At the same time in updating positions the knowledge of speed is necessary. When we are down to small time intervals (inevitable outcome of the geometric division) the (accumulated over many steps in time) uncertainty in the new position due to to the lack of knowledge of the speed will become important too (by that i mean DV*T(total) will become comparable to the small distances considered). We will have an inability to determine where the particle really is because of these 2 combined uncertainties (depending on their relative values in that problem one may become a problem faster than the other or both at the same time but the problem with one of them at least first is unavoidable).

That would not be a problem if these uncertainties were both 0 in some ideal world that this is possible. As a matter of fact whether we know the speed or location with the perfect accuracy or not in that ideal world it doesnt matter for the argument. The location and speed of the object in that world are independent realities. They are truths independent of the observer. They are objective realities. But experiments have shown that this is not the world we live in. You cannot measure both the location and the momentum of a particle with accuracies that violate the Heisenberg inequality. Dpx*Dx>=hbar/2. This is not an inability of the experimental process, this is a property of the system itself http://en.wikipedia.org/wiki/Uncertainty_principle

It will not prove intuitive problem in every day life though. If you treated a ball of 10 gr as a particle you would be able to know say its speed within 10^-6 m/sec (and therefore momentum uncertainty 10^-8 kgr*m/s and its location with 10^-6m accuracy and the uncertainties product would be 10^-14 far greater than the Planck constant/2 that is 1/2*1.054*10^-34 in the same units. Everyday life practical inaccuracies are mostly results of our non ideal technological approach to measurement. We can drastically improve them and still not hit any Heisenberg limit. Not only that but at the perception level we operate we would never find even those, far from ideal uncertainties i gave as example, problematic enough. Our eyes do not appreciate smaller distances than some not very small fraction of a human hair's thickness and our perception of time intervals isnt faster than say 0.01s (watching movies and feeling its close to reality in some high definition system would not be possible if that werent true). As a result we will never appreciate the quantum character of every day objects in terms of the inability to determine accurately both location and momentum/ie speed. Other macroscopic imprints of the quantum nature of the world are all around us though from the chemical bond and the rigidity of solids to the funny macroscopic liquid properties of superfluid Helium3 or 4. For our purposes it feels as if the above example of experimental uncertainties are sufficient to offer us the illusion we know all we need to follow the movement of these object (ball) with conviction. The Zeno argument however derived by eg the idealization of the ball observation experience above, with unrecognized to our consciousness naturally enforced uncertainties, eventually will test the limits of that approach. And then it will fail as an argument.

Spacetime isnt continuous although current established theories assume so and are formulated in this framework (ie the wavefuction is still a function in some continuous space, you have differential equations and integrals etc). Spacetime though loses its meaning as geometry if you insist to consider small distances and time intervals. The metric that describes the geometry (Pythagorean theorem say in flat spacetime) loses its character. Why? Because of quantum noise essentially. Only this is a noise that affects your ability to measure distances now and therefore establish a geometry through such tests/measurements. The spacetime itself becomes volatile, uncertain.

Without this being an argument delivered by a higher theory in place yet, one can already see problems in the merging of the classical general relativity and quantum mechanics regarding that very thing. General relativity views spacetime geometry as the result of the distribution of matter and energy. As the source of the field is quantum in essence, the output of that source ie the geometry itself will be forced to have a quantum character too. The inability to localize the particles will affect your capacity to have a rigid dependable geometry as well because these particles (the source) is what produces that geometry. So you anticipate that picture will fail eventually. In a loose way to visualize it, quantum vacuum fluctuations themselves will secure that the geometry is fluctuating too at some level. And when this happens you have lost the ability to talk about continuum in a reliable manner.

Great, it's true that people on the internet say things. Thanks for proving that.

How about a more interesting topic, was it the research or the bipolar disorder mixed with peers ridiculing his research that caused him to spend his later years in an asylum? And before you link a google search on the topic, don't.

I'm not sure that's true, or at least it's not obvious. The Planck length puts a limit to how finely an observer can measure things, but it's not a hard limit on the things themselves, which is why wavefunctions and the like are treated as continuous.

Uncertainty also seems like a red herring. Agree that we can see Achilles and the tortoise as wave packets and agree that there's some point where it's fundamentally indeterminate who's ahead, but in time the faster object will still overtake the slower one.

The Zeno paradox is a mathematical/logical construct, not a question about the physical makeup of the universe. If you like, we can reframe it as a purely mathematical question where continuously-varying functions are allowed, and all it still retains all the essential herbs and spices of Zeno's original question.