# Don’t Look Now, Nothing Will Happen –Zeno Of Elea

The Greek philosopher [Zeno of Elea] proposed that an arrow in flight was in fact not in motion and its visible movement is only an illusion. A simple example of this is to glance at an arrow in flight, doing this causes our mind to store a snapshot of a motionless arrow. [Zeno] further defended this argument by stating that if an object has to travel a finite distance to reach a destination then the finite distance can be divided in half and the object must first reach this halfway point before arriving at the destination. This process can be repeated an infinite number of times, creating an infinite number of points that the object must occupy before reaching the destination thus it can never arrive at the destination.

Whoa, that’s a bit heavy. Let’s take a second here to think about this and never arrive at the conclusion, shall we?

So what does a fancy mathematics parlor trick have to do with the fact that we have all seen an arrow arrive at its destination? Recent experiments conducted at Cornell University have in fact verified the Zeno Effect. Researchers were able to achieve this by having atoms suspended between lasers in temperatures ~1 nano degree above absolute zero so that the atoms arrange themselves in a lattice formation. As per usual in quantum mechanics when observed, the atoms had an equal possibility of being anywhere within the space of the lattice. However, when they were observed at high enough frequencies the atoms remain motionless, bringing the quantum evolution to a halt.

## 53 thoughts on “Don’t Look Now, Nothing Will Happen –Zeno Of Elea”

1. It sounds like [Zeno of Elea] was a first rate moron. In the macro world, such an argument is obvious bunk, and the macro world is the only one that Zeno could have had any knowledge of.

1. But that’s what was interesting about it. He realized that the ideal mathematics of it didn’t match up with what he was saying. I mean he was correct in his idea, that you could break down any distance in half. In fact, this was used to explain calculus to a degree when I was first learning it. So if you know the mathematical part is correct, that you can in fact half any distance, but the arrow still moves, what’s the answer? If you halve the distance, and speed remains constant, then the time it takes to achieve your destination also halves. Since you can divide the distance infinitely, you can also divide the time infinitely as well. This means, that when the delta of time is zero, the arrow isn’t moving because delta of distance is zero.

He definitely wasn’t a first rate moron. He was just observing an effect of calculus with change in time and distance and made a curious observation based on that fact. Just as Schrodinger wasn’t expecting his famous cat to be both dead and alive, Zeno wasn’t expecting an arrow to just stop in mid air.

1. Telimektar says:

Thanks for great explain :)

2. Dave says:

The problem with Zeno’s paradox is that it assumes that real space is continuous space and can be broken down into smaller and smaller pieces ad infinitum. Quantum physics tells us that is not so. On the other hand, the question of whether real space is continuous in some way or is inherently discrete is a much harder problem to address. At first sight, it would make sense that space should be truly discrete so that the amount of information in the universe is bounded but there are effects, such as action at a distance, that put that into question. How do two entangled particles know what the other is doing without passing information or without being “pre-programmed” in some way. Also, a much simpler problem, Pythagoras’s theorem is really difficult to address in discrete space. I have worked on and off on the later question for a while. It is surprising that the current approaches use “tricks”, such as giving the lines a width, to get it to work. More fundamental solutions are somewhat elusive.

1. Greenaum says:

Actually somebody, and I think it was Zeno, considered all 4 cases. That of space being continuous, or being made of fixed quanta. And also the same for time. There are 4 combinations of this pair of possibilities. He then explained logically how all 4 of them made movement impossible.

The answer isn’t obvious, in fact I’ll have to look it up to find out. Or I could just be empirical and remember all those arrows that moved, throughout history.

2. twdarkflame says:

“The problem with Zeno’s paradox is that it assumes that real space is continuous space ”

.The problem is it treats space and time as different.

If you acknowledge that as space is sliced, so is the travel duration you get “an infinitely small space traveled in a infinitely small time”

He was probably aware of this, but glossed over it or ignored it. I believe Prachett called this tendency “lies to children” – when the real ideas are too complex you sum it up with a sciency-sounding lie.

2. No he saw a problem in pure mathematics not in the macro world.

1. TheRegnirps says:

I would think there were enough engineers cruising this site who had real mathematics and recognize this fundamental example of a series that can be proven to converge. It is also put in terms of taking a step half way to a line then half that then half that. Do you ever get there? Or the bird (or light) flying back and forth between two trains on a collision course. How does the bird not have to fly an infinite distance? Show that the sum converges to a constant.

1. Chris says:

Convergence of infinite series is exactly right. Of course, the idea of convergence of infinite existence series did not exist at Zeno’s time (nor did a convenient number system for him to carry out such calculations…). In response to DainBramage, it is important to not jump to conclusions about the true nature of the world based on “common sense” observations. If we did this, we’d never get to the modern concepts of bent space and time (because common sense dictates that those are crazy ideas).

From Zeno’s point of view, either the arrow moves and his math cannot explain why (ie math needs to be improved), or the arrow doesn’t move and his model of reality needs to be improved. From a purely philosophical point of view, both interpretations were equally valid, and so the next step would have been to rule one of them out.

1. For those of us that are not philosophers, mathematicians, or engineers, we look at the arrow as an obviously moving object that has no trouble whatsoever reaching its destination, and conclude that the “deep thinkers” of the world are thinking too deeply and over-complicating a beautifully simple bit of energy transfer.

2. Josh says:

On deep thinking and mathematical abstraction versus everyday observations, the human brain lies. Your eyes, ears and skin take a thin slice of data and feed it to your brain which decides that you cannot survive if you know what the world really looks like so it fills in the gaps with a nice, comfortable narrative. Our truth is human truth, not absolute truth. This is not just philosophical voodoo but observable phenomena. Take aliasing of a car wheel viewed at certain rpm, zoetropes and the nature of free will (https://en.wikipedia.org/wiki/Neuroscience_of_free_will). We are alive because we ‘know’ that a pouncing lion will reach us after a given time, but that is just a survival reflex. These obvious details should not be ignored but nor should they be accepted as a complete explanation.

3. @Josh: If there’s an arrow flying toward me, I don’t worry about abstraction or anything else, I just try to get out of its way because I know that it make it to its destination just fine whether I understand the math or not.

I guess my point is that I keep hearing people try to break the world down into mathematical constructs, which as this article demonstrates, don’t actually work. I understand about human perception and have a fairly solid understanding of the brain, having studied about it for the last 24 years in an attempt to understand the effects of my TBI a little better (hence the alias). I’m terrible at math, but I know enough about it to catch a flying ball, and that fact alone proves that Zeno’s thought experiment was a total and complete waste of time. Perhaps if he put that much effort into understanding the very real science of ballistics, science may have actually been advanced a bit.

Am I a bit cynical? Certainly. I prefer reality (which my perceptions do a pretty darn good job of informing my brain about) to imagined constructs, no matter how impressive sounding.

4. Shawn says:

@DainBramage: “Perhaps if he put that much effort into understanding the very real science of ballistics, science may have actually been advanced a bit.”

The problem is, you don’t know what will pan out in advance. If you did, you wouldn’t have to think about it. If you discover a new paradox, determining where the assumptions supporting the paradox fail might tell you something new and wonderful (and possibly even useful) about the real world.

3. binaer says:

“””and the macro world is the only one that Zeno could have had any knowledge of”””
experimental knowledge maybe, but what about intuition ?
interestingly, the ancient Greeks coined first the word “atom” to conceptualize and idea that was impossible to prove or even test at that time.

“However, when they were observed at high enough frequencies the atoms remain motionless, bringing the quantum evolution to a halt.”
I feel this last sentence as highly disturbing. I don’t dare to begin to think to what consequences such a statement will bring to the fundamental of quantum physics. Is all that uncertaintity principle only about what appears to be a simple oscillation or movement ? So, Schroendiger talked just about a fast cat ?

1. Leithoa says:

No, the cat is not super fast. It could be more intuitively thought of as two identical cats superimposed in the same space. Until you open the box both dead and live states exist, but as soon as you observe one state the other one is destroyed (ie; the wave-function collapses).

But this is only a paradox in the Copenhagen interpretation. Other interpretations of QM handle this particular paradox more gracefully, but introduce other complexities.

Further complicating things is word replacement in an effort to avoid plagiarizing a sentence:
“… the atoms had an equal possibility of being anywhere within the space of the lattice. ” – HaD

“.. atoms are as likely to be in one place in the lattice as another.” – http://phys.org/news/2015-10-zeno-effect-verifiedatoms-wont.html

The HaD version has more in common with a description of electron orbital behavior than the quantum tunneling phenomenon that is described in the original article. If you can’t write the same thing in your own words, just quote the original article, QM is confusing enough without fighting the English language too.
As far as changing the fundamentals of quantum mechanics it doesn’t sound like it. According to the phys.org article what it may do is allow us to create new sensors that are extremely sensitive. Apparently, once you stop quantum tunneling behavior it doesn’t take much to disturb these ‘frozen’ atoms.

1. Pat says:

In fact, it’s stupider than that.

First, let’s stress – this is a magic box. This box cannot be built. It is completely impossible for it to be built, since in fact you would have to isolate gravity itself. The box’s walls are event horizons (yes, like the event horizon of a black hole).

Inside the magic box (which prevents all information about the cats demise from escaping), there’s a continuum of states superimposed. One where the cat is alive, and then others where the cat died 1 second ago, 1 minute ago, 1 hour ago, etc. All of those states exist inside the box. Until you open it. Then only one exists. Which one exists is probabilistic: some percentage of the time you’ll get a live cat, some percentage you’ll get “cat died 1 second after being enclosed”, etc.

The thing is, there’s nothing bizarre about this, once you realize the box has to be magic. It is literally like the box is inside the event horizon of a black hole – it’s completely separated from the rest of the Universe. So does it make sense to talk about the cat as being alive or dead? That’s the point – it doesn’t. It’s totally disconnected.

Quantum mechanics says “well, it’s some combination of alive and dead” – which is the same thing as saying “The cat could be alive or dead. I have no effing clue.”

Until you open the box. Then you reconnect it to the entire Universe, and you get the entire history as well, since there’s enough of a state space inside the cat system to encode *when* the cat died.

That’s the part people don’t understand. It’s just like ‘passing through the event horizon of a black hole.’ Nothing special happens to you then. It’s just the rest of the Universe that sees something special. Likewise, the cat doesn’t feel anything weird, and it definitely dies at a fixed time. It’s just the rest of the Universe that sees it as something different – something totally unknown.

1. pelrun says:

You’re reducing this to a classical physics interpretation which isn’t correct. The box isn’t recording the full history that you can just read out once you open the box. If that was so, then there’s a definite time t where the cat died, and opening the box gives you a 0% probability of a dead cat if opened before that point, and 100% after, and there’s no superposition. But in reality t isn’t decided until the box is opened! If you could go back in time and open the box again and again at the same time, then you’ll get a different outcome each time.

Also, you claim this magic box cannot exist, which is essentially true at the macro level, but not at the quantum level. I can fairly easily create a particle and send it off into a vacuum chamber (and physicists do it all the time). The only possible way to read information from that particle is to bounce another particle off it (this includes photons) – until that happens, it’s in a very real magic box.

1. Pat says:

“You’re reducing this to a classical physics interpretation which isn’t correct.”

It really is a classical physics situation once the box is opened. It’s only when the box is closed that you have any idea of a quantum situation, and even then the ‘quantum idea’ is stupid because the phase space is so huge.

“The box isn’t recording the full history that you can just read out once you open the box.”

Yes, it really is. Of course it is. How could it not? You think the cat just turns magic-wacko and suddenly freezes in time until the box is open? The cat is only in a superposition of states from the view of the outside Universe. Internally everything’s perfectly fine.

Remember: I’m talking about if you tried this in *reality*. Most Schrodinger’s cat examples that people talk about are talking about things as a thought experiment.

” If that was so, then there’s a definite time t where the cat died, and opening the box gives you a 0% probability of a dead cat if opened before that point, and 100% after, and there’s no superposition.”

Yup! That’s exactly right. Once you open the box, everything’s exactly fixed. But this is obvious. As soon as you interact with a system, it can only be in discrete states, not a superposition.

“But in reality t isn’t decided until the box is opened!”

This is, again, where most people stumble with understanding quantum mechanics. You’re right: t isn’t decided, *from the perspective of the outside universe*, until the box is opened. That’s because “inside the box” and “outside the box” are causally disconnected. They’re functionally 2 different universes.

If you’ve learned about relativity already, this idea is easy to understand: this is all a matter of perspective. Inside the box, *the rest of the Universe is in a superposition of states*. Outside the box, the “cat system” is in a superposition of states.

In the ‘real’ example, there’s nothing interesting you can do with that superposition of bajillions of states that the cat is in inside the magic box. There’s no way you can individually interact with the macrostate in such a way that it depends on all superposed states, like you can with an individual particle.

“I can fairly easily create a particle and send it off into a vacuum chamber (and physicists do it all the time). The only possible way to read information from that particle is to bounce another particle off it (this includes photons) – until that happens, it’s in a very real magic box.”

Yes! Exactly. And that’s because you *can* isolate that particle from all interactions (you cannot do that at the macro level – this isn’t ‘essentially’ true, it is true, thanks to gravity) And because it’s just an individual particle (or carefully contrived system depending on just one particle), you can interact with its superposed states all at once.

2. Pat says:

“I feel this last sentence as highly disturbing. ”

That’s because people use ‘observe’ when they should say ‘interact with.’ All the whole goofy Schrodinger’s cat, EPR experiment, etc. confusion goes away if you just think of it that way. Watch:

““However, when they were *interacted with* at high enough frequencies the atoms remain motionless, bringing the quantum evolution to a halt.”

Now consider that “motionless” also is poorly phrased – if you observe something *in motion*, that means you’ve observed it interacting *with something else*. So let’s replace that.

“However, when they were interacted with at high enough frequencies, the atoms stopped interacting with something else, bringing the quantum evolution to a halt.”

Suddenly the effect seems super-obvious. If you interact with something often enough, it doesn’t have a chance to interact with anything else. Not exactly spooky anymore.

1. Rob says:

Well put! I’m gonna have to remember this important distinction.

3. Smeeg says:

Sort of makes sense in an intuitive way, if the “quantum weirdness” of a system is determined by the density of interaction with the “outside world”.
With a low probability of interaction(measurement), the wave function of the observed system is decoupled from the observer, and evolves independently. With a high probability(repeated measurement), the observer and observed are linked, and maintain relative “phase”.
The macro-world is a special case of repeated measurement.

1. Pat says:

“The macro-world is a special case of repeated measurement.”

Yes! Exactly!

In fact, Penrose has stated that he thinks that gravity is the key to ‘wavefunction collapse’, and if you think about it, of course, he’s right. Macroscopically, you can’t shield gravity. There’s no way you can isolate a system from interacting gravitationally. So, obviously, everything macroscopic is continually interacting. Information leaks everywhere at the macroscale, regardless of how well you try to isolate it or couple it to a single interaction.

Why do we see ‘quantum weirdness’ at all? At the nanoscale, gravity is so weak that the chance of it interacting with the state enough to disturb it is practically zero – so you essentially prevent the system from interacting except in the way that you want.

The mistake in Zeno’s postulus was simple to explain. He chose the wrong window of time to observe. Allow me to explain. If we choose the window to be two set points (mind you this is a completely arbitrary distance set by the observer), then the arrow would in fact never arrive. Let’s assume the points were between the place at the moment an arrow was fired from a bow (as measured from the tip of the arrow) to the point where the surface of a target is located, some 50 meters away. We would want to measure the distance in time that it takes for the arrow to make contact with the target. Using Zeno’s method the time needed would be infinite. The arrow would never reach its target.
However, if we expand our time horizon somewhat and accept that the arrow not only reaches the target, but the enters it, and exits out the back, then Zeno’s problem suddenly disappears. Using the halving of time and the having of distance we never encounter unending problem, as our solution (tip of the arrow hitting the target face) happens at some point eventually.
Simply put, I believe the problem is not with the nature of the universe, but in how one chooses to define the problem so as to be able to apply a correct mathematical model.
This is not unlike the old statistical quandry where one is asked, ‘if you roll a six sided die six times and each time the number 3 appears, what is the likelihood that on the seventh to roll it will also be the number 3?’ The answer is 1 in 6. The probability of rolling the number 3 seven times in a row however (which this last roll will in effect bring about) is substantially smaller.

1. spritle says:

So when determining when the arrow will reach “the point where the surface of a target is located”, if that is not including the target but the space immediately before it then during that observation distance the arrow never does reach it.

5. m_wilson271 says:

I wish they would stop using the term “observed” and instead replaced it with “shot with a laser”. Observation in this case is adding some form of energy to the system. Using “observed” could imply that it takes an intelligence’s awareness to make things change.

1. dex drako says:

+1

6. TheRegnirps says:

A nit to science and tech writers. There are low temperatures but no such thing as “cold temperatures”. Extreme low temperature is preferred to “extreme cold” since coldness is not a measure and has no units. Same with high temperature versus hot temperature. Hot temperature is like saying “this morning the distances are extra long”. What does that mean? The meter got bigger overnight?

“Coldness” of atoms is tougher. In groups it can make some sense in terms if relative motion and kinetic energy. These LASER trapped atoms are very low temperature relative to each other as they are sorted and atoms that are not in step escape, leaving a bunch of nearly identical motions. But imagine a single atom in space going at any speed you wish. If your frame of reference is the same as the atom, what is the atom’s temperature?

7. Anonymous Coward says:

A possible problem I can see with the experiment is that 1) we are sucking most of the thermal and thus potentially kinetic energy out of the system and then 2) injecting unidirectional thermal and kinetic energy into it with the laser/s in order to “observe” the particle. We are hardly leaving it alone in order to see if “pure” observation effects it which seems to be the higher purpose of the (thought) experiment. It’s already been shown that lasers can trap particles in their interference pattern.

8. James says:

The linked article is kind of short on details but I would like to try this hack at home. I have some dry ice and a laser pointer so I’m good to go there. Do I have to count out the one billion rubidium atoms or can I just weigh them on my kitchen scale?

9. Tom says:

Some possible problems I can see with the experiment are that 1) we are sucking most of the thermal, and thus potentially kinetic, energy out of the system and then 2) injecting unidirectional thermal and kinetic energy into it with the laser/s in order to “observe” the particle. We are hardly leaving it alone in order to see if “pure” observation effects it, which seems to be the higher purpose of the (thought) experiment. It’s already been shown that lasers can trap particles in their interference pattern. Are we truly getting the experiment right?

1. Pat says:

Observation requires interaction. There is no such thing as ‘pure’ observation.

You might say ‘but do they have to interact with it so much?’ In fact, the fact that you see quantum interference patterns implies that they’re actually *minimally disturbing it.* They’ve reduced the system down to the point where it’s interacting only with the things they want it to.

1. Tom says:

True, but there is observation which requires less interaction. One form observes/recognizes when a state change has occurred in the original item being observed. (Example: A plucked string vibrates and produces a sound. Your ear detects the sound traveling over the transmission media, air which was already present, and you recognize the state the string is in as vibrating.) The other form introduces a new carrier media or some other circumstance which can also interact with the item being observed. (Example: A laser is shined onto the string. An observer watches the laser for interference patterns to determine if the string is vibrating. However the laser is now also interacting with the string and can change the state of the string.)

1. Pat says:

“True, but there is observation which requires less interaction.”

Not in this case. This *is* minimal interaction.

You might think removing thermal energy from the system is interacting with it. It actually isn’t: the ‘system’ that they want to observe are the atoms, alone. The thermal energy is actually the atoms interacting *with each other,* a lot. So you cool it down because you don’t want them interacting (‘observing each other’).

Then the lasers themselves are minimally disturbing. How do you know? Because the interference patterns resume after the beam. The interference patterns literally are the atoms saying “nothing else is interacting with me.”

1. Tom says:

The cooling is a good point but not what I was originally suggesting. By cooling it they are changing it from it’s normal state, so they are effecting the material prior to the experiment. The atoms are not “behaving normally in their normal medium”. However rereading the experiment, the description made says that they are cooling in order to isolate out non-kinetic motion by tunneling so your argument does bear weight. In addition it could argued that cooling is a form of observation. I really like Pat’s use of INTERACTION versus OBSERVATION.

As for the laser though, the interference patterns they are observing seem to be the ones caused by the laser shining through the matrices. From the description the laser seems to be on at all times and only varied in intensity: “The researchers observed the atoms under a microscope by illuminating them with a separate imaging laser. A light microscope can’t see individual atoms, but the imaging laser causes them to fluoresce, and the microscope captured the flashes of light. When the imaging laser was off, or turned on only dimly, the atoms tunneled freely. But as the imaging beam was made brighter and measurements made more frequently, the tunneling reduced dramatically”. It is true that this may not effect the tunneling behavior they were looking for but I’m not sure that is a proven fact. (And I’ll admit it’s been a while since I took Quantum… :) But I’d really need to read the abstract to know if the laser is being used to generate the pattern, or if they are using another method, and how much it effects the behavior of the atoms.

In the end I am not arguing for or against the experiment or the results so much as saying you need to carefully look at the experimental method. There can be unexpected interactions in any experiment, both from internal and external factors.

2. Pat says:

What they’re observing is pretty simple.

You have an atom, embedded in a lattice, at super-cold temperatures so it only really interacts with the lattice via tunneling effects – it can’t classically move around. But that tunneling can lead to lattice sites where 2 or more atoms get trapped. If those atoms get hit by a laser photon, they pop out, and you lose the atom. So the rate at which you lose atoms determines how much the atoms are tunneling around (e.g. how localized they are).

So then you measure the rate that you’re losing atoms, compared to how often you fire the laser. If the atoms are undisturbed, happily tunneling around, you’ll lose atoms at the same rate. Which is what they see. (So again, this is the point – the laser is not disturbing the atoms. If it was, they wouldn’t be able to tunnel, and you wouldn’t be losing them at that laser-off rate).

Then you slowly turn up the rate at which you hit them with the laser. And you find that at a certain point, you lose fewer, and fewer atoms. Which means that the laser is localizing them and preventing them from tunneling.

(Again, not terribly surprising, if you understand that ‘observing’ is ‘interacting’ – you’re whacking them with something that puts them in a known state, and then hitting them *again* soon after, so it’s not surprising that it doesn’t have a chance to end up somewhere else).

2. Tom says:

Or maybe I should have said “True as far as we know it”. If you take some of the more fun theories from Science Fiction then conception of an idea might be thought of as conception of reality. The mere thought of thinking/imagining something is enough to create it in the infinite universes and all the fictional worlds are true because they were imagined and thus perceived by their author. Which really sucks for most of the inhabitants because authors need to create conflict and tragedy most of the time to get interest from the readers.

10. Tom says:

“No fair! You changed the outcome by observing it!” – Hubert Farnsworth at a horse race

1. krinje says:

actual quote is “… measuring it!”

11. I see we are doing experiments in time travel as well. This same experiment was performed 25 years ago with a pot of Beryllium atoms (Search using your favourite search engine for “quantum pot never boils” http://articles.latimes.com/1990-03-17/news/mn-216_1_zeno-effect). It’s more a case of the collapse of the wave equation for a quantum system when it is observed; observe it often enough, the system returns to a ground state, and nothing escapes. Without an observer, the atoms can accumulate enough energy that there is a non-zero probability of their escape, and they do.

12. Heph says:

So Atoms are essentialy weeping angels and the camera shouldnt blink?

1. Skull says:

You sir have passed the sanity check. Now the next time you think you understand anything about quantum physics make another. You will eventually fail a check and become a great Quantum Physicist :p

13. jdarling says:

I’m no Physicist or Mathematician, but doesn’t the physical world come into play here at some point? There are known particle size limits and once you reach (say 1/2) the size of the particle limit then I would think that space/time is no longer divisible and thus movement would inherently (and observably) stop? This also seems to support Numberphile’s proof (via adding 1/2 S to S) that eventuality the problem resolves for well behaved values.

1. Skull says:

I think people are forgetting the purpose of a purely theoretical problem. Its not about the real world but rather about getting into a frame of mind for these kinds of mathematics To attempt to apply it to the real world is pointless and accomplishes nothing. Also mathematicians didn’t have YouTube or Hack a Day, so thought problems were the thing to do.

2. twdarkflame says:

There are (we think) limits to the size of things.

However at the moment (to my knowledge) physicists generally think positioning in space is continuous. It isn’t, like, pixels at some small level.

3. Tom says:

Physicists go back and forth on this. Quantums of space (discretely measurable chunks) have neither been proven nor disproved. Either would have interesting repercussions on our understanding of the universe. So far study has been concentrated more on quantums of particle size with the idea that once we find the smallest quantum of things we might have found a limit to the size of space. Vice versa, if something is large then it may not be possible for a quantum of space to be smaller than it unless the large thing is made up of smaller things. And that’s a VERY simplified version of the idea.

14. Sim says:

So the Weeping Angels from Dr. WHO are real ?

15. Greenaum says:

Just a note that the popular example of this is a tortoise racing Achilles. If the tortoise gets a head start, which is only fair, for every step Achilles takes to close in on it, the tortoise takes a further part-step, moving away from him.

16. Steel_9 says:

Just remember “it takes much longer to get up north, the slow way”

-Ian Dury

17. Leonard says:

It’s fine if you divide a distance by two, but please also divide the amount of time required to reach that destination by two. Some mathematical parameters will just go to either to infinity or to zero, that’s all.

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