Zeno’s Paradoxes of Motion

Zeno’s Paradoxes crop up occasionally. There are several paradoxes labelled as Zeno’s. Two of these are: Achilles and the Tortoise, and, the dichotomy paradox (or the race course paradox)

These are all basically the same sort of problem, about breaking down motion into an infinite series of ever smaller steps.

The runner of a race goes like this:

A runner starts running the length of a race track. At the half-way point he has covered half the distance, with half remaining. When he gets to half the remaining distance he still has the other half of that second distance remaining. At each ever shorter half-way point he has covered half the distance remaining, but always has half the distance to complete. This continues, with remaining half-distances remaining, indefinitely. He never completes the race.

What’s actually happening, with the runner of a race, is we tend to assume him to be traveling at a constant speed S, and yet the never ending half distances make it appear that he never completes the race. Paradoxical? Only if you think of a normal runner at constant speed, while thinking of this runner with each half length taking the same time.

Constant Speed

Supposing that we look at his position, x, at times, t, and X is the length of the track:

x0, x1, x2, x3, …, xn, x(n+1), …
t0, t1, t2, t3, …, tn, t(n+1), …

The difference between each is:
dxn = x(n+1) – xn: dx0 = x1 – x0, dx1 = x2 – x1, …
dtn = t(n+1) – tn: dt0 = t1 – t0, dt1 = t2 – t1, …

The paradox tells you that dx is being halved each time, but may neglect to tell you that if the runner is moving at constant speed then dt is also being halved each time:

dx1 = dx0/2, dx2 = dx1/2, …
dt1 = dt0/2, dt2 = dt1/2, …

The time intervals are halving too.

So, the speed is still constant: dxn/dtn = S, period to period.

Constant Time Intervals – Reducing Speed

The other view of the paradox, the one presented when looking at the detail, thinking only about the distances, leaves you with this impression:

dx1 = dx0/2, dx2 = dx1/2, …, dx(n+1) = dxn/2
dt1 = T, dt2 = T, … where T is constant. If the paradox is expressed with the intention of declaring a never ending race then this will be stated explicitly.

So, s the speed is halving each step:

sn = dxn/T = dx(n-1)/2T = s(n-1)/2

which would take infinite time to complete.

So, what you think you’ve got is a half-life-like expression: where the distance is asymptotic to the end point and never actually reaching it, in principle:

Stopping Time, or Never Ending?

Once the constant speed distance intervals become smaller than a part of the runner’s foot, and the time intervals become smaller than we can detect, he seems to become stationary. You’ve stopped time – it’s as if the runner can never complete the race. But of course you are supposedly looking at the runner in real time (in your mind’s eye), while being able to see these ever smaller distances and times, as if you see him stop. But in reality, at constant speed, these smaller intervals flash by in real small times and we do see the runner complete the race.

With the constant time intervals the runner actually runs slower and slower in real-time. Again he appears to become stationary because each interval (let’s say T = 1s) his distances are really getting smaller and smaller each second. Asymptotically, in principle, he never reaches the end – or, more specifically, the total time (the sum of all the T’s), extends indefinitely with the number of intervals. He would have to run so slow as he approaches the end that he would again appear to freeze.

Finite sum of a convergent infinite series

Another way of looking at the constant speed scenario is by summing intervals.

Assume the total distance is 100m, and the time taken to travel the distance is 100s. The speed S is:

S = 100m/100s = 1m/s (Ok, so he’s 10 times slower than Carl Lewis on his first 10s 100m)

Normalised to unitless 1 (i.e. 100m = 1 and 100s = 1):

S = 1/1 = 1

The paradox states each distance interval is halved, and in total you have a finite sum from an infinite series:

1 = 1/2 + 1/4 + 1/8 + …

The same must apply for the time intervals:

1 = 1/2 + 1/4 + 1/8 + …

So, again, the speed is 1/1 = 1.

If Ttot is the time to complete the race: Ttot = S/X

Then half the distance at constant speed: Th = Ttot/2

The series view is then:

Ttot = Th + Th/2 + Th/3 + …
Ttot = Th + Th(1/2 + 1/3 + …) = 2Th

Infinite sum of a divergent infinite series

For the constant time interval we have a divergent series:

So, the times at each x are:
x0, x1, x2, …
0, T, 2T, …


Ttot = T + 2T + 3T + … -> infinity

For a more detailed look at the maths of all this try these pages from S. Marc Cohen at Washington.


What we are led to believe is that we have a paradox concerning a single type of motion, with contradictory results. But it is simply two definitions of the motion of travel, neither of which is explicit in the statement of the paradox, and neither of which is problematic in isolation.

Some formulations may state explicitly that the time intervals are constant – in which case there is only one single description of motion, and you just have to avoid the trap of having the motion of a constant speed runner in the back of your mind to avoid coming over all paradoxical.

So, it’s not a real paradox:

“a statement or proposition that seems self-contradictory or absurd but in reality expresses a possible truth.”

The ‘paradox’ contains two actual truths masquerading as a self-contradictory single truth, or a single truth that you have mistaken for two. You have been conned.

Getting Real

Constant speed
We could object that it takes time to accelerate at the start of a race (even at 10m/s), but the example here just simplifies the case of a variable speed natural runner.

Constant Time Periods
The constant time interval is even more unrealistic. There is a difficulty with infinitesimal distances, even in the case of the constant time intervals. The atom on the tip of the runner’s nose is sufficiently large to jiggle back and forth across the finish line as the runner is halving his speed on approach to the finish. And good luck with dealing with this as you approach the Planck length.

Half Life:
In practice the half life of a piece of caesium isn’t infinite. A mole of caesium contains 6.022 × 10^23 atoms – which is a finite number, so it would eventually decay to zero.

But, it is a thought experiment after all, and philosophers, even back in Zeno’s time, weren’t too hot on thinking through thought experiments. Don’t get me started on ‘the heap‘.

It’s True! – Harry Redknapp

I posted on the nonsense that is the It’s True! claim, which theists are apt to use directly themselves, or which is the basis of the supposed truth of their religious books – that the books contain words proclaiming the truth of the books. LOL!

Well, of course religion isn’t the only source of this stunning[ly useless] argument. It turns out that it’s a favourite defence of the guilty in a court of law, and no doubt some innocent people too, when they become desperate.

Harry Redknapp used this very move in court yesterday. The story so far, from the Guardian piece

Redknapp and the former Portsmouth owner Milan Mandaric, who both deny two counts of cheating the public revenue, have told the court that the $145,000 paid in May 2002 and a further payment of $150,000 in May 2004, were paid by Mandaric as “seed money” for investments to be made on Redknapp’s behalf.

Here, from the BBC

Under cross examination on Thursday, Mr Redknapp said he lied to reporter Rob Beasley about the source of payments to the account because he did not want negative stories ahead of a cup final.

The Tottenham boss said: “I have to tell police the truth, not Mr Beasley – he’s a News of the World reporter.”

So, he lied to the reporter, but he didn’t lie to the police and he didn’t lie in court, honest. It’s true!

Remember, Hitler put his signature to the Munich Agreement, of which Chamerlain said on arriving home to England, “This morning I had another talk with the German Chancellor, Herr Hitler, and here is the paper which bears his name upon it as well as mine…“. Churchill didn’t buy it, “We have suffered a total and unmitigated defeat… you will find that in a period of time which may be measured by years, but may be measured by months, Czechoslovakia will be engulfed in the Nazi régime.

Of course, Hitler subsequently, metaphorically, wiped his own arse with his copy, and flushed.

All gullible people please take serious note! Liars will not only lie, they tell you they are telling the truth when they are not! It’s part of what they do. It’s what it is to be a liar! When someone says, “It’s true!“, without further justification, then you not only don’t have to believe them, I really recommend you ignore their proclamations and seek evidence of the truth.

Which of Harry’s truths is a lie? Because they are inconsistent. He claims he lied to the press. But if he admits to lying there, can he be trusted now?

Well, perhaps this is convincing, from the Metro newspaper…

Harry Redknapp - Not a liar
It’s true!

The Spurs football manager was even reduced to shouting from the witness box at prosecutor John Black QC.

“You think I put my hand on the Bible and told lies?” he exclaimed. “That’s an insult, Mr Black, that’s an insult.”

No Harry, it’s his job. The whole point of this hand on the Bible bollocks (funny how religion is in on the lies again) is that it doesn’t work, except when told to the gullible. If you are innocent of the charges Harry, the Bible won’t help, and if you’re guilty, the hand on the Bible isn’t working.

“Everything I have told you is the truth, the whole truth, and nothing but the truth, so help me God…”

“I am not a liar.”

Well, that’s that then. Case dismissed? FFS!

Post Script…

Harry is currently manager of Tottenham Hotspur Football Club, who denied my beloved Manchester City a place in the Champions League three years ago, though we turned the tables last year. They are now one of the serious contenders to our title hopes this year.

But, Harry is a great football manager, and Spurs are a great club that I actually like. I have no idea whether he’s guilty or not – that isn’t the subject of this post. So this post is in no way biased. Honest! It’s True, I tell you, it’s true!

It’s True!

Great Jesus & Mo cartoon again.

The best bit, and the most crucial bit that applies to all religious books is number 1:

1. This is true

This is the one and only necessary assumption in any religion to make it worthy of the name. It must declare its own truth. Of course (snigger 🙂 (more smugness to come) we all know this is pure bollocks don’t we.

I get regular visits from Jehovah’s Witnesses. They are really nice people (at least to ones who visit me are). I think they like me and return often for the following reasons:

(a) I don’t slam the door in their faces;
(b) they haven’t converted me yet, so I suspect, like a lottery rollover, my cache goes up with each rejection;
(c) the religious are masochists (why else invent sin and then admit to bing up to the neck in it).

Anyway, I keep two things handy for when they call.

The first is a copy of Darwin’s On the Origin of Species. They always quote from it incorrectly and it’s an easy book to show them they’ve been misled. They do go through some other topics, such as DNA, irreducible complexity, but I usually wing it once we get past Darwin, because it would take too long to go through any book to convince them (“Let’s start with basic chemistry…”). I know, it’s wrong of me, but I argue from authority of knowing just a bit more about this stuff than they do. As much as I can make points against Behe’s arguments generally, if they brought him along as a guest doorstepper I might be screwed on the biology, because I’m not a biologist.

But I digress, again. The second, and most important thing I keep nearby is a piece of paper, oh and a pen – so that’s three things I keep near by, but you know what I mean.

And on that piece of paper I write words to this effect:

This paper contains the word of God as revealed to Ron Murphy. If you’re here while he’s writing this you too must bear witness to this miracle. Now, as your God I command you to ignore the Bible, Koran, gold tablets and any other bollocks you may have come across telling absolute crap about me. I can swear by the way. I am God after all. Though the atheists got it wrong, they got it wrong for the right reasons – they don’t believe any old crap in a document claiming to be the revealed truth. What sort of fucking argument is that?! Anyway, on this occasion it happens to be the truth. However, I’ll forgive you not believing it if you throw it in the bin. On one condition: you throw your crappy book in the bin too and start thinking for yourself.

No, I don’t really write all that, I just feel as though I want to. A sentence or two is usually enough to make the point.

But, miracle of miracles, their faith survives even this cutting blow. What the fuck can I do?

I thought perhaps I should show them the fabulous Morwenna Banks, from that brilliant series Absolutely.

This for me says everything about religious imaginative invention. It encapsulates millenia of theology as Little Girl rationalises uncertainties and contradictions in what pops into her head.

And her punch line:

It’s true! I know because I do!

And Little Girl here provides us with the best accommodationism I’ve ever come across: Genesis plus Evolution! It’s true!


And this is what you get when It’s true! is put into practice. Enjoy!