«Even the longest journey must begin where you stand.»
Lao-tzu (604 BC - 531 BC), The Way of Lao-tzu
To Infinity ...
This article looks at one of the most famous ancient paradoxes - the Dichotomy paradox of Zeno of Elea - which claims that all motion is an illusion. One statement of this is that you cannot leave the room you're in.
Common experience shows that you can leave the room you're in so why does Zeno claim you can't, and why do people take this claim seriously?
Zeno produced several paradoxes to support the claim that motion is just an illusion and they have troubled philosophers for more than two thousand years. These paradoxes are usually expressed as 'thought-experiments' where we imagine a simple experiment that could take place in the physical world.
Zeno's Paradox The subject of this thought-experiment - you - is assumed to be standing exactly one metre from the door of the room you're in - the length of the fat line in the picture. (We assume that one can measure lengths with perfect precision - the sort of assumption that's allowed in a thought-experiment, but not in the real world.)
To get out of the room ...
you must first get to a point half way to the door (marked 1) - but when you do, you still have halfway to go ...
now you have to get to another point half way between the first point and the door (marked 2) - but when you get there you still have a quarter of the way to go ...
At the next step (marked 3), the distance and time are halved again leaving you still to go an eighth of the way ...
And so it continues - you are left with 1/16, 1/32, 1/64, ... to go so you never quite make it to the door.
The most popular resolution of this Zeno paradox is to point out that the first step is traversed in 1/2 second, the second in 1/4 second and so on. The total time taken to leave the room is therefore 1/2 + 1/4 + 1/8 + ... which gets as close to 1 second as we like provided we iterate often enough. But although we can get as close as we like to 1, we can't ever get to the end of adding an infinite number of steps, however small they might be.
And there is something quite compelling about this objection.
Even if you accept that the sum of all the terms of the geometric series a(1 + r + r2 + r3 + ... ) is a/(1-r), which is 1 if you take a=1/2 and r=1/2 as in our thought-experiment above, you are still left with the problem that you can never in practice add enough of them to reach it. In mathematical terminology, the limit of the sequence 1, r, r2, r3, ... is not a member of the sequence - there is no number n such that rn is equal to the limit.
Another Way to Resolve Zeno's Paradox
This seems to avoid the problem I described. It begins before anyone tries to leave the room.
And this is the point: in order for you to know you're 1 metre away from the door, someone must have already obtained that information and told you.
This must have happened before you start to move.
But if we agree with Zeno, then you cannot know how far your first step is to be, because that information can never reach you.
Thus Zeno is defeated by his own argument: his primary assumption - that you can know how far away you are from the door - is self-contradictory.
Enjoy your day on the beach!
... And Beyond
In a future blog I'll look at what happens when you extend Zeno's argument.
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