Racing tortoises
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Synonyms
Definition
Welcome to the Great Athenian Man–Tortoise Run-off. My name’s Zeno and I’ll be your commentator for the big race. I have to say, however, that the result is a foregone conclusion. Achilles has made the terrible mistake of giving Tarquin the tortoise a 100-yard head start. Let me explain.
Tarquin’s tactic is to keep constantly moving, however slowly. If Achilles is to overtake Tarquin, first he must get to where Tarquin is when the race starts. That will take him several seconds. In that time, Tarquin will have moved on a little and will then be a short distance ahead of Achilles. Now if Achilles is to overtake Tarquin, he must again get to where Tarquin is first. But in the time it takes Achilles to do that, Tarquin will again have moved forward slightly. So, Achilles once more needs to get to where Tarquin is now, in order to overtake him, in which time, Tarquin would have moved forward. And so on. You get the picture. It’s just logically and mathematically impossible for Achilles to overtake the beast.
Still, it’s too late to place your bets on the tortoise now, because they’re under starter’s orders, and … they’re off! Achilles is closing … closing … closing … Achilles has overtaken the tortoise! I can’t believe it! It’s impossible!
Source
The ancient paradox of Achilles and the Tortoise, attributed to Zeno (born c. 488 BCE)
Motivation & Background
Zeno’s explanation of why Achilles can’t overtake the tortoise is a paradox, because it leads us to the conclusion that two incompatible things are true. The argument seems to demonstrate that Achilles can’t overtake the tortoise, but experience tells us that of course he can. But there seems to be nothing wrong either with the argument or with what experience tells us.
Some have thought they can identify a flaw in the argument. It works only if you assume that time and space are continuous wholes which can be divided up into ever smaller chunks ad infinitum.
This is because the argument depends on the idea that there is always a length of space, however small, over which the tortoise will have moved on a little distance, however short, in the period of time, however brief, it takes Achilles to get to where the tortoise was. Perhaps this assumption is just wrong. Eventually you reach a point in time and space that can’t be carved up any smaller.
However, this in itself simply creates different paradoxes. The problem with this idea is that it claims the smallest unit of space essentially has no extension (length, height or width) because if it did, it would be possible to divide it up further and we’d be back with the problems of the race paradox. But then how can space, which clearly does have extension, be made up of units which do not themselves have extension? The same problem occurs with time. If the smallest unit of time has no duration and so cannot be divided any further, how can time as a whole have duration?
So we are left with a paradox of paradoxes: two paradoxes, both of which seem genuine, but which, if both are true, would make the only two possibilities impossible. Confused? Don’t worry – you should be.
There is no simple way out. Solutions actually require quite complex mathematics. And this is perhaps the real lesson of the tortoise race: armchair theorising using basic logic is an unreliable guide to the fundamental nature of the universe. But that in itself is a sobering lesson, because we rely upon basic logic all the time to spot inconsistencies and flaws in argument. It is not logic itself which is at fault: the more complex solutions to paradoxes such as these themselves depend on holding the laws of logic firm. The difficulty is rather with applying it.
Cross-References
Recommended Reading
The ancient paradox of Achilles and the Tortoise, attributed to Zeno (born c. 488 BCE)
