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Zeno's singularity
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Zeno's singularity
Zeno's prediction Zeno lived about 2400 years ago, and presented this story and its tricky puzzle. Achilles was fair-minded, and so when it came to a race with the Tortoise he could do no other than give the shelled one a good start. A little time into the race the distance between the two racers is half of the original. That is what we’d expect. A little later again the separation of the two has halved again. Then it halves again, and again, and again. The problem seems to be that the separation can always be halved, and each time Achilles gets closer. But if the distance between them really can always be halved then Achilles can’t catch up. The race, according to this analysis, might seem to be expected to go on for ever. Faced with such a clash between logical argument and knowledge based on experience, which should be abandoned? The prediction from experience We are very attached to our experiences, and they suggest that Achilles not only catches up with the Tortoise but runs on to win the race. A clock shows the start situation, the situation when Achilles has made up half of the ground on the Tortoise, and the situation as Achilles draws level and is overtaking the reptile. For neat simplicity, the clock has been set up so that Achilles reaches the Tortoise after exactly one complete turn of the hand. The graph shows what we would expect to see if we measured the distance between the racers and plotted it. The Tortoise’s lead starts off with a large value and steadily decreases until it’s zero, and then becomes negative once Achilles is in front. This is what might be called ‘sound common sense’. But cjommon sense on its own is not enough to defeat Zeno’s logic.
Time as well as distance For some people, a good puzzle is irresistible, and Zeno’s tale of Achilles and the Tortoise is as good a puzzle as any you’ll find. So various solutions have been suggested. Aristotle lived roughly one hundred years after Zeno, and pointed out that we need to remember that as the distances between Achilles and Tortoise decrease by a half, so do the time intervals. The clocks here show this. In this case Achilles’ narrowing of the gap with the Tortoise does not take all of eternity, but one turn of the hand of the clock. This is now fully in agreement with experience. The graph shows the same thing in different visual format. The halving of distances and of time intervals results in a straight line graph, exactly as experience predicts. But there is still a problem. Just as the paradox suggests that Achilles’ victory in the race is impossible, so it also tries to tell us that time itself is not possible. It seems to be saying that the hand of a clock cannot move any further than Achilles can. Any period of time from now until some imagined moment in the future can be halved, and halved, and halved, without end. Zeno’s ultimate and very rational conclusion is the Universe is a single timeless and changeless point. Nothing could be more beautifully simple, or more ideal. If rather boring.
A blurry Universe Is there any escape at all from the most boring Universe imaginable? It might be possible if we get rid of ideals like points and instants, and indeed infinity – which are describable in mathematics but only experienced through mathematics – then Zeno’s argument is built on fancy and we all get our Universe back. If there are no infinities or eternities, no points or instants, then times and distances get blurry when they are very small. That produces a fundamental uncertainty in distinguishing one very short time from another or one very tiny distance from another. The Universe when seen very close up, even to those with the very best spectacles, would be out of focus. It would blurry to the viewer and blurry in its nature (if those are different things). In that case, when Achilles is very close to the Tortoise he is effectively alongside. So, in fact, he can catch up.
Going round in circles Aristotle introduced the notion of time to try to show that change is possible. If part of Zeno’s conclusion is that time is impossible then Aristotle is not going to prove anything by arguing against it if he assumes the existence of time from the beginning. You can’t say that time exists and then argue from that to a certain conclusion that time really does exist. That would be going round in a circle, just like the hand of the clock (if only circular motion, or any other kind, were possible). Zeno destroys the Universe and replaces it with a single timeless point – an ideal that can only be imagined. No space, no time. A singularity. To do so he relies on the existence of other ideals – of purity of number, of the reality of points and instants, of zero and infinity. He uses ideals to create his idealistically simple world. He goes round in a logical circle. The paradox is self-consistent but proves nothing. It is, at least, no worse than that, to abandon ideals such as points and instants to show that the world is not ideal. Like Zeno’s argument, in such an approach the starting point and the conclusion are self-consistent. Which leaves me free to choose Zeno’s perfect point or a scruffy old world in which all kinds of events can happen. I don’t find the choice very difficult. But, like many choices, it has consequences. We have to walk away from the purity of mathematics into a fuzzy Universe. The ideals on which mathematics is built have been with us for a very long time. Mathematics has unmatched value in taking the human mind to places it couldn’t go on its own. But just because it helps us to describe and predict the world, often in new ways, it doesn’t follow that it can tell us everything. Mathematicians have developed highly sophisticated ways and means of escaping from the embarrassment of the endless race between Achilles and the Tortoise. But there is still debate amongst mathematicians and philosophers about whether these actually solve the problem, or whether they are merely self-consistent and only serve to take us in ever more elevated circles. The choice between the ideal and the non-ideal is still one that we are free to make.