24 Apr Is motion an illusion? (JIM AL-KHALILI)
The Dichotomy is the next of Zeno’s paradoxes and it is very simple to state:
In order to reach your destination you must first cover half the distance, but in order to cover half the distance you must first cover a quarter of the distance, and in order to cover a quarter of the distance you must cover an eighth of the distance, and so on. If you can keep chopping the distances in half forever, then you never reach that very first distance marker, and so you never actually start your journey. What’s more, this never-ending sequence of ever-shorter distances is infinite. So to complete the journey requires you to complete an infinite number of tasks. So you could never finish it. If you cannot start the journey and would never finish it, then motion itself is impossible.
We learn about this paradox from Aristotle, who knows that it is nonsense but searches for the logical argument with which he can refute it conclusively. After all, it is quite obvious that there is such a thing as motion. However, Zeno was applying a form of argumentation called reductio ad absurdum, which is the reduction of an idea to absurdity by demonstrating the inevitably absurd conclusion to which it would logically lead. We must also remember that Zeno was no mathematician. He was arguing with reference only to pure logic, and that is often just not enough. Other Greek philosophers resorted to a more direct and pragmatic approach in refuting Zeno’s arguments about the illusion of motion. One of them was Diogenes the Cynic.
Diogenes took the teachings of Cynicism to their logical extremes. He seems to have made a virtue of poverty and spent years living in a tub in an Athens marketplace. He became famous for being, well, cynical about everything, particularly much of the philosophical teaching of the time, even that of eminences such as Socrates and Plato. So you can imagine what he thought of Zeno and his paradoxes. On hearing about Zeno’s Dichotomy Paradox regarding the illusion of motion, he simply stood up and walked off, thus demonstrating the absurdity of Zeno’s conclusions.
While we may applaud Diogenes for his practical approach, we still need to investigate a little more carefully where Zeno’s logic is breaking down. And that turns out not to be so difficult—after all, we’ve had over two thousand years to figure it out. In any case, while you may feel that sheer common sense is sufficient to dismiss Zeno’s paradox, I do not. I have spent most of my life working and, more importantly, thinking as a physicist, and I am not satisfied with mere commonsensical, philosophical, or logical arguments that refute the Dichotomy. I need watertight physics—which, for me, does a far more convincing job.
What we need to do is to convert Zeno’s argument about distance into one about time. Assume you are already moving at a constant speed at the moment in time when you are at the starting point of the journey to be covered. The notion of speed, which Zeno would not have understood very well, means covering a certain distance in a finite time. The shorter the distance you must cover, the shorter the time interval needed to cover it, but whenever you divide the first number by the second they always give the same answer: your speed. By considering shorter and shorter distances that must be covered before you begin your journey, you are also considering shorter and shorter intervals of time. But time marches on regardless of how we might wish to split it up artificially into these ever-decreasing periods. Thinking about time, rather than space, as a static line that can be subdivided indefinitely is fine (and we often think of time in this way when solving problems in physics), but the crucial point is that the way we perceive time is not as a static line in the same way as we can view lines in space. We cannot take ourselves outside of time’s stream. Time marches on regardless—and so we move.
If we consider the situation from the point of view of someone not already moving, but starting from rest, there is just one more bit of physics we need to think about. This is something we all learn about at school (and most of us, no doubt, promptly forget). It is referred to as Newton’s second law, which states that to make an object begin to move, a force needs to be applied to it. This will cause it to accelerate—to alter its state from being at rest to being in motion. But once it is moving, the same argument applies: namely, that, as time goes by, the distances covered are based on the moving object’s speed, which need not be constant. The Dichotomy argument is then an abstract irrelevance that has nothing to say about true motion in the physical world.
I should make one final remark before moving on. Albert Einstein’s theory of relativity teaches us that maybe we should not dismiss the Dichotomy Paradox so confidently. According to Einstein, time can be regarded in a similar way to space—indeed, he refers to time as the fourth axis, or fourth dimension, of what is called space–time. This suggests that maybe the flow of time is just an illusion after all—and, if it is, then so is motion. But I would argue that, despite the success of relativity theory, this conclusion takes us away from physics and into the murky waters of metaphysics—abstract ideas that don’t have the solid backing of empirical science.
I am not suggesting that Einstein’s theory of relativity is wrong; of course not. It is just that Einstein’s ideas only really manifest themselves when things start to move very fast—close to the speed of light. At normal everyday speeds we are quite within our rights to ignore such “relativistic” effects and think of time and space in the familiar commonsense way we are used to. After all, if we push Zeno’s argument to its logical extreme, then it is in fact wrong to say that time and space are infinitely divisible into ever-smaller but still discrete intervals and distances. At some point things get so small that quantum physics comes into effect, when time and space themselves become fuzzy and indefinable and it no longer makes sense to chop them up into smaller pieces. Indeed, motion itself is a little illusory in the quantum domain of atoms and subatomic particles. But that is not what Zeno had in mind.