A few years ago, I had a dream. I don’t usually have a memory good enough to recall dreams from years hence. But this one registered with me, as I wasn’t fully asleep, and it’s content troubled me.
I was on a beach, barefoot, and walking toward a wooden branch protruding from the sand some distance ahead. As I continued, the distance from my starting point grew larger, and the distance to my target grew smaller. Yet I couldn’t reach it. As I tried to make sense of this I struggled with an abstract concept. Stuck in some sort of semi-conscious limbo, my waking mind fought with my sleeping one as I watched a seemingly paradoxical vision. In order to reach a given point, one must first complete half of the distance. Once complete, half the distance remains, and yet in order to complete the journey, half of the remaining distance must be covered again, and again, and again ad infinitum.
For reasons known only to my fictional psychiatrist, this both kept me from getting fully to sleep, and from realising I needed to wake up, and was one of the most uncomfortable experiences of my life. This was not the first or last time that a self-induced pointless confusion has kept me awake. I’ve described these dreams to friends in the past, and succeeded only in adding to their list of reasons to think I’m ‘not all there’. They may have a point.
This week, I described the dream to my brother, my twin, and probably the only other person I know who is similarly loopy (Like me, he would take such an insult as a complement). He is also a PhD student of theoretical physics, and likes to read. To my surprise, he not only understood my seemingly ridiculous quandary, but off the top of his head he was able to put a name to it; It is one of Zeno’s Paradoxes!
My dream was a visual interpretation of the dichotomy paradox.
That which is in locomotion must arrive at the half-way stage before it arrives at the goal.– as recounted by Aristotle, Physics VI:9, 239b10
Solutions to such a paradox may seem obvious, in that any of us has an innate understanding of the physics of the world around us, and we all know that we move analogously from one point in time or space to another. However Zeno’s paradoxes have long been the subject of debate whilst philosophers and mathematicians attempt to quantify and solve them. Some of the proposed solutions are covered in the Wikipedia article above, and make for interesting reading. The solutions mainly deny the paradox in physical terms by denying the possibility of a finite task being broken into infinite subtasks. In mathematical terms, Zeno’s Dichotomy can’t natively define a finite series. However, the faulty logic in Zeno’s argument is often seen to be the assumption that the sum of an infinite number of numbers is always infinite, when in fact, an infinite sum, for instance, 1 + 1/2 + 1/4 + 1/8 + 1/16 + 1/32 +…., can be mathematically shown to be equal to a finite number, or in this case, equal to 2. The simple calculus involved is currently beyond the scope of my brain, and therefore, this article.
It is unlikely that Zeno was trying to claim that motion itself is impossible. Just that an infinite series of acts cannot be completed in a finite period of time. I’ve only scratched the surface of this subject, but it’s a fascinating one.
More recently I dreamt that I crashed my submarine into the seabed, escaped the wreckage, and was washed ashore to an island occupied entirely by blind people. Closure on this one still eludes me…