Zeno’s Paradox and the errorless stay

The Greek philosopher, Zeno of Elea, gave us modern dog trainers the key to errorless training over 2000 years ago. Zeno is famous for several philosophical and mathematical paradoxes, but the most important to us is the Dichotomy Paradox, in which he argued that motion is an illusion. It goes something like this: imagine you are attempting to walk to your dog, who is 100 feet away. Before completing the 100 feet, you must first walk halfway there, or 50 feet. To walk the next 50 feet, you must first walk halfway, or 25 feet. To walk the 25 feet, you must first walk 12.5 feet. You must always walk half the distance before walking the whole distance, and therefore, your dog will remain forever out of reach. In dog training, we can apply this concept in a way which helps our dog to not be as concerned with distance as a factor in our work.

Consider the Dichotomy Paradox again, but this time, in reverse. We will start next to our dog, with the goal of having them stay, while we walk away 100 feet. What a mistake it would be to just walk 100 feet away and expect the dog to stay behind! We could, however, hope they will stay while we take half a step, then quickly return and reinforce the stay by rewarding. If that seems easy for them, we may take a full step, then return and reinforce. If they almost get up, or if they do get up, then go back to the half step and work there until it is truly trivial, then try the full step again. When a full step is easy, we take two steps, return, and reinforce by rewarding. If that is challenging, we work at that, but if it seems easy, we can make it 4 steps. The Paradox in reverse. ½ a step, 1 step, 2, 4, 8, 16, 32, 64, and Yay!  100 steps.

The original Dichotomy Paradox was about motion and distance, but the same principal applies to the parameters of duration and distraction, as well. 1 second, 2 seconds, 4, 8, 16, 32… obviously, we can always work in smaller increments than double – 45 seconds might be a better amount of time to practice than a full minute. Distraction is harder to measure precisely, but again, it can help to visualize walking 100 feet away from our dog. Does it make sense to ask for a behavior at a parade, and assume they can perform it just because they can in our living room? Of course that is not reasonable – to get from the parade from our living we might practice with friends and family milling about in our living room, in front of our home as people walk by, at the edge of a park, etc, etc… Half way there, and halfway to halfway, and halfway to halfway to halfway, and so on…

Zeno’s Dichotomy Paradox was meant to question our perceptions of motion and distance – or even to question the concepts of motion and distance altogether. When we apply these ideas to our training, we help our dogs succeed by removing the feeling that we are far away – we are only a little farther away than before, perhaps so little that the distance is not even perceivable. We will never actually be far away, because distance is relative – a relative illusion, if we proceed slowly enough. What would have seemed far away in the beginning becomes only a little further than the last interval. What would have been overwhelming becomes trivial. Distance is only a concept we use to measure motion. And according to Zeno’s Dichotomy, motion is an illusion.


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