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Tales of Zeno, Monty Hall, and Other Paradoxes

From infinite steps to game show tricks — The overlooked principles that change everything

Part One of Three: Zeno

At first glance, these problems seem completely unrelated. Zeno insists you can never reach your destination. A game show host somehow improves your odds by opening a door. Someone even claims π equals 4. And that strange staircase paradox.

Different stories yet all of them feel wrong in the same way. As if some principle or law exists… that we have yet to discover? Some aspect of reality we will someday uncover?

“All Tales Bode Well If Told Quickly”

My father knew how to attract my attention. One day he motioned for me to sit down. “Dad...” I objected. “I’m quite busy right now.” He announced “Perpetual motion!!” “Perpetual motion…you don’t say???” I had a seat expecting to hear another unique idea. He started off at a quickened pace “You have a wheel and another object that are let loose at the same time and same height. The wheel starts rolling and the object simply falls. Both move down a U shaped hill. Now both reach near the top of the other side of the hill, at the same time. The object has expended its kinetic energy but the wheel has energy stored in its rolling motion. Now with that imbalance of energy we attach a non rotating pivot…” I interrupted “A non rotating pivot —Explain!?!!…” My father cut in “Ahh, electromagnetics my boy, electromagnetics!!”…This was a simple decoy to prevent me from thinking about the rolling wheel.

When I was younger, my Father could foil me with his ideas but as I grew with age, I could recognize his slight, almost imperceptible grin. I knew to pay attention, there’s a catch somewhere.

The Paradox:

Zeno’s tales remind me of the stories that my father concocted. I could imagine Zeno at the Tavern talking to his friends at a quickened pace… He would begin “Now suppose a fellow walks halfway across a field, then he walks half of what remains. He continues this process, he’ll never get to the other side. Now supposing a wheel rotates halfway then half of what remains. It continues this process, it’ll never complete a full turn…Therefore I propose that movement is all in the imagination.” After all his friends have left mesmerized, I could see him declaring in a raucous voice “Fooled them again!!”

Below is a diagram of Zeno’s walk as he wants us to see it:

Zeno’s walk, This graph would have to extend forever to see all the steps: Mark Doknjas

What is that one piece of information he craftily ‘forgets’ to tell his listeners? Zeno divides distance into infinitely many parts or steps…but fails to account (purposely) that the time required to traverse each part also decreases proportionally.

Zeno’s paradox emerges when motion is indexed by steps, producing an exponential decay that never appears to finish. But when the same motion is indexed by time, the curve collapses into a straight line.

The paradox is not in motion, it is in the choice of parameters used to describe motion.

Below is a diagram how it would be when the walk is indexed by time:

Zeno’s walk indexed by time. The plot ends at the other side: Mark Doknjas

We have to hand it to Zeno. Very crafty! Zeno’s paradox could be a certain paradox if, and only if the same amount of time is required for each half distance of the preceding distance. Then Zeno would be like me after work, never quite making it off the sofa.

The Hill

But wait! The path is not always flat for Zeno.

While walking back from the tavern, the once flat ground begins to rise — slowly at first, then faster and faster still. The seismic upheaval rises exponentially toward some unknown height.

But yet, something strange happens.

Zeno finds new strength. His speed increases — also exponentially at the same instant the earth swelled upwards. Both the hill and Zeno’s speed have the same exponential growth factor.

A voice from somewhere above declares “If you are able to reach the summit, the mountain will stop rising.”

The hill, now a mountain rises ever more steeply towards a crest without bound!!

The Dual Between Speed And Height — Who wins??

Does Zeno ever reach the summit… or does he climb forever trapped between two equal infinities??? Perhaps as punishment for his original paradox LOL. You decide on his fate?

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Whether one is dividing steps, climbing mountains, or opening doors, there is always some forgotten or overlooked principle that, if found, would solve the paradox.

And yes Zeno does make it to the summit quite easily for one reason. Since Zeno is already moving at a walking pace of say, 3 Kph, his increase in speed will always be 3 times faster than the hills increase in height. Keeping in mind the hill started from 0 kph.