Saturday, January 12, 2019

Achilles and the tortoise

Achilles and the tortoise. Mathematics For Blondes.
Achilles and the tortoise

In the fifth century BC, the ancient Greek philosopher Zeno of Elea formulated his famous aporia, the most famous of which is the Achilles and the Tortoise aporia. Here’s how it sounds:

Suppose Achilles runs ten times faster than a turtle, and is a thousand paces behind it. During the time that Achilles has run this distance, the tortoise will crawl a hundred steps in the same direction. When Achilles runs a hundred paces, the tortoise crawls ten more steps, and so on. The process will continue indefinitely, Achilles will never catch up with the tortoise.

Another version of the aporia "Achilles and the tortoise":

In a race, the quickest runner can never overtake the slowest, since the pursuer must first reach the point whence the pursued started, so that the slower must always hold a lead.

This reasoning was a logical shock for all subsequent generations. Aristotle, Diogenes, Kant, Hegel, Hilbert ... All of them in one way or another viewed the aporia of Zeno. The shock turned out to be so strong that discussions are continuing at the present time; the scientific community has not yet succeeded in reaching a general opinion on the essence of paradoxes. Mathematical analysis, set theory, new physical and philosophical approaches were involved in the study of the question; none of them has become a generally accepted solution. Everyone understands that this is a hoax, but no one understands what it is.

From the point of view of mathematics, Zeno in his aporia clearly demonstrated the transition from quantity to an inverse quantity. This transition implies the use of variable units of measurement instead of constants. As far as I understand, the mathematical apparatus for using variable units of measurement has either not yet been developed or has not been applied to the aporia of Zeno. The application of our usual logic leads us into a trap. We, by the inertia of thinking, apply constant units of time to the reciprocal. From a physical point of view, it looks like a slowing down of time to its full stop at the moment when Achilles approaches the turtle. If time stops, Achilles can no longer overtake the turtle.

If we turn the logic we are used to, everything falls into place. Achilles runs at a constant speed. Each subsequent segment of its path is ten times shorter than the previous one. Accordingly, the time spent on its overcoming is ten times less than the previous one. If you use the concept of "infinity" in this situation, then it will be correct to say "Achilles will catch up to the tortoise infinitely quickly".

How to avoid this logical trap? Stay in constant time units and don’t go to inverse quantity. In the language of Zeno, it looks like this:

In the time that Achilles has run a thousand steps, the tortoise will crawl a hundred steps in the same direction. Over the next interval of time equal to the first, Achilles will run a thousand more steps, and the turtle will crawl a hundred steps. Now Achilles is eight hundred paces ahead of the tortoise.

This approach adequately describes reality without any logical paradoxes. But this is not a complete solution. On the aporia of Zeno "Achilles and the Turtle" is very similar to Einstein's statement about the irresistible speed of light. We still have to study, rethink and solve this problem. And the solution must be sought not in infinitely large numbers, but in units of measurement.

Another interesting aporia of Zeno tells of a flying arrow:

A flying arrow is motionless, because at each moment of time it is motionless, and since it is motionless at each moment of time, it is always motionless.

In this aporia, the logical paradox is overcome very simply - it is enough to clarify that at each moment in time a flying arrow is stationary at different points in space, which, in fact, is movement.

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