Philosophize mathematic-- Resolving the Zeno's paradoxes

Hey, this is Darren. As a person who has been passionate about both philosophy and maths for many years, I've barely seen connections between those two subjects. However, Plato, the founding father of western philosophy, vowed that mathematics is the indispensable prelude to philosophy and its highest form. He even put a sign over the doors of his Academy; “Let no man ignorant of geometry enter here.”

(The school of Athens, created by Plato, is the most prestigious academies among the whole European world. Plato inscribed the sign: “Let no man ignorant of geometry enter here” on the entrance of the academy.)

Knowledge comes from curiosity. I, therefore, assume that there could be a strong connection between philosophy and mathematics. So here I am, inquisitive and eager, opening my first blog dedicating to the Zeno's Dichotomy paradoxes; a mathematical riddle that has inspired many philosophers and mathematicians. 

Zeno is a pre-socratic Greek philosopher who was once described by Aristotle as "the inventor of dialectics." He is famous for inventing a series of arguments that seem to be logical but have a conclusion that is absurd or contradictory. One of the most famous ones is The dichotomy paradox, which means " the paradox of cutting in two" in ancient Greek. 

He proposes the problem like this:

 "Suppose Homer wishes to walk to the end of a path. Before he can get there, he must get halfway there. Before he can get halfway there, he must get a quarter of the way there. Before travelling a quarter, he must travel one-eighth; before an eighth, one-sixteenth; and so on." 

If there are infinitely many of these finite-sized pieces, shouldn't the total time be infinity, too? 

From Zeno's point of view, he thinks that "since there are infinitely many of terms and each term is finite, the sum should equal to infinity as well."

Zeno is making the claim that travelling from any location to any other location should take an infinite amount of time. which according to our common sense, is wrong. 

Zeno uses this paradox to support the assertion that all motion is impossible. However, this absurd proposition has drawn a lot of attention from mathematicians. And as a matter of fact, it has been refuted by mathematicians with sufficient proofs. 

There are two ways to counterprove the argument, I will present both of them below. 

The first solution is to use pure math, which according to the context, we can set the question up like this: 

Theoretically, the answer should be infinite as well based on the fact that there are infinite amounts of terms. 

Then we can do the math get the equation: 

  
Bingo! That's the first way to counterproof the paradox. 

The second way to counterproof Zeno's paradox is to use geometry, which I will present through video. Check it out: 

Therefore, Zeno's Dichotomy paradox is proved wrong through using mathematics! Take that, Zeno!

However, philosophers can sometimes be a little bit hard to deal with... For the sake of defending his idea that all motion is impossible, he asserted another example to support his idea; that is The Achilles Paradox. 

In the Achilles Paradox, Achilles races to catch a slow tortoise that is crawling in a line away from him. 

So if Achilles wants to overtake it, he must run as far as the place where the tortoise presently is, but by the time he arrives there, the tortoise has crawled to a new place. Meanwhile, Achilles has to continually catch up with the tortoise. But by the time that Achilles got to the place where the tortoise had been, the tortoise has moved into a new place.

Indeed it sounds very reasonable once we think of it. But according to our experience, we would know that Achilles can surely catch the tortoise in a short period of time. Then where does the flow of the logic come from? 

Oof, a tough question to answer. In the Achilles paradox, Zeno assumed that distances and durations can be endlessly divided into what modern mathematicians call a transfinite infinity of indivisible (further reading on this mathematical concept). However, it is still debatable on the validity of this concept. For Zeno's paradoxes, the standard analysis assumes that length should be defined in terms of measure, and the motion should be defined in terms of the derivative. 

But physical space is not a continuum because it is three-dimensional and not line although it has one-dimensional subspaces such as the paths of runners and orbits of the planet.

Anyway, long story short, a definite answer on The Achilles is still debatable in the fields of mathematics and physics. But the development of calculus was the most important step in the Zeno's paradoxes. The ultimate question for this paradox, in my opinion, is the debate on the existence of the World line.

Although we do not have a certain answer yet, but thinking and trying to tackle such an interesting paradox has given me pleasure that I would consider to be invaluable. I wish your experience of reading this blog to be similar with mine.

Further readings on Zeno's paradoxes.

https://www.youtube.com/watch?v=NCtw5f6XPF4
https://www.iep.utm.edu/zeno-par/
https://en.wikipedia.org/wiki/Monism