This is the answer to Zeno's famous paradox, about the Achilles hero racing against the turtle

Normally, Achilles will use his superior speed to run past the turtle, but philosophy says it is not that simple. The story of the turtle and the rabbit is also inspired by it?

Nearly 2,500 years ago, ancient Greek philosopher Zeno wrote a book about paradoxes. The nature of paradox is difficult to understand, but fortunately, we still have 'Achilles and the Turtle' in the easiest to understand.

Zeno of Elea shows the Youth the door to Truth and Wrong, a painting by Pellegrino Tibaldi.

Here are the basic elements that Zeno points out, although they have been recounted by generations in different forms but still retain their original values:

The famous hero of the Troia War, Achilles (we still know him and the "harsh" Sin's heel) races against a lowly turtle. Proudly, Achilles allows turtles to run first. The race is not difficult with a strong and agile warrior, but it is not so easy: to run past the turtle, he must catch up first.

When Achilles shortened the distance between himself and the turtle, the animal slowly made a new distance. Although the new distance is less than the distance between the turtle and Achilles, Achilles must run a whole new distance to catch up with the turtle.

He kept running, but in that time, the turtle created a distance again, forcing Achilles to run the latest distance to catch up.

This is an endless loop, showing that Achilles never catches the turtle. No matter how high Achilles' running speed is, the new gap always appears; although much smaller than before, this is still the distance that allows a turtle to run before Achilles.

Paradox readers will tend to deny Zeno's argument, but that response is based on two factors, either lazy or fearful.

1. Lazy, because just thinking about this paradox, we have the feeling of being able to solve it almost immediately, but actually never solve it. It is also the feeling of Achilles chasing forever without the turtle.
2. Fear, because I am being surpassed by an old man, an old man dies before people can figure out the divine zero. The intelligent person of modern times cannot be so poor!

But what if your children read this paradox, then ask their parents to explain? You can hardly argue in the way of 'Achilles running faster, obviously overtaking the Turtle '; The answer is not very interesting compared to the puzzle created by Zeno from 2,500 years ago. In the puzzle itself, Zeno also tells Achilles to run faster than the turtle: the distance of the new turtle is always much smaller than the distance between Achilles and the previous turtle.

The distance between the newly created turtle is always much smaller than the distance between Achilles and the previous turtle.

So we must rely on the help of some philosophers, mathematicians, to answer the question. Most of those brilliant minds think it is possible to write a book about this paradox (some writers already wrote it), but after asking for opinions, pen writer Brian Palmer, reported to the Slate newspaper, drawing again to divide the problem into three major sections as follows.

Step one: This is obviously a trick, but what kind of trick?

Zeno argues this paradox to support the thesis: change and movement are not real. Nick Huggett, a physicist philosopher at the University of Illinois, said that Zeno's 'disapproving point' is really crazy, but it is worse to accept it .

The paradox opens new points, showing us the difference between how people think about the world and the nature of the world itself. Joseph Mazur, emeritus professor of mathematics at Marlboro University, describes this paradox as ' a trick to distract your thoughts about space, time and movement '.

New challenge appears: exactly where did we think wrong? The motion is completely real, obviously, is the person running faster than the turtle? The hardship lies in the 'endless human concept'.

Step two: Recognize that there are many different infinite concepts.

The challenge of Achilles seems to be impossible, because he will have to 'perform infinitely many actions in a finite time,' said mathematician Mazur, referring to the distances Achilles must run to chase. turtle. But the way to create infinite not only one.

In Mathematics, we have two numerical strings that are convergence and divergence.

With obvious divergences like 1 + 2 + 3 + 4 ., we don't have the final result, or more precisely, the result is endless. If Achilles had to run all the small roads that were constantly created during the race, Achilles would never catch the turtle.

But now try to calculate the 1/2 + 1/4 + 1/8 + 1/16 sequence ., although the number sequence also runs to infinite, this is the series of numbers converging with the final result of 1. Achilles Just keep on running, constantly turning the new distance the tortoise is creating smaller and smaller, the famous war hero will catch the turtle in a certain time.

The secret of the puzzle lies in the magic of mathematics.

There are still cases where Achilles does not chase the turtle, even if he runs faster clearly.'Based on Mathematics, it is feasible for a fast-moving object to chase an object that runs slowly and infinitely and never catches up , ' said mathematician Benjamin Allen . keep moving slowly in a certain way '.

Again, the secret of the puzzle lies in the magic of mathematics , in particular, the series of convergent and divergent numbers.

For example, the string 1/2 + 1/3 + 1/4 + 1/5 . looks convergent, but in fact it is a divergent series. If Achilles runs the first part of the race at a speed of 1/2 km / h, and the turtle runs at a speed of 1/3 km / h, then slows down to a speed of 1/3 and 1/4 km / h, And then . the turtle will always run before Achilles.

Step three: This is not just a hypothesis

The minds of young children are difficult but unpredictable, the questions they may have. If they have read Zeno's difficult problem and we answer them like that, then the cunning child will continue to ask: why do we know the sum of 1/2 + 1/4 + 1/8 + 1 / 16 . is 1? Nobody can do this calculation, because it lasts infinitely.

In a certain way, the conclusion that an infinite series of numbers is a finite number is only a hypothesis, deduced and perfected by the great brains of Isaac Newton or Augustin-Louis Cauchy. , who find a way to apply a mathematical formula to assert a series of numbers is convergence or divergence.

But just consider it a hypothesis that is not worthy.

Augustin-Louis Cauchy.

' It's easy to say that a string of numbers combined into a finite number ,' said mathematician Huggett, ' but until you can prove it - rigorously - how to add a series of numbers Any endless, that's just a cliché. Cauchy himself gave the answer to humanity '.

Convergence series explains the myriad of things that exist in the present world. Not just how fast a person runs (like Achilles) can outdo a turtle, but:

Any distance, time interval or force that exists around us can be separated into an infinite number of series (just like the distance Achilles must run to catch the turtle), but decades of computing The physical and technical aspects have proved that the final result is still a single number, a finite result.

This answer may not satisfy Zeno, as many philosophers still have a way of thinking 'their logic is beyond reality'. But in the way that the mathematical and philosophical communities answer Zeno's riddle, using the observations we have to apply the reverse engineering to a hypothetical theory, is the most obvious example for us. see the importance of research and experimentation in unlocking the secrets of the Universe.

This is a critique for anyone to question the importance of studying science, philosophy, mathematics or any other area.

1. The truth behind three 1,000-year paradoxes no one could solve