Normally, Achilles would use his super speed to outrun the turtle, but philosophy says things are not so simple. The fable of the tortoise and the hare is also inspired by this, right?
Nearly 2,500 years ago, the great philosopher Zeno of ancient Greece wrote a book on paradoxes. The nature of the paradox is difficult to understand, but fortunately, we still have “Achilles and the Tortoise” among the easiest to understand.
Zeno of Elea shows Youth the door to Truth and False, painting by Pellegrino Tibaldi.
Here are the basic elements that Zeno raised, even though it has been retold in different forms for generations, still retain its original value:
Readers of the paradox will tend to reject Zeno’s argument, but that response is based on two factors, either laziness or fear.
But what if your children read this paradox and ask their parents to explain it? You can hardly argue with “Achilles runs faster, obviously will outrun the Tortoise “; the answer is not interesting at all compared to the puzzle devised by Zeno 2,500 years ago. In the puzzle itself, Zeno also implicitly tells Achilles to run faster than the tortoise: the newly created turtle’s distance is always much smaller than the distance between Achilles and the previous turtle.
The distance the newly created turtle is always much smaller than the distance between Achilles and the previous turtle.
So we have to ask for the help of some philosophers, mathematicians, to answer properly. Most of those brilliant minds think a whole book about this paradox could be written (and someone else has), but after consulting, writer Brian Palmer, reporting for Slate, sums it up. again to divide the problem into three major categories as follows.
Zeno argues this paradox in support of the thesis: change and movement are not real. Nick Huggett, a philosopher of physics at the University of Illinois, said that Zeno’s argument against animals ” is indeed crazy, but accepting it as fact is worse “.
The paradox opens up a new point, showing us the deviation between the way people think about the world and the nature of the world itself. Joseph Mazur, professor emeritus of mathematics at the University of Marlboro, describes the paradox as ” a trick to distract your thinking about space, time, and motion “.
A new challenge emerges: where exactly did we go wrong? The movement is completely real, obviously humans run faster than turtles? The conundrum lies in the “human concept of infinity”.
The challenge placed before Achilles seems impossible, because he will have to “perform an infinite number of actions in a finite amount of time,” says mathematician Mazur, referring to the distances Achilles must run to catch up. turtle. But the way to create infinite is not only one.
In Mathematics, we have two series of numbers: convergent and divergent.
With a series of obvious divergences like 1+2+3+4 …, we have no final outcome, or more precisely, an infinite outcome. If Achilles had to run through the continuous trails created during the race, Achilles would never catch the turtle.
But now try to calculate the sequence of numbers 1/2 + 1/4 + 1/8 + 1/16 …, although the sequence also runs to infinity, this is a series of numbers that converges with the final result of 1. Achilles keep running, constantly making new distances the turtle creates smaller and smaller, the famous wartime hero will catch up with the turtle in a certain amount of time.
The secret of the puzzle lies in the magic of mathematics.
There are still cases where Achilles failed to chase the turtle, even though he was clearly faster. “Based on mathematics, it is entirely possible for a fast object to chase a slow object to infinity and never catch up,” says mathematician Benjamin Allen, “as long as both things continue. continue to slow down in a certain pattern”.
Once again, the mystery of the puzzle lies in the magic of mathematics , namely, that series of numbers converge and diverge.
For example, the series 1/2 + 1/3 + 1/4 + 1/5 … looks convergent, but is actually a divergent series. If Achilles runs the first part of the race at 1/2 km/h, and the turtle runs at 1/3 km/h, then slows down to a pair of 1/3 and 1/4 km/h, and so on… the tortoise will always run ahead of Achilles.
Children’s minds are immature but difficult to predict the questions they may have. If they have read up to Zeno’s difficult problem and we answer them as above, the clever kid will continue to ask: how do we know the sum of 1/2 + 1/4 + 1/8 + 1/ 16…is 1? No one can do this calculation, because it goes on to infinity.
In a certain sense, the conclusion that an infinite sequence sums to a finite number is only a hypothesis, deduced and perfected by the great brains of Isaac Newton or Augustin-Louis Cauchy , who figured out how to apply mathematical formulas to determine whether a series of numbers is convergent or divergent.
But just treating it as a hypothesis is not worthy.
Augustin-Louis Cauchy.
“ It is easy to say that a series of numbers adds up to a finite number ,” said the mathematician Huggett, “ but until you can prove – rigorously – a way to add a series of numbers any end, it’s just a cliché. It was Cauchy who gave humanity the answer ”.
The convergent series of numbers explains the myriad things that exist in the present world. Not only how a fast runner (like Achilles) can outrun a turtle, but:
Any distance, time, or force that exists around us can be decomposed into an infinite series of numbers (just like the number of distances Achilles must run to catch the turtle), but decades of computing the The physical and technical aspects have shown us that the final result is still a single number, a finite outcome.
The above answer may not satisfy Zeno, as many philosophers still have a way of thinking “their logic is beyond reality”. But with the way the mathematical and philosophical community answered Zeno’s puzzle, using existing observations to apply reverse engineering to a deduced hypothesis, is the clearest example for us. see the importance of research and experimentation in unlocking the secrets of the Universe.
This is a counter-argument to anyone who questions the importance of the study of science, philosophy, mathematics or any other field.