Ted

Zeno paradox

This is Zeno of Elea

In ancient Greek philosopher famous for inventing a number of Paradoxes

Arguments that seems logical but whose conclusion is absurd or contradictory?，

for more than 2000 years Zeno s mind bending riddles have inspired mathematicians and philosophers to better understand the nature of infinity?

One of the best known of Zeno s Problems is called The dichotomy paradox .

Which means the paradox of cutting two in Asia in Greek it goes something like this:

After a long day of sitting around thinking, Zeno decides to walk from his house to the park The fresh air clears his mind and helps him think better he first has to get half way to the park ,this portion of his Journey take some finite amount of time. Once he gets to the halfway point, he needs to work half the remaining distance again this takes a finite amount of time, once he gets there he still need to walk, Half the distance that’s left which takes another finite amount of time, this happens again again and again you can see that we can keep going like this forever,

Dividing whatever distance is left into smaller piece each of which takes some finite time to Traverse so how long does it take Zeno to get to the park?

Well to find out we need to add the time of each of the pieces of the journey?

The problem is there are infinite many of these finite size pieces so shouldn’t the total time be infinite?

So shouldn’t the total time me infinity?

This argument by the way is completely general it says that traveling from any location to any other location Should I take an infinite amount of time , in other words it says that all motion is impossible . This conclusion is clearly absurd but where is the flaw in the logic？

To resolve the paradox it helps to turn the story to a math problem

Let’s post that Zeno‘s house is one mile from the park that’s when Zeno walks at 1 mile an hour

Common sense tell us that the time for the journey should be one hour

But let’s look at things from Zeno‘s point of view?

And divide the journey into pieces the first half to the journey takes half an hour,

This next parts takes a quarter an hour the??next part takes an eighth of an hour

And so on summing up all these times, we get a series that look like this

1/2+1/4+1/8+1/16+…

Zeno might say since there are infinite of terms on the right side of the equation,

And each individual turn is finite?

This sum should equal infinity this is the problem with Zenos argument.

As mathematicians have since realized it is possible to add up infinitely many finite size terms?

And do you get a finite answer

?how will you ask well let’s think of it in this way??let’s start with a square that has area of 1 meter now let’s chop the square in half

And then chops in the remaining half in half, and so on while we are doing this, let’s keep track of the areas of the pieces The first slice makes two pots, each was an area of 1/2 The next slice divides one of those half’s in half and so on but no matter how many times we slice up the boxes , The total area is still the sum of the areas of all the pieces.

Now you can see why we choose this particular way of cutting up the square, we’ve obtained??the same infinite Series as we had for the time of Zenos journey as we construct more and more blue pieces, to use the math jargon:

As we take the limit as N tends to infinite, The entire square becomes covered with blue but the area of square is just a unit and??the infinite sum must equal one.

Going back to the Zeno journey we can now see how this paradox be resolved, not only does the infinite series sum to a finite answer, but that finite answer is the same one that common sense tells us is true: Zeno’s journey takes one hour.