Zeno’s Paradox Solved By Calculus

Zeno is a Greek philosopher who lived around the time of 490 to 430 BC. His full name is Zeno of Elea. Sometimes, some people spell Zeno with an X as in Xeno.

He actually came up with many various paradoxes. So there is not just one “Zeno Paradox”, but “Zeno Paradoxes”.

The three of the well known one are …

  • Dichotomy paradox
  • Achilles and the tortoise paradox
  • Arrow paradox

Dichotomy Paradox

Perhaps the one that is most commonly provided as an example of one of Zeno’s paradox is the dichotomy paradox. And it goes like this…

Let’s say that I need to travel a distance of one mile. In order to get there, I must get to half-way. But in order to get to half-way, I must get to half of that (or 1/4 of mile). But in order to get to 1/4 of a mile, I must first get to 1/8 of a mile. Before that I need to go 1/16 of a mile. And before that, 1/32. Then 1/64 and 1/128 and on and on and on for an infinity. There is an infinite number of steps. So how in the world will I ever get to where I want to go?

Generalizing the problem… In order to travel a distance d, one must travel d/2. And before that, one must travel d/4. And d/8, etc.

The solution is resolved via calculus. In effect we have an infinite sum of a “geometric series”. In particular, we are summing (1/2)i as i goes from 1 to infinity. The answer to that sum converges to 1 and can be proven via calculus.

In short, it mean that the sum of an infinite number of “half-step” is finite. Therefore, you will get to where you will be going.

Math Joke

There is a math joke that is based off of Zeno’s paradox.

A group of boys line up at one wall at one end of the ballroom. A group of girls on the opposite wall. The two group walks towards each other. When will they meet at the center of the ballroom?

The mathematician says never, because it involves an infinite number of steps. The physicist says that they would meet when time equals infinity. And the engineer says that within one minute, they are close enough for all practical purposes. Hand it to the engineer for being practical.

Achilles and Tortoise Paradox

Achilles and tortoise are in a race. Since Achilles is a faster runner, he gives tortoise an 100 meter head start.

Therefore, it will take some time before Archilles reaches the tortoise starting point. But by that time, the tortoise (although slow) has since moved ahead. In order to catch up, Archilles must reach the spot where tortoise has already been. Thus, whenever Achilles reaches somewhere the tortoise has been, he still has farther to go. Because in the time it take for Achilles to reach there, the tortoise has progressed further.

So how in the world does Archilles catch up with the tortoise?

In the below video, watch as a math teacher explains this paradox…

Paradox Solved by Calculus

The solution is similar to the one before. The infinite number of catching up that Archilles has to do is counterbalanced the infinitely small time it takes for the subsequent steps. And therefore Archilles is able to catch up to the tortoise.

Calculus comes to the rescue by saying that it is possible to add an infinite number of steps. In fact, Calculus is the subject of adding, comparing, and manipulating infinities. But Calculus was not invented yet in the time of Zeno. That is why they were perplexed.

Arrow Paradox

In the book Physics written by Aristotle, the Arrow Paradox goes like this …

“if everything when it occupies an equal space is at rest, and if that which is in locomotion is always occupying such a space at any moment, the flying arrow is therefore motionless”

Confusing? Okay, here it is stated in another way by Wolfram Math World

“An arrow in flight has an instantaneous position at a given instant of time. At that instant, however, it is indistinguishable from a motionless arrow in the same position, so how is the motion of the arrow perceived?”

Basically, Zeno is saying that the flying arrow is motionless.

Of course that is ridiculous. And it is refuted by Aristotle when Aristotle writes …

“This is false, for time is not composed of indivisible moments any more than any other magnitude is composed of indivisibles.”



How Types of Dietary Fat Affect Risk of Heart Disease

Good article in Journal of the American College of Nutrition titled Types of Dietary Fat and Risk of Coronary Heart Disease: A Critical Review

It has a chart shows how LDL and HDL is affected by the various types of fat. Interestingly, certain saturated fats raises the good HDL cholesterol by the same amount that it raises the bad cholesterol LDL. So it balances out and may not be as bad as some people think.

Good reference.

Why Bloggers Write?

Writing is a Solo Pastime

There are some activities that we like to do with people. And then there are some activities that we like to do on our own. Some people get energy by working with other people. They are known as extroverts. And some people recharge their energy by spending time alone. They are known as introverts. Most people have both extroversion and introversion. It is just a matter of leaning towards one side or the other.

When we are in our introversion mode, writing is the perfect activity since it is a solo activity that is best done alone.

Writing Helps Practice Communications Skills

In order to write one must exercise their communication skills. This exercise hones and maintain one’s vocabulary and effective communication. It teaches us to write succinctly. It is more difficult to write succinct and precise. Sometimes we may be in a job where we are doing manual labor, or where the topic of conversation is within a narrow range of topics. By writing we expand our topic of conversation and broaden our range of topics to talk about.

For Money

And then there is the writing for money.  Some blogger write on their site because they constantly need fresh content to generate page view.  Pageview can be monetized by advertisement on the site such as Google AdSense.

But not always

However, that is not always the case.  Some do not write for money.  Leo Babauta is a blogger on a very popular ZenHabits.net. His site has no ads.  Why does he write?

In his article, Finding Your Voice, he says that he writes a lot and that is because …

“This is almost all I need to say, as nothing else matters without the constant practice of writing a lot. Write blog posts and letters, booklets and diatribes, letters to the editor and book reviews, love poems and short stories, novellas and manifestos. The sheer mass of your writing becomes the raw matter from which to chisel your voice.”

Barry Schwartz Explains the Paradox of Choice

Barry Schwartz is the author of the book, “The Paradox of Choice: Why More Is Less“. He explains the paradox of choice where in some situations, why less choice is sometimes better.  And that is why the subtitle of his book is titled “Why More Is Less”.

In July 2005, Schwartz gave a TED Talk which you can see linked here.  He first gives examples of how our modern world has given us too much choice. And two negative consequences of this.  First is decision paralysis.  An example he gives is when a retirement plan that has more mutual fund choices resulted in less participation because members find it to hard to choose.  The second negative consequence is that even after making the choice, you are less satisfied with the choice you made.  When there are lots of choices, it is just too easy to imagine that you could have chosen better.

Near the end of he talk, he gives the secret of happiness.  And the “secret of happiness is low expectation”.

Schwartz also gave a “Google Talk” on April 27, 2006 which you can view linked here.  Schwartz does have some humor in his presentations.  For example, 44 minutes into the Google Talk video, he says “Everything was better back when everything was worst”.

Everything suffers from comparison. People have higher expectation these days which makes them feel worst.

At 45 minutes into the video, an insightful comment he makes about money and happiness is …

“What is true is that once you cross subsistence, whatever subsistence is in your society, additional increases in wealth have virtually no effect on well-being. There is a huge steep curve going from zero to subsistence. But once you cross that line of subsistence, the curve flattens out. It is worth knowing, in case you have a choice between choosing x and making more money, almost certainly choosing x is what you should do.”