Here’s Mathematician KP Hart’s Math Question and Answer for Monday, January 11th!
What is wrong with the statement that something can stay divisible till infinity?
To quote Walter de Rochebrune: distance does not exist. In one of the legendary broadcasts by Van Kooten and De Bie he told a story about a gnome who used a tiny saw to cut a branch in parts until a smaller gnome with an even smaller saw took over. This can go on indefinitely. In other words: distance/length does not exist. I still have not found a satisfactory answer that disproves this statement.
Note to non-Dutch readers: Van Kooten and De Bie were a team of satirists and Walter de Rochebrune was one of their characters.
It all depends on how one interprets this story: in the physical world the process stops when the width of the blade approaches the diameter of the atoms in the wood molecules. In this, literal, sense De Rochebrune was wrong.
In mathematics we can tell an idealized version of De Rochebrune’s story. We do this by modeling the branch as a line segment of length 1 on the x-axis, from the origin to the point (1,0,0). The gnome stand at (½,0,0) and is sawing the branch in half. For every natural number n we can multiply this image by 2-n with respect to the origin. That produces an infinite sequence of gnomes, where every next gnome is half the size of the present one and its saw is half as wide. If the first saw is 1 mm thick then then all saws together turn 2 mm of wood into saw dust and so there is still 99.8 cm of branch left, in infinitely many pieces (and a piece of zero thickness at the origin).
This is an example of one of Zeno’s paradoxes and we have talked about these before. The solution of this problem is that infinitely many numbers can have a finite sum.
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Read all of KP Harts math questions here!
About Dutch Mathematician KP Hart: In the beginning of this year the Dutch government opened a website, The Dutch Science Agenda, where everyone could post questions that they thought were of scientific interest. This was an attempt to involve the whole country in determining what the Dutch science agenda should be in the coming years.
I looked through the questions and searched for terms like `mathematics’, `infinity’ … to see what mathematical questions there were and I noticed various questions that already have answers (and have had for a long time). On a whim I decided to post answers to those questions, in Dutch. For your edification I will translate these posts into English.
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