Limits and Liberation

Yesterday, a thick envelope arrived in the mail from overseas.  My name and academic title on the outside in straggling handwriting.  I knew what it must be before opening it.

It's from a circle-squarer.  He wants me to look at his "discovery" of a "new value for pi".

The manuscript contains some familiar themes.  The author is not trained in mathematics.  He has been working on this project for twenty years!  He has produced pages and pages of diagrams, dense with construction lines, circles, and letters. He claims the solution reached him by divine revelation.

The problem of "squaring the circle" using straightedge and compasses was proposed in ancient times. In the 19th century, Galois' theory of algebraic equations and Lindemann's proof that pi is not the root of any algebraic equation were combined to show that it is impossible to solve the problem on the terms originally proposed.

That word "impossible" is a red flag to the circle-squarer.  My correspondent wants to believe that it means simply that people did not try hard enough, or were not enlightened enough, or perhaps that the truth was deliberately suppressed for sinister purposes.

But it doesn't. In this context, "impossible" is a liberating word.  It signifies not ignorance, but enlightenment.  It signifies that we understand the limits of the tools that are available to us.

Such understanding will be needed in humanity's uncertain future.