The impossibility of squaring the circle

As the geometer his mind applies
To square the circle, nor for all his wit
Finds the right formula, howe’er he tries
-Dante, Paradiso, Canto XXXIII, line 133-135

Here, Dante makes reference to an ancient geometry problem. Can you, using just a compass and a straight-edge, construct a square with an area equal to that of a given circle. This is called “squaring the circle”. It actually can’t be done. The reason is complicated, but it’s related to the reason why pi has an infinite number of decimal places.

In his Patterns of Plausible Inference, Polya mentions this as an excellent of example on when it’s OK to give up.

Construct, by ruler and compasses, the side of a square equal in area to a circle of given radius. This is the strict formulation of the famous problem of the quadrature of the circle, conceived by the Greeks. It was not forgotten in the Middle Ages, although we cannot believe that many people then understood its strict formulation; Dante refers to it at the theological culmination of the Divina Commedia, toward the end of the concluding Canto. The problem was about two thousand years old as the French Academy resolved that manuscripts purporting to square the circle will not be examined. Was the Academy narrow-minded? I do not think so; after the fruitless efforts of thousands of people in thousands of years there was some ground to suspect that the problem is insoluble. (p.17)

We young people often suffer from what Lewis calls “chronological snobbery”. We are eager to discard the denouncements of the older generations. THEY couldn’t do it we say, but WE can. We are smarter and have better technology and are more enlightened. They are backwards, but we are forwards. We are always solving problems they had no idea how to tackle. Why not this too? But this is foolery. Some things cannot be solved.

Theodicy is a good example. If God exists and is good, why does terrible stuff happen? You will not find a resolution to to this. The digits of its solution extend into interstellar space along with those of pi. But they don’t stop. A crucial element of trust is required. Some rough frameworks can be contructed (and learning to understand these is a good idea), but they will never quite put their finger on the solution. We offer only an approximation, not a solution.

The post-modern deconstruction of language is a refusal to give up when one really should give up. Nobody tries to square the circle anymore, unless they desire to waste their time. Does that make you bristle? Some trust is required – trust in words, trust in meaning, and trust in our fathers.