Some features of the species

Our chief consultant writes:

Some weeks ago I mentioned paradoxers, those people from outside a certain science who come up with some amazing or important result that, sadly, is not accepted by those inside–mostly because it’s not true.  I promised to describe the outstanding characteristics of this fascinating species; here are two.

Most paradoxer characteristics don’t seem to have changed much since Augustus de Morgan named the species and described them in the nineteenth century (his work is conveniently collected in A Budget of Paradoxes, 1872), so I’ll begin with one of his examples.

As a preliminary, let me remind you of some things you may have forgotten from High School Geometry.  First, proof by contradiction: you assume the opposite of what you want to prove, and show that it leads to a contradiction.  Second, you can find an approximation for π (the ratio of circumference to diameter of a circle) by drawing a polygon inside the circle and adding up the lengths of its sides.  It will always give an answer smaller than π (each side goes straight across, while the circle goes around), but will get better as you add more sides.

One of de Morgan’s paradoxers sought to calculate π.  He assumed a value of 3 1/8, and showed that it was consistent with a value of 3 1/8.  He had a tight grasp of part of an idea: he knew he could use any hypothesis to start his proof; but he missed the part about the contradiction.  When it was carefully explained to him that the perimeter of a 24-sided polygon gave a value greater than 3 1/8, he stood on his “proof” and refused to listen.  He was unwilling or unable to follow someone else’s argument.  (I have adopted the label of “semi-paradoxer” for those who will accept an explanation.)

Current examples of these characteristics are easy enough to find.  I heard someone on radio who had heard about the law of conservation of energy, and understood part of it: something stayed constant.  He concluded that evolution violated that law, because it required things to change.  Complicated theories, like quantum mechanics, relativity, the Big Bang (to name only a few we’re familiar with) are very easy to understand parts of while missing others.

Now consider the position of de Morgan, or anyone defending the established value of π, if he had to explain the situation to someone with no knowledge of geometry.  How do you get across that this number is not just a conspiracy of mathematics professors?  This kind of thing often describes the position of scientists nowadays.

Most of you will not be facing paradoxers of science, at least as a professional task.  But you can find these characteristics in other sorts of people, and the effect on logic is the same.