# In The Armchair

## Mistaking Beauty for Truth

Posted in Armchair Ruminations by Armchair Guy on July 3, 2010

Karl Popper, in his 1953 lecture on science, described three pseudo-scientific theories thus:

I found that those of my friends who were admirers of Marx, Freud, and Adler, were impressed by a number of points common to these theories, and especially by their apparent explanatory power. These theories appeared to be able to explain practically everything that happened within the fields to which they referred. The study of any of them seemed to have the effect of an intellectual conversion or revelation, opening your eyes to a new truth hidden from those not yet initiated. Once your eyes were thus opened you saw confirming instances everywhere: the world was full of verifications of the theory. Whatever happened always confirmed it. Thus its truth appeared manifest; and unbelievers were clearly people who did not want to see the manifest truth; who refused to see it, either because it was against their class interest, or because of their repressions which were still ‘un-analysed’ and crying aloud for treatment.

I think Marx and Freud specialized in selling beautiful theories.  Their ideas had (have) a simplicity and basic unity that people found (still find) attractive.  Although the ideas were elucidated in pages upon pages of writing, there seemed to be some common underlying principles that were elegant and simple, yet possessed of wide explanatory power.  The ideas were beautiful.

A lot of people mistake beauty for truth.  I first came across this theme while reading blogs by self-styled economic liberals.  It seemed to me that many of these liberals are excessively concerned with coming up with pithy one-liner descriptions of reality.  A pithy, attractive, succinctly stated “theorem” that allows them to conclude major things or explain a wide variety of phenomena impresses readers and raises cachet. The problem is that the world usually doesn’t admit such oversimplified explanations, but some people are so in love with their one-liners that they continue trying to shoehorn every fact to fit such theories.  The problem isn’t just related to blogs; it’s widespread at all levels.

In fields like mathematics, beauty is an asset, because it’s often obvious what’s true. (Obvious in the sense that a trained mathematician can, with sufficient work, in most cases, correctly determine whether an argument is correct or not.)  In the other sciences, carefulness in establishing the truth usually trumps the coolness factor.  However, economics exists on the dangerous boundary between storytelling and empirical truth.  Economic theories are often grand, sweeping — and aren’t subject to the kinds of simple tests scientific theories are.  So beauty is an attractive feature of a theory, sometimes the most attractive feature.  This is perhaps why Karl Marx’s views are so attractive: they give a beautiful explanation that fits the facts.  Economists themselves recognize the problem.  I took the phrase “mistaking beauty for truth” from an article by Paul Krugman.

Scientists, like economists, are not immune to the fallacy.  Mathematicians often strive for elegance in their proofs, but they are supposed to.  Other scientists, however, may get swayed by such considerations too.  An interesting example of this is the canonical explanation for the reason moths are attracted to light.  For a long time, this was supposed to be due to confusing lights for the moon.  The theory went that moths fly in a straight line by keeping a constant angle to the rays of the moon.  The moon being a faraway object, its rays are almost parallel by the time they reach the earth.  So this is a good approximation for the moth; it wouldn’t deviate much from a straight line in a flight of several miles if it kept a constant angle from the moon.

However, if a moth sees a light that is much closer by and mistakes it for the moon, the situation is quite different.  The rays are no longer parallel, and keeping a constant angle to these rays will always result in nonlinear motion.  If the moth maintains a perfect 90-degree angle with the light rays, it will fly in a circle.  However, the theory goes, moths tend to maintain an acute angle to the light rays.  This causes them to move in a spiral that leads inward toward the light.  It’s fairly easy to calculate an equation for this spiral.  The polar equation of the curve followed by the moth when it maintains an angle of $\alpha$ to the rays, parametrized in terms of the angle $\theta$ subtended at the origin between the current position and the positive X axis, starting at the point $(d, 0)$, is given by

$r(\theta) = d e^{-\theta/\tan \alpha}$.

Here’s what it looks like in 2D:

Beautiful.  The only problem is, it isn’t true.  This isn’t why moths fly towards light, and that was shown by the first careful experiment to be done. Looking at the above graph, the moth circles the light several times before falling into it.  Henry Hsiao, a biomedical engineering researcher, studied moths’ flight patterns and found that they don’t fit this behaviour.  His alternate theory has to do with mach bands.  But why did this theory about the equiangular spiral survive so long?  It seems to me its mathematical beauty trumped considerations of its veracity.  It’s just another example of beauty being mistaken for truth.