Yogi Berra is famous for lines like “It ain’t over till it’s over,” “A nickel ain’t worth a dime anymore,” “When you come to a fork in the road, take it.” The trick is that the statements are literally false or empty or meaningless, yet suggest something true.
The challenge we more commonly face is statements of the reverse structure: literally true, yet suggesting something false. Here are two examples that blighted for a century their respective fields of fluid mechanics and numerical analysis.
(1) To test for instability of a 3D channel flow, it’s enough to look at 2D flow
perturbations. (Squire 1933)
(2) Given any system of interpolation points in an interval, there’s a continuous function f for which the polynomial interpolants diverge as n→∞. (Faber 1914)
Formulated precisely, (1) and (2) are as true as true can be: they are mathematical theorems. And yet, how misleading! What makes high-speed flows unstable in practice is not the theorist’s instabilities, but different 3D effects. Interpolants in Chebyshev points converge beautifully, so long as f is ever so slightly smooth.
You will find the wisdom of any field supported by confident assertions. It’s one challenge to spot the false ones, another to spot those that are true but misleading.
[8 May 2013]