I actually contacted John Brown in February while I was preparing my talk for CNY Skeptics. He was quite helpful in explaining what he meant and provided an example. He said it's important to be skeptical about even skepticism and that some hypotheses are quite legitimate but not testable. Here's his example:
Consider a beam of electrons which have a spectrum of energies F(E) - that is, There are F(E) electrons per second with energies in the range E to E+1. These are fired into a hot gas with which they collide and emit x-rays. This is how a medical x-ray machine works. There we KNOW F(E) and use the
x-rays to study bone structure etc. In the case of the sun, we do NOT know what F(E) is - only the spectrum G(X) of the X-rays we detect at earth, where X is the energy of an X-ray. That is, we detect G x-rays per second with energies between X and X+1.
The question is, given G can we infer F uniquely? OR, equivalently, can we use G to test between two hypotheses as to what F is?
In 1971 I proved that the answer is yes for a uniform hot gas. That is, data on G(X) will uniquely test between different F(E) if G is measured accurately enough.
In reality the sun's atmosphere is known not to be uniform but to have a sudden step in temperature in it. This is like an X-ray machine with the electron beam fired into a target comprising two successive metal blocks
of different materials. In 1998, much to my surprise, colleagues and I proved that, even when the form of that step is precisely known, it changes the mathematical character of the F(E) <-> G(X) relation in such a way that INFINITELY many different electron inputs F(E) produce EXACTLY the same X-Ray output G(E) (to infinite precision) and it is impossible using G to test the hypothesis of any of these F's being the right one.
VERY scary but well known to mathematicians and by no means a unique case.
