Comment by zmgsabst
I didn’t introduce it, you did — by positing that formal language is more objective, as you’ve again done here. My original point was that mathematics is human language.
> The further point is that the relative complexities of two systems will not switch orders regardless of basis, except perhaps in degenerate cases.
Two normal bases: Fourier and wavelet; two normal signals: a wave and an impulse.
They’ll change complexity between the two bases despite everything being normal — the wave simple and impulse complex in Fourier terms; the wave complex and impulse simple in wavelet terms.
That our choice of basis makes a difference is literally why we invented wavelets.
Yes, that is a degenerate case. We can always encode an arbitrary amount of data into the basis functions to get a maximally simple representation for some target signal. If the signal is simple (carries little information) or the basis functions are constructed from the target signal, you can get this kind of degeneracy. But degenerate cases do not invalidate the rule for the general case.