Comment by hackinthebochs 20 hours ago
Mathematical notation is a human invention, but the structure that mathematics describes is objective. The choice of basis changes the absolute number of terms, but the relative magnitude of terms for a more or less disordered state is generally fixed outside of degenerate cases.
This is getting tedious. The point about mathematics was simply that it carries and objectivity that natural language does not carry. But the point about natural language was always a red-herring; not sure why you introduced it.
>You can’t make the whole class simplified at once
Yes, this is literally my point. The further point is that the relative complexities of two systems will not switch orders regardless of basis, except perhaps in degenerate cases. There is no "absolute" complexity, so your other points aren't relevant.
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.
The structure that most words describe is objective, so you haven’t distinguished math as a language. (Nor is mathematics entirely “objective”, eg, axiom of choice.) And the number of terms in your chosen language with your chosen basis isn’t objective: that’s an intrinsic fact to your frame.
The complexity of terms is not fixed — that’s simply wrong mathematically. They’re dependent on our chosen basis. Your definition is circular, in that you’re implicitly defining “non-degenerate” as those which make your claim true.
You can’t make the whole class simplified at once, but for any state, there exists a basis in which it is simple.