Is Gabor uncertainty a feature of the Fourier transform, or of nature? By Gabor uncertainty, I mean the principle of uncertainty as applied to signals — with the result that you can't have arbitrary time and frequency localization. 

By way of background to my question, I wrote about this recently in a blog post. In the last paragraph, I stated:


  Signals do not have arbitrarily precise time and frequency
  localization. It doesn’t matter how you compute a spectrum, if you
  want time information, you must pay for it with frequency information.


The point here is that, as I understand it, this uncertainty is an irreducible property of signals. It is not a 'feature' (or bug) in the Fourier transform or mathematics generally. And you can't get around it by using non-Fourier methods like matching pursuit.

I'm now unsure this is really true. Someone wrote to me recently:


  ...the uncertainty is there because of the math, not nature. A
  'signal' is a mathematical construct... The timing information of a
  signal is in the phase, and for a short-lived pure tone you can
  exactly recover the timing and frequency from the temporal phase.


I guess I buy this. But when we're measuring unknown, real, broadband, noisy signals does it still hold?
Time/frequency uncertainty is a property of the meanings or definitions attached to the terms "time" localization and "frequency" estimation.  It also alludes to some apriori knowledge, or lack of it, about the characteristics of the signal in question.