Which is More Fundamental?

Some years ago I had the good luck to work with an engineer on the design of a modem. I was very familiar with time domain representations of the signal over a phone line and he with the frequency domain. At first I presumed that he agreed that the time domain was the real world and that he merely knew many useful theorems and had many useful intuitions about the frequency domain. It was soon evident that he thought the same about my view—that it was merely a peculiar formal way of looking at things. The good news is that many problems were obscure to one of us and entirely clear to the other. The most dramatic (in my favor) was a very peculiar property of filter that could be proven by fancy mathematics in the frequency domain. The same theorem in the time domain merely stated that a signal can’t begin to emerge from a filter until it has begun to enter it. There are many examples where the opposite applies and the frequency domain makes things clear. In this particular case one mathematical construct, the Fourier transform, mapped from either domain to the other. I found it to be dangerous to mix time and frequency domain arguments. I wish I could remember the nature of the pit-falls. An interesting situation here is that neither view claims to be the lower one. I learned this only after meeting someone that thought my view was not the fundamental one.