Often in mathematical writing one encounters texts like ''..we observe that this-and-that..''. Also one may find a review report basically saying ''..the paper is just a chain of observations…''.
What is an ''observation'' and how it differ from ''true results''? Is it possible to turn a nice theorem with a nice proof just to an ''observation'' with, say, some nice definitions? Does a ''real result'' turn to an ''observation'' if it is represented in a way that makes it easy to, well, observe?
I'm asking because I face papers that seem to be just ''chains of observations'' but in the same time I have seen some papers rejected for being ''just observations''. I start to feel it has something to do with the way the core ideas are presented. Or maybe some smart definitions make certain results somewhat ''too easy'' to be mathematically interesting? (Or maybe I'm just not capable to tell the difference between math and ''math''…)
Best Answer
Typically, one says that something is observed as a synonym for it being obvious or at least not something that they intend to prove. An observation isn't a result per se, but an implied result often left up to the reader.
Papers can contain a lot of observations, as long as the reviewers don't feel that this is being intellectually dishonest and skirting meaningful issues. That said, sometimes the technical issues result from the definitions themselves, and historically, many 'real results' have been made 'observations' by a better use of definitions, and sometimes this is exactly the point.
When doing mathematics research, I've found that discovering the definitions or frameworks that make the results somewhat "too easy" is really the crux of the issue and this is often the place where real progress is made.