One of the many articles on the Tricki that was planned but has never been written was about making it easier to solve a problem by generalizing it (which initially seems paradoxical because if you generalize something then you are trying to prove a stronger statement). I know that I’ve run into this phenomenon many times, and sometimes it has been extremely striking just how much simpler the generalized problem is.

edited Sep 26 2010 at 8:34
gowers

Great question. Maybe the phenomenon is less surprising if one thinks that there are ∞ ways to generalize a question, but just a few of them make some progress possible. I think it is reasonable to say that successful generalizations must embed, consciously or not, a very deep understanding of the problem at hand. They operate through the same mechanism at work in good abstraction, by helping you forget insignificant details and focus on the heart of the matter.

answered Sep 26 2010 at 10:27
Piero D’Ancona

— Generalizing a problem to make it easier

— MathOverflow

A general case has less information (details) than a special case. 

— Me@2012.03.10

2012.03.13 Tuesday (c) All rights reserved by ACHK