To solve problems it is necessary to identify them correctly. One way to define the term problem is as a contradiction between at least two desired outcomes. I like to categorize problems as being material, temporal, intellectual, or antagonistic.
Material problems include all manners of technical, biological, natural, and other spatial problems not directly caused by human interaction. A material problem is for instance one where there is a contradiction between the different properties sought in a material. One could for instance want a substance to be both flexible and hard, both of which might be impossible to get without some serious problem solving. The types of material problems I have dealt with include finding new ways of casting bronze.
Some contradictions occur within ourselves because we have different preferences at different times. What feels good in the short term might not be beneficial in the long term. We often end up following our short-term interests by habit. So how can we change that habit?
”We are what we repeatedly do. Excellence, then, is not an act, but a habit.”
(From Will Durant, apparently summarising Aristoteles teaching, The Story of Philosophy: The Lives and Opinions of the World’s Greatest Philosophers.)
By taking small steps that give in to our long term interests, we grow a habit to considers our long term preferences. Our will can be changed by our will – if we take small steps.
Off course, forcing our will towards long term interests is also dangerous ditch to fall in since it removes us from living here and now. Only thinking about what is best for the future is a sure way to get stressed out, highlighting that it is beneficial to find the balance between our long and short term preferences.
Mathematical problems can be closely linked to the material world but are clearly intellectual problems. Other examples include theological and philosophical problems. An intellectual problem is, for example, the supposed incompatibility between our responsibility for our actions, and the possibility that our actions are entirely determined by our genes and our environment.
These types of problems occur only because of human attitudes and behaviours. Some material, temporal or intellectual contradictions become problems only as a result of incompatibilities betweens individuals or groups. Sport competitions and civil wars are examples of antagonistic problems. Most estetic problems are also antagonistic problems, though often paired with intellectual (for instance colour theory) or temporal issues (fashion).
Problem solving involving more than one person will often be affected by antagonistic problems as we all have different pre-conceived notions and hang-ups of how problems should be viewed and solved.