Webster's (2010) definition of solution is “an answer to a problem”, the definitions of problem include “a question raised for inquiry, consideration, or solution ...”, “an intricate unsettled question”, “... a source of perplexity, distress, or vexation” and “... difficulty in understanding or accepting”. So when I ask –what appears to be the problem? I am potentially asking clients multiple questions.
If I say “I solved a problem” I could also be referring to any of those definitions. Sometimes a solution is as simple as explaining bugs and features or resetting expectations, sometimes it is as complex as analyzing system internals and addressing a misconfiguration or completely replacing non- or malfunctioning elements.
For me the first step of solving a problem is understanding why and to whom it is a problem. In my experience a great many 'problems' turn out to be misunderstandings and inaccurate expectations.
This is also why I don't see any problem as unsolvable, as long as one understands that an open-ended problem will require an open-ended solution. Gödel's incompleteness theorems (Wikipedia, 2010) don't say that solutions are impossible but that some elements may be unprovable. I find no better resource than logic for problem solving. 'Insoluble' problems come from framing (Wikipedia, 2010) or the belief that all solutions can be consistent and complete. In fact, in my experience the manner in which the statement of a problem is expressed gives the framework of, if not the solution itself. For example, when a client complains of a connectivity problem getting them to be specific about the symptoms often makes the solution apparent even to them.
However there are many problems for which the solutions are extremely complex or simply unacceptable. I don't see this much in the technical realm because it is usually possible to get people to understand facts and to act on them reasonably. But in the 'real world' it is very common for people to confuse belief with knowledge and to have emotional investments in unprovable or mistaken assertions. I'm going to refrain from using any political situations as examples.
Environmental degradation is not complex. While the environmental system is complex, that isn't the cause of the degradation. The cause is human behavior, which while also complex, degrades the environment in clear and simple ways. But I myself find it a challenge to walk everywhere I want to go and eat locally all the time without the use of non-renewable packaging and utensils. And I am well aware that my computer collection represents both a pile of toxins and much wasted energy. I content myself with the knowledge that some of us are making some progress and even if ecology collapses to the point that human civilization collapses life generally finds a way (Ben-Ari, 2005). So there's an open-ended problem with an open-ended solution.
Ben-Ari, M. (1995) Whose final hour? The problem of naïve egocentric catastrophism in doomsayers and catastrophists.', The Skeptic, 22 September, 2005. Excerpt available from: http://goliath.ecnext.com/coms2/gi_0199-5666301/Whose-final-hour-The-problem.html (Accessed 6 February, 2010)
Merriam-Webster Online Dictionary (2010) Problem [Online]. Available from: http://www.merriam-webster.com/dictionary/problem (Accessed: 6 February, 2010).
Merriam-Webster Online Dictionary (2010) Solution [Online]. Available from: http://www.merriam-webster.com/dictionary/solution (Accessed: 6 February, 2010).
Wikipedia (2010) Framing effect (psychology) [Online]. Available from: http://en.wikipedia.org/wiki//Framing_effect_(psychology) (Accessed: 6 February, 2010).
Wikipedia (2010) Gödel's incompleteness theorems [Online]. Available from: http://en.wikipedia.org/wiki/Godel's_incompleteness_theorems (Accessed: 6 February, 2010).