# Can a problem be solved if there exists no solution for it in any context?

Is finding a solution for a problem in a given context is an attempt to find a solution for the problem in another context?

For example, it seems some of the hardest problems of real analysis were not solved till their solutions were found in the context of geometry.

Update: I think it makes the question more clear if I provide an example. Let's say I want to prove if x+y=x+z, then y=z, where x,y,z $\in F$, where $F$ is a an algebraic field with operation "+". For example for this problem I can think of context where solution exits and that would be the ring of integers. If 1+2=2+1 then I can show 1=1. Now once this solution is found using this context of integers I can now apply this to the original problem in the context of field.

Now what I was asking was that should such a solution exists in some contexts (context as used in example above) for a problem to be solvable.

I do not know the background science behind this but it would be great if references can be introduced so I can get to know the right terminology.

• It may help to rephrase the question a bit. At the moment, it looks almost trivial: if a problem has no solution in any context – i.e., not anywhere – then it has no solution. Presumably, that’s not what you’re getting at, though; so, it would help if you could expand on what you mean by ‘context’, and on what it takes to have a solution in a context. Constructivists hold that you can’t prove ‘There's such-and-such a number’, except by producing a number that indeed is such-and-such. (In particular, you can’t prove it by contradiction.) Is your question related to that viewpoint in any way? – MarkOxford Nov 12 '17 at 13:47
• Do you mean : 'Is the non-solvability of a problem in any context in a given domain an indication that its solution, if any, is to be found in a context in a different domain ?' E.g., does the non-solvabilty of a problem in theology indicate that its solution - or dissolution - belongs to physics ? – Geoffrey Thomas Nov 12 '17 at 14:22
• @MarkOxford By a context I mean something like a domain and by domain I mean something that contains similar objects. For example, an algebraic filed is a domain where objects existing in it share certain characteristic. I think Geoffrey Thomas comment is very close to what I want. – Mathnewbie Nov 12 '17 at 19:52
• @GeoffreyThomas Yes very close to what I want but a solution to a non-solvable problem of theology could exist in physics or any other domain that has not been yet explored or discovered. Of course the domain must have certain characteristics for example there should exist a one-to-one correspondence between the objects of the two domains. – Mathnewbie Nov 12 '17 at 19:55
• I suppose I still don’t really understand the question. If a problem has no solution in any domain, then where would the solution be? Are there sets that aren’t domains, because their elements aren’t similar enough to one another? Or is the idea that there are ‘meta-domains’ or 2nd-order domains? If we find no solution in any of the 1st-order domains of a particular 2nd-order domain, then we might move on to a different 2nd-order domain and look through its 1st-order domains for a solution. – MarkOxford Nov 12 '17 at 21:00

To answer the second part of your question 'it seems that the solution was found and then the question answered' Yes, that is the way it's done most of the time because it is much simpler once you know the outcome. You can simply do Sherlock Holmes eliminate all the impossible until all you're left with is the possible. And that would be a good approach I would say it is a better way than how scientists handle problems today. They did not eliminate the possible they actually attempted to explain reality with the possible even though it is impossible, such as dark energy and basically everything else in science

If a problem is worth having, it can be solved. You seem to be talking about how new techniques have been needed to solve seemingly insoluble problems, like say Fermat's last theorem.

There seem to be mistakes in tense or spelling that make your question confusing, and I am struggling to understand what you are actually asking.

It puts me in mind of how for Newton it was essential to put his theorems and proofs in geometric forms, as geometry was considered most fundamental of maths by the Greeks. Even though, this was difficult and awkward to do and made it harder to see the implications of what he did. Soon after geometry lost it's 'supremacy'.

To what extent does the world divide cleanly into distinct contexts? In the end isn't there only one context, our minds, which unify all contexts? We switch language-game to adapt to each

If I correctly understand your mathematical example, then mathematics provides many examples of the following type:

A problem has no solution in the context, where the problem is stated. But the problem has a solution in a broader context.

For example take the problem:

1) Find all solutions of the quadratic equation x^2+1=0.

The problem is stated in the context of real numbers. But the problem has no real solutions. Instead the problem has two solution, namely +i and -i, from the context of complex numbers.

2) Similarly, take the problem: Find all solutions of the quadratic equation x^2-2=0. The problem has no solution in the rational numbers, but it has two solutions in the real numbers.