Equivalence of solutions Are all solutions to a problem Equivalent at some conceptual level?
An easy example,trying to solve a system of 2 linear equations has ways to be looked at:


*

*Graphs of lines and their intersection .

*A linear transformation from a vector to another.

*Using Euclidean geometry and line segments.
And...


All these examples have equivalent meaning at some level:
The degree of freedom must be zero in order for a Unique solution to exist.
Are there any counter examples?Where there are two solutions to a problem by they address different concepts?
 A: In general I would say that there exists problems to which there exists many ways to solve them. This is often caused by the different levels of abstraction. E.g. just as one could solve a differential eq. using differential calculus "abstractions", one could do so using algebra. This is because fundamentally most "higher level" constructs are constructed using lower level ones. Thus one can use any level to do the solving, but different levels might be trickier than others.
It's usual to progress in mathematics through lemmas and previous theorems so that one can produce more results by relying on the notions in earlier ones, without having to rewire everything again in some new proof.
Since the solving techniques are "equivalent" in the sense that all of them can produce a solution, then one can use any of them. However, generally one chooses a solution method that's the easiest, fastest or fits to the application.
For example, the field of functional analysis produces a lot of abstractions for some more common e.g. measure theoretic problems that make the proofs much easier to do.
