TITLE: Does algorithmic unsolvability imply unsolvability in general?
QUESTION [10 upvotes]: I recently found out that there is no algorithm which, given an arbitrary group presentation, will determine in finite time if it represents the trivial group*.  Additionally, in a lecture I recently attended, it was portrayed that given any desired property P of a group, there is no algorithm that will determine if an arbitrary group (presentation) has this property.  (For additional info, see the second answer to this question on MathOverflow).  
My question is about the scope of this notion of "algorithm."  With regard to group presentations, my intuition tells me that there must be some way to classify or otherwise look at these arbitrary group presentations that would enable us to determine if they are the trivial group (or have some property P).  However, I am not very well versed on what it means for something to be solved via an algorithm vs. other mathematical methods.
In general, if something has been shown to be unsolvable via algorithm, does it mean that it is entirely unsolvable (and thus futile to pursue in research)?  Or, are there cases where we have shown that despite being unable to use an algorithm to determine if a property exists, we have used other mathematical methods with success?
*Clarification by Arturo.

REPLY [4 votes]: It's actually even more shocking than what Arturo's answer might suggest.  Consider the following algorithm.  Given a group presentation $G=\langle S|R \rangle$, enumerate all possible proofs (say, using the axioms of ZFC), and check whether or not they give a proof that $G$ is trivial or not.  If $G$ is trivial, then this algorithm must finish.  Indeed, if $G$ is trivial, then we can write each element of $S$ as a product of conjugates of elements of $R$, and this constitutes a proof.
However, we know that there are inputs for which this algorithm does not terminate.  This implies that there are finite presentations that define nontrivial groups, but for which there is no proof that they are nontrivial.  This gives a concrete and down-to-earth example of Goedel's incompleteness theorem!