Mathematics (specifically, (computational) group theory and/or mathematical logic) actually has something called the word problem. As any frustrated high school student will testify, the word problem is unsolvable in a very real sense.
The word problem:
Let G=<Γ|R> be a finitely presented group (see generators and relations for groups to understand what this is). Since Γ generates G, any element of G can be represented as
g1k1
g2k2
...
gnkn
where all g
i∈Γ and k
i≠0 is an
integer.
We still haven't mentioned the set of relations R. What R does is make elements of G have more than one representation. For instance, if G=<a,b|ab=ba> is the free abelian group with 2 generators, then ab=ba=a17b-3a2a-18b4=...
The word problem is this: given Γ, R, and 2 words, determine if the 2 words are equivalent, i.e. whether they correspond to the same element of G.
The word problem is known to be recursively enumerable but not recursive: there's a (trivial) algorithm that will halts iff the 2 words are equivalent, but there's no algorithm to tell you (in the general case!) whether or not the 2 words are equivalent.