As far as my limited knowledge permits, it may be that meaning does motivate the creation of the said symbols. A problem therefore can motivate the creation of

As far as my limited knowledge permits, it may be that meaning does motivate the creation of the said symbols. A problem therefore can motivate the creation of a full set of symbols: Seven Bridges of Königsberg and Graph Theory; Calculus finds its root meaning in the necessity to formalize change. I also agree that the manipulation operates at an indirection but has meaning, I think the greatest example to that is problem reduction, that operates at a higher level of indirection but eventually maps back to a completely different meaning. But it is often times that the manipulation of the symbols is simply inspired by meaning, in fact, you can define abitrary operations on any system (by doing S -> S in a relation) without any useful semantic relevence to anything. Maybe we are cherrypicking the meaningful ones? But maybe the fact that you can do so implies a different meaning. Well I guess we will never know. If we define symbols to be part of nature then surely "The opposite of nature is impossible." Mathematics is effectively the study of things, therefore it has purpose, it can even study itself!