kickingcoin,
The mining computations are not computing anything useful beyond the bitcoin network, but they are necessary for the whole thing to work. As you know, bitcoin network is a so-called peer-to-peer network, where there are a lot of nodes, which can freely join or leave the network at any time, and no one is more authoritative than others.
Without those computationally difficult calculations, it would be technically possible that some "dishonest" node will join the network, running with the modified version of the bitcoin software, trying to generate some "fake" transaction, which would for example steal money from other address, etc. The mining prevents this from happening, because the dishonest node can't simulate the proof, that mining work was being done, so the other nodes knows that that transaction is fake and should be ignored.
You can read more about it here:
https://en.bitcoin.it/wiki/Proof_of_work