Would it be possible to replace random hashing calculations with say real world distributed computer calculation processing such as is from the BOINC client, World Community Grid or Folding@Home?
Why pay miners for calculations that don't secure the hash chain? These "random" hashing calculations are not random -- they add computational effort to the public hash chain, preventing double spending attacks.
I am thinking if this is possible this alone could be a HUGE boost to bitcoin (and bitcoin like) currencies as you greatly expand your interested client base and appeal to individual humanitarian causes as well as natural profiteering and liberty minded individuals.
You're welcome to do all those things if you want, but it doesn't help secure the hash chain. That's what miners are paid for.