Okay yeah, that makes sense. There needs to be a chain in the current system for it to work.
But say the UK and US governments decide that bitcoins are a threat to the national currency and ban them. Couldn't the existing infrastructure be modified and used as a kind of computational mechanical turk? Miners use their computers to solve problems, they are checked several times and then once verified miners are paid by the people who posed the problem based on the complexity of the problem?
? That would be a digital mechanical turk, not a decentralized currency. How would you actually make the payments? The WHOLE point of bitcoin is to create a verifiable history of transactions so that you can prove you own the coins you own. This problem solving system you propose has nothing to do with that. Awarding people bitcoins for doing work is COMPLETELY secondary to the main purpose of verifying (I like to picture it as shutting the lock on a clear box that you can look into, but not modify, like a growing matrioshka doll.). The awards are simply to incentivize people to actually do this verifying, to keep the system stable. The system could equally have been developed with all coins in existence in the genesis block, and awarded to one person, and the fundamental ability to TRANSFER value through them would remain unchanged (a far more inelegant solution, as satoshi has to spend all the coins for them to get into circulation, and noone has any incentive to run a miner, but the core double-spend prevention would STILL WORK.)