The proof of work needs to be tied to the block. It also needs to be very quick to verify but hard to compute (like getting a hash under a certain value). With scientific problems, you may need to solve a problem, then ever node on the network solves the same problem to verify what you did - which would waste a hell of a lot of resources!
We actually just had a double spend try and happen to us. I cannot show you the links, because the blockchain already ignored them.
What happened was:
We run a fundraiser for helping in the expansion of foldingcoin http://foldingcoin.net/fundraising/ and we use a company called Vennd.io to help in this fundraiser.
What happens is someone will send BTC to a certain address, and in return they are given an amount of 1 FLDC per every 0.00001 BTC they donated. Someone recently tried to double spend a BTC transaction to trick Counterparty into giving them double the FLDC in return for their BTC. However since the double BTC was denied, so was the double FLDC.
The proof of work needs to be tied to the block. It also needs to be very quick to verify but hard to compute (like getting a hash under a certain value). With scientific problems, you may need to solve a problem, then ever node on the network solves the same problem to verify what you did - which would waste a hell of a lot of resources!