The bitcoin network, including every miner, is the biggest computing project that humanity has created. Thus I ask: why, instead of using it to generate useless data, don't we use it to generate meaningful data, such as theorem proofs or something similar to folding@home?
Solving SHA256 hash problems on Bitcoin is useful in the sense that secures the Bitcoin blockchain, but if your question is: "why can't it do something computationally useful as a side-effect?", then I think the answer is "we don't know how". For Bitcoin to work, the proof-of-work that miners do must have the following properties:
Easy to verify solutions
Hard to find solutions
Difficulty of finding solutions can be precisely quantified
Provably inseparable from the block it secures
Cryptographic hash functions like SHA256 satisfy these four properties. I don't think automated theorem proving fits the bill because, as far as I'm aware, there is no way to prove how difficult it was to find the theorem that you proved. General purpose grid computing, like BOINC, doesn't fit the easy-to-verify requirement, at least in the context of Bitcoin. (In fact, I think this is an active area of research in grid computing, called the "cheating problem".)
If there some proof-of-work scheme that satisfies these four properties and also has some useful computation as a side-effect, that would be interesting. I'm not aware of any.
Well, I know of one altcoin that does useful work: Primecoin.
As Wikipedia describes it:
Primecoin (sign: Ψ; code: XPM) is a peer-to-peer open source cryptocurrency that implements a scientific computing proof-of-work system. Primecoin's proof-of-work system searches for chains of prime numbers.
Bitcoin mining is useful work, it secures the Bitcoin block chain. If you want to pay people to do other kinds of work, you can certainly do that, but that has nothing to do with Bitcoin mining. The work Bitcoin miners do is precisely the work needed to secure the Bitcoin block chain.
There is no known way to make something be both protein folding and also secure the Bitcoin block chain, any more than you can make doing your taxes also protein folding. Hashing wasn't picked arbitrarily, it was picked because it secures the Bitcoin block chain.
In addition to PrimeCoin, there was also PermaCoin proposed as a solution to archival of public knowledge:
Excerpt from the abstract:
We propose a modification to Bitcoin that repurposes its mining resources to achieve a more broadly useful goal: distributed storage of archival data. We call our new scheme Permacoin. Unlike Bitcoin and its proposed alternatives, Permacoin requires clients to invest not just computational resources, but also storage. Our scheme involves an alternative scratch-off puzzle for Bitcoin based on Proofs-of-Retrievability (PORs). Successfully minting money with this SOP requires local, random access to a copy of a file. Given the competition among mining clients in Bitcoin, this modified SOP gives rise to highly decentralized file storage, thus reducing the overall waste of Bitcoin.
The full paper can be found here: https://www.cs.umd.edu/~elaine/docs/permacoin.pdf