Mining in Crypto Currencies is the process to Verify, Order, and Secure the transactions in the global ledger. Without mining no one could send transactions, without transactions the currency does not function.

The hard problem that the mining algorithm computes is a Hash of a batch of transactions together with some meta data. This meta data also includes the Hash of the previous batch. This is creating a cryptographic chain of transaction batches (blocks). A Blockchain.

This is securing the transactions in the following way:

- You can not change a transaction without changing the hash of the batch the transaction was included in.

- This would also change the hash of all transaction batches that follow.

- The crypto currency miners and users follow the longest chain with the most work put into.

- To fake or alter a Transaction you would need to recalculate the hashes of the blocks that follow faster then the honest miners so that your new chain is the longest chain.

- More honest miners means it is harder to manipulate the blockchain since to do that you would need to control more then 51% of the global hash power of this currency.

Securing a Blockchain is not as noble of an effort then curing cancer with Folding@Home, but it is also not just wasting electricity.

You’re right; I phrased that poorly. What I intended was “the value derives from a proof that an amount of work was done, independent of the hard problem that necessitated the work or the resources expended to solve the problem.”

The value in proving some amount of work was done is that it’s a mechanism for defeating fraud.

What it does is prevent someone from fraudulently altering transactions, because to do so would require an enormous amount of energy.