I think you need to put that part of the abstract into context
payments to be sent directly from one party to another without going through a
financial institution. Digital signatures provide part of the solution, but the main
benefits are lost if a trusted third party is still required to prevent double-spending.
We propose a solution to the double-spending problem using a peer-to-peer network.
The network timestamps transactions by hashing them into an ongoing chain of
hash-based proof-of-work, forming a record that cannot be changed without redoing
the proof-of-work.
He is not referring to double-spending in the current economic system. Currently, they go through third party institutions that you trust that prevent as much as possible people double spending their money and over withdrawing their account, security, etc.
Digital signatures provide the part of solution of sending BTC from me to you, but they need a mechanism to prevent me spending those coins twice or more.
The mechanism to prevent double-spending in Bitcoins is the transaction block ledger, which is the peer-to-peer database that timestamps transactions by hashing them onto the ongoing chain.
Read the whole of the paper to understand exactly what he means, its not that long.
I don't think BTC was created with the idea in mind to prevent double-spending that may or may not be present in our economic system. Preventing double-spending is just an essential, most fundamental quality of any transaction.