Why it is not possible for a payee to verify double-spends of the coin? I'm reading Bitcoin paper of Satoshi Nakamoto (https://bitcoin.org/bitcoin.pdf).

It says:


  Each owner transfers the coin to the next by digitally signing a hash
  of the previous transaction and the public key of the next owner and
  adding these to the end of the coin. A payee can verify the signatures
  to verify the chain of ownership.


And the very next sentence is:


  The problem of course is the payee can't verify that one of the owners
  did not double-spend the coin.


That is, as I understand, after every transaction a new transaction hash is generated. So, first, it is not possible to double-spend a coin because it already has a new previous transaction hash. Second, why it says payee can't verify a double-spend if every transaction a coin has a new previous transaction hash?
As a participant in the peer to peer network, it is impossible for you to know that the state and timing you are seeing are consistent with what other people are seeing. You can't ask other participants in the network what they are seeing directly as they can lie, withhold information, or the request can be tampered with on its way across the network. The block chain and proof of work mining system are designed to ensure that in all probability, nodes in the network have consistent ordering and atomicity. Absent this, it would be very feasible for nodes to see a different state and make different decisions about validity.