Am I right in that Bitcoin isn't backed by computational complexity; but, instead the continued ability for machines on the network to communicate with one-another?
Bitcoin is backed by cryptography. The bitcoins in your wallet cannot be spent by anyone else.
Now if there were no network access, then there's a problem. I might accept your transaction but without access to the network I couldn't know that you didn't already spend those bitcoins elsewhere.
IANAeconomist, but a currency is back by trade. You and I accept a dollar or a gold coin because other people accept them at (usually) parity for value-- call it a network effect.
But, a dollar or a gold coin has limited value if you can't trade it. The same for a bitcoin; except, it can only be traded via coordination with many other actors in the economy. (The longest block chain is the "truth.") While a dollar or a gold coin can be traded physically, a bitcoin is never traded until the economy as a whole agrees-- the record of the transaction appears in the chain.
Therefore, the act of trading bitcoins, and by extension bitcoin itself, is backed by the presumption that Internet access will be pervasive and unfiltered.
surely the answer to that is that I shouldn't accept someone's bitcoins without being connected to the network to be able to check if they were already spent. this kind of "offline" transaction isn't allowed with regards to a credit card, why would it be allowed in our system?
Yes, for bitcoin there are no offline transactions. A transaction is recorded by telling everyone on the network it happened. Then everyone knows which addres spent and recieved bitcoins.
