Nick Rowe has an interesting incremental argument asking: when does the cash disappear? It's kind of another Sorites paradox, except it asks what properties must you take away for something to stop being "cash". In one sense, I agree: money seems to be better defined by symmetry properties, so whatever it is, it doesn't stop being money unless you take away those symmetry properties ... which means you've made the economy become a non-ideal information processor and lost lots of output.
But that's the key -- Nick Rowe's properties are mostly extraneous properties of money, not extraneous properties of "cash".
In the information transfer model, the key property that distinguishes what we call "cash" (bits of paper with e.g. Benjamin Franklin on them in the US) from other types of "money" like M1 or MZM is that "cash" (aka M0) is what anchors the core inflation rate as well as the NGDP path. That is to say the inflation rate (the change in the price level P) is set in the market:
P : NGDP ⇄ M0
M1 and M2 don't set the inflation rate (there are large fluctuations), but M1 and M2 are money in the sense that they can be used for transactions.
What is it about M0 that makes it have this property?
I don't know for certain. I think it may be the fact that central banks can't get a hold of your M0 to pay (positive or negative) interest on it. Things that are included in M1 and M2? Yes, definitely. I have some M1 in my bank right now gaining interest.
So when Nick says:
2. Muggers become a problem, so people put their currency in a box at the central bank with their name on it. ...
3. The bank notices it is now administratively easier to pay (positive or negative) interest on currency than it was when people kept their currency in their pockets. So the bank now has a second monetary policy instrument, in addition to Open Market Operations.
I think these steps are a bigger deal than Nick makes it out to be. The information entropy in the distribution of currency would suddenly vanish since the state of every dollar is now known (so that interest could be paid). Basically I have just acquired log2 C(n, k) ~ n H(k/n) ~
1 trillion 4 billion bits of information (k = people in the country 300 million, n = 1.3 trillion dollars). That may not sound like a lot in terms of electronic storage, but the entire annual economy (n = 18 trillion dollars) represents only about 2 trillion 5 billion bits. (Assuming indistinguishablility.)
In a thermodynamic system, knowing the state of every atom in an ideal gas would allow you to extract useful work out of the system. However, knowing each bit of information about the state of every atom costs kT log 2 in energy, so this is an impossible task.
There is no energy restriction in the economic case, but this still represents a major change in the properties of the macroeconomy.
Basically, labeling those boxes at the central bank is what "cash" does.