(Superseded by Metarepresented Money.)
We must consider public versus private money to learn how money can either be privately public, as in bank notes, or just publicly private, as in Bitcoin.
Publicly Private Money
In my pocket, I have an old leather wallet. It contains enough bank notes to buy a brand new wallet of a better model I saw in a magazine. This buying power belongs to just me: I am the only one who can use those notes to buy anything. Likewise, if I transfer them to another person, then instead of me, only this other person will own their buying power.
Yet even if my transferring away my bank notes can always transfer along their control, it could never transfer along their property, which is not only mine. The notes, as representations of money, do not belong to just me. For example, I have no right to create or destroy them: they are public. What belongs to just me or only to whoever else controls any such notes is rather their buying power, which hence must be private.
Indeed, if my bank notes were only mine, then I could transfer them away by selling them, not as money, but rather as purely concrete objects. However, this would prevent me at least temporarily from using those notes to buy anything. So by recognizing the buying power they then would lose as their monetary value and their instead keeping this value as its representation, we can conclude:
All monetary value must be private.
Its representation must be public, or unsellable.
Privately Public Money
Then, mistaking a representation of money for its represented monetary value makes that representation privately public. So any control of such a representational monetary value, whether centralized or decentralized, must also be privately public.
No commodity money can inherently distinguish between itself and its represented monetary value. Hence, all commodity money must be privately public. With directly monetary commodities (like sheer monetary gold), private control of public monetary representations is individual, or decentralized. However, with proxy representations of commodity money (like receipts for deposited gold), private control of public money becomes institutional, or centralized. Hence the privately public nature of central banks: any monetary authority must be as privately public as the monetary representations it depends on controlling. While conversely, any monetary representation controlled by a central authority must be privately public.
Purely Public Monetary Privacy
The Bitcoin monetary system represents any monetary value as a private key, then metarepresents it as the corresponding public key. Never before a monetary representation was inherently distinct from its represented monetary value: for the first time in monetary history, representing a private monetary value (as a private key) does not require any control of its public representation (as a public key). With Bitcoin, a public object can represent a private monetary value without ever becoming itself private—which makes its private control by any central monetary authority not only unnecessary, but also impossible.
Privacy Versus Anonymity
Monetary privacy means monetary control exclusiveness: the exclusive control of a monetary value and possibly of its public representation. It does not necessarily mean anonymity. Anonymous monetary control remains different from exclusive monetary control, even if helping protect it. This way, we can have monetary privacy without having monetary anonymity, despite not conversely.