Introduction:

The motivation for creating a type system for currencies is that it allows us to reason carefully about their underlying semantics which have political, economic and technological dimensions. What follows is a brief sketch that allows us to delineate important differences.

The five functions of money:

$$1.$$ Store of value: it must be possible to reliably save, store and retrieve money.

$$2.$$ Medium of exchange: when money is used as an intermediary between goods and services, it functions as a medium of exchange.

$$3.$$ Measure of value: Money serves as a standard numerical unit of measurement of the market value of both goods and services.

$$4.$$ Standard of deferred payment: an acceptable way to settle debt.

$$5.$$ Representation: money is designed to represent the government it serves. This is why each bill typically has ornate decorations.

Two types of currency:

It is worth noting that if a government is sufficiently powerful, relative to other governments, then its national bank may print as much money as it wants so its currency serves the role of fiat money. As fiat money no longer serves as a reliable store of value we may distinguish two types of currency:

Type I: satisfies all five functions.

Type II: satisfies the last four functions.

It is worth noting however, that the second type of money is separated from the first kind by a phase transition of sorts but they are still comparable. There is however a third, and fundamentally different kind of currency.

Cryptocurrencies:

Bitcoin and other cryptocurrencies attempt to dispense with governments as the minting process and verification processes are entirely algorithmic. In this setting, the strength of the currency represents the economic activity of a collection of entities without borders. Interestingly, the entities do not even need to be human.

In this sense, cryptocurrencies are a fundamentally different kind of currency.