Let there be two owners A and B of commodities x and y, respectively, of whom A wants y and B wants x. With no money and no third commodity, the only way for both owners to obtain their desired commodities is directly from each other:
| A ⇢ y | B ⇢ x |
| x | y |
| y | x |
Otherwise, A and B must delegate their commodity ownership to someone who then redistributes it between them. However, such a centralized solution would at least partially contradict the same ownership, by at least partially transferring it away from its rightful controllers. Hence, only a decentralized solution can preserve the whole commodity ownership underlying this exchange, by A and B exchanging x and y directly.
Still, direct commodity exchange poses two problems, either of which alone is enough to prevent it. The first problem has a subjective nature:
- To be exchangeable for each other, x and y must share the same exchange value.
- It can happen that every exchangeable quantity of x has a different exchange value to that of any exchangeable quantity of y.
The second problem has an objective nature instead. Let (as below) A, B, and C own commodities x, y, and z, respectively. If A wants y, B wants z, and C wants x, then direct exchange could not give those three owners their desired commodities—as none of them owns the same commodity wanted by who owns their wanted one. Moneyless exchange now can only happen if one of those commodities becomes a multiequivalent: a simultaneous equivalent of the other two commodities at least for the owner who neither wants nor owns it—whether the other two owners also know of this multiequivalence or not. For example, A could obtain z in exchange for x with C only to give it in exchange for y with B, this way making z a multiequivalent (as asterisked):
| A ⇢ y | B ⇢ z | C ⇢ x |
| x | y | z* |
| z* | y | x |
| y | z | x |
Still, this individually-handled multiequivalence poses a second pair of problems:
- It enables conflicting indirect exchange strategies. In this last example, A could still try to obtain z in exchange for x with C (only to give it in exchange for y with B) even with B simultaneously trying to obtain x in exchange for y with A (only to give it in exchange for z with C).
- It not only allows—again—for all mutually exchangeable quantities of two commodities to have different exchange values, but also increases the likelihood of that mismatch, by depending on additional exchanges between different pairs of commodities.
Social Multiequivalence
Fortunately, all those problems have the only and same solution of a single multiequivalent m becoming social, or money. Then, commodity owners can either give (sell) their commodities in exchange for m or give m in exchange for (buy) the commodities they want. For example, again let A, B, and C own commodities x, y, and z, respectively. Still assuming A wants y, B wants z, and C wants x, if now they only exchange their commodities for that m social multiequivalent—initially owned just by A—then:
| A ⇢ y | B ⇢ z | C ⇢ x |
| x, m | y | z |
| x, y | m | z |
| x, y | z | m |
| y, m | z | x |
With social (rather than individual) multiequivalence:
- There are always two exchanges for the owner of each commodity (who either sells or buys it before buying or after selling another one, respectively), with any number of such owners, in a uniform chain.
- All commodity owners exchange a common (social) multiequivalent, which eventually returns to its original owner.
Additionally, with a social multiequivalent (money) divisible into small and similar enough units, even if all mutually exchangeable quantities of two commodities have different exchange values, these two commodities will remain mutually exchangeable. For example, let two commodities x and y be worth one and two units of a social multiequivalent m, respectively—x(1m) and y(2m). Then, let their owners A of x and B of y be also the owners of three m units—3m—each. If A and B want y and x, respectively, but always exchange their commodities for m units—x for 1m and y for 2m—then:
| A ⇢ y | B ⇢ x |
| x(1m), 3m | y(2m), 3m |
| y(2m), 2m | x(1m), 4m |
Finally, with social multiequivalence thus making, as only money does, commodity exchange always possible, every social multiequivalent is money, which is conversely any form of social multiequivalence.
Money as Decentralization
Even so, historically, despite preserving the decentralized ownership of commodities during their exchange, money has itself become rather centralized. Indeed:
- It must represent the same decentralized ownership it preserves.
- It must be concrete for all commodity owners to share it.
However:
- Its concreteness to each among those owners requires its private control by a public authority—whether over selling, buying, creating, or destroying it.1
- Its then-centralized control at least partially prevents it from still representing a decentralized commodity ownership—thus defeating its original purpose.
Fortunately, despite necessarily concrete to all people, or socially concrete, a monetary representation can be rather abstract to each person, or individually abstract. For example, cryptocurrencies—like Bitcoin—use public-key cryptography to represent both money as a private key and this private key as a public key, so money becomes metarepresented, or metamoney. Then, despite remaining socially concrete as a decentralized network, any such metarepresentation of money becomes individually abstract as a monetary—meta—unit, which preserves its decentralization, by preventing any public authority from privately controlling it.
