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Avebury, here is the reasoning behind my statement that the value of money is ultimately controlled by armies.
One of the roles that money plays is as a denominator of debt. In a world without money (your immediate family and close friends, for example), if someone gives you something or does something for you, you feel obligated to return the favor.
But if your neighbor or a stranger or a person whom you will never see again does some work or service for you, you might give them money in return for their work or service. Why would they accept money (in any form; gold, paper, sea shells, etc) instead of requiring you to immediately give or do for them something of equal value? Because they look at that "money" as a contract which can be called in later. Perhaps, if the money is widely-enough distributed, they can present it to a third stranger and demand some service or some product.
That money, or piece of paper, is a written promise saying that whoever holds it is entitled to demand a product or a certain amount of service from someone in exchange for it. It is an IOU for a given amount of tangible, real-world value.
Why would a stranger be willing to take a green piece of paper in exchange for, say, a perfectly good car or a day's worth of hard labor? Because the paper has printed on it the words "This note is legal tender for all debts, public and private", and is signed by the secretary of the Treasury of the United States.
Now, I could print up some Adam dollars and try to buy lunch with them, but the lunch staff might not want to accept such things, no matter how attractive I made them, or what substrate I printed them on. Why not? I could print the same "legal tender" message as the US government does on my own Adam dollars.
The reason they wouldn't want Adam dollars is that other people might not accept them as redeemable debts. As curiosities, perhaps. But not as reliable stores of callable-value. And that's because I don't have an army or a court system which forces people to accept payment of debts in Adam dollars, but the US government does.
Ultimately, where ever you find money (account ledgers between strangers), you will find armies (to enforce the accounting).
You mentioned that the Euro is not backed up by an army, but it is. That army is the US military, which pledged to defend Europe against any attack. The US maintains over 800 military bases around the world in order to ensure that every government with which we deal will accept dollars. It bombs countries and initiates coups when they don't play ball. Maintaining the value of the US dollar is no joke to the US government.
Here are two cases in point.
The CIA promised a tribe in Arabia that they would support them in their bid for rule, if the tribe would always price the oil that the country produced in dollars. Since the US prints dollars, that makes the price of oil surprisingly affordable to the US. That tribe was the Saudi tribe, and when they got control of the country, they re-named it Saudi Arabia, which is kind of like the Bushes renaming America "Bush America".
This works out very well for the US, as you can imagine. Keeping the Saudi's happy is worth a lot to the US. All the other oil-producing nations also price their oil in dollars.
However, in September 2000, Saddam Hussein stated that he would no longer accept dollars for oil, but would accept Euros instead. In March of 2003, about two and a half years later (it takes time to move equipment), the US started bombing Iraq.
In 2010, Libya's Gaddafi was sitting on 144 tons of gold and initiated a movement in Africa to refuse both the Euro and the dollar for oil and to initiate a new currency, the gold dinar. The bombing started in March, 2011, and one of the first things the NATO-supported rebels opposing Gaddafi did was to establish a central bank.
You know the standard cry of freedom fighters everywhere: "Freedom and central banking!"
The US government really doesn't tolerate competing currencies. It may ignore them for a while, but eventually, the treasury agents show up and haul everyone off to jail. Or the bombing starts. Right now, bitcoin is flying under the radar. If it ever gets to the point where it is a credible currency, it will be outlawed, and that outlawing will be enforced by an army.