The Story So Far
The Ankh-Morpork Mint is named in Going Postal as an organisation in as much trouble as the Post Office was before Moist von Lipwig took over. Reacher Gilt had a chance of freedom while re-organising it, but he chose the wrong door. More was expected about the Mint in 2007's Making Money.
Physically, the Mint takes the form of quite possibly the ugliest building never to have won a major architectural award. It is described as a gaunt brick and stone block, its windows high, small, many and barred, its doors protected by portcullises, its entire construction saying to the world: Don't Even Think About It. Sticking out of its roof, giving the impression of a coin half-inserted into a very large piggy bank, is the Bad Penny.
On the inside, the "factory floor" is surreally composed of randomly built ramshackle wooden sheds, which are the preserve of the Men of the Sheds.
A hint of the future may be gleaned from the Patrician's satisfaction at Moist von Lipwig's achievement at the Post Office. For the Mint has a problem: the currency it makes in various combinations of metal is costing too much. For instance, the cost of making one small coin is nearly two hundred times the face value of that coin. Multiply this by several tens of thousands, and the Mint itself can be seen to be one enormous drain on...well, the Mint. Obviously this state of affairs cannot be allowed to continue for much longer - or as Vetinari phrases it - "no great rush, 'do' take your time".
Moist, in introducing stamps to pay for the cost of a letter, has inadvertently created a crude but serviceable form of paper currency (described by Mavolio Bent as the revolutionary unsecured one-penny note) which passes all the standard economic tests: ie, it represents the full economic value of the service provided and may be exchanged for an equivalent value of goods. And, crucially, people have now been conditioned by Moist to believe a piece of paper does have a monetary worth, far beyond that of the paper it's printed on.
It is not a long step from a dollar stamp, redeemed at any time, will buy you a dollar's worth of postage, backed by our absolute promise to deliver to a more universal I promise to pay the bearer on demand. If people believe a piece of paper is worth its face value in gold, and everyone confidently participates in the illusion, then we have a paper-based currency.
Which is just as well. Reflect on what has been unkindly (but maybe accurately) said about the Ankh-Morpork dollar: that it contains less gold than an equivalent volume of seawater. Such a state cannot go on indefinitely in any economy. People might lose confidence, which, in the last chapters of Going Postal, we see is the trick of managing an economy.
...Reality is what you can get away with.
Which, when you think about it, explains paper money. Otherwise, why else would a farty bit of paper with some (generally ridiculous) squiggly lines on it actually be worth the equivalent in commodities? Swapping $20-worth of beer for a piece of paper with "$20" written on it seems ridiculous - and in fact is ridiculous unless the whole community buys in to it.
Hence the Goon Show joke (for our non-English readers, the Goon Show was a ground-breaking radio show in the 1950s, '60s and '70s which operated on the outer edges of the realms of the possible, and to whom all subsequent "alternative" comedy owes a huge debt - one that TP himself readily acknowledges):
Shopkeeper: "That'll be £3.50."
Customer: "OK - here's a photograph of a £5-note."
Shopkeeper: "Thanks. Here's a drawing of £1.50 change!"
Which is both a very clever social comment on the perception of values (why should a photograph of a $5 note be less valuable than the note itself?) and a relatively funny joke in its own right.
For an example of how a whole community might buy into an idea and give it collective legitimacy, consider the case of Emperor Norton.
In most societies, a patently insane person proclaiming himself Emperor of the United States would either be sectioned, or ignored and left to wander the streets ranting and raving. There must be something in the air in San Francisco, as the populace fell in with the joke and humoured him, to the extent that the currency issued by His Imperial Majesty Emperor Norton I was honoured - even by the banks - as if it were real. Which of course, in San Francisco at that time, it was as everyone colluded to treat it as such. Reality really is what you get away with - and what other people are prepared to treat as real.