A lot of excellent other answers, so I hesitated to chip in but I think there is a useful way to think about this. If the unit of exchange was always physical

A lot of excellent other answers, so I hesitated to chip in but I think there is a useful way to think about this. If the unit of exchange was always physical gold coins then yes, there would be a finite amount of money. However the moment we turned to promissory notes, cheques, bank notes and financial instruments such as in fractional reserve banking the link to physical gold was a fiction. Such notes have been around a very, very long time although of course increasing in prominence over time. It was a fiction even for precious metal coins as well because the moment a government started running low on the metal, or just wanted to inflate it's currency, it would just debase the coins.