Inflation should be tied to population to prevent countries from devaluing threi money to compete. For example, country A has 10 people and for each person the national bank can print $100. This means the total money in circulation is $1000. If country A's population falls, spending will usually go down so it would match. This also makes money predictable and a better investment. Why would I buy US dollars if they are just gonna print more devaluing what I had bought. It's like buying a bag of apples and the store clerk coming up to me and dumping out half my apples.
Firstly, cash isn't the only kind of money. Money supply is made of two components: cash in public circulation and deposit money. In a fractional reserve system, deposit money is must be greater than the cash reserves in banks (as long as the required reserve ratio is not 100%), so there's always more money going round than cash, on which you focused. Secondly, an increase in money supply isn't the only cause of inflation anyhow. There are other causes, like cost shocks or increase in foreign demand for your goods, etc.