In a capitalist economy, money is the accounting identity Y * P = M * V, which is true by definition; in English, the real Gross Domestic Product (Y) times the price level (P) equals the quantity of money (M) times the velocity of money (V). By definition, then, all economic activity (worth counting) involves the transfer of money. There must be a finite, measurable amount of money in existence. Money must move; money that doesn't move isn't really there; if the velocity of money were zero, then M * V = 0 for any quantity of money, even an infinite amount. As a corollary, the amount of money by itself doesn't determine anything; what matters is how much money is actually moving. On the other hand, because there are physical limits on how fast money can move, and because there are physical limits on how much the price level can change, economic activity can be limited by just the quantity of money.
With a little algebra, we can say that Y = (M * V) / P; real Gross Domestic Product is money in motion (M * V) divided by the price level. We can arbitrarily choose a unit of measure in which P = 1, therefore Y = M * V: money in motion is therefore a symbolic representation of real productivity, which is just labor, energy, and matter in motion.
At the macroeconomic level (the economy as a whole), what is produced must, for the most part, be consumed; actually producing goods and storing them in a warehouse is a tremendously inefficient activity. It is much more efficient to simply not produce goods that won't be immediately consumed: we could use the resources to produce something that would be consumed, or we could just produce less and have more leisure.
Similarly, human labor, the sine qua non of productivity (we have free energy from the sun, and anything can be produced given enough human labor) must be used or wasted: the hour of labor lost by some individual who could work for an hour but doesn't cannot be stored and used later.
All three of these observations entail that saving money is, in general, if not an entirely bad idea, then a fairly limited one. We cannot save what is produced, and we cannot save potential labor that is not used, so saved money doesn't symbolize anything actually being saved. Money saved is money that is not in motion, and thus not "really" money.
Saving money can (and does) symbolize is virtually shifting consumption in time by physically shifting consumption in space. If I want to spend a lot of money next year, I let someone else spend my money this year, and I spend her money next year. If I just save my money and don't let someone else spend it this year, then we simply reduce real productivity, since there's no reason to produce something this year that will not be consumed this year. And that lost productivity is really lost: If we can produce 100 cars a year, but only produce 99 this year, we cannot therefore produce 101 cars next year*. Indeed, if we lower productivity enough in amount and time, then we can actually lose the capacity to produce: if we can produce 100 cars a year, but we only produce 50 cars per year for a few years, then we might find that we can produce only 90 cars per year. And now because my money is still there in some sense, we have more money chasing fewer goods, which is the definition of inflation.
Instead of saving money, I can invest money: instead of using labor (and energy and matter) to produce goods for consumption, I can allocate my money to produce goods for increased efficiency of production. In exchange for not consuming the 100th car (which is the most expensive car to produce), I can build a car factory, which will allow us to produce 110 cars per year, and I get to keep or trade 10 cars per year. Win all around!
But there's a catch, of course. Obviously, in a large economy with many goods, investment is both risky and uncertain. I might build a car factory only to have it burn down or get hit by a meteor (risk). I might build a car factory only to find out that people don't want that 110th car enough to justify its production (uncertainty).
But there's another, subtler, catch. Increased consumption lags economic growth by a considerable margin, and the effects of any one factory on wages diffuses through the whole economy. If I build a car factory, I will produce my 110th car before the workers who actually built the cars receive their wages. Furthermore, these workers aren't going to actually buy all ten cars: they will buy more shirts, shoes, dishwashers, movie tickets, lattes and yoga lessons. I have to hope that the shirt makers, shoe manufacturers, etc. have also increased production and wages so that my workers can buy more shirts, etc., and their workers can buy my extra cars.
There's yet another, even subtler catch. Capital costs are sunk costs. If I have a car factory that produces 10 cars per year, I can sell all ten cars only for what it costs me to produce the tenth car. The cost of the factory is embodied only in the cost of the first car. As Michael Perelman describes at length in Railroading Economics, capital as a sunk cost (mostly building railroad tracks and steam engines) was the primary cause of the Long Depression of 1873-1879. In a pure free market with perfect competition, capital costs cannot be recovered.
Because of these catches, a complex industrial economy requires a degree of coordination impossible in a free market with perfect competition. If we're going to increase production, we have to increase production across the board, so there's more of everything. If we're going to increase production, we need to make sure that consumers have the money ahead of time, so they can buy the products. We must make sure that sunk capital costs are included in the marginal costs of all the goods. By definition, the institution that provides this coordination is the government; in other words, if we look around and we see that some institution is coordinating investment and consumption, we slap the label "government" on that institution.
A coordinating institution, a "government" in the above sense, must exist for an industrial economy at a technological level greater than that of the early 19th century to exist. Without this coordinating institution, there's just no reason to invest in steam railroad or higher levels of capital; doing so at best pays off more for everyone other than the actual investors and at worst pays off for no one.
There are, at least superficially, four ways of coordinating investment and consumption, corresponding to the four terms of the fundamental equation of money, YP=MV. We can try to directly control production, by just having the government build and operate factories. We can try to directly control the price level by having the government set prices. We can try to affect the velocity of money, making money move faster or slower. And finally, we can try to affect the quantity of money.
The government concentrates its actions on directly affecting Y, real GDP and M, the quantity of money; P and V change indirectly.
A government changes Y and M in tandem in two ways. First, the government creates new money, changing M, and spends that money directly for public works, directly changing Y. Second, the government creates new money, changing M, and loans it to businesses for investment, changing Y. Both measures overcome diffusion, because investment is spread across industries. Creating new money to loan also overcomes the capital as sunk cost problem, because the "sunk" cost is virtual, and becomes baked into the marginal cost of all the products through periodic loan payments. Finally, consumer lending, again creating new money, changing M, and loaning it to consumers who buy things with it, changing Y, overcomes the lag problem. To offset all this creation of new money, money must also be destroyed, so the government destroys money by collecting taxes and loan payments. It's tricky, but with only four major variables, keeping everything in balance is a tractable problem. Fundamentally, then all coordination of an industrial economy requires the targeted creation and destruction of money, so that money can lead productivity rather than follow it.
It doesn't matter what we call the institutions that create and destroy money: they act like a government, therefore they are a government even if we call them a "private" banking system. Because it is impossible to maintain cooperation in a purely uncoercive competitive environment, cooperation must have some coercion built into it. Again, calling coercion by another name, like the justified enforcement of contracts, doesn't make it any less coercive. And if we went on a pure gold standard, we would just create some other virtual something which would fulfill the same role as money does.
The question is not whether we have a government, but who gets to be the government. And the only options are monarchy, oligarchy, or democracy, or some mixture of the three; no political philosopher has found an alternative that is more than a renaming of one of those three.