Thus, I have been forced to build an understanding of my own around these matters. It was -- or rather, is -- quite a struggle, as anyone who has been into monetary economics must surely understand. How I ended up where I am in my thinking today is an interesting story in itself, but I'll leave that for later. What eventually emerged from this struggle is a view, or perspective, more or less consistent with Hyman Minsky's view which characterized the economy as a system of interlocking balance sheets (Thanks, Michael Pettis!), with the difference that what I suggest is both more concrete and, arguably, more radical.
We could call it the "Bookkeeping View". Simply put, I view our current economy as a "pure credit economy": something Wicksell, Hawtrey and Schumpeter, even Jevons and Hicks, imagined and perhaps thought possible, but nevertheless considered non-existent as long as "fiat money" existed. The Bookkeeping View defines even this "fiat money" in bookkeeping terms, and thus effectively gets rid of it. What we are used to call "money" (the non-abstract part of it) is just a credit balance in a ledger. For instance, physical cash refers directly to, or acts as, a credit balance in the central bank ledger. This view describes the entire monetary system simply in terms of credit and debit balances, and changes in these balances. These changes (as do these balances to start with) arise from credit and debit entries that are made on various accounts, and in various ledgers, in course of buying and selling. *
According to this view, banks (the central bank included) and other financial institutions can be considered "bookkeepers". They don't record their own claims and liabilities (Schumpeter, too, has suggested that deposit holders should not be viewed as creditors of the bank), but the claims and liabilities -- credit/debt relationships -- of the entities found on both sides of the bank balance sheet.
We are all used to think that what these credit balances refer to, i.e. what it is that is owed, is “money” (ultimately “cash”). But the Bookkeeping View challenges this. According to this view, these credit balances are just denominated in an abstract unit of account (numéraire; e.g, USD). In other words, they refer to a price. †
A price of what? A price of whatever is bought and sold. It might go like this: A creditor (owner of a credit balance, often in a bank ledger) goes out on the market, looking for goods or services, and chances upon a debtor (owner of a debit balance). The debtor is selling and the creditor is buying. (Of course, it is likely that they don’t know each other’s balance, nor should they know.) Assuming the creditor is interested in buying what the debtor is selling, they agree on a price. A transaction is made. And it is this transaction that is recorded in our monetary bookkeeping system. By selling, the debtor has redeemed (part of) his debt (to the amount of the agreed price), and that will be reflected in his reduced debit balance in the bank ledger (assuming we are dealing through a bank; it is easiest to think of this in terms of overdrafts instead of "traditional loans", because it gives a more accurate picture of the reality. What is essential is the net balance, which overdraft gives us automatically.). Similarly, the creditor’s credit balance is reduced. Now — only ex post — we know what was it that was owed. It was the object of the transaction, often a good or a service. And only ex post can we establish a direct link between a single creditor and a single debtor, brought together most likely by the “invisible hand” (or, market forces).
Think of this as an alternative way to view the world. One cannot prove this view wrong by just positing that "money as a means of payment" must exist because, well, we are used to agree that it exists. Naturally, adopting this view makes great demands on our cognitive flexibility -- it's been a tough journey for me: the view itself is simple, but getting rid of "money" is mentally hard --, but the rewards in the form of new insights on the most pressing problems in economics, and the economy, are, I believe, well worth the effort.
UPDATE, Oct 20, 2015:
Here is a quote from R.G. Hawtrey ("Currency and Credit", 1919) which sums up quite well how I view our contemporary economy (for Hawtrey, this was an imaginary economy):
Suppose then that society is civilised, and that money does not exist. Goods are brought to market and exchanged. But even though there is no medium of exchange, it does not follow that they must be bartered directly for one another. If a man sells a ton of coals to another, this will create a debt from the buyer to the seller. But the buyer will have been himself a seller to someone else, and the seller will have been himself also a buyer. The dealers in the market can meet together and set off their debts and credits. But for this purpose the debts and credits, which represent the purchase and sale of a variety of goods, must be reduced to some common measure. In fact a unit for the measurement of debts is indispensable. .... Where there is no money, the unit must be something wholly conventional and arbitrary. This is what is technically called a ''money of account". ... This is an approximation to the state of affairs which we are assuming. But however conventional and arbitrary the unit may be, once it is established as the basis of the debts and prices and values of a market, it is bound to assume a certain continuity.
Each dealer in the market calculates his own command of wealth in the same unit; it affords the basis for his valuation both of what he wants to buy and of what he wants to sell, and he looks for only such divergence from the previous prices as variations of supply and demand will justify. The total effective demand for commodities in the market is limited to the number of units of the money of account that dealers are prepared to offer, and the number that they are prepared to offer over any period of time is limited according to the number that they hope to receive. Therefore, arbitrary as the unit is, capricious variations in its purchasing power will not occur.
* I believe I'm very close to the (Steuart-)Mcleod-Innes "Credit Theory" line of thinking, and the way I interpret for instance Alfred Mitchell-Innes is such that he seems to be in agreement with me. Some contemporary writers, for instance Richard Werner, seem to follow this line of thinking, but the propositions they present differ from mine to such an extent that we seem to draw different conclusions.
† For an overview of the “abstract unit of account” concept, see, for instance, "Is Numérairology the Future of Monetary Economics?" (2007) by Willem Buiter. But note that what Buiter says doesn’t correspond with the “Bookkeeping View”. In his thinking, currencies like USD and GBP are “money” which serves as a “means of payment”, in addition to its function as numéraire (abstract unit of account). The "Bookkeeping View" position is that USD and GBP are only abstract units of account that are used to express the nominal value of a credit balance. These credit balances refer to prices denominated in USD or GBP.