All this happened, more or less.

The idea was that the seal should bear a clear mark, identifying its owner. Like the clay envelopes studied by Schmandt-Bosserat, seals were instruments of economic control, guarteeing the supervision of proceedings, or confirming that a transaction had tken place. In practice, the Sumerians produced some very imaginative devices with which to identify owners: worshipping at a temple, procession of boats, prisoners efore a ruler, feedin of animals. The were, in effect pictographical signatures. Later, a new type of seal emerged, produced by cutting machines. This clearly suggsts that trade was increasing and that the need for identifying marks was likewise growing. ibid, p. 80.

It's very informative to see how much faith a transition from concrete to abstract requires from its participants. For example, the transition from a physical "clay" representation of animal count to the more abstract inscription on a clay tablet, representing this count, must bear a seal of a trusted person. Otherwise, how one is to know that the writing actually denotes what it writes about. No wonder, first mathematicians were priests. The belief in the abstract must be supported by either strong evidence delivered by science, or strong faith delivered by religion. Money, as an abstract representation of value, must bear the seal of a trustworthy government. God, as an abstract representation of higher powers, has to be supported by miracles of faith.
Also of note would be the emergence of abstract models where the scalability of operation is concerned.
cf: Aristotle's classification work.