Has a joker ever tried to show you that instead of having the obvious 10 fingers, he actually has 11? This is done by bending down and counting fingers on one hand backwards from 10: "10, 9, 8, 7, 6...." The joker then says "... and 5 makes 11".
Counting on one's fingers seems a natural way to compute numbers, but the Greco-Romans didn't just count "on" their fingers. They counted with their fingers, and not to be quick and accurate with the finger symbols could be embarrassing.
Evidence, particularly from the Imperial Roman world, comes from archaeology and ancient writers. Archaeological evidence comes from tombs, mosaics, and tesserae (dice), which last show finger symbols on one side and Roman numerals on the other.
Burma P. Williams and Richard S. Williams, in their article "Finger Numbers in the Roman World and the Early Middle Ages," from Isis, Vol. 86, No. 4 (Dec. 1995), pp. 587-608, state that Quintilian said an orator would be considered uneducated were he hesitant in calculating or used the wrong gestures with his fingers while making a calculation. This shows that fingers were probably not just used to show numbers, but to perform operations like addition and subtraction on them. However, as the authors state, we don't know whether the ancients used their fingers merely to keep track of intermediate stages while making mental calculations or whether they used their fingers actually to perform the computations.
This finger number system appears to have been in use throughout the Greco-Roman world and well into the Middle Ages. That the Greeks used fingers for computations seems implicit in the Homeric term for counting, pempathai, which means 'to count by fives'.
The article "Finger Numbers in the Roman World and the Early Middle Ages" mentions a riddle (from Symphosius and Alcuin) which seems to show something about how counting was done:
How can 7 from 8 equal 6? Since the number 8 was represented by lowering the little and ring finger of the left hand and the number 7 was represented by lowering just the little finger, if you start with the 8 -- i.e., little and ring finger lowered -- and then raise the little finger to subtract the 7, the ring finger remains lowered and that was the symbol for 6. Obviously, the wrong answer, but perhaps a cautionary tale for those learning the finger calculator.