Let’s imagine that you have a toolbox in your garage, full of all sorts of different useful things, and I’m your annoying neighbor. One day I drop by while you’re working. I rummage around, pick up a screwdriver, and say to you, “Gosh, that’s not a very good hammer, is it?” Naturally, you protest that it isn’t a hammer at all. Next, I hold the screwdriver by the head instead of the handle and say, “Well, of course, you *could* use it like *this* to bang in nails, but it would be very cumbersome.” You look at me, wondering whether I didn’t hear you properly, and say, “No, really. It’s not a hammer. I have a hammer, but it’s in the trunk of my car, and that’s not it.” I turn to you and say, “Well, I’ve never seen your hammer, and it would really be a lot easier if you just used the handle of a screwdriver to bang in nails. Except that it’s no good for that.”

Now let’s turn from this surreal Pythonesque world to another scenario.

You’re an epigrapher and you find some inscriptions with some Roman numerals. You look at them and say, “Gosh, those things aren’t very good for math, are they?” Of course, the writer is dead, so he/she doesn’t say anything. Next, you fiddle around with the numerals and think to yourself, “Well, look at that! You *could* use those for arithmetic if you wanted to, but it would be very cumbersome.” Again, the writer is not around to protest, although as it turns out, someone else dug up an abacus a few kilometers away. You think of that, though, and say, “Well, it would really be a lot easier if they had just used numerals to do arithmetic, except that their numerals are no good for that.”

So this is the world I live in, and this is the battle I fight.

The problem is a cognitive and ideological one. We are so attached to the idea that *numerals are for arithmetic* that it’s very hard to stop and ask whether number symbols were actually used for doing calculations in a given society. There’s essentially no evidence that Romans or anyone else ever lined up or computed with Roman numerals on papyrus or slate or sand or anything else, while there’s abundant evidence that they used an abacus along with finger-computation. This should give us pause, but our cognitive bias in favour of the numeral/math functional association overpowers it. For almost all numerical notation systems used over the past 5000 years, there’s precious little evidence that numerals were manipulated arithmetically. You might have a multiplication table, or you might write results, but you wouldn’t line up numbers, break long numerals into powers to work with them, or anything of the sort. And since we don’t know that much about abaci and other arithmetic technologies, even though they *were* obviously used for arithmetic, we assume (wrongly) that they certainly could never be equally good as written numbers. And thus we conclude (finally, wrongly, again) that Romans were hopeless at arithmetic. We might even blame their (purported) lack of mathematical proficiency on their lack of a ‘good’, ‘efficient’ numeral system.

It’s a casual, all-too-easy ethnocentrism, and hard to detect. It’s not the nativistic, “our ways are good, your ways are bad” ethnocentrism that we mostly know to avoid. Because arithmetic as it is presently taught almost everywhere relies on the structure of the positional decimal numerals, lined up and manipulated as needed, it takes on a naturalness that is deceptively difficult to untangle. Yes, the Roman numerals are quite difficult to use if you presume that the way to use them is to break them apart, line them up, and do arithmetic in something like the way we were taught. This isn’t to say that the functions of technologies aren’t relevant, but if we decide in advance what their functions must be, we are likely to miss out on what they* *actually were, and our judgements will be compromised.

To hammer the point home: if we do that, we’re screwed.

Did you really have to trot out that old saw?

What a gut-wrenching pun.

(PS: Good to see you back!)

For some reason your post reminded me forcibly of the following clip, even though it’s related to language rather than number systems…

Haha! Clearly a brilliant insight. In fact, I mentioned this very sketch a couple weeks ago on a different post in an unrelated context: https://glossographia.wordpress.com/2013/06/02/good-graffiti/

I have always found Roman engineering achievements to be one of the great historical mysteries of all time, given their demonstrated lack of arithmetic technologies. ☺

The traditional response is, “Oh, well they had Greek mathematicians and engineers to help them!” which neglects two important facts: first, the Greek numerals were also non-positional and had no zero, and second, for this to be plausible, every Greek person in the Roman Empire would have to have been employed building roads and aqueducts and temples.