On numerals

Having started with ‘one, two, three, many’, humans learned to count properly. Possibly at the same time, possibly later, they learned to write numbers down.

Maybe things started with something like the five bar gates illustrated above, maybe once used by ancient warehousemen to tally the arrival of tribute in the form of slaves or of sacks of barley, and probably still occasionally used by clerks in offices – they certainly were when I was young. With five no doubt being the result of our having five digits, including here thumbs and big toes, at the end of each limb.

In which there is a fairly straightforward relation between the tally sheet for the number of sacks and the sacks themselves. One could, for example, almost cut the tally into suitable pieces and stick exactly one piece on each and every sack.

The Romans used letters, a system which had emerged from prior Etruscan efforts. A system which, in the event, ran in parallel with the Arabic numbers which followed, for some time. Even now, for example, we still use Roman numbers on the faces of quite a lot of watches and clocks. 

With first two lines of the table above giving the first fifteen numbers. And the last two lines giving a selection of larger numbers

So the Romans first counted above zero, then below five, then above five, then below ten and so on. A system focussed in this alternating way on zeroes, fives and tens. We still have some connection with counting sacks. The I’s for ones are quite like the strokes of five bar gates. The adding of I’s on the right and the taking away of I’s on the left are quite like adding sacks and taking sacks away. Nevertheless, the connection has been weakened. It is all much more symbolic. Organised even. And rather simpler than what the Greeks went in for.

And while the Romans settled on using existing letters to make their numbers, this was a change from Etruscan usage which involved some special characters, for example ‘↑’ for 50, rather than the Roman ‘L’. The Romans just repurposed some of their pre-existing letters, relying on the context to make clear whether a ‘D’ is 500 or something else. Noting that ‘C’ for 100 went with its name ‘centum’ and ‘M’ for a thousand went with its name ‘mille’. This doesn’t work with the other letters, and in any case it would be difficult to insist that the symbol for a number was related to the name for that number in a systematic way. And noting also that the Romans only had upper case letters, no lower case letters.

While our Arabic numerals – actually invented in India along with zero – are nearly completely symbolic. A number like ‘4’ is not visually related to that number of sacks at all, any more than the sight or sound of the word ‘four’ is, let alone a number like ‘1,529’. They are also ten new characters, which worked better in the long run that recycling some letters.

And while ‘4’ may not relate to sacks, a number like ‘423’ does relate to four hundreds, two tens and three units in a simple way. We know straight away that our number is something more than four hundred and less than five hundred. And we know straight away that a short number with a small number of digits is a small number and that a long number with a large number of digits is a big number. Perhaps more straight away than would have been the case with Roman numerals

Which leads to what is perhaps the key advantage, in arithmetic. It was much easier to do sums with numbers written the Arabian way than with numbers written the Roman way. One might have thought that they made the once pervasive abacus possible, although Wikipedia claims at reference 5 that the abacus predates the use of written Arabic numbers, otherwise decimal notation.

So like letters and words, while numbers may have started out by trying to copy the real world, they are now almost completely symbolic. A world of their own and we have to learn how to get from one to the other. To be able to account for the sacks coming in and to be able count them out again. Maybe not so different really to teaching the computer that the pixels making up the character ‘4’ it sees are to be mapped to the eight bit Unicode byte ‘00110100’.

Another way of looking at things might be to say that nouns came first in human history: tiger, bear, tree, rain. These related to the world in a reasonably direct way, a matter touched on at reference 6. Then adjectives: heavy, red, soft, hot. Again, related to the world in a reasonably direct way, but now depending on observing that some of the things identified in the first phase had properties in common. Then after that came numbers. Not quite a property of a thing, in the way of ‘red’, a property which individual things might share, rather a property of a small collection of things, a property which small collections might share. Together with a recognition that these things have something in common, have been brought together into a place where they can be counted. One can also say that one number is bigger than or smaller than another, something which does not come so easily with some other properties. It is not obvious that red is bigger than green, although nowadays, with all our science, we can make it so. On this account, on this story, a rather more complicated business than distinguishing a lion from a tiger. Perhaps a plausible story - but not a story which is given much support at reference 8, where the relationship between words and numbers turns out to be both intimate and complicated.

At which point, it seems best to stop.

PS: the abacus above lifted from Bing is a 15 digit Japanese Soroban. Very similar to my 23 digit version, which has survived many culls and which I could once use, after a fashion at least. I could still put my hand on it today and ones very like it are still to be obtained from the Soroban Company at reference 10.

References

Reference 1: https://en.wikipedia.org/wiki/Etruscan_numerals. 

Reference 2: https://en.wikipedia.org/wiki/Greek_numerals. 

Reference 3: https://en.wikipedia.org/wiki/Roman_numerals. 

Reference 4: https://en.wikipedia.org/wiki/Arabic_numerals. 

Reference 5: https://en.wikipedia.org/wiki/Abacus. 

Reference 6: https://psmv4.blogspot.com/2020/10/the-power-of-word.html. 

Reference 7: http://www.child-encyclopedia.com/numeracy. 

Reference 8: Numerical Knowledge in Early Childhood – Catherine Sophian – 2009. To be found at reference 7.

Reference 9: Numerical Notation: A Comparative History - Stephen Chrisomalis – 2010. May well be the last word on the subject but Amazon, Abebooks and Ebay all want £100 or more and even Google want £50 for an electrical version.

Reference 10: https://www.soroban.com/.