The new mayor of NYC recently apparently trolled MAGA people by saying that elementary schools would be teaching Arabic numerals. Some people were in on the joke, others reacted because they didn’t realise what they were and thought they were associated with the current Arab world, seeing it as political correctness, or perhaps correctitude, gone maaaaaaaaaaaaaaaaaaaaaaaaaaad!
I probably don’t need to say this but just in case I do, Western Arabic numerals as they are sometimes called to avoid confusion are simply the digits “0” to “9” used in most Western countries and probably to some extent everywhere. They tend to be associated with Latin script and Arabic script itself has a different set of numerals, though somewhat similar in form. They’re called “Arabic numerals” because they entered Europe via the Moorish kingdom of what’s now Spain, in I think the tenth century CE. Their form was quite different than it is today, but the basic principle of place value and a symbol for zero was established. Perhaps surprisingly, they actually have their own names in the same way as the ampersand has even though it just means “and” nowadays, but these are never used and haven’t been since the middle ages: 1 Igin, 2 Andras, 3 Ormis, 4 Arbas, 5 Quimas, 6 Caltis, 7 Zenis, 8 Temenisa, 9 Celentis. Presumably 0 is “zero”. I don’t think their etymology is known.
When Arabic numerals first appeared in Europe, they were indeed regarded with suspicion. For instance, it was forbidden to use them in accounting along similar lines to “you can prove anything with statistics”. Because there was no apparent physical correspondence between their form and the quantities they represented, unlike Roman numerals with their I, II, III, IIII, XX and so on, it was felt that underhand surreptition would be easy, and in fact maybe it was. There was also apparently a period during which place value wasn’t used and the numbers were just written down in any order or something.
Up until that point there’d been a direct link between integers and the Latin alphabet, as Roman numerals had long since been adapted into a form where each was identical to a letter. In fact it extended beyond this, with new “Roman” numerals being introduced such as N (I think), representing ninety. The subtractive principle was less widespread at the time. Something interesting about that is that it’s how Etruscan spoken cardinal numbers work as well, and also the Finnish numbers for eight and nine. The reason for clock dial numerals being different is to balance the dial visually and because it made them easier to cast in batches.
Just for the sake of completeness, the Eastern Arabic numerals are: ٠١٢٣٤٥٦٧٨٩ . It probably doesn’t need explaining that they were originally from South Asia, where they have been ०१२३४५६७८९ . Other Southern and Southeast Asian scripts have adapted these numbers too. But there have been other systems. In particular there’s been a tendency to use letters as numbers, alphabetically, with the first nine as numbers one to nine, the second nine as ten, twenty and so on and the third nine as one hundred, two hundred and the rest. Larger numbers can just be written out as words. This approach was used by Greek, Cyrillic, Gothic, Hebrew, Arabic before the invention of the numerals and possibly others, I’m guessing runes for example. An acrostic approach was also taken in Greek with the initial letters of the number words standing for the numbers themselves.
Regarding arithmetic, it’s clearly easier to use the alphabetic system than Roman numerals, and still easier to employ the place value system with a zero, but one thing the alphabetic system does is allow numerological analysis of words. For example, the Hebrew word יד means “hand” and adds up to fourteen supposedly because there are fourteen bones in the digits of the hand. The number 666, likewise, stands for NRWN QSR or “Neron Caesar”. Interesting things can be done with personal names more generally.
There were still other systems, but many of them have been rendered obsolete. Maya numerals were as sophisticated as Western Arabic ones, included a zero and were vigesimal, which also occurs in French, Danish and Celtic counting but in the European cases below the century, i.e. there’s no special word for 400, 8000 and so on. The most successful system other than the ones mentioned is probably the one originating in the Far East and exemplified by Chinese numerals. This system starts off the same as Arabic numerals, including a zero: 零 or 〇、 一、 二、三、四、五、六、七、八、九、十. Then it’s 百 for a hundred, 千 for a thousand, 萬 for ten thousand, which is simplified as 万, which incidentally is it’s said why you’re supposed to do ten thousand steps a day because it looks like a runner on the display of a pedometer, and the higher numbers proceed in the same way. Chinese has a separate set of numbers used for retail purposes on invoices and price tags, called “rod numerals”: 〇,〡,〢, 〣, 〤, 〥, 〦,〧, 〨, 〩, 〸to ten and then 〹 and 〺 for twenty and thirty. These were actually the first Chinese numerals I learnt, before I started to learn Mandarin itself, and I seem to remember they ended up getting used in the TV series ‘Sherlock’. There’s a third set of numerals used for financial considerations, which are considered harder to forge and have a large number of strokes: 壱, 弐, 参, 肆, 伍, 陸, 漆, 捌, 玖, 拾, 陌, 阡, 萬. The idea behind these is that they can’t be easily added to so as to read differently. 一、 二 and 三 can clearly be altered and an 八 can easily be turned into a 九, and so on. I seem to recall these are also called “capital” numerals, so there is a point to capital numbers after all. There are also Western capital numerals but that’s another story.
Chinese even has a character for a hundred million: 億, simplified as 亿. A big strength of this system is that because Chinese words tend to be monosyllabic (there’s a whole tale to be told there too), it’s very succinct. You can say “seven hundred million” in two syllables rather than seven.
Indian numerals are also different from the West, with “lakh” and “crore” used for a hundred thousand and ten million respectively. This leads to the commas being put in different places than Europe and the Americas et caetera. I use spaces rather than numbers in commas, but of course in this part of the world either arranges digits in sets of three, but in South Asia the approach is to use a group of three for the lowest orders and two for all the others. In the Far East, digits are arranged in groups of four. One thing I found very surprising when I first moved to an area where South Asian languages were widely spoken was that they would say ‘phone numbers, house numbers, postcodes and prices in English but other numbers in their own language. This seems to be because they perceived those digits as foreign and therefore to be pronounced in English due to their own system being native to them, but it still seems odd to me.
This grouping into threes brings up the second issue with numerals which used to have a nationalistic tinge: the short and long scales. Western integers and decimals use two different systems, one based on powers of a thousand and the other on powers of a million. There are also some separate numbers used to a lesser extent. In these isles, and I don’t know how long this has been declining compared to the American usage, the numbers million, billion, trillion and so forth go up in powers of a million, so a billion, for example, is 1 000 000 000 000. In America, and in fact the Americas, a billion is 1 000 000 000 and the numbers go up in thousands. It’s been said that this bucks the trend because everything else in America seems to be bigger than in Britain. This also transcends languages, as Spanish and Portuguese also use the short scale in Latin America and the long in Europe. Canadian French uses the long scale in formal language but can use the short scale informally. Another change which has happened in my lifetime in English is that we used to use the word “milliard” sporadically to refer to the short billion and French has that and also billiard, which we presumably never used because of the game. I’m not aware of the number “trilliard”, but maybe. Apparently yes, and in fact they can go all the way up but are rarely used. The point being that the short scale is redundant and unnecessary because there are already perfectly good words used for that purpose, which makes me wonder how this situation ever arose. The long scale is a fifteenth century French invention, whereas rather more surprisingly, the short scale was also French, invented a century later. I find the existence of the short scale very irritating. I’m also surprised that English and other nationalists here aren’t more exercised about it.
This has created a fair bit of ambiguity and confusion, and the metric system of prefixes helps to address this. Instead of talking about “a million kilowatts”, which is a billion watts in the short scale, that quantity can simply be referred to as a gigawatt. The SI multiples such as “mega-“, “giga-” and “tera-” have the same intervals as the short scale numbers but there are smaller steps closer to the units themselves, two of which, “myria-” and “hebdo-“, seem to have been retired. Also, references to digital data storage use a superficially similar system which however goes up in powers of 1024 and has slightly different prefixes such as “mebi-” rather than “mega-“. This is because the sizes of memory and storage devices gets increasingly out of kilter the larger the prefixes get, with a petabyte, for example, being almost an eighth larger, so sticking to the metric prefixes makes them sound more generous than they really are if a gigabyte is literally 1 000 000 000 bytes.
Of the other traditions, Maya and South Asia probably took numbers the furthest. The Maya had the Long Count calendar, which has a maximum date equivalent to the number of days since the extinction of the non-avian dinosaurs. South Asia went a lot further than that, ultimately finishing with the number referred to as असंख्येय, asankhyeya, which is 1 followed by 140 zeros. However, the large numbers in ancient South Asian texts don’t seem to be meant literally and seem to have a kind of hyperbolic or rhetorical value. It’s also notable that Jain texts in particular use larger numbers than others. In Jainism, the same word is taken at face value, and it literally means something like “uncountable”.
Now at this point I could go on about the vast numbers used in mathematics such as those resulting from the Ackerman Function, Skewes’ Number, TREE(3) and of course the famous Graham’s Number, which is a predictable direction for me of course. Instead of that, I want to use the rhetorical and hyperbolic usage found in ancient South Asia as a jumping off point for a related subject: finitism.
Finitism is the belief that infinity does not objectively exist but is merely the product of the human mind. An extension of this idea is that even the large finite numbers used in mathematics are non-existent. It’s a little difficult to word this properly because the words “real” and “imaginary” both refer to kinds of numbers widely agreed to exist objectively, or at least as valid concepts within maths. Ultrafinitism further holds that no numbers which can be constructed by human activity exist. Viewed in this context, perhaps our own use of extremely large numbers is in the end no more literal than the Indian use, but simply of a different quality. It enters into the issue of platonism versus formalism.
Platonism is the view that mathematics exists regardless of human activity. This is mathematical Platonism incidentally: there are other views also called Platonism which are quite similar but aren’t focussed on maths. It’s a little like the view that if a tree fell in a forest with no hearkeners, it would still make a sound. Suppose there’s a planet somewhere, gigaparsecs away, with three peaks within a few kilometres of each other on a level plain in an approximately equilateral triangle. We’ll probably never know about this, but many would say that whether or not this is ever seen by any sentient being, there would still be three peaks arranged in an approximate equilateral triangle on that planet and therefore that equilateral triangles and the number three do exist objectively.
If you know much about theoretical particle physics as a lay person as opposed to being a physicist, it’s tempting after a while to wonder if it isn’t all just a massive flight of fancy, particularly when you consider quantum mechanics and its apparent lack of commitment to anything being real. Certain views on this can be asserted which have a bearing on this which have a philosophical basis and don’t require extensive knowledge of physics, but my point here is mainly that what we see as concrete may in fact not be. The basic question is whether mathematics is invented or discovered.
To be fair, although this is an interesting subject, I think I should probably leave it for now as it’s very long and involved, so I’ll cover it next time.
A Box Of Chocolates 