Arabic And Other Numbers

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.

Pythagoras

I’m currently sitting on our favourite couch. It is in turn sitting in a room downstairs in our house in Scotland. We bought it in England and tried to get it up the stairs of our English house because our living room was upstairs there. We had enormous trouble getting it past the bends in the stairs and eventually I decided to measure the bend and the couch, so I measured the depth and height of the couch and then the three dimensions of which the bend consisted. Using the well known right angle triangle equation a²+b²=c² and taking the square root of c, I was able to calculate the hypotenuse of the couch. I then made the slightly more complex calculation of using the hypotenuse of the dimensions of the stair bend with the height of the ceiling above the stairs to work out the maximum length of an object which could be fitted through the gap, and since that second figure was smaller than c, I was able to prove, and I have to state this carefully to be precise, that the couch would not be able to fit into the space on the stair bend, and therefore it would be impossible to take it up the stairs and put it in our living room, so it remained downstairs. Now there could’ve been some other approaches, such as taking the feet off or the banisters down, but in fact both of those were part of the objects concerned and it wasn’t going to happen because I’m not Bernard Cribbins.

This is of course Pythagoras’s Theorem. People often say they never apply anything they learnt in maths to their lives after leaving school, leading me to conclude that either their lives are unnecessarily hard or that they don’t realise they’re using it, because this kind of problem comes up all the time in everyday adult life and I can only surmise that people think really strangely in this area. I scraped an O-level pass in maths and this is obvious to me. In fact I almost stayed in the CSE group and was the lowest grade person to go “up”. I should also mention that there is a famous Moving Sofa Problem in mathematics, but this isn’t that. The moving sofa problem is the question of which rigid two-dimensional shape of the largest area can be manoeuvred through an L-shaped planar region with legs of unit width. It didn’t help us because the stairs were three dimensional, i.e. they went up diagonally, turned through two ninety degree angles while continuing to ascend and the ceiling of the ground floor was in the way too. There migh be some couch-stair combinations which it could’ve been useful for, but not this one.

Most people know one thing about Pythagoras, and that’s that he’s responsible for Pythagoras’s Theorem that the square of the hypotenuse is equal to the some of the squares on the other two sides of a right angled triangle. This also brings up the issue of the square root of two being irrational, i.e. not being expressible through a ratio, i.e. a fraction, because an isosceles right angled triangle with unit opposite and adjacent sides will have a hypotenuse length equivalent to the square root of two in units. As a child I thought this proved that units of measurement didn’t exist, but obviously that was my child’s mind failing to grasp things properly. The only thing is, Pythagoras probably didn’t think of his theorem. It’s more likely that in order to give it some kudos, people decided to attribute it to him, and it was known about before his time.

Unfortunately I don’t seem to be able to satisfactorily answer the question of whether Pythagoras existed. He may well not have done. I want to start by mentioning a few other figures: Nero was the Roman emperor who fiddled while Rome burned and rebuild the city in a much improved condition; George Washington was the guy who cut down the fruit tree as a boy and admitted to it, saying “I cannot tell a lie” and Archimedes was that bloke who got in the bath which overflowed, giving him the inspiration to tell whether a crown was solid gold, and shouted “Eureka!”, running down the street naked. Or maybe not. I haven’t checked these and they’re very likely to be just stories, and actually the question of whom we refer to when we tell stories like this is a modern philosophical problem. So Pythagoras, by the same token, was an ancient Greek philosopher who discovered something important about triangles, was vegetarian, wouldn’t eat beans and thought numbers were very important to the nature of reality. That’s probably more than most people “know” about him.

So I’m going to start with the question of whether he existed. At least three other important Greek men wrote about him and his life: Aristotle the philosopher, Herodotus the historian and Iamblichus the Neoplatonist philosopher. There was a whole school of philosophy named after him which he’s said to have founded, although that doesn’t mean he existed. That school of philosophy has a consistent belief system rather than just being arbitrary unconnected beliefs, so there is such a thing as a Pythagorean philosophy. However, no writings at all can be attributed to him because Pythagorean philosophy was an oral tradition. It was passed on by word of mouth long before it started to be written down, and this of course means it could’ve ended up being distorted even if he did exist. There was also a tendency in the Greco-Roman world for people to attribute ideas and quotes to people to make them seem more important and respectable than they would’ve been perceived as otherwise, rather like how lots of quotes today are attributed to Churchill and Einstein that they never said.

And the thing is, Pythagoras as he was understood in ancient Greek sounds absolutely bizarre. He had a thigh made of gold, was able to be in two places at once and could converse with non-human animals, and there were a few other things about him which were odd-sounding. He comes across as a kind of magical cult leader and demigod, perhaps a shaman or a sage rather than a philosopher. This partly reflects how philosophy was not neatly parcelled off from religion and spirituality as it is today, at least in academia, and what we separate today was actually considered together until at least the time of Newton. The difficulty, in fact, is similar to those of establishing the nature of the real Jesus and Socrates. So we’re in a situation where the one thing everyone thinks they know about him isn’t true and he was seen as some kind of superhero with incredible psychic powers. But in a way the question of whether he existed or not is the most boring thing about him. Everything I say about him from this point on has therefore to be attributed to some kind of possibly mythical or otherwise fictional figure rather than any real person called Pythagoras living in Ancient Greece.

He was seen as an expert on the soul. In Ancient Greek times before him, nobody thought there was a separate soul which survives death. This was more an Ancient Egyptian thing, and for all we know that’s where it originated. Because of this expertise, combined with his belief in reincarnation he was said to be able to remember his past lives. He once got someone to stop beating a dog because he recognised the cries as those of a dead friend reincarnated in the dog’s body. This is also why he was able to talk to members of other species. And whether or not he existed, there was clearly a cult based on his apparent beliefs, and this cult was also rather strange. They believed that the right shoe should always be taken off before the left one but that the left foot should always be washed before the right, that no-one should eat anything red, and they were seriously into numerology and vegetarianism. In fact, before the invention of the English word “vegetarian”, we were called “Pythagoreans”. They also included both women and men, which seems to have been unusual at the time. We may assume that the idea of an institution which admits women to be the exception back then but we don’t actually know. You also had to be silent for five years once you joined. Returning to the vegetarianism, although they did believe in it, justified through the idea of human souls being reincarnated in other forms, they also believed in sacrificing animals to deities. There’s even a story that Pythagoras was once seen eating chicken and replying to the objection that he was supposed to be veggie and not eat live animals by saying that the animal he was eating was dead, and this makes me wonder if they were actually vegetarian or simply sacrificing animals so they could eat them. Even so, many veggies do have stories like that made up about them, and most surviving records about Pythagoras are about criticising him and his followers or lauding him and them. There isn’t much attempting to be objective. Incidentally, although he had a religious cult of his own, he still worshipped the Greek deities of the time and what they did was “extra”: it was still dodekatheism, as it’s known nowadays, but a kind of denomination of it rather than a separate religion.

Pythagoras was of course into maths, which he combined with numerology because at the time there was no distinction. He seems to have been the first person to connect mathematics to an attempt to explain the world. This particular notion has been extremely influential. Even today, a hard science has to include maths to be taken seriously. One of the reasons psychology emphasises statistics so heavily is that it wants to be a “proper” natural science. However, the way Pythagoreans approached maths and its relationship to the physical world back then seems quite different to how they’re approached now. For instance, even numbers were considered female and odd numbers male, and since the number 1 wasn’t considered a number at all because it didn’t have a beginning and an end, five was considered the number of marriage, as it was the union of the first female number with the first male number. The number seven was considered sacred because, being prime, nothing could make it up and it could make up nothing. Two was considered the number of justice because it enabled things to be divided equally into two halves. Three was considered to sum up the whole Universe as it was the first number to have a beginning, middle and end. He also discovered triangular numbers. The number three was considered to represent a human being, and was of course male, representing the threefold virtues of prudence, good fortune and drive. That almost sounds like it’s out of a contemporary self-help book.

Although the links Pythagoras made between numbers and the Universe were peculiar, he also connected geometry and arithmetic more thoroughly than his predecessors, because of the hypotenuse connection with the square root of 2, and because of his theorem, although that had been known to the Babylonians. He was the first person to come up with a method for constructing a dodecahedron, and connected many shapes to the Cosmos, bringing me to what ought to be the most famous thing he was known for: he was the first person to claim Earth was round. Remarkably, although this has turned out to be incorrect, his reasoning had no connection to any observations because science wasn’t there yet. In addition to that, he came up with the idea that Earth and other planets moved in orbits, although oddly not around the Sun but a central fire, and also that there was a counter-Earth, required to make up the numbers in the system. There are convoluted reasons for all this.

This initially peculiar link between the Universe and mathematics, once forged, has stayed ever since and may not in fact be as obvious as it seems. I have suggested before that one solution to the Fermi Paradox (“where are all the aliens?”) might be that they’re all really bad at maths compared to humans, but another solution may be that although they’re perfectly good at maths, they never had a Pythagoras to make a link between the two and it’s never occurred to them to apply maths in this way. Hence their science is still Babylonian in nature, or even less like Western European science than that. They never got any further. If that’s true, it makes Pythagoras, even if he never existed, an incredibly important figure.

Another aspect of all this is that we can look back from our own “rational” viewpoint and poo-poo the idea that he was an ancient Doctor Dolittle, could be in two places at once and remember past lives, when actually maybe he could do all of that and it’s our own restrictive mind sets which have stopped that from happening. This doesn’t sound sane, but when we consider what many Christians believe about Jesus it becomes more a case of us simply having decided that one ancient semi-mythical person has such attributes rather than the other. It only sounds crazy today because we chose to retain the deification of Christ rather than Pythagoras, which could be seen as practically a coin-toss. There is a world not far from here where many millions of people still believe Pythagoras had something in common with C3PO.

Another numerological aspect of Pythagoreanism was that nobody should gather in groups of more than ten because the number ten was 1+2+3+4, so ten in particular was a sacred number to them. This extended to them composing prayers to that number, and I find this interesting because it creates a link between mathematical entities and deities and other spirits. Platonism and intuitionism are two opposing views of maths. Intuitionism holds that humans invent maths as we go along, i.e. it’s a creation of the mind just like a poem might be, whereas Platonism holds that maths is discovered. It’s already out there before we get to it. So for example, there are considered to be eight planets in this solar system. Assuming there are no others, there were also eight planets when the first trilobites appeared 521 million years ago. In fact, at that point there was a number representing the global population of trilobites, as there still is today: zero. So does that mean that the number eight exists independently of human consciousness or, more precisely, the ability to count? I have a strongly atheist friend who is also a Platonist, and she acknowledges that it’s an odd position to be in. The Ontological Argument for God tries to bootstrap God into existence from the concept of God, and this perhaps reflects the notion that God exists as a concept in a more objective manner than an atheist or agnostic would usually be expected to think. The concept of God is “out there” in the Cosmos in some way, and maybe in the same way as maths is said to be by Platonists. But this, well, I’m going to have to use the word “idea” at some point, of deities existing abstractly is usually considered separately nowadays from the idea that squares or numbers exist. We have a partition in our thoughts which Pythagoreans had yet to erect.

This can be directed back on Pythagoras. Clearly the idea of Pythagoras does exist, although it seems to have varied. We have Pythagoras as the triangle guy and the first person to suggest that the world is round, although actually that might’ve been one of his successors. But Pythagoras himself may not have existed in the same sense that Elizabeth I of England did, and as such this accords quite well with the general attitudes of the time and the problems of ancient history. Also, back at that time and place, the Greeks seem to have taken their religion quite literally so for them Zeus was as real as Pythagoras whether or not we think of him as real.

On consideration though, I do think he existed in the way we generally understand existence today, i.e. not just as an abstract or mythological entity. The reason for this is that his cult existed and was quite forceful and distinct in nature. It seems to me that a requirement for a large group of people to avoid speaking for five years and never to eat beans sounds like the kind of thing a charismatic leader would get their followers to do, and it really sounds like cultish behaviour by today’s standards. It makes cults seem like constant fixtures in human life rather than phenomena characteristic of the modern world. This is probably not terribly surprising, but maybe this assumes too much, because it might be that cults with leaders are more recent developments connected to individualism and a tendency for people to seek complete answers to life’s problems. I haven’t checked, but I don’t think the Essenes had a founder or leaders.

Here’s the weird bit though. As I’ve said before, although Pythagoreans seem to have been the first people to link maths and science, from today’s perspective they seem to have come up with a list of arbitrary superstitions and ideas without a thorough connection to reality. But despite this, somehow they were able to assert the correct idea that the world is round, which to us seems to depend on observation rather than philosophical or mathematical abstraction. Nobody seems to have had that idea before. Later Greek philosophers came up with ways of testing this and measuring Earth’s size, but it wasn’t those careful tests which led to the initial thought. What are we to make of this? Maybe the idea crept in from somewhere else.

We still have the metric system. Does that maybe represent a similar superstition about numbers? We happen to have ten digits on our hands and it’s led to us producing a system which is easier to use than imperial because of how we count, but are we also partaking of Pythagorean mysticism there? We’ve put that into the box of rationality, but maybe it’s more to do with custom. Also it seems that the real mystery is how maths actually manages to engage with the world at all. Why would this be?