‘It’s surreal to me that it’s 2022 and there are still people out there who think 2 + 2 = 4 is an objective truth that was true before humans even existed and not just like a thing society agreed on because it’s useful’ (culled from Twitter, where people say the most extraordinary things out loud)
Let us start with big sister and little brother minding their flock of twenty sheep. Little brother hugs one of the sheep and says, ‘Methera is my favourite!’ Big sister asks, ‘Why do you call her Methera?’ Little brother looks surprised. ‘Because that’s her name,’ he says. ‘you call them all by name at the end of the day, but Methera’s the only one I can pick out – O (pointing to another sheep) that’s Bumfitt!’
Big sister realises that little brother has misunderstood. She explains that she isn’t calling the sheep, she’s counting them. She shows him using her fingers as she says ‘Yan, tyan, tethera, methera, pimp!’ (she holds up all five fingers on one hand, then moves to the other) Sethera, lethera, hovera, dovera, dik! (she hold up all ten fingers of both hands, then curls one into her palm as she continues) Yanadik, tyanadik, tetheradik, metheradik, bumfitt! (little brother laughs) Yanabumfitt, tyanabumfitt, tetherabumfitt, metherabumfitt, Giggot!” And when she reaches ‘giggot’ she makes a score on the ground with her crook. ‘there, you see – we have one score sheep: I counted them. Now you try!’
It takes little brother a few attempts to get the sequence right – because that’s the important thing – but luckily the rhythm, rhyme and pattern all help: ‘dovera hovera’ just doesn’t sound right the way ‘hovera, dovera’ does, and in the same way his ear and tongue tell him it’s ‘sethera lethera’ not the other way round. Once he’s got the sequence, big sister tests him on the rule.
‘The rule is ‘add one’ you see – each number in the sequence is one more than the number before it. That’s why you need to know the sequence: the value of each number depends on its place.’
She tests him by holding up different numbers of fingers and having him count them, then adding one. He gets the hang of it quickly and pretty soon if she holds up seven he says ‘lethera’ right away and if she adds another three he counts ‘hovera, dovera,’ in his head then says ‘dik!’ out loud. Then big sister gathers up some pebbles and sets out all the numbers up to twenty – a group of one, then two and so on – so that he can see them all side by side. They play around with the pebbles and see how they can make up the numbers in different ways: Yan and tethera give you methera, but so does tyan and tyan. By the end of the day, he’s able to count the sheep all by himself and he knows what numbers are.
When he’s a bit older, little brother goes into the town and meets a merchant with an abacus. He watches him for a bit and the merchant, aware of his interest, asks, ‘can you count?’ ‘Yes!’ Says little brother, and slides each bead across on the abacus as he says ‘Yan, tyan, tethera, methera…’ ‘Oho, a shepherd boy!’ says the merchant. ‘Here in the town we have another way of counting, but it’s the same really.’ And he writes out the numbers 1-10 in the dust as he counts on the abacus. ‘Now, we say ‘one, two, three, four five’ but you can say ‘yan, tyan, tethera, methera pimp’ – the names might be different but the numbers are still the same.’ He teaches the boy to count using numbers – 1,2,3 – and shows him how he can go beyond a score: 21, 22, 23. They pass a pleasant day playing around with numbers and translating the merchant’s numbers into shepherd’s numbers and back again.
Another day, when he’s older again, the boy goes to the city and meets a philosopher. ‘Do you know what numbers are?’ The philosopher asks. ‘O, yes,’ says the boy, ‘I can count.’ And to show him, he counts to twenty the shepherd’s way and then the merchant’s way. ‘Indeed, you can count,’ says the philosopher, ‘but that wasn’t what I asked – what are numbers?’ The boy is puzzled a moment, since he thinks he has just shown him, but then he writes out the figures 1 to 10 in the dust. ‘I suppose you mean these? That’s what numbers are.’ ‘But I could do that a different way,’ says the philosopher. ‘here’s how the Romans used to do it.’ And he writes out in roman numerals, I, II, III, IV, V and so on. ‘And do you know your alphabet?’
The boy recites it for him.
‘Well, you could use that too,’ says the philosopher. ‘any sequence you know can be used to count if you follow the rule of ‘add one’ : so a is 1, b is 2, c is 3 etc. Or d is methera, e is pimp, f is sethera and g is lethera, if you like.’
‘I see that,’ says the boy. ‘They’re just different names for the same thing, or different ways of doing the same thing.’
‘But what is that thing?’ asks the philosopher, ‘that’s what I’d really like to know! If seven and lethera and 7 and VII and even g are all the same thing, what is that thing? And where is it?’
The boy shrugs. He can sense the philosopher’s excitement, but he doesn’t share it. It does not seem necessary to him to know these things.
‘What does it matter, as long as you can count?’ He asks. ‘Isn’t that the important thing? If you follow the sequence and apply the rule your sums will always work out. Tyan and tyan will give you methera, two and two will give you four, 2+2 will always =4.’
‘But isn’t that the wonder of it?’ says the philosopher. ‘Here are these things – numbers – and we can call them by all sorts of different names, but they always add up, and you know you can rely on that. Suppose someone came up and said, 2+2=5 – how would you react?’
‘I’d tell him he was wrong, that he couldn’t count.’
‘But suppose he insisted? How could you show him that he was wrong?’
After some thought, the boy says, ‘I’d ask him to count to ten. If he did it right, then I’d show him using pebbles for numbers, and he’d see that 2 and 2 couldn’t make 5 but had to be 4.’
‘but what if he did it wrong? What if he counted 1,2,3,5,4?’
‘Then I could show him that we both agreed, but that we used number-names differently: what he called 4, I call 5 and the other way about. I’d like to see what he did with the higher numbers, too – like 14 and 25 and 44 – but as long as we both used a consistent sequence, even in a different order, we could make our sums work out, because it’s the place in the sequence that determines the value, along with the rule – go to the next in the sequence, you add one, go back, you take one away.’
‘And isn’t that marvellous? Suppose someone else came up and said ‘for me, two and two is seven, and two and three is eleven, and eleven and seven is nine, and nine and two is one?’
‘Well, I wouldn’t trust him to count anything, that’s for sure. But I could ask him what rule he’s following, what sequence he’s using.’
‘And if he says, ‘O, I don’t follow any rule (and you can’t make me!) I just use the numbers in any order I like – 7,4,5,8,2,3 one day and 6,1,7,4,9 the next. I’m a free spirit. If I say 2+2=7 then that’s what it equals, for me. After all, numbers are just something we’ve invented: you can use them any way you want.’
‘Then I’d ask him ‘but what do you use them for?’ I use mine to count sheep.’ I suppose I might try to diddle him, just to teach him a lesson, but that would hardly be fair, since he clearly doesn’t know what he’s talking about: he can’t count, he doesn’t know what numbers are.’
‘Which brings us back to my original question,’ says the philosopher. ‘Just what are numbers? There’s something mysterious about them. They seem to be the same for everyone, though we can call them by different names. Once you apply the rule of ‘add one’ to a sequence, you always end up with the same numbers, no matter what you call them, because they always add up the same way, and you know they always will, as long as you’ve got the sequence right. Yet they don’t seem to be anywhere: I mean, you can’t pick them up, or go and look at them or show them to someone else – but you know they must exist, and that they’re infinite, because no matter how high you count, you can always add one more. It’s amazing!’
The boy shrugs. ‘Maybe numbers aren’t a thing at all,’ he suggests. ‘Maybe they’re something you do, like – like playing the piano.’
This brings the philosopher up short but seems to please him. ‘What made you say that?’ he asks. ‘I don’t know,’ says the boy, ‘it just came to me. I suppose because there isn’t a ‘what’ you can ask about – it’s just playing the piano: it’s something you do. So’s playing with numbers.’
‘But what are you playing with?’ demands the philosopher afresh.
The boy shrugs. It does not seem necessary to him to know such things. It is only when he is an old man that he one day says to his big sister as they sit by the fire, ‘I see it now – he was trying to fit them in to his scheme of things. Only I didn’t have a scheme of things, so it didn’t matter to me. He wanted to find a way to think about them, to connect them up to a bigger picture so that it all worked. I suppose that’s what philosophy is: trying to fit everything into the same big picture. I seem to have managed without one all this time. What about you?’
But his sister is already asleep.