And on some nights, I try to tie some old strands together. Tonight, I said, I want to talk about exponential growth.
“We’ve talked about exponential growth before, Daddy,” she said. “I hate it – it runs into big numbers really quickly.”
It can easily go out of control, I said, in a positive feedback loop. But it never lasts long – what stops it?
“Negative feedback?” she said, remembering the lesson.
Right, I said. I’ve told you the basics of exponential growth, and now I wanted to talk about some of the ways it affects the world. For example: I’m going to make beer this summer, and I do that by cooking grain and mixing it with yeast. Cooking turns the starch to sugar, the yeast eat the sugar, and they multiply exponentially. They turn the sugar to alcohol, and when the food is used up, they die and the beer’s done. You got it so far?
She nodded – we’d been through this before.
Let’s say the yeast double every hour, I said –
“So two yeast, then four, then eight,” The Girl said, trying not to sound too bored. “Exponential growth.”
Right, I said. Let’s say they take 100 hours before they’re done – before the saturation reaches 100 per cent. So here’s my question: at what point is the vat half done? When is it 50-per-cent saturated?
The Girl started to say “fifty,” but then stopped. “Wait – that would be regular growth.”
She thought a moment, trying to winkle out the trick. “Ninety hours?”
You’re on the right track, I said – very good. It’s 99 hours – one hour before it’s done, it’s only half full.
“Toward the end, it starts growing much more quickly,” she said. “That’s how you know it’s exponential.”
Right, I said -- the rate doesn’t change, but the amount does. So here’s the next question: what if the growth is something very small, like one per cent a year?
The Girl smiled. “You can’t fool me, Daddy,” she said. “It’s still exponential. If it’s just one per cent a year, it … “ She paused, struggling to find the words. “Pretty soon it’s double what it was, and then double that, and so on.”
Very good, I said – I’m proud of you. Things like this will be important when you’re older and handling money – you know those adverts on the telly, for quick loans?
“I’ve seen them,” she said.
They’ll lend you money, I said, but you have to pay it back plus seven per cent – that means that in ten years, you owe twice as much. In 20 years you’ll owe four times as much, in 30 years eight times, and so on.
“That’s huge!” she said. “Well, depending on how much it was originally.”
Well, that’s why you shouldn’t borrow money unless you have to, and then pay it back right away, I said.
“Am I going to see a lot of exponential growth when I’m a grownup?” she asked. “’Cause I’m really hating exponential growth more all the time.”
I’m afraid you will, I said – but that’s why I tell you these things, so you’ll know it when you see it. Remember the lesson about carbon in the air? That’s been increasing exponentially. A few weeks ago I told you about how a lot of the things around us are powered by petroleum -- our petroleum use has doubled every 20 years or so. A lot of things are growing exponentially.
She looked at me gloomily. “But those are all terrible things,” she said.
I know, I said gently – they’re all in positive feedback. They all started small, and grew before anyone realised what was happening. But there are people making things better, and they’re growing exponentially too --- remember the parable of the mustard seed?
She looked puzzled for a moment, before calling up the memory. “It starts small, but…”
I smiled. What stops positive feedback? I asked her.
“Negative feedback,” she said, her expression brightening.
Right, I said – if people learn how to cope with the changes, and they teach others, and they teach others …
“It’s a positive feedback too,” she said. Yes, I said – but in the other direction, to make things better. They’re mustard seeds.
“Are we those people?” she said. “We’re not teaching anyone else.”
I am, I said, right now – and she smiled. And other people are teaching me, and you’ll teach others.
After we curled up and read together --- one of C.S. Lewis’ Narnia books – I kissed her goodnight and asked her: what did you learn tonight?
“How to make myself beer,” she said slyly.