What is a mathematical model? It is simply a thought experiment that helps you make a prediction.
Sometimes models are very complicated: if you want to land a rocket on the moon and bring it home again, you need to take into account many different things, and you necessarily must use a complicated model.
But simple models can also be very helpful, and we use them all the time.
When I was 5, I got pocket money of 5d a week. (The abbreviation for penny was d. 12d made a shilling. 20s was one pound.) My memory of prices is a little hazy, but I think I could get a very small bar of chocolate for 2d and the sweets I really liked – Mars Bars and Crunchies – were 6d each. I could walk to the shop on the corner (which didn’t require crossing a street, so I could go on my own), pick out the sweets I wanted, and show the
shop assistant how much money I had to see if I could afford them. Or, I could do a calculation in advance, counting up my money. Maybe my Uncle Billy had paid a visit – he always slipped me a half-crown, which was a coin worth 2 shillings and 6 pence. Or maybe I owed my brother 3d, so all I could spend was 2d. With a little arithmetic, I could work out what I could afford to buy before I went to the shop.
This is an example of a model. It ignores many features of the objects – what they are made of, what color the wrapper is, how heavy they are – and just abstracts the price. And to count my money, it doesn’t matter how many individual coins I have, or their shapes and sizes (the big copper pennies, the dodecagonal bronze threepenny bit, the silver sixpence) all that matters is how much each one is worth.
The first great mathematical idea was the abstraction of the concept of number. This means that if you can count ten pebbles, you can also count ten apples, or ten sheep. In elementary school our teacher told us that shepherds in the morning would pile a stone up for every sheep they let out of the fold. In the evening, as each sheep came back, they would take a stone away from the pile. If all the stones were moved, no sheep were left outside. I thought at the time this meant they couldn’t count, but I now realize it was just a version of a clicker – very useful if you have to keep interrupting your count to chase wayward sheep. The underlying idea is that you can set up a correspondence between sheep and stones, the latter being much easier to manipulate.
It takes a while to learn how to count. Arithmetic, as every elementary school student can assure you, is even harder to learn than counting. (I found 6 times 7 particularly difficult to remember). But we spend years learning arithmetic, because it is so useful. Once I know how to add 3 and 2, I don’t need to learn a different rule if it is 3 nuts plus 2 nuts, or 3 dogs plus 2 dogs.
A model is like a map. If I want to tell you how to get to my house from my workplace for a drink after work (ah, the good old days!) I can sketch a very simple map that suffices. If an architect is designing an extension of my house, they need a more detailed (and more localized) map. But there is no point constructing a map that is 1:1 scale and includes every detail
(you already have one!) Similarly for models – you do not want to include every detail, just enough for the model to be useful. Our sheep/stone model will not answer the question of whether Betsy the ewe is sick, but it will tell me if she and all her flock are in the fold.
Choosing what to include is tricky – it is hard to make a simple model! Include too many details, and it becomes unmanageable. Include too few, and it is too inaccurate.
In my next post, I’ll describe some simple models that can be used to draw non-obvious conclusions.