Addition and Abstraction
BLOG: Heidelberg Laureate Forum
In a previous post, I talked about the mathematical idea of generalisation – taking a system or idea and extending it in as many directions as possible to see what other interesting things we can make, and what these will tell us about the problem in hand. Another concept that’s crucial to mathematical thinking is the idea of abstraction – taking a concrete, real-world idea and finding an abstract, pure version of it, free from all the messy details, to which you can then apply some mathematical thinking.
For many people, this process is the first step and one of the things some people really dislike about mathematics. They can understand a physical real-world problem – how do I get these things to there, and how much of this do I need to make those – but turning this into a cold mathematical equation can cause difficulty. It’s actually one of the most useful skills you’ll learn by doing mathematics, and it involves carefully thinking about the problem, deciding which aspects of it you can afford to ignore and which are crucial to the question being asked.
Mathematicians who do modelling are faced with this problem all the time – and the decisions they make about how to model a problem can mean it’s an inaccurate model, if they lose too much detail, or impossible to compute, if they try to capture too much.
On a simple level, abstraction can be used to simplify a problem. For example, say you’re given the challenge of colouring in a diagram so that no two regions that are adjacent are the same colour. The one pictured below is one I often show to people, and challenge them to find a way to colour it using only three different colours – or prove that it can’t be done.
The diagram you’re looking at might contain a lot more information than you actually need to solve the problem – the shape of each region, or how big they are – when all you really need to know is how they’re connected together. Mathematically, we can abstract a diagram like this by replacing each region with a single point, and joining two points with a line if they’re regions which touch – then we just need to colour the points so they’re different colours whenever they’re connected.
This simpler diagram contains only the information we need, and can be studied more easily.
Let’s Play A Game
This type of abstraction can be applied to any structure – a puzzle, a system, or perhaps a game. Let’s play the following game: we place the numbers 1 to 9 in a pile on the table, and two players take it in turns to pick one up and put it in a separate pile in front of them. The aim of the game is to find a set of three numbers in your own pile which add up to the total 15 (and it has to involve three of the numbers – you can’t claim a win holding 7 & 8, or 9 & 6). Not all of the numbers in your pile have to be used, but the winner is the first person to be holding any three numbers which add up to 15.
If you’d like to have a go at the game, I’ve made a slightly terrible version of it here which you can play against a friend. The game can be surprisingly difficult – it feels like there’s a lot of things to keep track of at once, and it’s easy to miss an opening that would allow your opponent to make a winning set (while at the same time, you’re also worrying about the numbers you need to make your set).
Once each player has picked up a few numbers each without making a set, you might find yourself stuck – if there’s two numbers your opponent could make a set with, using different ones from their pile, you can’t stop them from getting both before they grab one and win.
In order to analyse the game mathematically, we can consider it in a more abstract form. The game involves a set of nine objects, and players take it in turns to pick up an object and add it to their own pile, claiming it for themselves. The condition for winning the game is that there are specific sets of three objects from the pile which have a ‘winning’ property, and if one player manages to get all three of the objects from one of these sets and claim all three of them for their pile, they win.
If we wanted to analyse the structure of the game a little more, we could also notice that the sets of three that count as a win aren’t all separate – some of them contain objects that are also part of different winning sets. In fact the winning sets are:
These are all the winning combinations. You might notice that no two numbers (objects) appear together in more than one sum (set of winning objects) – which makes sense, as if you add two numbers together, you’re left with only one number and they need to add up to 15, so there’s only one value which will work.
This abstract version of the game still has all the same properties as the real version of the game – the same strategies, the same impossible situations and the same possible games will exist in both versions. So, abstracting it doesn’t necessarily give us any advantage. But what if the abstract version is a clue to something that will help us play the game better?
Hey, wait a minute…?
If you’ve got a feeling that this game seems somewhat familiar, think hard about this game structure. Nine objects to choose from, and you take it in turns… eight sets of three things that count as a win, and no two sets share more than one of the objects… a game you’ll almost certainly have played before?
How about tic-tac-toe? Noughts and crosses, as it’s also known, has a set of nine things you take it in turns to choose (by drawing Xs and Os), and there are eight specific sets of three things (three rows, three columns and two diagonals) getting all of which count as a win, if you can do it before your opponent does. You can also see that any one square in the grid can be part of multiple winning lines, but once you’ve picked two squares you’re committed to a specific third, so no two winning lines share more than one of the squares.
If you’re still not convinced these two games have the same underlying structure, this should seal it for you: write the numbers 1-9 in a 3-by-3 square grid, as shown:
This is known as a Magic Square, and it’s a special arrangement of the numbers 1-9 such that any row, column or diagonal of the grid will add up to the same magic total, which in this case happens to be 15. So any valid noughts and crosses line on this grid will also have this property. If you’re playing the 15 game against someone, and can secretly memorise or hold a copy of this grid, it will give you a huge advantage.
Even though the 15 game is a mild challenge to play when you’re thinking purely about numbers, it turns out to be exactly identical to noughts and crosses, which is a game many find pretty easy. But would it be as easy if the squares weren’t arranged in a nice neat grid, and you had to remember, or work out, where each one was in relation to the others before making each move?
Maths allows us to reduce complex ideas to simple abstract ones – and this means that we can apply things we know from past experience, like how to play tic tac toe, to new situations, without having to re-learn all the things in this new context. This is part of the beauty of it – tools we develop by studying one thing can then be applied to something else. And, we have a new sneaky way to beat our friends at a game!
„The most important problem for AI today is abstraction and reasoning,” (Francois Chollet, AI researcher at Google).
Why is that? Probably because only a deep understanding of the relationships of a real object (the inner and outer relationships) makes it possible to peel out, to prepare out, a meaningful and powerful abstraction. She who abstracts dissects. And dissecting can only be done by those who understand what needs to be preserved and what needs to be cut.