The edge of reason

What maths can and can't tell us about the world

The central idea of my book is that mathematical ideas can help us understand the world better. But actually, it’s important to know how far to take it - I’m certainly not claiming that the world is entirely predictable! So what might maths tell us - or not tell us! - about Trump, Ukraine and monkeypox?

On behalf of the group

The most basic maths you can imagine is probably just adding numbers together. It’s maybe second nature, but you might notice certain things about doing this:

1. When you add two numbers together, you get another number (kind of obvious!)

2. There’s a special number, zero, that if you add it to anything then nothing changes — 5 + 0 = 5 and so on.

3. Every number has a partner, where adding the number to its partner takes you back to zero — 3 + (-3) = 0, (-8) + 8 = 0.

4. If you have three numbers to add, it doesn’t matter how you bunch together the adding — (3 + 5) + 4 = 8 + 4 = 12 is the same as 3 + (5 + 4) = 3 + 9 = 12.

Ok, but so what? Well, it turns out that these four properties crop up for lots of different systems, not just for adding numbers. For example, the same is true for moving a minute hand round on a clock - doing two moves is the same as one single move, there’s a kind of move (do nothing!) that leaves things where you are, an anti-clockwise rotation cancels out a clockwise one of the same size. Or think of shuffling cards.

In fact, these properties crop up in so many different contexts, that mathematicians give them a name: any structure satisfying them is called a group. It turns out that distilling out these kinds of essential properties is a way to be simultaneously smart and lazy.

For example, suppose we want to show that there’s only one special “leave-it-alone” move (as in point 2). We could find a separate way to argue that’s the case for adding, for the clock and for the card shuffling. Or, we could instead show that it’s true for every group. If we do that, it takes care of all those settings, plus settings we haven’t even thought of yet!

It turns out this way of reasoning, of finding the essential properties of a system and hence simultaneously studying all systems that satisfy them, can be immensely powerful. Of course, assuming more properties lets you prove more, but makes the results more restrictive. For example, if we only study systems with the “adding numbers” property that order doesn’t matter (so 5 + 4 = 4 + 5) then we would miss out on card shuffles, which don’t have this property.

In general though, finding the right level of abstraction is exactly the kind of thing that mathematicians are good at, and people have played similar games with distances and geometry for example.

A cautionary tale

However, this kind of reasoning can also be deceptive, and definitely requires care! For example, it’s very easy to come up with properties which seem useful but are hard to check. Worse still, these properties might not even be true!

Take Gabriel Lamé. Like many other mathematicians, he wanted to prove Fermat’s Last Theorem. As you may know, this is the claim that you can’t add two perfect cubic numbers together to get another perfect cube, you can’t add two fourth powers to get a fourth power and so on. Lamé did a lot better than most people: in 1839 he was the first person to prove rigorously that you couldn’t add two seventh powers together to get a seventh power.

But Lamé took it further than that. He claimed to have proved the whole theorem, that his methods would work rigorously not just for seventh powers, but for eighth, ninth, tenth powers and so on. As we know, this was wrong - Fermat’s Last Theorem wasn’t proved until the work of Andrew Wiles in the 1990s - but Lamé’s mistake was a subtle one, and he wasn’t completely wrong.

His methods worked by studying what are called the complex numbers, and unfortunately there his intuition let him down. Lamé assumed that, like regular numbers, these complex numbers could only be split into prime factors in one way. However, this turned out to be wrong. While it was true a lot of the time, and Lamé’s methods were developed by Kummer to hold for much more than just seventh powers, they couldn’t prove the whole of Fermat’s Last Theorem.

It didn’t end up too badly for Lamé. He’d made a big contribution to the problem, and his name ended up engraved on the Eiffel Tower, but he hadn’t gone all the way. Even within the rigorous world of pure mathematics, where it is possible to distil out the key properties and make deductions from that, you have to be really careful!

Back in the real world

As I say, mathematical ideas are very powerful. By building and understanding models we can hope to explain and predict the world. But it’s equally important to understand the limitations of these models and the uncertainty around them. And, just like Lamé, it is possible to distil out the wrong lessons and draw the wrong conclusions.

A particular example of this, much beloved by opinion columnists, is to look for historical analogies to present situations, and imagine they will pass over exactly. (It’s even more prevalent among people who have been wrong once, and over-compensate for that by moving their predictions too far).

For example, because Donald Trump was underestimated in the 2016 election many people clung to the idea that he was the favourite in 2020 despite all the polling evidence to the contrary. Similarly, the fact that Russia had managed to annex Crimea relatively easily in 2014 helped lead to over-confident predictions that they would also seize the whole of Ukraine. COVID took over the world and is likely to infect almost everyone in due course, monkeypox didn’t.

In each case it seems that the right approach is not simply to find a close analogy, but also to kick the tyres a bit, to see why things might be different this time, and to think about what effect this might have. For each of these scenarios it’s relatively easy to find these kinds of reasons (albeit perhaps with hindsight!) and see why the story didn’t play out in the same way a second time.

So, I think the lesson of the groups and of Lamé is that looking for rules and patterns is definitely worth doing, but it’s important to not be over-confident in those rules and to expect that everything will be as clean and as simple in the real world as in mathematics.