Beautifully simple mathematics for everyday life
Maths gets a bad rap. It’s boring. It’s arcane. When am I ever gonna use this junk?
I disliked maths for most of my early schooling. I found it hard, often boring, and I often wondered about its usefulness. I remember telling my parents that maths was my weakest subject, and that I was really more of an ‘english’ person than a STEM kid. How things have changed..
I’m just weeks away from graduating with a degree in mathematics, and I want to share some of the most beautiful and intuitive and most importantly, useful, ideas that I have come across in the world of maths.
Some of these ideas do have scary names, but I’m not going to talk about integrals or derivatives or any complex gobbledygook. There won’t be a single formula. So don’t fear, stay calm and keep reading :)
Idea #1: The statistical power of storytelling
How often do you tell stories?
I’m personally not very good at storytelling. I tend to get stuck on unimportant details, and people’s eyes glaze over. But I do like listening to stories.
When I think of stories, this is what I picture:
Background is given to start, and then events build, one after another, until the climax. Sometimes the order’s a bit different. But in general, stories look like this.
In some ways, a story is like a big, multi-part statistical model. One event happens in some way, following some probability distribution of outcomes. This leads to another event, with it’s own distribution.
One of the reasons we tell stories is that, like statistical models, they can be used to predict. I was reminded of this very poignantly, when I read this fascinating story on Medium recently: Do Not Fall in Love with a Smart, Introverted Man.
When you can find patterns from your own life that mimic the patterns of a story, you can use the story’s ending to consider the likely ending of your own tale. This is the power of correlation. In statistics, when two quantities move together in some pattern, we call it a correlation:
In the case of the introverted man article, the man that runs away reminds me a bit of myself. I see a correlation, between the characteristics of the story and those of my own life. And I wonder whether I would run away from a healthy relationship one day, in much the same way?
The responses to that article were absolutely fascinating to me, because they demonstrated the limitations of correlation. Some of the comments were hilarious. Basically: this story is very similar for me, but the ending is wrong! Because I didn’t leave, they did!!
Such is the weakness of stories, and of statistical correlation. Just because two variables move with one another, following some complicated dance of events, does not mean they will follow each other forever. Stories can only be right on average. Life is chaotic.
Idea #2: Chaos is everywhere
One of my favourite little metaphors for chaos is that of the butterfly which flaps its wings in Brazil, disturbing some air flows. Like a ripple in a pond, the disturbances in the air move outwards, playing off one another. The effects build, and by freak chance, eventually lead to a tornado in the USA.
In many ways, this metaphor is illustrative of the chaos that we see in life. Not only is it illustrative, but in many senses it holds truth.
Even incredibly simple physical systems can be extremely chaotic. I remember early in my maths degree having to simulate a double pendulum. I lost my simulation long ago, so I had to go and make a new one:
Each of the double pendulums above start in almost identical states, just positioned slightly apart by a few degrees. Such unnoticeable changes to a double pendulum can completely and utterly change the path it follows. Make the string just a centimeter shorter, or move the bob just an extra fraction of a degree, and within a few swings its path can totally diverge.
This is just a silly little pendulum, described by four fairly simple equations. Nevermind the ridiculous and messy complexity of life, with variables numbering… infinity.
Amongst all those variables, plenty of things behave chaotically — not just double pendulums. The stock market is a classic academic example.
We often think that if we can just build the perfect plan, we can live our best lives. But unfortunately this is often just not true. Tiny changes in the world around us can lead to profound changes in our lives.
Is this to say we shouldn’t plan? No. Planning can be incredibly useful, when we recognise the uncertainty in the world around us and plan with flexibility. It is important to accept that sometimes life takes you where it takes you.
Fortunately we are given pretty powerful machinery to tackle the complexity of life. Our brains can draw on huge quantities of information and experience to form plans and make decisions under the radical uncertainty we face every day. Unfortunately, our brains are often badly misled by biases.
Idea #3: Biases abound
Admittedly this is more of a psychological topic. But then again I’ve strayed away from the maths I originally thought I’d be talking about, so let’s have at it.
Our brains take in information from our eyes, ears, skin and so on. Neurons activate, like little beacons, and send electrical energy onwards into the brain. Neurons that fire together, wire together.
This means we often learn by association. When you touch something hot, it hurts. Hurting is bad. Therefore touching something hot is bad. Association.
Unfortunately this process of learning is very prone to bias. Perhaps one of the greatest sources of poor decisions is omitted variable bias (also called many other things depending on the field, but I happen to like statistics, so that’s the name we’re going with).
Take a look at this drawing I whipped up earlier:
Suppose you’re standing at the running track every afternoon, watching me first eat lunch, and then run (please don’t!). On the days when I eat a curry, you would see me run faster, and you would begin to associate my curry consumption with speed.
If you were statistically-minded, and recorded data on my eating habits versus my running speed, you might see the following:
But in fact, curry does not mix well with running (in my experience). What’s actually happening here, is that I eat curry when the temperature is cooler. I also tend to run much faster when it’s cooler.
My curry consumption coincides (correlates) with the temperature. The temperature itself causes my running speed. But if you were just focusing on me eating lunch, and then running, it seems that curry-consumption makes me faster!
Of course that’s quite a trivial example. But it illustrates one of the many ways that we go wrong. We often only see one side of the story, and it’s easy for our understanding to be biased by that which we don’t observe.
I’ve run out of steam at this point. There are still so many cool ideas to talk about, but I hope you can see now why I like maths.
It’s not all funky equations and scary symbols. In fact, the true beauty of mathematics has little to do with algebra or polynomials or the equals sign. These are just tools (albeit very clever ones) that we use to condense our understanding of a complex and messy world onto a two-dimensional bit of paper. The true beauty comes in the understanding we obtain along the way.
