I have always been a big fan of the idiom, ‘necessity is the mother of invention’. It’s the kind of phrase that just *sounds* wise and true. Whenever somebody says, ‘necessity is the mother of invention’, you can’t help but nod along in silent agreement.

If I had but one criticism, it would be that it puts forward the idea that our greatest discoveries come at our time of greatest need, and therefore all the most worthwhile inventions solve some kind of urgent crisis. Is this really how we want people to think about science?

To give you an example, when I was in school, I thought the main purpose of Biology was to help us cure disease. I even considered applying for medicine, motivated in part by the prospect of curing some disease and forever impacting the world in a positive way. I hadn’t settled on which disease yet, but my sister has diabetes, so maybe that one would do. Y’know, a healthy, realistic goal.

Aspirations are, of course, important, but I have found that science has impacted my life in much *much *smaller positive ways; it is a vehicle for exploration that need not necessarily lead to earth breaking discoveries, but is enjoyable in its own right.

This leads me on to running in the rain.

A few years ago it was raining in London. I’m pretty sure it has rained since, but this particular day I was going to meet a friend from university. We’d since graduated and our lives had diverged, so this was a rare opportunity to catch up. We had both studied maths (medicine was not for me as it turns out – sorry, diabetics) and in particular were interested in a part of maths called ‘fluid mechanics’, which is pretty much trying to figure out how fluids move – most prominently, water.

Back to the rain: it was heavy, it was cold, it was the kind of rain you really don’t want to be in. As I picked up the pace to a slight jog as I got nearer the bar, the thought crossed my mind that I might just be causing the rain to hit me faster than if I had kept my leisurely stroll, with my chin tucked into my coat. On the other hand, my suffering would be over sooner, which seemed appealing.

After the quick initial catch-up with my friend, conversation somehow moved onto this exact idea. It’s probably one that many people have had, actually. Who likes getting soaked by cold miserable rain? Nobody. However, I imagine that the extent of the conversation is usually noting that the idea is interesting, shrugging and moving on. Not us, though. Like the shameless mathematicians that we are, we got out the notebooks and set to work

This is what most mathematicians would recognise as a classic optimisation problem. There are two warring principles that oppose each other:

- Running in the rain gets me to my destination faster, therefore I get
*less*wet in the process. - Running in the rain causes more rain to hit me per second, therefore I get
*more*wet in the process.

Which of these wins out overall? It’s not really very obvious. There are 3 possible solutions to this problem:

- Walking as slow as possible gets us the least wet (this one seems unlikely)
- Running as fast as possible gets us the least wet
- There is some kind of ‘optimal’ speed at which to run in order to get the least wet.

The last one is not entirely implausible and could probably use some further investigation. Fortunately, when it comes to optimisation, calculus is there to help. Whenever you want to find the minimum or maximum of something, you are talking the language of calculus. In this case, we want to minimise how miserable I get walking from A to B.

Okay, so in order to model this situation mathematically, we had to make a few approximations. Notably, we approximated the shape of a person to be perfectly spherical (yes yes, I know, but be grateful that it’s at least three dimensional – earlier iterations of this model had our person as a single 1D line). We also assumed that the rain was vertical, continuous and not split up into individual droplets (this one is fairly reasonable, since any approximation errors should average out nicely).

So what did we find? Well, it looks like the faster the better. However, it’s not possible to run so fast that you remain completely dry. There is a sort of minimum wetness you must incur travelling from A to B, which is roughly equal to the amount of rain in the given volume that you travel through on your journey.

I’m not sure how long it took us to work through the problem – time flies when you’re having fun and all that – but I’m pleased to say that, once satisfied with our answer, we did proceed to have an enjoyable evening like two ordinary human beings.

The reason I bring up this episode at all is because, whilst this small bit of maths we did together was utterly useless and will in no way benefit humankind or cure my sister’s diabetes, it was still an enriching experience. Much like reading a novel or listening to music can be an enriching experience, maths too ought to be judged by that same metric. When we discuss the motivation for studying maths & science in school, I think this philosophy should be a consideration.

I did a little research to see if I was the first person to think this way. Completely unsurprisingly, I’m not. In fact, I rather enjoyed this quote from Alfred North Whitehead, an English mathematician active in the early 20th century, well known for his work with Bertrand Russell:

*“Necessity is the mother of invention is a silly proverb. Necessity is the mother of futile dodges is much closer to the truth. The basis of growth of modern invention is science, and science is almost wholly the outgrowth of pleasurable intellectual curiosity”*