By the time you read this… I’ll probably be presenting at a conference. The particular conference I am at this week is the 2013 conference on the Mathematics of Planet Earth, which is being held because 2013 is also the international year of the maths of planet earth.

Rather than going into details of either my presentation or the conference itself, I want to take the soap box for a bit to talk about the ways in which maths is an essential tool to do good science. I also want to ask the question: “how does maths help you with your science / what sort of maths do you need that might make your science better?”. Also note I’m not going on about pure maths, I’m talking about maths the tool rather than maths the subject. I’m also keeping it fairly generic: no huge discussion on second-order Bessel functions despite the fact I use them daily at the moment.
Apologies in advance for any minor gremlins in this post, editing on a small tablet is… interesting.

There are some key areas in which maths can really solve your issues when it comes to science, and potentially in daily life, too. I’ll illustrate some of these below and then open it up to you guys to talk about what maths might help you and also what sort of maths you use in your daily science or life.

1. Statistics (and hence risk)
Front and centre, I think the biggest mathematical tool is statistics. Unfortunately, this can be routinely misused or outright abused. Mark Twain famously once said “there are lies, damned lies and statistics”. Shows the credibility most people associate with statistics. Stats, like ’em or hate ’em are one of the key ways to obtain clarity in terms of what results mean. One has to be careful to select a useful statistic though – not all stats are created equal, and that is where people seem to struggle.
Statistics, and by extension probabilities, are also how we understand risk and uncertainty. So, it is crucial people understand this better.

2. Numerical approximations, and understanding where they fall down
A lot of modern science is done using numerical approximations. People may not even be aware of it, but the minute you represent decimals in a computer you are dealing with approximations. The maths behind these is poorly taught, and yet is crucial to truly doing good science.
An excellent example are the stability criteria for models. There are ways to establish if you are taking too large a step in time and/or space in your models, yet very often people don’t quickly evaluate these to see if their approximations will still be valid.

3. The maths of change: calculus
Calculus is one of those topics which really scares people. It shouldn’t! Calculus is the maths of rates of change and we actually experience it daily. People stare at a speedometer in their car not realising it is representing a rate of change (… The units it represents gives a hint). Lots of science, again, is done using derivatives or integrals. The key to good science here isn’t necessarily knowing the ins and outs of specific analytic solutions but understanding how they work to then understand how that influences your science. Given that derivatives and integrals underlie a lot of modelling, this is crucial to good modelling science.

Thats probably enough justification on why at least knowing about some key mathematical concepts is important. Maths unfortunately is poorly taught and too often only viewed as a subject on its own rather than a key enabling tool for science.

Over to you guys now – what are some of the key mathematical things you have had to grapple with in your own research? What sort of maths would you like to understand/do better to help your own science?