Scotland’s independent think tank
Scotland’s independent think tank

The Joy of Mathematics – Professor Paul Hare


Although an academic economist for most of my adult life, I started off as a mathematician.

At secondary school down in England I took to mathematics from a very early age, always loved it and mostly found it fairly easy. Hence no surprise when it became the subject of my first degree. Back in the early 1960s, when my serious interest in maths gained momentum, everything was enormously different from nowadays. We had no calculators, of course, they weren’t invented. So partly as a result, we had no option but to become pretty good at mental arithmetic. From an early age, too, we had to learn how to use log tables to perform routine calculations – I still have the four figure tables I used at school, they still ‘work’! At the same time, some of the maths we learned was a bit different from what young people do now. Thus we had a lot more geometry, to do with lines, triangles, circles and the like, and probably more basic algebra to keep us on our toes, and for fun.  Until not so very long ago, all this was considered an essential part of what it meant to be reasonably well educated.

In my last year at school I was very lucky with one of my maths teachers. On quite a few Saturday mornings, the three of us in my class who were especially good at maths and keen to try harder problems, would cycle out to our teacher’s house; by the time we got there, she would have baked some scones or something else nice, and we would spend the morning scoffing these treats and struggling with some of the problems she had found for us. It was very challenging, and a highlight of the week. It was only much later in life that I fully appreciated what a special privilege all this extra tuition was. Before starting university in October 1964, I was also lucky enough to get into the early days of computing. I worked for nine months as a mathematician at a firm in my home town, Hull. Nowadays that might be thought of as a form of ‘gap year’, though such things were unheard of then. My work initially involved some statistical calculations using mechanical calculators (the firm had such things, though my school had nothing of the kind), then the firm asked me if I wanted to learn computer programming and do some work on sales forecasting. I knew nothing about either of these, so I said ‘yes’ to both. I had not even seen a computer before. Soon I read up on sales forecasting methods and taught myself an early programming language. Then I would cycle across Hull to use the University’s computer – it only had the one in those days. My program was on a roll of punched paper tape, and the output was another roll of paper tape. Somehow I got my program working and wrote my first ever business report, an amazing experience between school and university. During this period, I was also, periodically, a baked bean taster, one of a panel of tasters. Once we had collected lots of data from tasters, I had to do the statistical analysis to identify the characteristics of the ‘ideal’ baked bean. I don’t think our findings were a big success in marketing terms.

One of the puzzles these days is that somehow, mathematics is widely thought of as a ‘hard’ subject. This seems a real shame, and mostly unfair. For even getting to a good basic level in maths, something like a Scottish Higher, for instance, can already open up lots of doors to a wide range of jobs and career paths, and solving tricky problems is immensely satisfying, especially when done without the aid of a solution or even any hints towards a solution. Getting to this point has a huge positive impact on one’s self confidence, regardless of whether you carry on doing much maths.

In my own case, after graduating  I worked for a while as a mathematician in the chemicals industry, then went back to university to make the career shift into economics. After that I mostly worked in Scottish Universities, first Stirling, then Heriot-Watt. As an academic I did now and again make use of my maths background, writing some technical papers and reports on economic modelling, including a good deal on Eastern Europe where I did a lot of work on various countries. More recently, I’ve done less maths, but have written various reports, mostly for the EU, on several of the UK’s Overseas Territories, including the Falkland Islands,  the Turks and Caicos Islands and Montserrat. Partly because we’ve now left the EU, partly because of the current Covid-19 crisis, this sort of work has largely ground to a halt for now.  However, I am still doing some online maths teaching using Skype, and that is very enjoyable.

It’s especially satisfying working with young people who want to improve their maths, and even have a go at some tough problems, as I don’t know of a better mental exercise. Along the way, I also try to teach my students a bit of the history of maths, as hardly anyone these days seems to know who the great mathematicians were, or what they did, rather a shame. So at the very least I make sure that folk know about Euclid, hugely important as various editions of his Elements were for several centuries the best-selling books in Europe after the Bible.  Everyone was expected to learn a good deal of basic geometry. And what about Edinburgh’s own Napier, the inventor of logarithms, though unfortunately no one cares so much about him these days, as our calculators make everything so easy, but all Advanced Higher students do have to learn about Colin Mclaurin and his famous series. Mclaurin enrolled at Glasgow University at age 11, and was appointed professor of mathematics at Aberdeen University at age 19, quite impressive. Even today, I suppose everyone who takes maths to Higher or Advanced Higher levels knows a bit about Newton and his famous apple tree (which I visited recently in Cambridge) – (or the tree I saw was said to be a descendant of the original one); and they probably know of Euler, the greatest mathematician of all time, in my view, inventor of the number, e, and for a long time official mathematician at the Court of Catherine the Great in Russia. For some reason, I don’t think our Queen even has an official mathematician, not sure why not.

There are a few movies which give a feel for the power of mathematics, so I’ll mention just two: first, The Dish, about a radio telescope in Parkes, Australia, and the problems of tracking the first moon landing. Second, Hidden Figures, about some of NASA’s early work and the struggles faced by some coloured women to be accepted as mathematicians and early computer programmers and for their work to be fully recognised – a really inspiring movie – and in the background some very clever maths, mostly done by hand, without computers.

Yet not only is maths enormous fun, hugely interesting and wonderful  exercise for the brain, even still for older folk like me, but it is also of massive practical importance. For now, and to finish off, let me just mention three important, topical examples.

  1. Internet shopping. We all do this nearly every day and take it for granted. But some very smart maths is involved in the coding algorithms that keep our shopping secure, much of it based on  pure number theory  to do with large prime numbers, developed – largely by folk who thought of maths as fun, and who had no interest in its practical utility, back in the 1930s.
  2. Covid-19. The virus crisis is going to be with us for a while, and some very interesting mathematical modelling is being done by various teams now, to understand virus transmission and to get a better feel for the likely scale and timing of the crisis, how many people might get ill, how many might die, and what practical steps can we take to limit the harm? This is not easy, and much is not yet fully understood, but I’m doing my best to read as much as I can to learn more about the whole thing, and the mathematical models that can help us.
  3. Last, global warming.  Despite the distractions of the virus crisis, and other issues the government has to deal with, we should not forget about this major problem. There is loads of science and various economic and mathematical models to help us in understanding the challenges. Yes, much of it is quite difficult and it’s hard work to keep up with the latest research and policy advice, much of which will force us to accept quite big changes in our way of life in coming decades. So especially for the young, it’s important for them to get a good basic understanding of all this, as part of their core education. This includes, too, getting a feel for some of the most useful mathematical methods and tools.

Hence for young people missing out on schooling while we get through the Covid-19 crisis, especially those in the later years of secondary school, there’s loads of maths they could be doing, for a mix of fun, stimulating their developing minds with tricky problems, and just building up their basic knowledge and understanding. All this is important, and if a scheme is developed to offer online tuition and problem solving in maths, I’d be delighted to be a part of it.

Paul Hare is an Emeritus Professor of Economics, Heriot-Watt University