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Mathematics is vital to our world, and has played a major role in bringing it about. Yet it remains widely underestimated and undervalued. It is much broader, vital, and creative than most people think, because they base their opinions on the very limited mathematics that are taught in schools.
By its nature, mathematics at research level is highly technical, and it’s hardly surprising that these misunderstandings occur. The best way to put the importance, and fascination, of real mathematics over to the general public is to show them what’s going on. And the only way anyone can be expected to pay attention is if the ideas are comprehensible without specialised knowledge. In short: we have to popularise mathematics to make it accessible to non-specialists, and then they can make informed judgements about its worth, beauty, interest, creativity, and utility – a utility which can be seen in mathematics’ application in the ‘real’ world.
For the past 200 years the main interaction between mathematics and the wider world has been in the physical sciences: physics, chemistry, astronomy, engineering. Now it is beginning to look like something similar might be happening for the life sciences. In my book, The Mathematics of Life, I’m not suggesting (as one of my reviewers assumed) that mathematics can unify the whole of biology and turn it into a deductive science like physics, but that increasingly, mathematical models and methods are becoming essential to our understanding of biology.
Mathematics has one major virtue: it makes the concepts under discussion precise. That can also be a vice, because precision relies on specific modeling assumptions. The existence of living creatures is often discussed as if it is some kind of rare miracle, requiring extraordinarily special conditions. However, the mathematics of ‘complex systems’ — which are often quite simple, despite the name — shows that as soon as a reasonably rich mathematical process gets started, it typically produces enormously complex behaviour. The complexity of a complex system lies in its effects; the causes can be very simple.
This strongly suggests that the organised complexity we call ‘life’ is actually a very robust phenomenon. Life on this planet evolved to suit the conditions on this planet, so it is hardly surprising that it relies on those conditions for its existence. But it’s a logical error to deduce that those are the only conditions that can support that level of organised complexity. So mathematics, as well as giving us a chance of understanding the details of the origins of life here, puts the questions in perspective.
My book is organised around five revolutions in biology, with a possible sixth on the way. For the first five, I chose innovations that completely altered our view of living creatures: taxonomy, the microscope, evolution, genetics, and molecular biology (especially DNA).
Does mathematical biology count as revolution six? Not yet, but it has the same potential to totally change how we think about the life sciences. The precedent in the physical sciences shows just how revolutionary mathematical ideas can be: we simply can’t do physics or astronomy without them.
Now, mathematics has been a tool in biology for years: statistical analysis of experimental data, for example. But that’s a kind of off-the-peg use of mathematics that already existed. What’s really new is the way biological questions are stimulating the creation of really new mathematics, leading to advances in mathematics itself. Now the trade runs both ways – math to bio and bio to math, to quote an American colleague – and that’s exciting for mathematicians as well as biologists.
I firmly believe this trend will continue. Worldwide, governments are starting to invest more and more money in biomathematics. Many aspects of biology will remain outside the scope of mathematics, but the scope of mathematics itself will become wider because of problems imported from biology. In particular, anyone who imagines that biology is too complicated for mathematics to handle has failed to understand what real mathematics can do. If it is too complicated for mathematics to handle, it is far too complicated to understand any other way.
Ian Stewart is a Mathematics Professor at Warwick University. His books include ‘The Mathematics of Life’, ‘Professor Stewart’s Cabinet of Mathematical Curiosities’ and ‘The Science of Discworld’ trilogy with Terry Pratchett. He is a Fellow of the Royal Society, appears frequently on radio and television, and does research on pattern formation and network dynamics.