Why should we care about this mathematics thing anyway?

25 Feb 2013

Mathematics is poised to become one of the greatest tools for the further development of the biomedical sciences in the twenty-first century. Particularly due to the continual rise in available computational resources, mathematics is in a strong position to lend its precise, predictive power to the kind of problems that face modern biology. The whole discipline must grasp at this opportunity and become fluent in mathematics as applied to biological problems.

The current status of mathematics in the biomedical sciences stands in stark contrast to its continual role in the physical sciences where it has been one of the most invaluable tools in determining the nature of our universe. However leading scientists have long theorised that mathematics will someday take a similar role in biology too. In 1901 the statistician Karl Pearson wrote: “I believe the day must come when the biologist will — without being a mathematician — not hesitate to use mathematical analysis when he requires it.” But a push by any influential figures seems on the most part to have failed. Some limited progress has occurred, for example in ecology, but mathematics has yet to grasp the imagination of the biomedical sciences as a whole.

Further progress will require more of those who work with traditional methods to connect with the theoretical results arising from contemporary work in mathematical biology. At present this connection is far from evident — in fact citations have been shown to drop with the inclusion of a mathematical equation (Fawcett 2012). And unfortunately we can’t in general use the power of mathematics while hiding the equations! Mathematical literacy must increase among those who currently avoid it.

I would posit a single reason why mathematics has so long been fundamental in the physical sciences, and why it must now become so for the biomedical: quite simply mathematics can never be wrong. Mathematical facts are proven through manipulating a set of predetermined assumptions. If the assumptions are correct and the logic is correct then our result must also be correct. The only barrier to their validity can be human error, and the only barrier to their meaning, their metaphor.

What do I mean by metaphor? A bit like building a model aeroplane to put in a wind-tunnel, we can use mathematics to build small models that can predict real world consequences. To do this we must make simplifications — cut down on detail — but leave enough of whats important to answer our original question. Never need we attempt to design a ‘theory of everything’, all we need is a ‘theory of useful’. The inherent metaphor is simply how the inputs and outputs to our model represent the respective features in the real situation. This metaphor can take many forms depending on both the biology and the mathematical technique, for example the concentration of a molecule represented as a real number. Throughout, a successful model is simply one which simply suggests to us, or attempts to predict, something new about the world using whichever metaphor.

Experimentation will never stop being the ultimate technique for validating our theories as nature alone knows her true secrets and a model is only as good as its predictions. However we can easily learn to combine mathematical and experimental thinking to increase the rate of discovery. Models can be built relatively quickly (and cheaply!) to represent novel biological hypotheses, and permutation and repetition are easy to perform in silico. The predictions from these models can then be validated through experimentation, which can drive further modelling, ad infinitum.

The biomedical sciences themselves have recently entered the new paradigm of complex biological networks. No one has the right to assume that they can understand the behaviour of such a network with only their own brain power. Classically, understanding the behaviour of complex networks requires an abundance of careful observations, requiring an equivalent abundance of time and money. Mathematical models — more precisely, computer simulations of these models — are a better solution. Observation of the network under normal and abnormal conditions is then trivially easy, and can, for example, assist in the identification of upstream targets and downstream indicators.

So far, one of the most fundamental tools in the toolbox of mathematical biology have been differential equations, which excel at modelling the rate at which variables change. As mathematical biology becomes more mainstream, I expect that we will begin to see new links between biology and a greater variety of fields of mathematics. However where the fields will really begin to flourish in this partnership is when novel mathematics begins to arise with regularity from biological problems. In this sense I believe that biology can be to mathematics in the twenty-first century, what physics was to it throughout the nineteenth and twentieth centuries: a major driving factor behind new theorems.

The biomedical sciences need more than ever to embrace the power that mathematics can provide. Relevant mathematics simply cannot be ignored, and should not be distrusted any more than any other work published in the scientific literature. Once mathematical literacy improves, modelling should become a part of the standard biologists toolbox. This will give us countless opportunities to save time and money, and hopefully to drive discovery in both fields.

Fawcett, T.W., and Higginson, A.D. (2012). Heavy use of equations impedes communication among biologists. Proceedings of the National Academy of Sciences 109, 11735–11739.