Approximating the living

Is a herd of wildebeest better thought of as a series of individual animals, each with its own glorious and unmanageable volition, or as a field of moving arrows? Are the morphogen gradients that set up the coordinate systems for embryonic anterior–posterior patterning a smooth and continuous concentration field or instead a chaotic collection of protein molecules each jiggling about in the haphazard way first described by Robert Brown in his microscopical observations of pollen? Is water, the great liquid ether of the living world, a collection of discrete molecules or instead a perfectly continuous medium with a density of $\approx$1000 kg/m${}^3$? In this article, I will argue that these questions pose a false dichotomy since there are many different and powerful representations of the world around us. Different representations suit us differently at different times and it is often useful to be able to hold these seemingly contradictory notions in our heads simultaneously. Indeed, mathematics is not only the language of representation, but often is also the engine of reconciliation of such disparate views. In a letter to Alfred Russel Wallace on 14 April 1869, Charles Darwin noted that Lord Kelvin’s “views on the recent age of the world have been for some time one of my sorest troubles”. Here, I will argue that one of the highest attainments of the scientific enterprise is a coherent picture of the world, a picture in which our stories about the geological age of the Earth are coherent with our stories of how whales populated the oceans, our understanding of the living jibes with our understanding of the inanimate, our insights into the dynamics of genes and molecular structures are consonant with our physical understanding of the laws of statistical physics. The underpinnings of such coherency are often best revealed when viewed through the lens of mathematics.