Maths' relationship with other sciences
Ever since the work of Isaac Newton (right), mathematics has been the language and the basic tool of theoretical physics. Mathematics models the motion of the planets, the flow of heat, the propagation of electromagnetic waves, the quantum behaviour of electrons, the curvature of space and time, the flow of liquids and gasses, and much else besides.
The last few decades have seen a hugely fruitful interaction between pure mathematics and the more speculative end of theoretical physics. There are fascinating parallels with the situation in the last century, when Joseph Fourier made spectacular advances in the theory of heatflow using some rather strange manipulations with infinite series, which made no sense when interpreted literally. Nonetheless, it later proved possible to reinterpret Fourier's arguments rigorously, vindicating his confidence in the essential correctness of his arguments and leading the way to many generalisations and extensions of his theory. A similar story could be told about the "delta-function" introduced by the physicist Paul Dirac, which inspired and was made rigorous by Laurent Schwartz 's theory of distributions.
Today we have a similar situation on a much grander scale, where physicists (exemplified by Edward Witten, right) have used their intuition about possible models of quantum mechanics (with neither a solid mathematical basis nor a realistic possibility of experimental test) to suggest profound conjectures about higher-dimensional geometry. This has inspired a great deal of new mathematics designed to address these conjectures more rigorously; they have proved to be correct in a remarkable number of cases. A part of this story is surveyed at the Cambridge relativity home page.
A more recent development is the use of mathematics in biology. One very early example was the use of statistics by John Snow in the mid 19th century to show that cholera was caused by contaminated water; this story (and much, much more) is told in Tom Körner's book "The Pleasures of Counting". The first half of the 20th century saw work of Fisher and Haldane using probability and statistics to study the spread of genes in an evolving population. There are also mathematical models of many biological processes: the conduction of signals along nerves, the flow of blood through the heart, the formation of scar tissue and so on. Some mathematical theory of the topology of knots has been used to study knotted strands of DNA. Some scientists have been inspired by this to construct strands of DNA knotted in bizarre and complicated shapes.