Mathematics and physics have always been joined at the hip, so to speak. In ancient Greece the two were inseparable; the natural philosophers studied geometry as much as they studied (what we now call) physics, biology, and astronomy. Throughout the ages, an advance in one field typically led to an advance in the other. Unfortunately, the classic philosophy of science depended on teleology, a belief that there is a purpose (called logos) behind existence. The rebirth of science in the past, say, three centuries has led to a crusade against teleology. Biology has been most resistent to these changes; see, for example, the outcry against teaching evolution over intelligent design which has been going on in American schools ever since the Scopes Monkey Trial.
People ask me, "What is the use of all this mathematics?" about once a month. Physics is the natural response to these questions, but it presents a problem as neither the mathematics nor the physics are accessible to someone who hasn't devoted, say, at least a year of their life to the material. My new example comes from Emergence, and it involves the most curious behaviour of leaf slimes.
Apparently, these icky creatures are composed of several independent cells, and can both split into individual cells and regroup into a single entity at will. Biologists have known that the regrouping process is controlled with a specific chemical trigger. It was thought that some of the cells were special, "commander" cells whose purpose was to begin this chain reaction. Some mathematicians studying partial differential equations and other abstract nonsense like that found that some of their simulations mimiced the behaviour of the leaf slime. Their results did not rely on the assumption that there were "commander" cells to start the reaction. This discovery was met with disapproval until later experiments showed that the "commander" cells couldn't exist at all.
Mathematical biology is just that — the removal of teleology from biology by means of mathematical simulation. It views biology as a deontological process where the end result is determined not by the organism's desire (or some higher power's desire) to achieve a given purpose, but instead as an intrinsic property of the organism. Traditionally this has been limited to population prediction of the sort that tells us the Earth's population will be seven billion by 2030¹, but advances in computing have made other branches of mathematical biology more viable.
One fundamental result in this area is a paper (The Chemical Basis of Morphogenesis) written by Alan Turing on the mathematical structure of flowers, a few years before his death². Other people involved in mathematical biology include John Conway (Game of Life), and John von Neumann.
People interested in studying mathematical biology should take a healthy dose of both in college, with a touch on mathematical modelling, differential equations, game theory, computer simulation, computational theory, and physical chemistry thrown in for good measure. Some undergraduate schools are now offering "Mathbio" degrees (instead of a double major in mathematics and biology) tailored to people looking to become mathematical biologists. These are usually either applied maths degrees with a biological focus or biology degrees with a few computational mathematics requirements, depending on which department is more dominant.
- http://en.wikipedia.org/wiki/Earth
- http://www.swintons.net/jonathan/turing.htm is one reconstruction of that work.