Maxwell's Demon

This week, my apologies, the post is a little late.  However I have an excuse!  I wanted to put the piece below out, but wanted to make sure that it had been published (in the December issue of Mathematics Today, the magazine of the Institute for Mathematics and its Applications/IMA) first.  So apologies again and enjoy…

The world’s financial markets are in trouble, leading the whole world into crisis.  The causes will probably remain murky; economists still debate what caused the Great Depression.  However it cannot be disputed that complicated mathematics has become an increasing part of the business of large financial institutions.  Furthermore the models produced by this mathematics are often used primarily for profit rather than understanding.  Maybe we can no longer hide behind G.H. Hardy’s claim that:

it would be paradoxical to suggest that mathematics of any sort does much harm in a time of peace. 

A Mathematician’s Apology

  The use of mathematics for financial modelling is a modern application that is now widely accepted.  However many other new areas are opening up to quantitative modelling, and thus to mathematics, but mathematicians are not very active in this process.  Perhaps it is time to reassess the priorities of our subject and ensure that mathematics is indisputably a force for good, or at least for progress in the world.

I am a mathematician at Imperial College London.  In September I attended the World Knowledge Dialogue in Switzerland, a meeting intending to bridge the gap between the natural and the human/social sciences.  From Nobel laureates to former UN High Commissioners, the speakers laid down a clear challenge to all intellectuals to become engaged in addressing the worlds problems.  The central issue was described by the Harvard biologist E. O. Wilson as how humanity can deal with having:

Stone-age emotions and medieval institutions combined with god-like technology.

I was presenting a poster on how mathematical art could be used to inspire people to put in the work required to learn mathematics.  A worthy cause, maybe.  It was, however, flawed by a false assumption: that the delegates believed that mathematics and mathematics education had a key role to play in solving the world’s problems.  Many did not, due in part to a perception, extending even into the life sciences, that the work of mathematicians is esoteric, and only useful in limited situations.  For this we must accept some portion of the blame.

Up until the middle of the 19th century mathematicians, like Gauss and Euler, were primarily motivated by the quest to describe and resolve physical problems. Today, however, the primary goal for many mathematicians has moved from utility to the sense of mathematical beauty described in Hardy’s A Mathematician’s Apology.  This beauty is a wonderful concept and it cannot be denied that there have been great discoveries that have later found practical applications.  The importance of Hardy’s number theory in internet commerce is a prime example.  In the quest for beauty, especially when combined with powerful tools for generalisation from Hilbert on, the wonderful practicality of mathematics has not quite been lost, but has been buried. 

To me mathematics is the study of ideas without reference to the real world.  This is part of the problem, mathematics is indeed esoteric.  However consider an effective model of the world.  To have predictive power this model must be able to run without constant reference to the situation it is modelling.  Mathematics, therefore, is the only language in which we can model anything.  While the success or failure of a model depends on its design not its mathematics, without mathematics there can be no quantitative models.  In the sixteenth century the telescope allowed many measurements to be made that had previously been impossible.  This led to the first great discoveries of physics: simple mathematical models to make sense of the observations.  We are seeing a similar process in the explosion of data available through computers and the internet.  This has opened many areas to quantitative study and potential modelling.  We are also discovering the rich data within DNA (and protein) sequences in biology.  In just ten years it is predicted that sequencing a whole genome will cost just £10.  The areas in which complex mathematics is of use are therefore widening rapidly.

The practicality of mathematics can only be revealed through communication.  It is no longer true, if it ever was, that anyone with a problem that could benefit from mathematics will search to find the correct mathematical language and approach mathematicians with the problem formulated for us.  It is therefore our responsibility to become active.  To remember that utility, not just beauty, should be the goal of good mathematics.  To go into the common rooms of universities, to international conferences and into the world and make contacts in other subjects.  To listen to the problems that other intellectuals are tackling.  To find where there are mathematical solutions and explain the things that mathematics can (and cannot) do.  Essentially, to find mathematical applications outside the traditional areas, other than those motivated by a desire to make money.

I admit that this pushes mathematicians into something at which we are famously bad.  Too few of us are able to give a good picture of our research to other mathematicians, let alone general intellectuals or the public. In many ways we live up to the public perception of being brilliant but engaged in puzzles that are hard to understand and only of moderate use.  We must try to address this and not accept the public perception as fact.  As well as our personal responsibility to learn to communicate, we need to increase the importance of communication in the mathematical education.  To initiate teaching of communication and send the message, to undergraduate and graduate students alike, that communication is a necessary and honourable part of our profession.

It is true that there are many mathematicians pushing the limits of the subject’s applications already.  However in many cases the drive comes more from universities and the need for funding than professional interest.  The area of mathematical biology for example often carries a slight stigma.  This is made worse by the fact that much of the research carried out here is not good mathematics but more importantly is not good biology either.  The research driver is too often old mathematical questions that can be written in a biological language.  The challenge is, therefore, not to look for new applications for current research but to be willing to listen to the problems in other fields.  In some cases a solution will be ready, perhaps with an imaginative application of classic mathematics.  In others mathematics will be of no use.  However a few such problems might call for the development of entirely new mathematical models.  Who knows, they might even be beautiful!