The relationship between the applications of mathematics and rigorous proofs is something which I find mysterious. For many non-mathematicians proofs probably seem superfluous and a kind of fetish of the mathematicians. At the same time the endeavour to develop such proofs lies at the heart of the subject. I feel that it is here that the ultimate strength of the subject lies. My starting point in writing about this here is that, rather than having a well-formed opinion on the subject, I am trying to develop one. I want to try to get useful input from whatever source. The basic direction is to concentrate on the way that mathematics can be used in practise rather than on building philosophical theories on the subject. It would be nice to have some positive examples of what mathematics can contribute.
An example of this kind is the development of multi-drug therapy for AIDS. Some relevant information can be found in an interview with David Ho for the Academy of Achievement. He was one of the key contributors to this breakthrough and his work was recognized by his being chosen as Time Magazine’s Man of the Year in 1996. He started out his scientific career as a physicist before going into medical research. He says in the interview, ‘…this is where my physical science background really came in useful, having a strong background in mathematics and applying it to biology…’. In the highly cited paper by Ho and collaborators related to this development (Nature 373, 123-126) mathematics plays a rather low-key role. There are numbers and graphs of data but the only equations are confined to the captions of the figures. A lot of background about the applications of mathematics to AIDS and related problems can be found in the book Virus Dynamics by Martin Nowak and Robert May. On the first page of this book there is the following interesting quote: ‘ … mathematics is no more, but no less, than a way of thinking clearly’.