# At the cutting edge

Hooked on mathematics since reading a book on elementary calculus as a schoolboy in Edinburgh, University of Canterbury mathematician Professor Douglas Bridges’ enthusiasm for his subject has seen him become one of the leading authorities in his field.

Bridges, who has held the title of Professor of Pure Mathematics at the University since 1999, is recognised worldwide for his work in constructive mathematics, in which the focus is on finding algorithms for constructing mathematical objects.

“In standard mathematics a typical proof of the existence of an object goes like this: assume the object does not exist, derive a contradiction, and conclude that the object exists after all. The trouble with this type of proof is that it doesn’t give any clue about finding the object. In contrast, an existence proof in constructive mathematics must provide the information that enables the mathematician to find — or, usually, to compute/construct — the object in question. Thus every constructive proof embodies an algorithm that could be implemented on a computer. Moreover, the existence proof also shows that the algorithm is correct — in computing parlance, ‘meets its specifications’.”

Although he has a particular interest in constructive functional analysis, Bridges says he does not work in any individual branch of mathematics, tending instead to look at constructivity over a wide range of areas. These have included algebra; apartness and uniform spaces; topology and analysis in metric, normed and locally convex spaces; constructive methods in the foundation of physics and economic theory; and mathematical logic. He is currently working in a relatively new area of mathematics, constructive reverse mathematics, one aspect of which involves finding the principles that are necessary and sufficient for constructive proofs of individual results.

The work he is most proud of, however, has been the development of the axiomatic theory of apartness spaces, which he started working on in 2000 with former University of Canterbury mathematician Dr Luminiţa Simona Vîţă. This theory, which has subsequently been investigated by mathematicians in Japan, Sweden, Germany and the United States, resulted in his latest book, Apartness and Uniformity: A Constructive Development (Springer, 2011), written with Vîţă. It is the first book to deal with the apartness approach to the mathematical discipline of constructive topology.

“This theory is absolutely our baby and I think it is the most significant thing I’ve done in the last 10 years,” says Bridges.

“We’re particularly proud of the whole thing because the theory was initiated and developed by us, rather than being based on someone else’s work. What we developed is an approach to topology that provides a fairly general framework for large parts of mathematical activity. It is based on five axioms that encapsulate the notion of objects being apart, and it encompasses both point-set topology and the theory of uniform spaces.”

Apartness and Uniformity is the latest in a prolific output of published work. Bridges has written more than 170 research papers as well as eight books. Two of these monographs — Constructive Analysis (1985, with Errett Bishop), and Varieties of Constructive Mathematics (1987, with Fred Richman) — are regarded as indispensable references for those working in the field of constructive analysis. A third, Techniques of Constructive Analysis (Springer, 2006), written with Vîţă, is the only book to highlight the developments in constructive analysis over the preceding 20 years.

Bridges’ work has seen him dubbed the successor to American mathematician Errett Bishop, the father of constructive analysis. It was Bishop’s 1967 book, Foundations of Constructive Analysis, which got Bridges into constructive mathematics when he was a graduate student at the University of Newcastle-upon-Tyne.

“I was studying operator algebra theory when I came across Bishop’s book, and when I read it, I was gripped by it. I’ve been interested in constructive mathematics ever since — but I wouldn’t call myself Bishop’s successor, more an acolyte who, together with others in various countries, has been keeping Bishop’s ideas alive.”

A graduate of Edinburgh, Newcastle and Oxford universities in the United Kingdom, Bridges has received numerous accolades both nationally and internationally. He was appointed a Fellow of the Royal Society of New Zealand in 2000 and, four years later, was elected a Corresponding Fellow of the Royal Society of Edinburgh, one of only 61 such fellows worldwide. In 2006 an international conference was held in Bavaria in his honour and further tribute was paid through a special edition of the Journal of Universal Computer Science. However, perhaps the strongest international endorsement of Bridges’ work came in 2000 when he was awarded a higher doctorate, a Doctor of Science, from the University of Oxford.

The latest accolade came in 2011, when he was awarded the University of Canterbury Research Medal during the December graduation ceremonies. The medal is awarded annually and is one of the University’s highest honours, recognising academic staff who have made an outstanding contribution to academic and scholarly research.

In support of Bridges’ nomination for the medal, one of the referees, fellow mathematician Professor Michael Rathjen from Leeds University in the United Kingdom, said that the medal would be “a very well-deserved tribute to this outstanding mathematician and most eminent authority in the world of constructive mathematics”.