One of my own favorite themes for over a decade-and-a-half now is that, as an academic community, we mathematicians tend to be too insular--an insularity that I think is as harmful to mathematics as to others.
Before I give a specific example--my own pet example to do with my own pet research project (and if a columnist is going to be self-indulgent there is no sense in doing it by half measures)--I should make it clear that I am not saying that there is anything wrong with some mathematicians, indeed many mathematicians, working exclusively on pure mathematics. Nor am I arguing that we should all adopt "goal-oriented research" of the kind demanded by legislators ignorant of the way that almost all the major advances of mathematics and science are made. Nor am I saying that our colleagues in other disciplines should dictate what is taught in our college and university mathematics courses, or how it is taught.
What I am saying is that mathematics as a whole should be prepared to embrace as a useful and bona fide part of the enterprise, collaborative investigations involving mathematicians together with experts in other disciplines. What is more, I am also saying that such collaborations should not be restricted to mathematicians peddling their wares on their own terms, or entering a collaboration with the intention of simply "doing the math parts" or providing expertise in the application of established mathematical techniques. Again, there is a place for such kinds of collaboration, and I think that by and large the mathematical community has always been very good at that kind of thing. But such collaborations are essentially risk-free for the mathematician (and often pay dividends in the form of funding). The mathematician engaged in such a collaboration might have to become familiar with part of another discipline, but in the final analysis what the mathematician does is what he or she is familiar with: mathematics. What is missing, I would claim--or at least, what is relatively rare--is for the mathematician to make a full-blown attempt to meet his or her collaborator(s) on the no-mans-land between their disciplines. To do something that is not (yet?) regarded as 'mathematics'.
In such a collaboration, the mathematician brings his or her experience in mathematics into the project, but the work done is probably not going to constitute mathematics in any currently recognizable sense, and is unlikely to lead to publications in mathematical journals. Let me give you an example of what I have in mind.
It seems to me that the world of information management is an area where significant progress will not be made until there is a radical new mix of mathematics and other disciplines (notably psychology, sociology, and linguistics). What do I mean by information management?
A modern company has many assets that have to be properly managed. The most obvious--at least from a traditional perspective--are physical plant, personnel, and the company's financial assets. Management of each of those assets requires different kinds of expertise. The appropriate expertise may be provided by the company's regular employees, or it may be outsourced, either in the form of subcontracting or by the use of expert consultants.
In today's commercial environment, information is another asset that requires proper management. Of course, information has always been important to any organization. But it is only within the last fifty years or so that it has become a clearly identifiable asset that requires proper management. The reason for the growing importance of information within the organization is the growth of computer and communications technologies, and the increasing size and complexity of organizations that has in large part been facilitated by those technologies. It is both a cliche and a fact that information is the glue that holds together most of today's organizations.
Actually, that last metaphor is often apt in a negative way. In many cases, information acts as the glue that causes things to stick fast when it should be the oil that keeps the wheels turning. A familiar scenario in the industrial world of the late twentieth century is for a company to introduce a new computer system to improve its information management, only to discover that, far from making things better and more efficient, the new system causes an array of problems that had never arisen with the old way of doing things. The shining new system provides vastly more information than was previously available, but it is somehow of the wrong kind, or presented in the wrong form, at the wrong time, or delivered to the wrong person, or there is simply too much of it for anyone to be able to use. What used to be a simple request for information to one person over the phone becomes a tortuous battle with a seemingly uncooperative computer system that can take hours or even days, eventually drawing in a whole team of people.
Why does this happen? The answer is that, for all that the newspapers tell us we are living in the Information Age, what we have is an information technology, or rather a collection of information technologies. We do not yet have the understanding or the skill to properly design or manage the information flow that our technologies make possible. In fact, it is often worse. In many cases, companies are not even aware that they need such skill. Faced with the persuasive marketing of ever-more powerful and glitzy computer systems, there is a great temptation to go for the 'technological fix'. If the present information system is causing problems, get a bigger, better, faster system. This approach is like saying that the key to Los Angeles' traffic problem is to build even more, and still bigger, roads.
The solution? Just as the company has experts to manage its other assets, so too it needs experts to manage its information assets. Alongside the lawyers who handle and advise on the company's contracts and the accountants who handle and advise on the company's financial assets, should be the 'information scientists' who handle and advise on the company's information assets.
But there is one problem. There are, at present, no such 'information scientists'. The world of information flow does not yet have the equivalent of a lawyer or an accountant. There is not even an established body of knowledge that can be used to train such people. To become a lawyer, you go to law school and follow a well-established educational path. To become an accountant, you learn about the various principles and theories of accounting and finance. But there is no established 'information science' (in the sense I am using this term).
It is, I would suggest, only a matter of time before such a science develops. The real question, I think, is what will that science look like? My guess is it will not resemble any of today's sciences.
It may be presumptuous of me, as a mathematician, to say that I think mathematics will play a role in the development of this new 'science of information'. On the other hand, I suspect it will not look very much like anything we see in today's mathematics books. I can't see it coming from the highly constrained cross-disciplinary research collaborations I described at the start of this column. What is needed, it seems to me, are radical, new interdisciplinary efforts, where mathematicians leave behind practically all the mathematics they know, taking with them only their mathematical skill and a pair of mathematician's eyes.
Given the constraints on our careers, I doubt that many of us in the profession already could take the necessary steps. But we can help to prepare some members of the next generation of mathematicians to do so. They'll do it anyway, of course. New generations always march boldly onto the ground their predecessors thought inaccessible. But if we help them, they'll do it faster. And we'll all benefit from that. So too will the discipline we call mathematics--though we will change the borders of mathematics in the process.
Devlin's Angle is updated at the start of each month.