Mathematical Modelling in the Two Cultures

Models, mostly based on mathematics of one kind or another, are used everywhere to help organizations make decisions about their design, policies, investment, and operations. They are indispensable.

But if modelling is such a great idea, and such a great help, why do so many things go wrong? Well, there’s good modelling and less good modelling. And it’s hard for the consumers of models — in companies, the Civil Service, government agencies — to know when they’re getting the good stuff. Worse, there’s a lot of comment and advice out there which at best doesn’t help, and perhaps makes things worse.

In 1959, the celebrated scientist and novelist C. P. Snow delivered the Rede Lecture on ‘The Two Cultures’. Snow later published a book developing the ideas as ‘The Two Cultures and the Scientific Revolution’.

A famous passage from Snow’s lecture is the following (it can be found in Wikipedia):

‘A good many times I have been present at gatherings of people who, by the standards of the traditional culture, are thought highly educated and who have with considerable gusto been expressing their incredulity at the illiteracy of scientists. Once or twice I have been provoked and have asked the company how many of them could describe the Second Law of Thermodynamics. The response was cold: it was also negative. Yet I was asking something which is the scientific equivalent of: Have you read a work of Shakespeare’s?

‘I now believe that if I had asked an even simpler question — such as, What do you mean by mass, or acceleration, which is the scientific equivalent of saying, Can you read? — not more than one in ten of the highly educated would have felt that I was speaking the same language. So the great edifice of modern physics goes up, and the majority of the cleverest people in the western world have about as much insight into it as their neolithic ancestors would have had.’

Over the decades since, society has come to depend upon mathematics, and on mathematical models in particular, to a very great extent. Alas, the mathematical sophistication of the great majority of consumers of models has not really improved. Perhaps it has even deteriorated.

So, as mathematicians and modellers, we need to make things work. The starting point for good modelling is communication with the client.

Good modelling is partly an art, a craft skill that requires talented mathematicians to be done well. But there are good guiding principles. Here are some important ones (with apologies to Albert Einstein and Alfred Korzybski).

  • The modeller and the client should have a common understanding of what they are trying to achieve. What are the questions to be answered? What parts of the system are relevant? Often such an understanding is best achieved by an iterative process constructing a sequence of models, each new model being informed by the observations and challenges of the client on the previous model.
  • Models should be as simple as is necessary for their intended purpose and no simpler (Einstein).
  • The modeller and the client should gather as much evidence as possible to support the assumptions used in the model.
  • Where the model requires parameters for which the available evidence is weak, the range of possibilities should be thoroughly explored experimentally.
  • It’s a good idea to build models compositionally; that is, build up larger, more complex models from smaller, simpler ones.
  • The map is not the territory (Korzybski): models do not contain all of the detail found in the systems they represent.
  • Many models are better than one. Models built using different mathematical approaches can reveal strengths and weaknesses in one another.
  • It can be better to produce a crude model quickly than a detailed model more slowly, if the results are needed quickly.
  • Large amounts of data don’t necessarily lead to good models. The provenance and structure of the data must be understood.
  • The context in which a model was constructed must be remembered: the model won’t necessarily be applicable on other contexts.

Here at UCL we’ve been working on mathematical modelling for computer and other information systems for some while. We’re particularly interested in modelling how security policies fit together with the systems — including the people — to which they’re supposed to apply. We’ve done a lot of theoretical work — involving logic, discrete mathematics, probability, and economics — and we’ve worked with a lot of managers and engineers.

Recently, we’ve written an article for a magazine called IEEE Security and Privacy for a forthcoming issue about the economics of security. It’s intended to be accessible. It’s not completely free of mathematics, but the mathematics is minimal, kind of like knowing about mass and acceleration. And we use a lot of pictures to explain things. We know it doesn’t go far enough, but both cultures need to work on communication.

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