Soft Mathematics:
The Mathematics of People

by Keith Devlin

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This year is the 400th anniversary of the birth of Rene Descartes. To future historians, this might well be seen as the period when the Cartesian domination of science came to an end. Having long underpinned Mankind's attempts to comprehend the physical world, mathematics has failed to achieve comparable success in our attempts to understand the human world of people and minds. The failure of mathematics to dominate these areas is not a failure of mathematics as an enterprise. It is only a failure if your goal is to develop a 'science' of mind or society analogous to, say, 'physics' as a science of matter and the physical universe. There are alternatives.

Long used to a position of domination (it has been called the 'Queen of the Sciences'), mathematics now has to adjust to being just one of a number of ways of understanding how minds work, how people communicate, and how societies function. The blending of mathematics with other disciplines that is increasingly being adopted in the study of people and their actions is giving rise to a fascinating new way of using mathematics - what I have elsewhere referred to as 'soft mathematics'.

Soft mathematics is not the same as 'applied mathematics'. Nor is it the same as 'mathematical modeling'. Mathematics can indeed be 'applied' to the study of human activities (often with terrible results). And mathematical models can be useful in the study of human activities. But soft mathematics is something quite different - a genuine attempt to blend mathematics with other approaches.

The legacy of Descartes

The modern 'scientific method', based on observation, mathematical measurement and description, and logical analysis, is largely the product of three individuals: Galileo Galilei, Francis Bacon, and Rene Descartes. In the words of Galileo, "The great book of nature can be read only by those who know the language in which it is written, and this language is mathematics." In a similar vein, Descartes wrote that he "neither allows for nor hopes for principles in physics other than those that lie hidden in geometry or in abstract mathematics, for in this way all phenomena of nature will yield to explanation, and a deduction of them can be given."

For Galileo and Bacon, the role of the scientist was focused on measurement and the discovery of descriptive, quantitative formulas, rather than the formulation of causal explanations obtained by philosophical reflection, which had been typical of earlier work in 'science'. In many ways, Galileo and Bacon were each early forerunners of the 'no-nonsense, down-to-earth, practical scientist' of the twentieth century.

Descartes' major essay on the scientific method, Discourse on the method of properly guiding the reason in the search of truth in the sciences, was published in 1637. In essence, Descartes' 'method' consists of (1) accepting only what is clear in one's own mind, beyond any doubt, (2) splitting big problems into smaller ones, (3) arguing from the simple to the complex, and (4) checking when one is done.

Like Plato and Aristotle before him, Descartes believed that his method, the method of science and mathematics, could be applied to the inner world of the mind as well as to the outer world of the physical universe. Four hundred years later, after decades of failures in artificial intelligence and mathematical linguistics, it would finally be realized that, in this belief, Descartes had been wrong.

It's a mathematical world - or is it?

Mathematics, as everybody knows, lies at the heart of the physical and life sciences - physics, chemistry, and biology. More recently, mathematics led to, and again stands at the heart of, computer science and the closely related new information sciences.

But what of the non physical world, the world of human thoughts and human actions? What is the role of mathematics in these areas? These are the areas in which the so-called 'soft' sciences have sprung up-psychology, sociology, linguistics, management science, political science, and economics. Since the very phrase 'hard science' is more or less synonymous with 'mathematically-based science', it seems to be an accepted 'given' that these newer, social sciences, are intrinsically non mathematical. (economics and linguistics may be partial exceptions.)

Of course, social scientists of all kinds frequently use statistical techniques to collect the data for their study. There is no question that practically all sciences, soft or hard, rely on mathematics in one way or another when it comes to data collection. The issue is, do the newer, social sciences make genuine use of mathematics when it comes to analyzing the data, to formulating and testing hypotheses, and to formulating and evaluating theories? To put it bluntly, could any one of these newer sciences be called a 'mathematical science?'

The answer is no. Enter soft mathematics.

Soft mathematics

The idea behind soft mathematics is to blend the use of mathematics - often little more than a few mathematical objects, the use of algebraic notation, the occasional definition - with other descriptive and analytic techniques, techniques from the various social sciences. Such an approach is becoming increasingly common in economics, management science, and political science.

A recent illustration of soft mathematics in action is provided by the new book Language at Work, by myself (Keith Devlin) and Duska Rosenberg. In this book, Rosenberg and I study the way that different contexts affect the way a particular kind of document is understood by different people in a large company. The manager in the office, the expert at the front desk, and the engineer in the field, are all working in a different context, and they all understood the document in ways that are sometimes the same and sometimes different. They then use the information they acquire in order to make decisions and adopt various courses of action, also within their own context.

Our analysis begins with the observation that the contexts are important. And so we denote them by algebraic symbols, s, u, e, etc. Never mind that it would clearly be impossible to specify these contexts in any precise way. By denoting the contexts by symbols, we are doing two things that are important for the analysis. First, we are making a definite statement that the contexts are significant features in our analysis. (The significance of context might seem obvious. But many, many attempts to develop computer systems have come to grief because the designer failed to take account of the context of use.) Second, we are providing ourselves with the facility to manipulate and use those contexts in our analysis without getting bogged down in their specifics. (Physicists and economists do this kind of thing all the time, of course.)

Our goal is to analyze the way the documents convey information around the company. We want our analysis to have enough mathematical precision to be of use to an engineer trying to design a computer support system and yet is flexible enough to reflect the subtle observations that would result from a more traditional social science investigation. Since we have no way to decide on the balance between the sparse mathematical formality and the sociological richness (and fuzziness), we let the data make the decision for us.

LFZ analysis

To carry out our analysis, we developed an analytic technique called 'layered formalism and zooming' (LFZ analysis). In a nutshell, an LFZ analysis of a given body of communicative data proceeds like this. You start by making an initial analysis of the data in a fashion that is largely non mathematical, but which makes use of mathematical formalisms. (The formalism we used for our work is situation theory, a new branch of mathematics developed in the early 1980s, and discussed in my book Logic and Information.) Then, you subject that initial analysis to a process of stepwise refinement and increased formalism. Whenever a problem is encountered, you increase the mathematical precision as it applies to the problem area. That is, you use mathematics to 'zoom in' and examine the problem in detail. When the problem has been resolved, you zoom out again.

At each step of the refinement process, you adopt the minimal possible level of formalism and the minimal possible level of precision, thereby minimizing the likelihood of any inadvertent alteration to the data under consideration. (Unintentionally changing the data is a common problem with mathematical analyses.) For the same reason, the analysis is checked against the data after each stage in the analysis refinement cycle. As a result the balance between the mathematical and the sociological aspects of our analysis is determined not by us as analysts but by the data we are working with.

In short, we use the process of formalization as an analytic technique. The aim is not to produce a formal theory. Indeed, there can be so many symbols floating around that denote decidedly 'soft' entities (such as contexts), it would take a lifetime to come close to anything that might resemble a 'formal system' in the mathematician's usual sense. Just as the mathematician R. W. Hamming once said of computing that

The purpose of computing is insight, not numbers
so too we say of our use of the LFZ technique,
The purpose of formalization is insight, not a formal theory.

Publication notes

A longer version of this article is available on request from the JPBM or directly from the author.

The book Language at Work, by Keith Devlin and Duska Rosenberg, will be published in late spring, 1996, by Cambridge University Press and CSLI Publications, Stanford University.

In fall 1996, John Wiley and Sons will publish Keith Devlin's Goodbye Descartes, a general audience trade book that describes the entire history of Mankind's quest for a mathematical science of human reasoning and communication.

Dr Keith Devlin is Dean of Science and Professor of Mathematics at Saint Mary's College of California, in Moraga, California, a Senior Researcher at Stanford University's Center for the Study of Language and Information (CSLI), and a Consulting Research Professor in the Department of Information Science at the University of Pittsburgh.

He is the author of over sixty research articles and numerous books, including Mathematics: The Science of Patterns, a Scientific American Library book published by W. H. Freeman in 1994, and Logic and Information, published by Cambridge University Press in 1991. He is also the editor of FOCUS, the newsletter of the Mathematical Association of America.

Keith Devlin, School of Science, Saint Mary's College of California, Moraga, CA 94575.

Phone: (510) 631 4409; Fax: (510) 631 7961; e-mail:


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