How does mathematics influence the society?

The Mathematics-Society System

This approach to looking at mathematics enters not at the level of mathematical knowledge but at the level of the social systems in which that knowledge is created and applied. The social system of science refers to patters of employment, funding, communication, training, authority, decision making and so forth. The aim here is to look at the way systems of production and application of mathematics relate to social interests. To do this I select out some salient features of the social systems associated with mathematical expertise.

Sources of patronage

Most of the money for mathematics research -- which is largely for salaries, but also for offices, libraries, computing and travel -- comes from governments and large corporations. The source of funding inevitable has an influence on the areas of mathematics studied and the types of mathematical applications undertaken. As argued by Hodgkin (1976), much of the stimulus for work in computational mathematics also comes from actual or potential military applications.

At the detailed level of application, the formulation of mathematical problems is strongly influenced by funding and opportunities for application. In manufacturing industry, mathematical problems grow out of the need to cut costs, improve technologies or control labour. A mathematical model for the rapid cooling of a metal bar without cracking is tied to an immediate problem. The mathematics of light transmission in optical fibres is driven by interest in application in telecommunications. The number of examples is endless.

What happens in many cases is that a practical problem, such as modelling air pollution dispersion or the trajectories of missiles, leads to a more esoteric mathematical project in numerical analysis or differential equations. The applications, and thus the funding, in these cases have an indirect influence on the type of mathematical problems studied and thought to be 'interesting'. That particular types of parabolic partial differential equations become whole fields of study in themselves is not due simply to some abstract mathematical significance of these equations, but to their significance in practical applications, even if at several stages removed.

Professionalisation

Today, most mathematicians -- taking a mathematician to be a person who creates or applies mathematical knowledge at a high level -- are full-time professionals, working for universities, corporations or governments. There are few amateurs, nor do many mathematicians work for trade unions, as farmers, in churches, or as freelancers. Mathematics, like the rest of science, has been professionalised and bureaucratised. The social organisation of mathematics influences the ways that ambitious mathematicians can pursue fame and fortune (Collins & Restivo1983)

Mathematicians have a vested interest in their salaries, their conditions of work, their occupational status and their self-image as professionals. Their preferences for types and styles of mathematics are influenced by these factors.

Judith Grabiner (1974) argues that there have been 'revolutions in thought which changed mathematicians' views about the nature of mathematical truth, and about what could or should be proved'. Grabiner examines one particular revolution, the switch from the1700s when the main aim of mathematicians was to obtain results to the1800s when mathematical rigour became very important. Of the various reasons for this which Grabiner canvasses, one is worth noting here. Only since the beginning of the1800s have the majority of mathematicians made their living by teaching. Rather than just obtaining mathematical results for applications or to impress patrons, teachers need to provide a systematic basis for the subject, to aid students but also to establish a suitable basis for demarcating the profession and excluding self-taught competitors from jobs. This is an example of how the social organisation of the profession of mathematics can affect views about the nature of mathematical truth.

Gert Schubring (1981) has argued that in the professionalisation of mathematics in Prussia in the early1800s, the 'meta-conception' of pure mathematics played an important role. By defining 'mathematics' as separate from externally defined objectives, the mathematicians oriented the discipline to internal values that they could control. To do this, support from the state had to be available first. Given state patronage for academic positions, the mathematicians could proceed to establish a discipline by establishing training which channelled students into the new professional orientation, reducing the number of self-taught mathematicians obtaining jobs in the field and socialising students into the meta-conception of pure mathematics. This account meshes nicely with that of Grabiner.

This process continues today. Especially in universities, the home grounds of pure mathematics, mathematicians stake their claims to autonomy and resources on their exclusive rights, as experts, to judge research in mathematics. This is no different from the claims of many other disciplines and professions (Larson1977). The point is that if mathematicians emphasised application as their primary value, their claims to status and social resources would be dependent on the value of the application. The conception of 'pure' mathematics enables an exclusive claim to control over the discipline to be made.

Herbert Mehrtens (1987, p.160) develops the thesis that 'a scientific discipline exchanges its knowledge products plus political loyalty in return for material resources plus social legitimacy'. He shows how German mathematicians in the1930s were able to accommodate the imperatives of the Nazis, especially by providing useful tools to the state. The adaptability of the German mathematics community grew out of its social differentiation, specifically the different functions of teaching, pure research and applied research. Mehrtens' study provides an excellent model for analysing the interactive dynamics of the two factors of patronage and the structure of the profession.

Male domination

Most mathematicians are men, and mathematics like the rest of natural science is seen as masculine: a subject for those who are rational, emotionally detached, instrumental and competitive. Mathematicians are commonly thought, especially by themselves, to have an innate aptitude for mathematics, and claims continue to be made that males are biologically more capable of mathematical thought than females. The teaching of pure mathematics as concepts and techniques separated from human concerns, plus the male-dominated atmosphere of most mathematics research groups, make a career in mathematics less attractive for those more oriented to immediate human concerns, especially women.

Male domination of mathematics is linked with male domination of the dominant social institutions with which professional mathematical work is tied, most notably the state and the economic system, through state and corporate funding and through professional and personal contacts (Bowling & Martin1985).

The high status of mathematics as a discipline may be attributed in part to its image as a masculine area. Mathematical models gain added credibility through the image of mathematics as rational and objective -- characteristics associated with masculinity -- as opposed to models of reality that are seen as subjective and value-laden.

Specialisation

There are various ways in which mathematicians shape and use their expert knowledge to promote their interests vis-à-vis other social groups. If mathematical knowledge was too easy to understand by others -- both non-mathematicians and other mathematicians -- the claims by mathematicians for social resources and privilege would be harder to sustain. Specialisation enables enclaves of expertise to be established, preventing scrutiny by outsiders. In applications work, specialisation ensures that only particular groups are served. In all cases, specialisation plus devices such as jargon prevent ready oversight by anybody other than other specialists. Since hiring professionals to understand specialist bodies of knowledge can be afforded on a large scale only by governments and large corporations, specialisation serves their interests more than those of the disabled or the unemployed, for example.

The role of these factors is particularly obvious in mathematical modelling. A mathematical model may be a set of equations, which is thought to correspond to certain aspects of reality. For example, most of theoretical physics, such as elementary theory for projectiles or springs, can be considered to consist of mathematical models. In most parts of physics the models are considered well established, and physicists work by manipulating or adapting the existing models. But in other areas the choice of models is open. Various parts of reality may be chosen as significant, and various mathematical tools may be brought to bear in the modelling process.

Many people who have been involved in mathematical modelling will realise the great opportunities for building the values of the modeller into the model. I have seen this process at work in a variety of areas, including mathematical ecology, game theory, stratospheric chemistry and dynamics, voting theory, wind power and econometrics.

A good example is the systems of difference equations used in the early1970s to determine the 'limits to growth'. The choice of equations and parameters more or less ensured that global instability would result (Cole et al.1973). When different assumptions were used by different modellers, different results -- for example, that promotion of global social equality would prevent global breakdown -- were obtained, nicely compatible with the values of the modellers. Another example is the values built into global energy projections developed at the International Institute for Applied Systems Analysis (Keepin & Wynne1984).

Mathematical models are socially significant in two principal ways: as practical applications of mathematics and as legitimations of policies or practices. Most models are closely tied to practical applications, such as in industry. The narrow specialisation involved in the modelling ensures that few other than those developing or funding the application would be interested in or capable of using the model. This sort of applied mathematics is closely linked to the social interests making the specific application. Whether the application is telecommunications satellites, anti-personnel weapons or solar house design, one may judge the mathematics by the same criteria used to judge that application. It is not adequate to say that the killer is guilty while the murder weapon is innocent, for in these sorts of applications the mathematical 'weapon' is especially tailored for its job. Certainly applied mathematicians cannot escape responsibility for their work by referring to 'neutral tools', whether this refers to their mathematical constructions or to themselves.

Models serving as legitimations are involved in a more complicated dynamic. In many cases such as limits-to-growth studies the models do no more than mathematicise a conclusion which would be obvious without the model. But the models are seen as important precisely because they are mathematical, thus drawing on the image of mathematics as objective. A mathematics-based claim also has the advantage of being the work of professionals. Anyone can make a claim, but if a scientist does so, relying on the allegedly objective tools of mathematics, that is much more influential. Although exercises in mathematical modelling are often shot through with biases, for public consumption this often is overlooked; the modellers draw on an aura of objectivity which is sustained by the more esoteric researches of pure mathematicians.

What then of pure mathematics? There are two major ways in which a link to social interests can be made. First is potential applications. These are not always easy to assess, but a good guess often can be obtained by looking at actual applications in the same or related specialities. If any new application turns up, it is likely to be in the same areas and to be used by the same groups.

It is a debatable point whether mathematics should ever be evaluated separately from applications. Arguably, the study of nature is the primary motivation for the development of and importance of mathematics, and the 'correctness' of pure mathematics should be judged by its ultimate applicability to the physical world (Kline1959,1980). The primary reason for the ascension of pure mathematics, namely mathematics which is isolated from application, is the social system of modern science.

This system -- including funding, professionalization, male domination and specialization -- in which claims to sole authority over areas of knowledge are used to claim social resources, is the second way that pure mathematics is connected with social interests. Even if some bit of pure mathematical research turns out to have no application, it is still usually the case that social resources have been expended to support professional workers who are mostly male and who produce intellectual results of interest only to a handful of others like themselves. Furthermore, the work of pure mathematicians, and indeed their very existence, helps legitimate the claims of mathematics to objectivity.