Mathematics, Quantification and Social Change
Review of Sal Restivo, Mathematics in Society and History, Kluwer Academic Publishers: Dordrecht 1992, and Theodore M. Porter, Trust in Numbers, Princeton University Press, Princeton, 1995.
Mathematics and number-worlds are very influential in our daily lifes. We all have to learn some mathematics at particular moments in time, so we can buy chicken at fixed prices per kilo, work with grade point averages, organize schedules, understand policies established for noise levels at work places, etc. The use of numbers in the world around us seems so obvious; numbers seem so logical in the way they are established as well as in the way they function that its use doesn’t need any explanation or further exploration. Do we ever doubt that 2+2=4? Do we ever check whether a liter of milk bought in the supermarket actually contains one liter? The truth of mathematical representations seems to leave hardly any room for the study of the relation between mathematics and society. This review will show that the opposite is the case. Sal Restivo and Theodore Porter even have opposed views on the relation between mathematics, quantification and social change.
Restivo’s and Porter’s books on mathematics and quantification are very different in focus, argumentation and style. Restivo is the sociologist, Porter an historian of science. Restivo studies the development of mathematics as a discipline. Porter studies the role numbers play in bureaucratic, political environments and in scientific communities working under pressures from the outside world. Both tell big stories; taken together they ambitiously grasp the whole world except Africa and Australia. Restivo and Porter share a perspective on the making of knowledge: both stress that they are somewhere in the middle between the realists and the relativists. By starting with the statement that scientific knowledge (about objects and processes) is a social process through and through, they both stress that this does not mean that the knowledge is random and completely arbitrary. Knowledge can’t be made whichever way scientists would like. Having said this, both focus on the question as to which social processes create knowledge.
David Bloor was one of the first in STS to open up mathematics to social enquiry. He used the mathematical discipline perceived as the ultimate producer of ‘truths’ to show the possibilities of the ‘strong programme’ (Knowledge and Social Imagery, London: Routledge 1976). Bloor situated the locus of sociological enquiry in the selection of possible mathematical orderings of (physical) objects. This implied some form of ‘alternative mathematics’ which Bloor exemplified by comparing Greek ideas about ‘number’ with our own. Oswald Spengler, author of Decline of the West (first published in 1918), spent one chapter on the Meaning of Number which apparently inspired Bloor to think about an alternative mathematics (p95).
Restivo presents his book as ‘the first by a sociologist fully devoted to a sociology of mathematics.’ Restivo has also been inspired by the writings of Spengler on whom he spends his first pages. Spengler has been cited as an important though neglected source of inspiration for sociologists by Randall Collins and Restivo in the “Development, Diversity, and Conflict in the Sociology of Science” (The Sociological Quarterly, Spring 1983, pp185-200): “Oswald Spengler did treat science and mathematics as culturally related outgrowths of the historical ethos (“soul”) of each civilization. But he had little influence on the emerging sociology of knowledge, in part because he did not link ideas to specific social classes or institutions” (p187). In his book Restivo tries to rehabilitate Spengler by exploring his thesis ‘that there are as many number-worlds as there are cultures’. To many (social) scientists mathematics is a unique mode of knowing that has produced universal orderings. Spengler, however, attacked this privileged status of mathematics as an intellectual or scholarly discipline:
‘We find an Indian, an Arabian, a Classical, a Western type of mathematical thought and, corresponding with each, a type of number - each type fundamentally peculiar and unique, an expression of a specific world-feeling, s symbol having a specific validity which is even capable of scientific definition, a principle of ordering the Become which reflects the central essence of one and only one soul, viz., the soul of that particular Culture. Consequently, there are more mathematics than one.’ (Spengler, Cited in Restivo, p8)
In part II of his book, Restivo works out what he calls the weak interpretation of Spengler, i.e., the consideration that mathematics is a social and cultural phenomenon. He aims at developing a sociology of mathematics: ‘Numbers are social through and through.’ In my opinion he is most successful in chapter 7, ‘Conflict, Social Change, and Mathematics in Europe’, which is both consistent and humorous.
“The ideals or norms of science do not cause scientific behaviour, but emerge from the struggle for individual success under different conditions of competition.” This is Restivo’s structuring argument in writing his history of the emergence of a European mathematical discipline. The controversy between two Italian mathematicians, Cardan and Tartaglia (in the first half of the sixteenth century), reflects a transition from a situation in which solutions to mathematical puzzles were kept secret to one in which it was normal to share intellectual properties. Cardan’s advantage was the result of his decision to publish his solution to cubic equations. The contest between Cardan and Tartaglia was settled by a mathematical duel, which was a traditional way to earn your money among sixteenth-century mathematicians. Cardan came out as the winner because Tartaglia had withdrawn.
The next controversy Restivo goes into is the one between Leibniz and Newton, a priority-dispute about the infinitesimals (second half of the seventeenth century). This controversy, Restivo claims, reflects the shift from informal message centers, where knowledge was personally delivered from by one scholar to another, to organisations (e.g., the Royal Society) with printed knowledge in the form of journals. ‘Leibniz must rank as one of the most successful organisation builders in the history of science’ (p71).
The chapter ends with a critique on Merton and Kuhn in favour of his own argument. “In no case do we find a mathematical change centered in a struggle between rival traditionalists and innovators. Moreover, the long-term trend in Western mathematics has not been towards a single, dominant paradigm, but rather towards rival schools at odds over fundamental questions about methods and knowledge” (p85-86). In the case of mathematics the state of the discipline resembles more the social sciences than Kuhn’s image of normal science. Therefore, the dynamics of mathematics can best be analysed in terms of the changes in organisational forms.
Another very nice piece in Restivo’s book is his attempt to develop a sociology of 2+2=4. This ‘representation’ seems to be the exponent of the kind of universal truths mathematics produces. He starts with citing some mathematical realists, one of whom claimed that 2+2=4 accurately described the encounter of two dinosaurs with two others, although no one has ever observed this event. There cannot be any culture- bound answer to the question of 2+2. It’s always 4. Considering these attributions of truth to this expression, it is remarkable that within the mathematical community Whitehead and Russel (who are the exponents of the relation between mathematics and logic) wrote a lengthy book of 800 pages to prove the truth of 2+2=4. How true actually is 2+2=4? Restivo argues that there must be more at stake. For him, the fundamental question here is what these kind of number-truths do in social relations. “It is not just that something is or is not logical in some absolute sense. It is that logic - and certainty relations in general - are cultural resources that can be used to defend or attack a social order by affirming or denying self-evident statements” (p114). To rephrase this we could say that these kind of mathematical truths function as resources of power.
Restivo goes on by empirically problematizing 2+2=4. “Adding is in general empirically problematic.” What happens when we add two cups of rice and two cups of water? The ones absorb the others. This example may look like a misunderstanding, referring to the general truth that we can’t add apples and oranges. However, whenever we add we make abstractions and generalizations. If we add two books we could ask questions about the numbers of pages or the authors. When we chose to add two books with the same number of pages, we could ask questions about the sort of paper the book is printed on, etc. To rephrase it in constructivist terms, adding implies similarity, and similarity is problematic.
One of Restivo’s main claims is that the competitive structures of the mathematical discipline forced mathematics into an ever ongoing process of abstraction and generalisation. The more mathematics developed, the more mathematics itself formed the basis for new mathematical knowledge.
The idea that mathematical representations are made into certainties by people outside the mathematical community in the first place raises a question which happens to be central in Porter’s book, namely “How are we to account for the prestige and power of quantitative methods in the modern world?” (pviii). Porter is an excellent story-teller; his book consists of stories of accountants in the US, engineers in France, statistical bureaus, cost-benefit analysis and many smaller histories like the development of the metric system. All the stories are about knowledge developed by experts under more or less bureaucratic and political control. The moral of these stories is, to put it bluntly, that quantification supersedes elitism. The less experts derive their authority from the institutional context the more their work is organized along strict rules (which Porter calls mechanical objectivity) in order to produce controllable numbers.
The stories carry one main lesson, and serve to illustrate one main point and one main answer: numbers are technologies of trust. The development of capitalism and the emergence of national states with one central government resulted in a replacement of face-to-face relations by relations at a distance in the domains of trade and politics. In a period in which personal trust could no longer be the basis for negotiation, numbers were given this role. The amount of trust attributed to and built in numbers is however a matter of degree. Porter distinguishes France and England from the United States on the question where experts derive their authority from. In France, experts were educated at the cole Polytechnique which was very close to the French state bureaucracy. Bureaucrats and politicians imputed the value of experts’ personal judgments to their Bildung; discretion was highly appreciated. In England, it was not so much the education as the ‘mobility’ of the experts through bureaucracy which had given them their authority. Experience in different domains - instead of specialist experience only - resulted in trustful personal judgments of experts. Democracy in the US resulted in a need for impersonal judgments; expert discretion could always be disputed in court. This provided an impetus (not always to the expert’s pleasure) to quantify their knowledge. Porter ascribes to numbers a kind of objectivity which can be characterised as ‘impersonal knowledge’, the exclusion of subjectivity. Therefore, in Porter’s opinion, there is a relation between quantification and democracy, and the expert Bildung and technocracy. “The pursuit of rigor flourishes mainly in conjunction with democracy. … The regime of calculation involves a bid to empower experts who have at most a limited ability to subvert democratic control. Technocracy presupposes relativily secure elites” (p146).
It seems to me that Porter is arguing that quantification tends to break down elite cultures. He mentions quantifrenia in the bureaucratic management of diversity (p76). Quantification and impersonal knowledge go hand in hand; not the blacks themselves but the figures tell us about their oppression. Porter considers quantification a liberating and emancipating force, supporting social change. Restivo, makes a completely different plea. “The realm of the ‘logical’, ‘rational’, ‘scientific’, ‘objective’, and ‘quantitative’ is, among other things, a realm of ideas that symbolize the reigning social order, and inevitably become targets of opponents of that order” (p135).
While Porter claims that striving for a specific sort of objectivity and a changing society went hand in hand, Restivo claims, as we have seen, that objectivity and a stable order go hand in hand. These opposed views seems to suggest that the relation between quantification and social change/stability is a dynamic one. In the late nineteenth century there was a great concern about the extreme poverty of the working class. This led to the idea of measuring nation’s national incomes to be able, among other things, to gain insight into the distribution of this national income. In the Netherlands, for example, it was the well-known socialist W.A. Bonger who tried to establish a national income figure. In the 1920s, this led to the phenomenon of the ‘budget statistics’ that required a huge system of measurement. We could argue, at this point, that the quantification of nation’s wealth was at least partly an attempt to show how small laborers could share this wealth. Although the poverty of the working class must have been very obvious in terms of bad clothing and housing, a quantified proof of this poverty made the problem ‘objective’. One could no longer deny the problem. Poverty was redefined as a problem of the distribution of national wealth. This part of the history supports Porter.
After the 1940s, these budget statistics were one of the roots in the later development of national accounts and macro- econometric models in the Netherlands. During the second half of this century, government increasingly used economic data in the interpretation of economic events and in the preparation of economic policies. These numbers became part of a consensus on how to think about economic life of the established bureaucracy. One consequence was that left-wing parties and progressive movements had more and more problems getting their different views accepted as at least a possible viewpoint on the economy. All the other parties accepted the economic data produced which increased the ‘objectivity’ of these numbers. As Restivo argues, “In general, the wider and more diffuse the social interests embodied in a representation, the more it qualifies as objective” (p125).
This Dutch example of the history of quantification in economic policy suggests that questions about quantification need to be historicised as well as sociologised. It suggests that we can’t speak about numbers per se, but that we always have to ask ‘whose numbers?’. And then we come back to Restivo’s point about the limited empirical relevance of numbers. We have to ask ourselves what these numbers embody and for what purpose they were constructed. The problem then becomes who is actually able to deconstruct numbers. Everybody who has ever watched a quiz on television knows how scared people are when a numerical math problem is to be solved.
In my opinion, Porter’s argument about the relation between ‘openness’ and quantification misses precisely the fact that numbers also have a mystifying role. Who is actually trained enough to deconstruct numbers when the need arises? For many people the origins of numbers, let alone when mathematical symbols enter the arena, are highly obscure. People believe them or they don’t, but seldom do they themselves check the numbers about the economy’s growth. Porter himself must have recognized that it would have cost him quite a lot of readers if he had included complex nineteenth century mathematics used by the Ponts engineers. He tries to verbalise some quantitative arguments, but never writes down any number relation, let alone a symbolic equation. Doing this would have closed his text for those not very educated in numbers and symbols. The same holds for Restivo’s book: why does he mention, for example, ‘transcendental numbers’ without explaining them?
Let me finish with a comparison with technology studies. During its history we have had advocates of “technology as emancipating force” and advocates of “technology as oppressing force”. Nowadays we know that both positions are as true as they are ridiculous. Sometimes technology is emancipating, sometimes it is not; sometimes it emancipates the one group, while it oppresses an other. Sometimes technology emancipates in one period while it oppresses in another period. We, who study the role of mathematics and quantification in society, and the way society and culture have shaped mathematics, should take over these lessons.