Is there such a thing as a social construction of mathematics? Over the past two decades, constructivist science and technology studies (STS) have questioned traditional concepts of objectivity, examining some significant fields of modern sciences, from 17th century chemistry to 20th century physics. Generally, these studies argue that scientific truth is not inherent to the objects that are studied in research but rather depending on external, sometimes rather unscientific, influences like experimental arrangements, social networks or public policies. Within STS, some sciences have been particularly popular topics, most notably modern physics and modern biomedicine, whereas others have attracted rather little attention. Surprisingly, until recently, modern theoretical mathematics remained largely unnoticed despite several calls, by David Bloor or Bruno Latour for example, to question the 'holy of the holies'.

In her habilitation work, sociologist Bettina Heintz (University of Mainz, Germany) has now taken on this task. A social construction of mathematics? The unconventional answer of Heintz is both, yes and no. No, for theoretical mathematics as it is done today largely resists a sociological scrutiny; but also yes, because the history of the discipline shows that theoretical mathematics has emerged as a 'technology of trust' to unify the diversities of modern mathematics. In this sense, the rise of theoretical mathematics is indeed deeply affected by the history of modern societies. As most of recent scholarship in STS is dominated by models of social constructivism, Heintz' book, by exploring the uses and limitations of these models, reads like a critical account of the different STS approaches - from the classic laboratory studies to the recent interest in the cultural history of scientific practice - thus also offering a vision for future projects in STS.

The trajectory of Heintz refers to the two main methodological achievements of the sociology of scientific knowledge after the late 1970s, the laboratory studies and the controversy studies (often represented by the works of Karin Knorr Cetina, Bruno Latour, Steven Shapin and Simon Schaffer). For four months, Heintz sneaked into the Max-Planck-Institute for Mathematics in Bonn, with its over two hundred mainly visiting scholars one of the world's leading research institutes for theoretical mathematics. The core of Heintz' argument is based on the material she gathered during this period of field research: some twenty open interviews, and plenty of observations of the talks and debates in the seminars, of the daily work in offices and library, or of informal chats at coffee brakes.

As a starting point, Heintz critically discusses the works of David Bloor, Sal Restivo, and Eric Livingston, who laid out two distinctive programs for examining mathematics. Bloor and Restivo represent the externalist view, following the strong program of the sociology of scientific knowledge and its controversy studies. The examination of controversies and alternative approaches in the history of mathematics, they argue, would show that mathematics is a science determined by processes of social interaction, of negotiation and conventions, instead of transcendent, objective laws. Mathematical knowledge, in the view of Bloor and Restivo, is fundamentally contingent. Livingston shares this conclusion but differs from the externalist approach by taking a closer, internalist look at the topic, and by focusing on the rules that govern mathematical practice. By means of a close reading of a mathematical proof, Livingston succeeds in showing how the locally situated work of individual mathematicians makes its way to end as a contribution to universal knowledge.

Heintz aims to make good for the shortcomings of both of these approaches by combining them: undertaking a close examination of social practices in current mathematical research (what Bloor and Restivo suggest but fail to do), and by explaining not only the techniques but also the reasons for the production of universal knowledge in mathematics (missing in Livingston's approach).

However, this task has given Heintz a hard time. One of the most striking conclusions of her work is that "the characteristics of modern mathematics hardly offer any room for a sociological analysis" (p. 274, my translation). For this conclusion, the study gives several reasons.

First, mathematics is no laboratory science. Different from scientists in biomedical or physical laboratories, mathematicians rarely work in long-term social cooperations. Moreover, their crucial skills are often thought of as non-communicable gifts. When the interviewed mathematicians describe their key abilities to be successful, which is to know how to find the appropriate hypothesis to solve a mathematical problem, they speak of enlightenment ("Erleuchtung"), of aesthetical feelings, or even of some mysterious unexplainable mechanism: 'suddenly you see the picture' (p. 148, 150). How hard it is to communicate these creative skills is shown by the fact that several times the interview partners, especially the younger ones, turned the tables and started to ask Heintz to teach them on what they thought she must have discovered during her research: the secrets of the mathematician's success (p. 147).

Second, modern mathematics, according to Heintz, does not offer the same paradigmatical controversies as other sciences. Heintz concedes that there are fundamental controversies over the epistemology of mathematics (on a meta-level), but never within mathematics itself: a new mathematical theory may make an older one superfluous, but never wrong (p. 236). The formalities for mathematical proofs go back at least to the 17th century, and their rules have hardly changed since. To rely on the otherwise so successful approach of the controversy studies seems to be a vain venture with mathematics.

Instead, in a long final chapter, Heintz offers an historical explanation of theoretical mathematics. With the rise of the disciplines in academia since the mid-nineteenth century, mathematics has become one of the most diverse sciences, split into several independent sub-disciplines from actuarial science to theoretical physics. Drawing on the work of Niklas Luhmann, Heintz interprets this history of mathematical disciplines as a process of differentiation with a growing need for integration (p. 246ff.). Describing the cultural, social and institutional changes of mathematical practice in the 19th and 20th century, Heintz argues that in this process theoretical mathematics has played the part of the symbolically generalized communication media (Luhmann), offering a highly formalized language to unify the diversity of mathematical disciplines. In his book on "Trust in Numbers" (Princeton 1995), Ted Porter has argued that the rise of quantification in economic and social sciences since the end of the 19th century can be seen as a method to defy cultural and political diversities within academia and society. Heintz makes the same case for mathematics, interpreting the rise of theoretical mathematics as a 'technology of trust' (a concept of Porter) unifying the multitude of mathematical disciplines.

This trajectory of Heintz' book is symptomatic for some current trends in STS. In recent years, studies on mathematics and statistics exeeded the boundaries of the disciplines with a growing interest in the social and cultural implications of scientific practice. Mary Poovey (A History of the Modern Fact, Chicago 1998) has studied the use of numbers and statistics in bookkeeping and policy making from the 17th to the early 19th century; Ted Porter, as already mentioned, has written on the rise of quantification, especially its relevance for engineering and accounting in 19th and 20th century; and Leigh Star and Geoffrey Bowker have examined the social consequences of classification and standardization mainly in the afterwar period (Sorting Things Out, Cambridge MA 1999). These studies all agree that since the end of the 19th century statistical and mathematical practices have become pervasive both in science and society. However, we have just started to understand the implications of this process. We do know quite a bit about the history of statistics and mathematics as academic disciplines, but we still miss a general picture of their social and cultural impact. Against this backdrop, the study of Heintz points to the huge potentials of future studies in this field.

This is just the main achievement of the book. The reader will also find a handful of other inspired arguments or extended marginal notes, although this sometimes turns the book into a demanding read. Worth mentioning is the passage in which Heintz uses G.H. Mead's pragmatist concept of action to enlighten the terms of "scientific practice" and "material agency" that have become so central in STS, notably with the work of Andrew Pickering and Ian Hacking (p. 127ff.). Also, in the second chapter, Heintz engages in a comprehensive and critical account of the debates on constructivist approaches in philosophy of mathematics, from David Hilbert to Imre Lakatos and beyond. Finally, the summary of recent theoretical and methodological debates in sociology of science (chapter 3), combined with an excellent bibliography, offers probably the best short German-written introduction to STS so far (p. 104ff.).

There are but a few shortcomings. One whishes to know more about the gendered aspects of the topic. Though Heintz offers some hints (on the barriers that might lead to the misrepresentation of women in mathematics, p. 254, 257), her remarks are limited to the footnotes. Considered that Heintz has also widely published in gender sociology, it is unfortunate that she did not write a chapter on, say, 'mathematics and masculinity'.

Also, given the long trajectory of the book, it is unavoidable that the last chapter, where Heintz offers the main line of her historical argument, raises more questions than will be answered. Heintz is fully aware of this; she announces the chapter as an admittedly programmatic argument still in need of systematic empirical study. It would be of particular interest to compare the history of theoretical mathematics with that of applied mathematics, and thus to bridge the gap between the study of Heintz and the works of Porter, Poovey, Star and Bowker that are focusing on modes of application. It is a puzzling paradox that mathematical applications have been so successful in engineering, insurance, accounting, and other social contexts, and still theoretical mathematicians are keen to stress that their discipline is purer than others and must be detached from any association with applied mathematics. Historically, theoretical mathematics has its roots in applied mathematics. It is tempting to think of the proclamations about the purity of theoretical mathematics as attempts to deflect from some of the more controversial application of mathematical knowledge for military or industrial research. We probably need somebody writing a sort of conspiracy theory of modern mathematics.