Bringing together Philosophy and Sociology of Science

Vrije Universiteit Brussel

Centre for Logic and Philosophy of Science

October 21-24, 2008


Sabine Ammon (Berlin).

Reconstruction versus Construction — The Context Distinction and its Impact for Philosophy and Sociology of Science

Abstract: The separation into context of discovery and context of justification became a paradigm for Philosophy of Science in the 20th century. Famous through the writings of Reichenbach it opted for the method of rational reconstruction when science and knowledge are investigated epistemically. But the formal method of reconstruction causes many problems and inconsistencies when it is used exclusively. Only when ways of constructing come into play, the problems begin to vanish. The paper uses the positions of the early Carnap (reconstruction) and late Goodman (construction) as examples to show that reconstruction and construction must be considered as a joint interplay. But this new perspective does not only solve many problems of theory of knowledge. It is also a starting point for a new relationship of Philosophy and Sociology of Science, as construction implies a reassessment of individual and social factors in theory of knowledge.

Alexandre Borovik (Manchester).

Science Wars: a Time for a Truce

Abstract: I discuss dramatic changes in socioeconomic environment for mathematics as a cultural system and argue that perhaps we are entering an epoch when mathematics will be treated as part of humanities.

Matthew Brown (San Diego CA).

Science as Socially Distributed Cognition: Bridging Philosophy and the Sociology of Science

Abstract: A wide range of scholars from philosophy, cognitive science, sociology, education, and other areas have been attempting to use or assess a family of theories which we might call "socially distributed cognition" to study different aspects of science. This family of theories regards cognition as not always located "in the head" of an individual cognizer, but also in some cases in the collective activity of social groups together with artifacts. Applying a social-cognitive approach to science holds great promise for analyzing the social dimensions of scientific inquiry that, unlike many traditional approaches in the social studies of science, allows for normative assessments of scientific reasoning in a non-skeptical, non-relativist fashion, because of our ability to connect an analysis of cognition to our theories of rationality and knowledge. But unlike most philosophy of science, it can do so in a way that fully respects the complex socio-cultural, material, technological conditions in which scientific activity occurs. In this talk, I will examine the prospects of applying the distributed cognition framework to the study of science as a way of productively connecting the sociology and philosophy of science, with special attention to illustrations dealing with mathematical tools in science and the formal sciences themselves.

Filip Buekens (Tilburg)
Maarten Boudry (Ghent)

Institutional Facts or Social Constructions? A Searlean Reconstruction of Psychoanalytic Facts

Abstract: We propose to explain some intriguing features of psychoanalysis with the help of a theory of institutional facts originally developed by John Searle (Searle 1969, 1975, 1995), and further explored and modified by philosophers like Tuomela (2002) and Lagerspetz (2006). We hold that the Freudian technique issues in the unintended introduction of institutional facts, in a precise technical sense to be explained. Freud consistently presented his findings as discoveries of natural or brute facts. The continued existence of institutional facts requires the creation of a tight pattern of shared beliefs which contribute to the truth of claims about these institutional facts. This explains (i) the closed character of psychoanalysis (no independent evidence supports it, no independent theory fruitfully exploits psychoanalytic findings) and (ii) the distinctively social characteristic of psychoanalysis as a tightly controlled thought system, with various competing schools internally characterized by strong hierarchical relations. Our assessment of psychoanalysis will be contrasted with a social-constructivist approach to psychoanalysis. We argue that a classic social constructivist account of science lacks the explanatory tools to explain these distinctively social and theoretical features of psychoanalysis.

Bernd Buldt (Fort Wayne IN).

Husserl's Theory of Objectivity

Abstract: From the first systematic exposition in the Logical Investigations up to the posthumous Experience and Judgment Husserl claimed that his analyses of intentional epistemic acts do also account for the intersubjectivity and objectivity of the formal sciences, logic and mathematics in particular. Recently (see, e.g., Lawlor (2002, pt 4), Husserl's approach met with renewed interest due to Derrida's exposition (published 1962, English translation in 1989) and the English translation of Derrida's doctoral thesis in 2003.

In its first part, the paper gives a survey of how Husserl's ideas on intersubjectivity/objectivity evolved and matured over time. In its second part, the paper gives a critical assessment of Husserl's arguments and seeks to answer the question whether, by modern lights, his approach provides a viable theory of the objectivity of the formal sciences. Special attention will be given to social/sociological aspects of Husserl's theory, which are so much more promiment in his late philosophy.


  • Derrida, Jacques, Edmund Husserl's "Origin of Geometry," Lincoln : U of Nebraska P 1989.
  • Derrida, Jacques, The Problem of Genesis in Husserl's Philosophy, Chicago: U of Chicago P, 2003.
  • Husserl, Edmund, Logical Investigations, New York, Humanities P 1970.
  • Husserl, Edmund, Cartesian meditations, The Hague: Nijhoff, 1960
  • Husserl, Edmund, Experience and judgment, Evanston, Northwestern UP, 1973.
  • Lawlor, Leonard, Derrida and Husserl, Bloomington: Indiana UP, 2002.
Jessica Carter (Odense).

The use of diagrams in mathematical reasoning

Abstract: Topics addressed in current so-called philosophy of mathematical practice are that of visualization in mathematics as well as use of representations in reasoning. This talk will discuss the role that diagrams play in mathematical reasoning.

I will present part of a case study from contemporary mathematical practice (free probability theory), where certain combinatorial expressions were first proved using diagrams. The authors found that the proof  based on diagrams was not rigorous enough. Thus in their published proof, the pictures were replaced by reasoning about permutation groups. Although the diagrams are removed from the paper, I argue that they still play a role. In the talk I will present these roles and propose reasons why the use of these diagrams is successfully.

Helen De Cruz (Leuven)
Johan De Smedt (Ghent)

Cognitive and cultural factors influence the spread of mathematical concepts: The case of zero

Abstract: Although the historical development of mathematical concepts is being fleshed out in increasing detail, we still lack a plausible mechanistic explanation for their evolution. Why are some mathematical concepts successful, such as zero, which spread from an isolated area - the Jain community of India in the 5th century - to the rest of the world, while others never quite catch on, such as the intuitionist number theory? Models of concept change in mathematics are lacking partly because many philosophers of science, following Kuhn, still deny that conceptual change occurs in mathematics. When concept change is acknowledged, many sociologists picture it as a simple progressive change, where elementary mathematical concepts are abandoned in favour of more complex and efficient ones. Through anthropological and historical examples, we show that mathematical concepts do not unfold according to a simple progressive scheme of which modern Western mathematics is the alleged summit. In this paper, we develop a tentative general model of mathematical concept spread, drawing on theoretical work in the epidemiology of culture by Sperber, Nichols and earlier work by one of the authors. According to this model, the success of mathematical concepts depends both on intrinsic cognitive factors and socially constructed culture-specific values. Both of these function as 'attractors' that channel the reception and spread of new ideas. We apply this general framework to the concept of zero, thereby gaining an improved understanding of both its invention and its spread (see also extended abstract).

Liesbeth De Mol (Ghent).

On the use and (interactive) role of computers in computer-assisted proofs

Abstract: In this paper we will study the use and role of the computer in computer-assisted proofs. We will explore several different features of such proofs and the consequences they have for mathematics through a detailed analysis of three concrete cases (not avoiding some of the technical intricacies of the proofs discussed). We will not consider the well-known proof of the four color theorem but shift attention to one lesser known and younger (Hales' proof of the sphere packing problem) and two hardly known and older examples (a proof on cubic residues by D.H. Lehmer et al. and several proofs related to Ràdo's Busy Beaver Game). Our main emphasis will be on the interactive aspects of computer-assisted proofs because they are often neglected. On the basis of our analysis we will discuss the question whether the computer can be regarded as more than just a tool within the context of mathematics.

Till Düppe (Rotterdam).

Listening to the Music of Reason: Nicolas Bourbaki and the Phenomenology of Mathematical Experience

Abstract: Central theme of a phenomenology of science that is inspired by the late work of Edmund Husserl is the "intentional correlation" between the object of science, the theory, and its practice. What describes the reality of science (experience), and the reality that is claimed in science (truth) are correlated by means of a "history of sense". This history of sense describes the phenomenological constitution of science. With such approach to science phenomenology is in principle capable of combining, if not mediating between the two main paradigms of the commentary of science of, the philosophy of science that is limited to the justification of theory, and the sociology of science that is limited to the social reality of science.

Critical for this phenomenological approach is the role of mathematics in modern science. For Husserl the mathematization of science amounts to the same as the "oblivion of the life world" because in mathematics the validity of its theory depends on the absence of the mathematical experience: In mathematics, so Husserl, "experience does not function as experience". Mathematics does not seem to be constituted by its experience, so that it "history of sense" is a supplement of its validity, as Derrida emphasized later. The mathematical experience is thus of transcendental significance since the status of experience as such is questioned. This character of the mathematical experience is, however, mitigated as long as it is embedded in a philosophical world-view or in a pragmatic context of applied mathematics. Its transcendental significance comes only to the fore within the tendency of the isolation of mathematics from both philosophy and the sciences. Just this happened with a rather obscure, and yet experientially and socially peculiar school of mathematics, Nicholas Bourbaki (1935).

Nicholas Bourbaki is a suitable case for phenomenology of mathematical experience because this collective did not hold an explicit philosophy of mathematics, a lack that they excused with a naïve pragmatism ("handbook for the working mathematician"). In the history of mathematics, the group (still existing today) stands for the mathematization and thus disciplinary isolation of mathematics in its pretence of "founding the whole of modern mathematics" along the axiomatic method, "laying claim to perfect rigour". Their project was naïve in that it simply ignored the challenges of axiomatizations from Cantor to Hilbert and Gödel, but was also decisive in that it prepared the ground for category theory (Eilenberg, Grothendieck). The crucial experience that gave the group its peculiar verve, as I argue, was thus the liberation from the meaning of mathematics. In Bourbaki mathematics itself became an object of experience. And this peculiar status of Bourbaki is pointedly described in Dieudonné's expression of Bourbaki being the "music of reason".

This metaphor I exploit within a transcendental discussion of the mathematical experience. I compare the experience of following a proof with the experience of "listening to a melody" that was one of the examples for Husserl's account of the constitution of the inner time consciousness. Treating a mathematical proof as a temporal object (objects that are constituted not through time, but in time) allows to explicitly expose the aesthetical, or better: affective appeal of proofs as opposed to their discursive embedding (to which hermeneutists would point). I thus highlight the pathos of following a proof as the experience of epistemic necessity freed from the burden of reference. The experience of a mathematical proof points to a layer of intellectual life that is not reducible to the active synthesis of judgements. Instead it shows the passive synthesis and thus participation of intellectual life in sensual life.

The punch line of this account of the mathematical experience is that it leads to a necessary clash with the discursive reality of mathematics. As apparent in particular in Bourbaki, their discursive reality almost ridiculed the elevated experience that draw them into mathematics. The process of writing the proofs, the problems of sticking to their "mother-structures", the pragmatic decisions which proofs to include etc., let to tensions among the members and after all to the failure of the project in the 1960s. The pathos of the mathematical experience was overweighed by the reality of the world that surrounds it.

Karen François (Brussels)
Bart Van Kerkhove (Brussels)

Ethnomathematics as an implicit philosophy of mathematics (education)

Abstract: This paper consider the field of enquiry called ethnomathematics and its role within the philosophy of mathematics (education).

We elaborate on the shifted meaning of the concept "ethnomathematics". Until the early 1980s, it was reserved for the mathematical practices of 'nonliterate' "formerly labeled as 'primitive'" peoples (Ascher and Ascher, 1986). What was strived for were detailed analyes of sophisticated mathematical ideas among these, ideas "it was claimed" akin to and as complex as those of modern, "Western" mathematics. D'Ambrosio, who was to become the intellectual father of the ethnomathematics program, then proposed "a broader concept of 'ethno', to include all culturally identifiable groups with their jargons, codes, symbols, myths, and even specific ways of reasoning and inferring" (D'Ambrosio, 1985). As a result, today, within the ethnomathematics discipline, scientists are collecting empirical data about the mathematical practices of culturally differentiated groups, literate or not. The label 'ethno' should thus no longer be understood as referring to the exotic or as being connected with race.

This changed and enriched meaning of the concept 'ethnomathematics' has had its impact on the philosophy of both mathematics and mathematics education. Within the philosophy of mathematics, it has contributed, as part of a broader and quite diverse (or even dispersed) sociological movement, to giving critical mass to studies of mathematical practice. Within the field of mathematics education, ethnomathematics clearly gained a more prominent role, as now also within Western curricula, it became meaningful "and indeed appeared relevant" to explore various aspects of mathematical literacy (D'Ambrosio, 2007). We discuss a number of possibilities and dangers this has opened, and on the basis of this present ethnomathematics as an alternative, implicit philosophy of professional and school mathematical practices.


  • Ascher, Marcia; Ascher, Robert [1986] (1997). Ethnomathematics. In Powell, Arthur B.; Frankenstein, Marilyn (eds.) Ethnomathematics, Challenging Eurocentrism in Mathematics Education, State University of New York Press -SUNY, Albany, pp. 25-50.
  • D'Ambrosio Ubiratan [1985] (1997). Ethnomathematics and its Place in the History and Pedagogy of Mathematics. In Powell, Arthur B.; Frankenstein, Marilyn (eds.) (1997). Ethnomathematics. Challenging Eurocentrism in Mathematics Education. State University of New York Press, Albany, pp. 13-24.
  • D'Ambrosio, Ubiratan (2007a). Peace, Social Justice and Ethnomathematics. The Montana Mathematics Enthusiast. Monograph 1, 25-34.
  • D'Ambrosio, Ubiratan (2007b). Political Issues in Mathematics Education. The Montana Mathematics Enthusiast. Monograph 3, 51-56.
Norma B. Goethe (Cordoba).

Modes of Representation and Working Tools in Leibniz´s Intellectual Workshop
Abstract: My aim is to look at the inter-connection between Leibniz's theory of signs and his actual practice in writing science during the Paris years (1672-1676). Taking advantage of recently published material, as well as more recent scholarship, I propose to take a fresh look at some of Leibniz's most striking insights concerning 'tangible' signs, the ideal of a 'universal character', and his view of the essence of science and the growth of knowledge.

Mathematical understanding begins with seeing; and the modern view that understanding and the advancement of learning more generally require perceptible signs or forms of expression can be traced to Leibniz. Leibniz's insight is that language is a human creation that does not merely record our thought but is instead an embodiment of understanding. But at the turn of the 20th century, the focus on the study of Leibniz philosophy pointed in other directions. Bertrand Russell's book on Leibniz (1900) with its emphasis on traditional logic flatly ignored most of Leibniz's mathematical innovations. And in the influential neo-Kantian historiographic perspective developed by Ernst Cassirer (1907), Leibniz's philosophy was depicted merely as the 'culmination of rationalism' that –gracefully removed from the 'world of the senses'– Cassirer takes to be one of the 'dialectical' paths leading into Kant´s critical philosophy. A third difficulty has been that Leibniz's work had been thus far only fragmentarily, and selectively, published; it is only with the 20th century that a sustained effort has been made to make available the complete works and correspondence.

From 1672 through 1676 Leibniz pursued his mathematical studies in Paris. He studied Descartes´ work in geometry, and building on it developed his own methodological views, which would prove most fruitful, leading him to valuable results in the new mathematical sciences. At the same time, his conception of signs underwent an important transformation. Later in his career, in response to Locke's Essay (1696) Leibniz expressed his conviction that having joined 'theory and practice' so as to make 'many discoveries which have manifested themselves useful', he was in more of a position than Locke to discuss the 'fundamentals of the investigation of truth'. To the fundamentals of the investigation of truth belongs Leibniz's scheme of universal character, which he identifies with the science of forms (of the similar and dissimilar), or Ars combinatoria. Just as the compass played a central role in the successful navigation of the oceans, so Leibniz's Ars Combinatoria was to play a central role in the successful navigation of (what Leibniz often thought of as) the oceans of human knowledge.

On Leibniz's mature views there is no abstract human thought that does not require something sensible. He describes his character in visual terms: just as in mathematical symbolic writing, his symbols or characters are to provide with the 'tangible' thread necessary to develop and fix our thoughts. Despite our limitations, tangible characters give us 'the means of being infallible' because, by rendering our reasoning sensible, we can easily recognize errors at a glance and rectify them. In a similar way, to write out a proof provides us with a way to 'see' whether the results hold and to communicate them to others.

On the other hand, symbols as employed in mathematics as well as any other form of characters (and other instruments) are the products of human industry. We are cognitively "bounded" agents who constantly aim at improving our means of investigation, our tools and instruments in order to gain a better understanding of things. Because, in particular, everything conceivable in nature is accessible to numerical determination, Leibniz held that we can always extend the 'horizon' of what mathematics is capable of in order to further our understanding. Time and again, Leibniz compares the realization of a 'general character' with the advantages offered by other scientific instruments, claiming that it would 'bring more use to those who traverse the oceans of research than the magnet ever gave to seafarers'.

In my paper I start out by focusing on previous results by E. Knobloch (2004), which concern the process of writing in scientific practice in the case of Leibniz's use of manuscripts. Knobloch tries to illustrate the many different ways in which Leibniz carried out his mathematical research step by step thus obtaining his results 'by thinking in writing'. Knobloch's study shows to what extent Leibniz's mathematical thinking is deeply intertwined with writing and, in particular, with the genesis of his mathematical texts. The expression 'text' is used in a broad sense so as to include conventional one-dimensional discursive structures (mostly in Latin but also in French), but also formulae, tables, figures, drawings, and different forms of illustrations.

For Leibniz, mathematical thinking unfolds and goes along with writing and in this sense, as Knobloch emphasizes, "Leibniz's posthumous writings provide a unique insight into his intellectual workshop".
Writing played a most crucial role in Leibniz's mathematical thinking. In order to make this point, we may describe Leibniz's mathematical practice by selecting four aspects under which 'texts' are being used by Leibniz in his manuscripts:

  1. 'Texts' serving the art of invention.
  2. 'Texts' serving the visualization of thoughts, theorems, and proofs.
  3. 'Texts' as fixation of insights and elaboration of treatises.
  4. 'Texts' as transcribed discussion and argumentation.
Christian Greiffenhagen (Manchester).

Formal versus Practical? Opposition to formalism in the sociology of science and mathematics

Abstract: The formal character of the formal sciences has been a longstanding fascination and puzzle. Mathematical proofs, for example, seem to have a transcendent, universal, and necessary character. Since the Ancient Greeks, philosophers have tried to understand and specify the nature of formal knowledge. The result has been a variety of 'formalist' pictures of the formal sciences.

It was largely in reaction to such ?formalist? conceptions of the natural sciences in general, and the formal sciences in particular, that the 'new' sociology of scientific knowledge was developed. In effect, sociologists read formal representations (e.g., methods reports) as intended descriptions of the day-to-day investigative practices of scientists and then set out to establish, by various means (including anthropological field studies of scientists at work), that such representations did not capture the realities of scientific practice and, more importantly, that it was in principal not possible that they could do so. The aim of these studies was, in part, to establish the practical, contingent, and 'negotiated' ways in which formal representations are composed in order to demonstrate that these constructs therefore could not satisfy the ?formalist? requirements (which are essential to endow them with the transcendent, universal, and necessary properties they supposedly possess).

With respect to mathematics, philosophers and sociologists have been particularly fascinated with mathematical formalism (proof theory, metamathematics), which seems to conceive of mathematics as merely the manipulation of formal symbols according to certain formal rules, and which is often seen as the main alternative to a Platonist or empiricist philosophy of mathematics. Mathematical formalism invites questions about the relationship between 'formal' proofs and 'ordinary' proofs (those published in academic journals), and have led for example Eric Livingston to start his sociological study of mathematics trying "to review what a formal logistic system looks like and to examine whether or not, or in what sense, such a system is descriptive of the material details of mathematical practice" (The Ethnomethodological Foundations of Mathematics, 1986, p. 25).

In this paper, I want to revisit the interplay of 'formal' and 'practical' features in the context of the presentation of mathematical proofs. I will use video-recordings of graduate lectures in mathematical logic as a means to assess the strong and weak points in the opposition to formalism in the philosophy and sociology of science.

Ari Gross (Toronto ON).

Feynman Diagrams and Visual Reasoning

Abstract: Since the late 1940s, subatomic physicists have developed the remarkable ability to reduce complicated particle interactions to a compact collection of intersecting lines. These drawings, known as Feynman diagrams, occupy a fascinating place in scientific practice: they are neither physical theories nor symbolic mathematics, but simple, versatile ``paper tools'' which have become a near-essential component of contemporary subatomic physics.

My paper examines the role that Feynman diagrams play in physicists' reasoning. Drawing on recent academic interest in visual reasoning and David Kaiser's work on Feynman diagrams, I characterize two distinct ways in which Feynman diagrams are used: as images capable of radically facilitating calculations by mediating between theoretical characterizations of a subatomic event and its associated mathematical description, and as powerful heuristic tools used to generally enhance one's understanding of either a particular interaction or of the nature of subatomic physics in general. Referring to these uses as ``formal'' and ``informal'' diagrammatic reasoning, I explicate the relationship between Feynman diagrams and other scientific concepts, such as physical theories and symbolic mathematics, and highlight the importance of visual reasoning in scientific practice.

The primary goal of my paper is simple, yet ambitious: to assist in the overall characterization of how high-energy physicists reason, that is, to explicate the manner in which individuals, both alone and in groups, arrive at their scientifically-relevant conclusions. In doing so, I challenge traditional conceptions of ``reasoning'', adopting a naturalistic, sociological perspective for this traditionally philosophical topic. This approach is not without its precedents, as the past decade has seen an increased interest in scientific visual reasoning; images, not just words or arithmetical symbols, are increasingly being recognized as fundamental, perhaps indispensable, constituents of the manner in which scientists reach their conclusions. In general, by understanding reasoning as a rich and textured, socio-psychological, primarily (but not exclusively) cognitive process, I hope to renew debate over what it means to reason and confront the implications of such reevaluations.

Albrecht Heeffer (Ghent).

On the curious historical coincidence of algebra and double-entry bookkeeping

Abstract:Algebra was introduced in medieval Europe through the Latin translations of Arabic texts between 1145 and 1250 and Fibonacci's Liber Abbaci (1202). Algebraic problem solving was further practiced within the so-called abbaco tradition in cities of fourteenth- and fifteenth century Italy and the south of France. From the sixteenth century, under the influence of the humanist program to provide new foundations to this ars magna, abbaco algebra evolved to a new symbolic algebra with François Viète (1591) as the key figure. This is a brief characterization of the current view of scholars on the history of European symbolic algebra.

Now consider the following statement: The emergence of double-entry bookkeeping by the end of the fifteenth century was a consequence of the transformation from the traveling to the sedentary merchant, primarily in the wool trade situated in Italy and Flanders (van Egmond, 1976). Given the vast body of evidence from Renaissance economic history and the evident causal relationship, not many will contest the relevance of merchant activities on the emergence of bookkeeping. What about the mitigated statement: "The emergence of symbolic algebra in the sixteenth century is to be situated and understood within the socio-economic context of mercantilism". Philosophers of mathematics who believe in an internal dynamics of mathematics will not accept decisive social influences as an explanation for the emergence of something as fundamental as symbolic algebra. At best, they will accept social factors in the acceleration or impediment of what they consider to be a necessary step in the development of mathematics. Also it seems difficult to pinpoint direct causal factors within economic history for explaining new developments in mathematics. However, the relationship between bookkeeping and symbolic algebra is quite remarkable. Many authors who have published about bookkeeping also wrote on algebra. The most notorious example is Pacioli's Summa, which deals with algebra as well as bookkeeping, and the book had an important influence in both domains. But there are more. Grammateus (1518/21) gives an early treatment of algebra together with bookkeeping. The Flemish reckoning master Mennher published books on both subjects including one treating both in the same volume (1565). So did Petri (1583) in Dutch. Simon Stevin wrote an influential book on algebra (1585) and was a practicing bookkeeper who wrote a manual on the subject (1608). In Antwerp, Mellema published a book on algebra (1586) as well as on bookkeeping (1590). While there is no direct relationship between algebra and bookkeeping, the teaching of the subjects and the books published often addressed the same social groups. Children of merchants were sent to reckoning schools (in Flanders and Germany) or abbacus schools (in Italy) where they learned the skills useful for trade and commerce. There is probably no need for algebra in performing bookkeeping operations but some basic knowledge of algebraic rules was very useful in complex bartering operations or the calculation of compound interest.

While scholars on the abbaco tradition, such as Jens Hĝyrup (2007) maintain that the problem solving treatises written by abbaco masters served no practical purpose whatsoever, we will argue that their activities and writings delivered an essential contribution to Renaissance mercantilism, namely the creation of value. According to Foucault (1966, 188) the essential aspect for the process of exchange in the Renaissance is the representation of value. "In order that one thing can represent another in exchange, they must both exist as bearers of value; and yet value exists only within the representation (actual or possible), that is, within the exchange or the exchangeability". The act of exchanging, i.e. the basic operation of merchant activity, both determines and represents the value of goods. To be able to exchange goods, merchants have to create a symbolic representation of the value of their goods. All merchants involved must agree about this common model to complete a successful transaction. As such, commercial trade can be considered a model-based activity. Given the current global financial market and the universal commensurability of money we take for granted, we pass over the common symbolic representation as an essential aspect of trade. However, during the Renaissance, the value of money depended on the coinage, viz. the precious metals contained in the coins which differed between cities, and varied in time. As the actions and reciprocal relations of merchants, such as exchange, allegation of metals and bookkeeping became the basis for the symbolic and abstract function of money, so did the operations and the act of equating polynomials lead to the abstract concept of the symbolic equation. Both processes are model-based and use the symbolism as the model. Therefore, we have to understand the emergence of symbolic algebra within the same social context as the emergence of double-entry bookkeeping.

Herbert Kalthoff (Mainz).

Doing/Undoing Calculation: Sociological Insights from Risk Management

Abstract: From small enterprises to financial markets, economic calculation is a daily routine activity in economic life-worlds. The paper argues that calculation is situated in the practice of the participants involved and the technological tools used. It advocates a sociological analysis of the world-constituting character of calculation. As empirical examples the paper discusses risk management strategies, the embeddedness of calculation practices in banking infrastructure, and the process of internal rating procedures. Arguing against the idea of the omnipresence of calculation, the paper analyzes the social phenomena of the neutralization of calculation. The notion of undoing calculation is introduced and outlined theoretically and empirically.

Jouni-Matti Kuukkanen (Durham).

Inevitable or Contingent History of Science? Re-specifying the Difference between Scientific Realism and the Sociology of Scientific Knowledge

Abstract: Scientific realism and the Sociology of Scientific Knowledge (SSK) have typically been taken to offer two radically different perspectives on science. The latter generally explains the emergence of scientific knowledge by social factors, while the former attributes a central role to non-social determinants. However, this polarized account fails to capture the complexities of the interactions between social and non-social factors in the development of scientific knowledge. It seems that the evaluation of their relationship is entering a new stage. It is possible to find textual evidence from the representatives of both these traditions to conclude that they both accept that both social and non-social factors play a role in belief formation in science. For this reason, we need either to accept that there is no principled difference between them on this matter or to re-specify their relationship. I suggest that this newly specified relationship can be captured by two modal concepts - inevitability and contingency - and consequently, by two contrasting modal views of the history of science. The crux of the matter is the status of the current body of scientific knowledge: whether we should take is as the only possible outcome or an end result of various possible ones. I will also spell out in what ways the history of science can be seen as contingent or inevitable.

Brendan Larvor (Hatfield).

Mathematics, Phenomenology and Social Cognition

Abstract: The French philosopher Albert Lautman (1908-1944) studied the most advanced mathematics of his day from a philosophical perspective rooted in the later dialogues of Plato and the early works of Heidegger. He was, therefore, free from two of the characteristic limitations of most philosophy of mathematics; his mathematical examples were not limited to elementary or foundational topics, and his philosophical viewpoint could accommodate the mathematician as a questing, embodied agent rather than as a disembodied "ideal enquirer".

This talk will present Lautman's philosophy of mathematics and assess the extent to which it offers a way out of the impasse between current rival approaches to the philosophy of mathematics. Neither Lautman nor Heidegger emphasised the social aspect of scientific work. However, Lautman's deep engagement with Plato's dialogical methods may suggest a means to recognise the role of social cognition without losing sight of the objective content and structure of mathematics, and the phenomenology of mathematics as a mode of human experience. Lautman's own dialectical model is in some senses too rigid, but nevertheless his overall approach may suggest a way forward.

Frank Linhard (Frankfurt).

Formal and non-formal approaches to the notion of "Risk" — a historical perspective

Abstract: One of the problems modern theories of risk have to cope with is the variety of approaches to the subject. While the interest in solutions for applied fields like risk management and financial mathematics in general increased during the last years due to systems of regulations like Basel II and Solvency II, particular problems were discussed employing mathematical approaches since the 16th century. The variety of the particular problems investigated and the multitude of perspectives and interests in the related formulations of the questions lead to miscellaneous approaches during the centuries. These were the more or the less applicable to specific problems under consideration and had to be completed and supplemented and often additional aspects were added.

The multitude of isolated considerations and solutions of problems connected with the notion of risk led to a development of many single and equally isolated approaches and nowadays there is still no conclusive and integrated theory of risk in sight.

The paper will focus on the early texts of Galileo Galilei (1613~1623) and Daniel Bernoulli (1738) as examples of pure combinatorical analysis and perspectively considerations within the mathematical discipline of probability theory. It is argued that Bernoulli's approach needed to be developed further in order to achieve a successful and satisfactory theory of risk. In modern economy the need for a proper definition of a notion of risk is seen and currently discussed within the frame of ISO standards. But as already mentioned this interest is mainly owed to the governmental demands of the Basel II and Solvency II standards and therefore an external demand. On the other hand an intrinsic understanding of the meaning of risk, as could be provided by a conclusive theory, could lead to a better success in modelling various risks and help to achieve better prognosis.

Geerdt Magiels (Brussels).

Towards an ecological understanding of science. The case of the discovery of photosynthesis

Abstract: Who discovered photosynthesis? Not many people know. Jan IngenHousz' name has been forgotten, his life and works have disappeared in the mists of time. Still, the tale of his scientific endeavour shows science in action. It opens up an undisclosed chapter of the history of science which can help to shine some light on the ingredients and processes that shape the development of science. The fate of IngenHousz and of photosynthesis research can be better understood if one takes into account as well the individuals involved, as their social, cultural and historical context, interwoven in social interactions and interconnected by the theoretical, instrumental and practical requirements of their scientific research. This paves the way for a fresh multidimensional approach in the philosophy of science, taking steps towards an "ecology of science".

Eric Oberheim (Berlin).

Incommensurability and Reconciliation

Abstract: In order to facilitate reconciliation between these different approaches in 20th century science studies wars, I offer an analysis of one of the causes of the tensions generated between sociological and philosophical accounts of the natural sciences: Incommensurability. Incommensurability results because of the logical limitations imposed by the use of competing, incompatible taxonomic classifications within the natural sciences, and between their sociological and philosophical interpretations.

Incommensurability has been a key notion to the Western analytic tradition ever since its inception. This idea is widely recognized as having played a pivotal role in the initiation of this tradition ever since Pythagoras and Aristotle's discussions of it. It is often taken to be the first application of this traditions central method: the use of logical analysis to make an analytic argument, as in Aristotle's demonstration of incommensurability by reasoning logically from premise to conclusion to prove that in Euclidean Geometry, there are incommensurable magnitudes, a fact that is now represented with irrational numbers.

But the idea of incommensurability was not only central to revolutionary developments in mathematics necessary to the natural sciences and the ensuing analytic method that continues to dominate the natural sciences and their philosophical, sociological and historical interpretations. It has also now come to represent the very irrationality and misunderstanding that fuels the science wars, ever since it became a controversial keyword both in philosophy and in sociology of science in the wake of Thomas Kuhn and Paul Feyerabend's influential 1962 popularisations of it.

Kuhn and Feyerabend used the mathematical idea metaphorically to apply to historical, philosophical and sociological explanations of the natural sciences. Kuhn was drawing on Ludwig Fleck, an early pioneer of epistemological sociology (1927, 1935), who used the term "inkommensurable" to apply to conceptual incompatibilities in the development of medical concepts. Fleck bridged the gap between sociology and philosophy by using analytic methods to investigate science as a social phenomena, taking incommensurability to be a central feature of scientific advance. Feyerabend developed his notion of the incommensurability of scientific theories from Albert Einstein, who was the first to use the term incommensurable to apply to physical theories (1949).

This paper will examine this controversial notion in order to show how it has fueled misunderstandings in the science wars, with the aim of alleviating the tensions that it has caused by suggesting how becoming bilingual can overcome the challenges that incommensurability poses to philosophical and sociological accounts of the sciences.

Hauke Riesch (Cambridge).

On the philosophical talk of scientists

Abstract: This paper reports on a study on how scientists themselves talk and write about philosophical topics, and how these topics get used in scientific thought. 30 popular science books were analysed for how they treat philosophical topics on the nature of science. 40 academic scientists were then asked in a series of semi-structured interviews questions based on the philosophical topics that were found discussed most often in the books.

In interpreting the books and the scientists' responses on these topics, I use the concepts of boundary work and boundary objects, and social identity theory: The study demonstrates how philosophical topics can be used to draw boundaries and to define social identities around science or various disciplinary affiliations. Philosophies and famous philosophers like Popper also act as boundary objects facilitating scientific communication across boundaries. The talk surrounding the various philosophical categories however often hides a big variation in actual philosophical opinion, which is set slightly apart from how the philosophy itself is discussed.

Through the analysis of the philosophical talk of scientists, this study represents a way in which the philosophy of science can be studied through sociological methods. I will therefore discuss what I think are some of the implications of this study for the philosophy of science and its efforts of making itself relevant to scientific practice.

Georg Schiemer (Vienna).

Carnap's early semantics: models, isomorphism and categoricity

Abstract: In recent years one was able to witness an intensified interest in the historical details of Carnap's technical contributions to logic and the philosophy of mathematics. In this talk I take up this line and focus on his early, formative contributions to a theory of formal semantics around 1928. Carnap's manuscript "Untersuchungen zur allgemeinen Axiomatik" includes early formal definitions of the logical concepts of model, logical consequence and satisfaction (see Carnap 2000). In my talk I intend to present a conceptual analysis of the technical details of these explications and to contextualize Carnap's results in their historical and intellectual environment. Certain interpretive issues concerning his tacit assumptions underlying the concepts that are left implicit in the text will be addressed. Specifically his notions of satisfaction and truth of a formula in a model as well as his conception of the universe of a model leave room for further interpretation.

I will claim that from a modern perspective of model theoretic semantics Carnap's notions are non-standard in several ways.

Marc Staudacher (Amsterdam).

Around 20 notions of conventions and still no clue?

Abstract: Since David Lewis' groundbreaking work on conventions in 1969, the literature on conventions has developed into a maze. Around 20 notions have been proposed by different authors. Though little has been said about their similarities and differences. Rather, the differences led to heated debates in philosophy and sociology about what the "true" notion of a convention is. In this article, another approach is chosen. I propose a taxonomy and a logical framework to represent and reason about some key features of the different notions. This approach allows us to give a precise formulation of the disagreement and agreement among the proposals of the different authors and helps us to choose the right notion for the job at hand. In particular, the job conventions are assigned in philosophy of language will be discussed. It will be shown that linguistic conventions have to fulfill a specific job profile which helps us in constructing formal models about them but this job profile makes this kind of convention also quite special when we compare it to other kinds of conventions which are common in the social sciences.

Renate Tobies (Braunschweig).

Career Paths in Mathematics: Women and Men by Comparison.

Abstract: In western industrialized countries the following stereotype still persists: "Mathematics is nothing for women". In Germany, the number of female professors in mathematics is indeed exceptionally small. In order to discover the (historical and current) causes of this phenomenon, we analysed the career paths of several thousand individuals who successfully completed their studies of mathematics at the beginning and at the end of the 20th century. This presentation introduces the methods and sources of our interdisciplinary research project and presents the main results which were yield by comparing career paths of female and male mathematicians.

Ana Teixeira-Pinto (Berlin).

The philosophical concepts of autonomy and automatism and the process of mathematization of mechanics.

Abstract: “To the average educated person of the present day, the obvious starting point of mathematics would be the series of whole numbers:
1,2,3,4, ...etc

Probably only a person with some mathematical knowledge would think of beginning with (...) the series of natural numbers:
0,1,2,3,4, ...etc

It is only at a high stage of civilization that we could take this series as our starting point. It must have required many ages to discover that a brace of pheasants and a couple of days were both instances of the number two: the degree of abstraction involved is far from easy. And the discovery that 1 is a number must have been difficult.

Roy Wagner (Tel Aviv).

Mathematical variables as indigenous concepts

Abstract: The suggested paper discusses the semiotic status of mathematical variables. It will describe a mathematical case study, and then proceed to analyze the case study using a theoretic framework that crosses anthropology, philosophy and semiotics.

The case study to be discussed belongs to the theory of generating functions. I will show that variables in generating functions take several formal roles: variables to be substituted for by real or complex values, formal variables (which are in fact not variables, but lambda-operators), and transcendental extensions of the real or complex field (which are not variables either, but constants). Furthermore, I will show that these roles do not exhaust the field of possibilities. Generating functions variables take other formal and semantic roles as well, some of which are formally rigorous, and some which hare not.

The main point of this case study is that one needn't decide in advance how to interpret variables when working with generating functions. Practitioners can and do switch interpretations and make things up as they go along. They rely on authority and experience to assume rigorous justifiability, but do not actually set up an a-priori, rigorous theoretic framework that would cover all their interpretations and manipulations.

This situation will be analyzed using a tradition of semiotic analyses of "indigenous concepts" going back to Marcel Mauss. I will show how Mauss' analysis of the concept of Mana in his famous work on magic, the subsequent critique by Levi-Strauss, and further elaborations by Derrida can help us understand the role of variables in mathematical practice. I will demonstrate how notions such as the social a-priori system of signs, supplementary signifiers, zero symbolic value, and mimesis and iterability as constitutive of signs can lead to a sociologico-philosophical understanding of mathematical practice.

The point is not to claim that variables are identical or isomorphic to esoteric concepts of magic — this is obviously false. The point is to show how the interpretive and theoretical framework of structural and post-structural anthropology can shed light on mathematical variables, and explain their functioning. This will hopefully bring up the relevance of the discourse bridging anthropology, philosophy and semiotics to the study of mathematical practice.

Paul Ziche (Utrecht).

The multiple discovery of logic around 1900 — Interactions between Philosophy, Mathematics and the Cultural world

Abstract: Multiple discoveries, especially when the individual pathways leading to the new discovery seem mutually incompatible, have the potential to highlight massive shifts in the system of the sciences (in the broad sense of the Dutch wetenschappen or the German Wissenschaften) as a whole. An especially dramatic and intriguing instance of a multiple discovery can be found in the development of modern logic, i.e. a logic that is based upon or orientated towards mathematics and the natural sciences, at the end of the 19th century. Players in the quest for priority in the discovery of modern logic are, first of all, a group of mathematicians, mathematical logicians and mathematico-logical philosophers that form the standard ancestry of modern logic: Boole, de Morgan, Frege, Peano, Schröder, Whitehead, Russell, and many others. There are, however, rival claims that, interestingly, to us appear almost invariably obscure, though they were raised by scientists of the highest esteem; among the more vigorous, and today more obscure, claims of this sort are those brought forward by the chemist Wilhelm Ostwald and the biologist Hans Driesch, but one should also think of more moderate figures such as the psychologist/philosopher Wilhelm Wundt or of a central figure of the philosophical discourse such as Edmund Husserl. These groups are in several interesting and rather confusing ways related, as some examples may highlight: The French mathematician and logician Louis Couturat reviews Whitehead, explains Russell, and works together with Ostwald in projects aimed at establishing new artificial languages; Whitehead, Ostwald and Driesch are united in not alone being exponents of an allegedly new logic, but also of another by then new discipline, namely a new philosophy of nature; most important, perhaps, is that the mathematical background of which theories did inspire the new logic is universally shared among all these researchers: relevant are, especially, new algebraic theories such as Grassmann’s calculus of extensions, the theory of groups, and generalizations of number systems.

These multiple claims to the discovery of modern logic, in each case based upon a shared background in mathematics and upon strictly scientific criteria, clash with our intuition of what has to count as "logical" or "scientific", thus indicating that the fundamental intuitions as to what has to count as scientific have changed, but not simply as a consequence of the invention of modern logic; rather, it is the other way round: this invention itself is a factor, i.e. a result and a movens, in the changes that lead to these thoroughgoing revisions of our notion of what a science is.

As already the inclusion of such popular and controversial figures as Ostwald and Driesch in the list of pretended discoverers of modern logic shows, this development was by no means restricted to a purely academic discussion, but could rather see itself as entwined with broader developments in the notion and position of science.

A first systematic summary could stress the relevance of generalizations in the period in question. It is no accident that one of Frege's great achievements was a (logical) theory of abstraction, and that Husserl devoted great care to a precise distinction between abstraction and formalization. The role of formal languages might be a central rest case for distinguishing between the genuine mathematical logicians and the rivals that typically do not employ fully formalized languages. On a more general level still, generalization is itself intimately linked with broader, and typically ambivalent, cultural tendencies of this period that run customarily under headings such as a "cultural crisis" or "disenchantment". That mathematical ideas and the development of a new logic played a role in these tendencies can be seen via a particularly salient case study: the widespread unease about the introduction of complex numbers. Though well established in both mathematics and physics, the complex numbers began to cause new problems both for mathematicians (e.g. Hermann Hankel or Frege), philosophers such as Husserl, but were also used by writers such as Robert Musil to illustrate cultural tendencies that might be subsumed under the problem of an increasing uncertainty about man's position in the world. It is also obvious that there is a particularly intimate link between (some of) the relevant new mathematical discoveries and borderline sciences (e.g. in the almost instantaneous employment of theories of multi-dimensional mathematical structures for a "scientific" explanation of parapsychological experiences).

Logic itself, the very paradigm of an abstract science, thus finds itself related to completely different (but in the period itself these differences need not necessarily be maintained!) types of human endeavours, and the history of logic itself proves to be, as the incompatible rival claims show, decidedly not logical.

Investigating these philosophical, mathematico-logical and cultural developments in their interrelation leads not only to a better understanding of a period that proved formative for our understanding of the sciences, but can also contribute to a more differentiated way of looking at the relation between various forms of sciences (in the broad sense) and between the sciences and the cultural world.