Abstracts.
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 socialcognitive 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
nonskeptical, nonrelativist 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 sociocultural, 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 socialconstructivist 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 
CANCELLED!
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.
Literature.
 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 socalled 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 culturespecific 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 computerassisted
proofs 
Abstract: In this paper we will
study the use and role of the computer in computerassisted 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 wellknown 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 computerassisted 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 worldview 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
"motherstructures", 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.
References.
 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. 2550.
 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. 1324.

D'Ambrosio, Ubiratan (2007a). Peace, Social Justice and
Ethnomathematics. The Montana Mathematics Enthusiast. Monograph 1,
2534.
 D'Ambrosio, Ubiratan (2007b). Political Issues in
Mathematics Education. The Montana Mathematics Enthusiast. Monograph 3,
5156.

Norma B. Goethe
(Cordoba).
Modes of
Representation and Working Tools in Leibniz´s Intellectual
Workshop 
Abstract: My aim is to look at
the interconnection between Leibniz's theory of signs and his actual
practice in writing science during the Paris years (16721676). 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 neoKantian
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 onedimensional 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:
 'Texts' serving the art of invention.
 'Texts' serving the visualization of thoughts, theorems, and
proofs.
 'Texts' as fixation of insights and elaboration of
treatises.
 '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 daytoday
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
videorecordings 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 nearessential 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 highenergy physicists reason, that is, to
explicate the manner in which individuals, both alone and in groups,
arrive at their scientificallyrelevant 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,
sociopsychological, 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 doubleentry
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 socalled
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 doubleentry 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 socioeconomic 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 modelbased 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 modelbased 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 doubleentry 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 lifeworlds. 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 worldconstituting 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.
 JouniMatti Kuukkanen (Durham).
Inevitable or
Contingent History of Science? Respecifying 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
nonsocial determinants. However, this polarized account fails to capture the
complexities of the interactions between social and nonsocial 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 nonsocial 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 respecify 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 (19081944) 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 nonformal 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 semistructured 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 nonstandard in several ways. 
Marc
Staudacher (Amsterdam).
Around 20 notions
of
conventions and
still no clue? 
CANCELLED!
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 TeixeiraPinto
(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 lambdaoperators), 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 apriori, 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 LeviStrauss, 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 apriori system of signs,
supplementary signifiers, zero symbolic value, and mimesis and
iterability as constitutive of signs can lead to a
sociologicophilosophical 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 poststructural 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 mathematicological
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 Grassmanns 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
multidimensional 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, mathematicological 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. 
