Arvanitis, Rigas

Grossetti, M.

Revue d'Anthropologie des Connaissances

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Counting, demonstrating, formalizing

Rigas Arvanitis

Michel Grossetti

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Revue d anthropologie des connaissances

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Revue d’anthropologie des connaissances – 2009/2 a
SPECIAL ISSUE “COUNTING, DEMONSTRATING, 
FORMALIZING”
COUNTING, DEMONSTRATING, 
FORMALIZING
RIGAS ARVANITIS AND MICHEL GROSSETTI
Calculation and mathematical formalization is an important aspect of knowl-
edge. For this reason, our journal is devoting a special feature to this theme 
which is in all probability only the fi rst of a series to come. 
For this feature we have chosen: two authors of reference in the sociology 
of sciences, Bruno Latour and Donald McKenzie; an author on the way to 
becoming an international reference on the question of demonstration, Claude 
Rosental; two young researchers who have written some remarkable theses, 
the fi rst, Jérôme Lamy, on the history of an astronomical observatory and the 
second, Hélène Guillemot, on climate modelling in France. These diverse arti-
cles enable us to give a picture of the multiple facets of the activities of calcula-
tion, demonstration and formalization and the way to approach them in the 
social sciences.
Bruno Latour presents the recent book by Reviel Netz, The Shaping of 
Deduction in Greek Mathematics: A Study in Cognitive History (2003, Cambridge, 
Cambridge University Press), which he considers as “the most important book of 
science studies to appear since Shapin and Schaffer’s Leviathan and the Air-Pump”, 
and “offers for the origin of formalism what Shapin and Schaffer have done for 
the origin of experimental science”. What interests Bruno Latour in Netz’s 
book is the central place it gives to graphical (diagrams) and written (math-
ematical language itself) formalizations, the materialization of relationships that 
mathematicians endeavour to preserve in whatever support they are using.
Donald McKenzie reviews the notion of “fi nitism” put forward long ago 
by Hilbert (which can generally be summarized by saying that mathematical 
entities can only result from fi nite developments) and taken up again by histo-
rians and sociologists of sciences including in particular Barry Barnes, in the 
sense of an impossibility of identifying general properties that would stand up to 
the effects of context (which sums up well the quotation from Barnes pointed 
up in McKenzie’s article (“[N]othing in the rule itself determines its applica-
Revue d’anthropologie des connaissances – 2009/2b
tion in a given case” (Barnes, 1995, p. 2002)). McKenzie uses the perspective 
of fi nitism to shed light on mechanisms of accounting starting from the case 
of a company quoted on the stock exchange using an integrated management 
software (Enterprise Resource Planning, ERP). Interviews with members of the 
fi nancial department of this company enabled him to show that in spite of the 
constraints imposed by ERP, the application of nomenclatures is always contex-
tual. That enables him to argue that the regulation of accounts by an ever more 
precise formalism is doomed to failure. 
Claude Rosental has conducted a considerable amount of research on the 
question of demonstration, delimiting a space which embraces formal math-
ematical demonstration as much as the demonstration of technical artefacts 
similar to the demonstration method concerning household electrical appliances 
(Rosental, 2003). He aims to show in this article, which refers back to several 
of his previous research studies, that “the demonstrations cannot be reduced 
to a description in terms of proof and persuasion, (…) the latter elements can 
for example play a role of observation points for the demonstrators of trans-
actional support, as an instrument for project management, a capitalization tool 
or indeed a means of establishing relationships”. He then arrives at a broad 
defi nition of the demonstration: “it appears in fact that as a general rule, the 
term ‘demonstration’ can be usefully employed to refer to a written or audio-
visual development, a progression, whose avowed purpose is primarily one of 
examination and/or of argument, or even instruction, but which can just as well 
play other roles”.
Jérôme Lamy focuses his attention on the calculation practices of Toulouse-
based astronomers, from the aristocratic scholars of the XVIIIth Century to 
the university academics of the early XXth Century. He brings out the impor-
tance of those “invisible technicians” to whom from the outset the astronomers 
subcontracted a portion of their calculations. In the beginning it involved a small 
number of associates with varying degrees of versatility. However, in the most 
recent period of his investigation, the beginning of the XXth Century, it had 
become a complete department operated exclusively by female assistants (the 
“ladies’ offi ce”) with a strictly defi ned division of labour, the women concerned 
carrying out basic operations, under the supervision of others with the role of 
“checkers”, and paid according to the degree of diffi culty of their tasks.
Hélène Guillemot examines exchanges between the mathematical models for 
climate and the diffi culties raised by their validation by climatologists’ observed 
data. She centres her examination on modelling practices, studying the work 
of the major climatology laboratories in France. In the case of climate models, 
given the importance of the stakes involved in climate change, the economic and 
social controversies are sometimes conveyed as being as a calling into question 
of the models themselves. She examines the practices adopted by modellers 
as an autonomous activity, distinct from that of experimental physicists and 
climatologists who gather data yielded from observation. Her fi ndings show 
Presentation c
that the forms of validation, or more modestly of assessment of the large-scale 
climate models, become defi ned through a continual to-ing and fro-ing between 
observed data and information gleaned from modelling, but at smaller scales 
than those brought into play by the models. 
Some classical questions of the philosophy of science and epistemology are 
found in these works. The question of the validity of data links back into that 
of their construction − in the true sense, in other words to the construction of 
hypotheses, equations and other formulations which can feed the demonstra-
tion, model or vision of the skies. Correspondence between the reality and 
its model rests on observation points − highly tangible for the astronomical 
or climatological sciences, more dematerialized in the case of mathematical 
demonstration, in Netz or Rosental − and on an intense exchange of data and 
accounts relating to the subjects dealt with. There is a continuum between the 
production of calculations and the observation, an account which links up the 
specifi c formulations necessary for the demonstration, and a network which is 
both social and semiotic in which ideas, models and demonstrations circulate. 
Certain philosophers and social sciences researchers, basing their approach 
on an abstract, ideal conception of scientifi c activity, sometimes fi nd it diffi cult 
to accept this relative fl exibility and fl uidity of scientifi c activity and in particular 
that of scientifi c “data”. Nevertheless researchers, as Hélène Guillemot neatly 
reminds us, accept this more easily when they talk for example of “compensa-
tions for errors” or “assessment of a model” to show that it is “good enough to be 
useful”. Or again, when they admit that this assessment “[…] requires astuteness, 
creativity, more than adherence to a method”. 
All the articles in this special feature emphasize above all, from the diagrams 
of Greek mathematicians to laborious operations of the female calculation assist-
ants of Toulouse observatory, by way of McKenzie’s management programs, 
the robots that supposedly incorporate mathematical results of Rosental, or 
the reworked and “parametrized” data of climate modellers, that mathematical 
activity is not a solely a mental, solitary practice; it is also collective work that 
makes use of material devices and a wide range of supports. 
Latour even maintains that it is the concentration of this collective work in 
mathematical forms which gives it its specifi city. The “fl at laboratory” of math-
ematicians is made up of those fi gures, diagrams and data that have meaning only 
in the mathematical world. Permanent tension is created around this materiality 
between the contextual and the universal, between the “here and now” and the 
“at any time and in any place”. 
Therein lies the second “oversight” of a certain philosophy of sciences, which 
in the end always thinks of a solitary person doing calculations, alone before the 
problems to solve. Yet if there is one strongly distinct characteristic of calcula-
tion, demonstration and modelling activities it is their collective nature. In both 
the senses we could give to this term: by the manipulation of subject matter 
Revue d’anthropologie des connaissances – 2009/2d
which is always the result of a great number of activities conducted by many 
different people, but also in the sense of a collective system of referencing. The 
example of Elkan’s theorems which Rosental brings up is particularly illuminating 
in this respect. There is indeed a collective process of elaboration, with succes-
sions of proofs, and series of demonstrations. But it would be stopping in mid 
path if only this aspect were to be stressed, essential as it may be. These articles 
show also that these chains of operations − by their ever-increasing complexity, 
involving at each stage new “specialities”, specifi c points of expertise, data that 
are increasingly sophisticated to the point of overturning the very meaning of 
the word data (as is the case in climatology where the data become almost more 
malleable than the results from model simulations) − are themselves becoming 
the focal point of scientifi c effort. The data, normally taken to be a kind of 
expression of the observed reality, are the input for, as much as they are the 
result of, this intense effort of a scientifi c world and the subject it studies. This 
reciprocal knowledge held by the actors involved, founded on intense inter-
action around scientifi c topics and resources, should be at the centre of any 
discussion on science and its ability to describe the world.
It must also be pointed out that the subjects of scientifi c study in the 
pages of these articles are indeed of the strangest sort: calculations of celestial 
mechanics, calculations of reduction, geometric fi gures, mathematical relations 
expressed by letters, models, simulations and demonstrations. Our authors set 
out the types of knowledge used, their materiality, their attachment to prac-
tices, their constant fl uidity. What is relatively evident in experimental fi elds 
is more surprising in the more formalized activities: modelling, geometry and 
fi nally the demonstration itself, whether it be in the modern (Rosental) or clas-
sical (Netz) form. 
For our authors, the reality supposedly described for example by the models 
is carried in the data. The latter are not the result of some “magical” inference 
from data towards the model − or the inverse, of a “pure” deduction from 
the model towards the results − but are the product of incessant circulation 
of data, constrained experiments and verifi cations performed on sections of 
models. There is nothing more malleable than the “data”, which only take on 
their formal role as such once integrated in the unceasing fl ow of exchanges 
between distinct mathematical worlds, either in the general space or within the 
speciality.
The data (in the broadest sense, both resulting from models and demon-
strations in the form of results, or as input to tools for calculation, observation 
and simulation) and the more general concepts used to describe the world are 
constructed at the same time, locally. In the scientifi c fi eld of climate, as Hélène 
Guillemot stresses, “the referent with which the simulations are compared is 
not ‘the climate’, it is a set of restructured data selected from the hundreds 
Presentation e
of thousands provided by the instrument networks on the earth, seas and in 
space. Only a world fi tted with instruments, duly prepared for this, is capable of 
providing the data that will put the models to the test.”
The extremely short life of the ethnology of mathematical practice has 
already detected a high degree of variability. This diversity is the most persistent 
result, the one most invariably repeated by the anthropology of knowledge in 
a number of fi elds (see previous issues of our Journal). Diversity of techniques, 
diversity of forms of calculation, diversity of artefacts. This diversity is what 
allows the world to be divided into smaller spaces, which can for example be 
perceived by measurement and observation tools, delimited for experimenta-
tion or defi ned in demonstrations. 
Mathematicians, calculation specialists and “demonstrators” perform an 
enormous effort to keep a world which seems ever determined to escape from 
them within the narrow limits of their models, graphical representations and 
data. Claude Rosental devotes a large portion of his analysis to observation 
of this will to master the “variable docilities and resistances which greet [the 
demonstrations]”. It is this same proliferation of connections with the “real 
world” which has made climate modellers particularly cautious about the possi-
bility of truly “demonstrating” climate change.
Going further into the detail we see a complex world unfurling, one made 
not of the logical and epistemological simplicity that would be the preroga-
tive of mathematics but arrayed with tools, developed, or armed should we 
say. The Greek mathematicians’ graphic representations and formalism were 
some of the fi rst elements of such developments. The assistants assigned to 
help astronomers perform ever more complex calculations were another mani-
festation of this before the arrival of calculating machines and then, of course, 
computers. The “fl at laboratory” of mathematics thus becomes a distinctly 
more strongly dimensioned world. The very notion of calculation sees itself 
transformed because it generates universes that are constructed, virtual, collec-
tive and extremely complex that open up possibilities that are always new to 
research. These universes are moreover international and represent referents 
for political and social discussions with a scope previously unknown, as the 
example of the IPCC shows (Intergovernmental Panel on Climate Change).
All the articles furthermore raise the question of our approach to complexity 
in the social sciences: in the end a core question in these articles. One way is 
proposed by Claude Rosental, who immediately calls it into question, as he 
brings our attention to the stage management aspect of the demonstration. 
The demonstrations Rosental presents are in fact set-up pieces, and for each 
representation, creation of connections with and within specifi c communities 
stem from discussions. They are, Rosental says, the very place for social linking: 
“For researchers reputed to be loners, whose activity proceeds solely in their 
Revue d’anthropologie des connaissances – 2009/2f
mind, and who for the sociologist constitute a priori the extreme fi gure of the 
individual, the demonstration activity often represents one of the main sources 
of social liaison.”
Even stopping at this aspect − and we encourage the reader on the contrary 
to continue this refl ection and read these articles which emphasize many other 
dimensions besides − these are pioneering articles. They testify to the develop-
ment of a fi eld founded on the history and sociology of mathematical forms, a 
fi eld which fully belongs to the anthropology of knowledge that we uphold in 
this journal. 
BIBLIOGRAPHIE  
Rosental C. (2003). La Trame de l’évidence. Paris : P.U.F.  
Barnes B. (1995). The Elements of Social Theory. London : UCL Press.
Rigas Arvanitis, research director at IRD, sociologist, member of the 
research unit « Savoirs et développement ». He studies scientifi c communities 
in the developing world and technological learning in particular in Venezuela, 
Mexico and China. He co-direct recently a special issue on science in non 
hegemonic countries in this journal (RAC 2008/3). He now works on European 
projects on science policy in the Mediterranean region 
ADDRESS:  IRD - Savoir et développement
   32, avenue Henri Varagnat
   93143 Bondy cedex FRANCE
E-MAIL:  rigas@option-service.fr  
WEB:  http://rigas.ouvaton.org
Michel Grossetti, research director at CNRS, sociologist, is member 
of the Laboratoire Interdisciplinaire Solidarités, Sociétés, Territoires 
(LISST). The study of scientifi c institutions is among his research topics. 
He has published among other books, Science, industrie, territoire (1995, 
Toulouse, Presses Universitaires du Mirail) and directed a collection of 
essays with Philippe Losego La territorialisation de l’enseignement supérieur 
et de la recherche. France, Espagne, Portugal (2003, Paris, L’Harmattan). 
Presentation g
ADDRESS:  LISST-CERS, Université de Toulouse le Mirail
   5, allées Antonio Machado
   31058, Toulouse, cedex 9
E-MAIL:   michel.grossetti@univ-tlse2.fr


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