Sciences, Philosophie, Histoire – UMR 7219, laboratoire SPHERE

Starting from the case of the edition of a Babylonian mathematical procedure by several scientists in the middle of the twentieth century, I would like to show how the modern mathematical formulas appearing in the commentaries to explain the ancient text, far from constituting transparent readings, concentrate numerous tacit assumptions underlying the interpretation. They are thus a sort of historiographical revelation that should be handled with caution at a publishing company. We will evaluate with the audience the possibilities of generalization and circumvention.

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– Mourtaza Chopra : The mathematical astromomy we found in Babylonia, southern Mesopotamia, is apparently in the form of recipes. Is it a kind of extreme positivism and pragmatism or can we think that this is only the tip of an iceberg? and if so, what can we know or say about what motivates it or, at least, directions that preside over its constitution?

– Sarah Hijmans : An expression found in many historians and philosophers of chemistry is that chemists think as much with the hands and the five senses as with the brain. Thus, chemistry is not limited to propositional knowledge, since a large part of chemical knowledge can only be learned in the laboratory. We will compare the notion of Polanyi’s tacit knowledge with the conception of knowledge as allowing us to act as we can find in Peirce, and we will ask ourselves how these ideas are adapted to chemistry. How could the professional chemists’ experiences of Peirce and Polanyi have formed their conceptions of knowledge?

– Benoît Duchemann : An introduction to the "tacit dimension" of Polanyi and the sociocultural character that knowledge of the same name can assume: In what way can the tacit knowledge and powers defined by Polanyi apply to artificial intelligence and more specifically to neural networks? How then to differentiate these "thought techniques" from specifically human tacit knowledge? Can they then be commensurable?

– David Waszek : Unspoken knowledge and use of reasoning supports in mathematics: To speak of tacit knowledge in mathematics may seem paradoxical: it is often imagined that this discipline approaches, more than any other, an ideal of complete explanation of reasoning and procedures. Yet such a notion is often used by recent authors (eg Netz 1999 or Ferreiros 2016) to argue that mathematical knowledge is embedded in shared, nonverbal material practices and, as Netz writes, "’Invisible’ to practitioners" (p.2). I will be particularly interested in one of Netz’s suggestions, which, in the case of ancient geometry, places such nonverbal practices in the use of figures. I will discuss his thesis and compare it with other simple historical examples of the use of reasoning supports in mathematics (Euler diagrams, symbolic writing of differential calculus).

References: Netz, Reviel (1999). The Shaping of Deduction in Greek Mathematics, Cambridge: Cambridge University Press / Ferreirós, José (2016). Mathematical Knowledge and the Interplay of Practices. Princeton and Oxford: Princeton University Press

From the concept of "paper tools", developed by Ursula Klein, we will deal with the question of the manipulation of diagrams in chemistry, physics and mathematics. Based on the development of molecular formulas in the 19th century, Klein argues that these representations are comparable to laboratory tools rather than objects of study: just like laboratory instruments, diagrams can be manipulated to study indirectly objects, give results that confirm or refute theories, and open up new perspectives. This idea of ​​operations or manipulations that are done on paper (or other material supports) will allow us to take a particular look at the diagrammatic representations found in the different sciences and to question the boundary between theoretical practice and experimental practice. In particular, we will ask to what extent we can apply the notion of operations on paper, forged on the basis of examples from chemistry, in the case of mathematical figures and diagrams used in particle physics.

Preparation session for the day of the Doctoral School 400. This session of the PhD students’ seminar will be devoted to the preparation of the scientific day of the ED 400, which will take place on March 13, 2019, the theme of which will focus on the issues related to the proof and the result in the different disciplines of the ED. Based on reflection materials prepared by the speakers of this day, we will discuss some questions, such as: What are the limits of the evidence in didactics and by extension in our respective disciplines? What are the strategies that researchers can adopt to circumvent and / or surpass these limits? Given the variety of disciplines doctoral students in this doctoral school, the goal will be to take advantage of our respective knowledge to feed a reflection on the issue of evidence.

Mary Somerville (1780-1872) was an active member of both the London- and Edinburgh-based communities of scholars who publicly advocated for the adoption of ‘French analysis’ into British mathematics in the nineteenth century. Her most influential contribution was the 1831 English adaptation of Laplace’s Mecanique Celeste, in which she showcased the fecundity of analysis as applied to celestial mechanics. However, this work was deemed inaccessible and useful only to the most high-achieving Cambridge students. Hence in 1834 Somerville produced a complementary treatise entitled Theory of Differences, in which she gives an introduction to the differential calculus which had been assumed in her earlier book.
Left unpublished, an extant manuscript of Theory of Differences is held by the Bodleian Library, Oxford. The treatment of emergent ideas on infinitesimal quantities, power series, and functions contained therein can be seen as Somerville’s own novel synthesis of work by concurrent mathematicians, consciously written so as to appeal to a British readership. For example, she utilises the nomenclature of Lacroix, the notation of Lagrange, but makes no reference to the arithmetical limits of Cauchy. Early drafts of passages, contained in a separate notebook, contain criticisms of both her own and Lagrange’s work which further illuminate Somerville’s re-writing process. In this talk I will examine these two archival resources and highlight the new insight they offer into the accessibility of, and contemporary attitudes towards, the differential calculus in 1830s Great Britain.

In the 1860s, French geometer Michel Chasles (1793-1880) created a ‘theory of characteristics’ that, after German mathematician Hermann Schubert’s 1879 Kalkul der abzählenden Geometrie, soon became the bedrock of a branch of geometry known as ‘enumerative geometry’. In the last years of his life, Chasles expanded on these results and published long series of propositions, each expressing a property of a geometrical curve. These propositions were merely listed one after the other, and sorted into various categories, with very little in the way of commentaries, proofs, or examples. Schubert’s book inherits from Chasles’ list-making practice, but also alters it, as the lists it displays consist of huge tables of numbers and symbolic formulae, given without verbal descriptions or explanations.
Chasles’ and Schubert’s lists aim to address the same geometrical problems, but they differ both in their textuality and in the epistemic tasks they fulfil. Indeed, Chasles’ must be read against the backdrop of a specific epistemic culture, wherein the generality of a method is demonstrated by the fact that large numbers of propositions can be derived from its systematic and uniform application. Schubert, instead, viewed Algebra as a free human creation, bounded only by the requisite that certain symbolic forms, drawn from the realm of concrete, natural numbers, be regarded as valid when extended to more abstract entities. Thusly, Schubert’s lists of formulae express the formal rules of a geometrical calculus. In both cases, different epistemological virtues were expressed through different list-making practices. As such, the transfer of these lists between epistemic cultures resulted in their rewriting. This transformation operated at the levels of both the textuality itself and the values of generality they expressed. By untangling the complexities of this transformation, this case-study illuminates how the literary devices used to structure and convey mathematical knowledge change according to the concerns and values of different scientific cultures.

Mathematicians and historians use techniques to produce knowledge. These techniques are the result of a technology, writing, which allows just a whole lot of ways to organize the world on written support. For example, the list allows classification, and thus could be analyzed among Mesopotamians as an indispensable tool for the development of the administration, both men and goods. The painting is another way of synthesizing representations, and thus of classifying, of ordering observations, results or data. These intellectual technologies, which allow a memorization of the intellectual operations as much as a synthesis of scattered elements, are vectors of science as much as scientificity, but they are also tools which are transmitted within the scientific communities. They can still be the vector of problem-solving methods, and are a great object for the historian of science who is interested in modes of reasoning and algorithms.
After a quick presentation of the pioneering work of Jack Goody and Daniel Bell on intellectual technologies, we will take a closer look at the changes involved in the application of new intellectual technologies to problems already known by the actors, in mathematics and history.