Organisation : Pascal Crozet, Vincenzo de Risi, (CNRS, SPHere), Angela Axworthy (Université de Milan), Sandra Bella (Archives Poincaré, Nancy)

**PROGRAMME 2023-2024**

Friday, 13th October 2023, 10.30 am - 1.30 pm (followed by lunch at 1.30 pm), location : room Internationale (324) - Archives Henri Poincaré, 91 av . de la Libération, 3° étage, 54 000 Nancy

**Analysis in Greek mathematics.**

**Felix Zheng** (Budapest)
*What is a proposition in Euclid’s Elements ? — From shape point of view*

The aim of this research is not to provide an overview of Euclid’s Elements, but to offer a delimited investigation with precise and well-demonstrated results, primarily achieved through the analysis of the structure of a proposition in the Elements. Firstly, I will revisit a deficiency in the Proclean division for the structure of propositions in Euclid’s Elements and provide a much more detailed formal description of the type of problems. Using my formal description as an analytical tool, I will reveal the differences between a proposition in the Data, which appears as a problem, and one conforming to the problems in the Elements. This formal comparison will also help us better understand why Pappus ranks Euclid’s Data first in his enumeration of Greek works in the field of analysis.

**Gianluca Longa** (Clermont-Auvergne)
*Can we do without Pappus to understand ancient geometric analysis ?*

The description of the method of analysis and synthesis proposed by Pappus in the introduction to Book VII of his Collection has served as the starting point for almost all modern interpretations of this method in Greek geometry. We believe this to be a mistake. Indeed, we will demonstrate that Pappus’s description should not be regarded as ’the most elaborate utterance on the subject’ (Heath), but rather as a source that attempts, certainly with great stylistic skill, to assemble and interpret the texts at his disposal with rhetorical and didactic objectives that do not necessarily align with the intentions behind the original texts. Consequently, this presentation aims to show that taking Pappus as the starting point for the interpretation of ancient analysis ultimately leads to constructing a partial, or even inaccurate, image of the practice of analysis and synthesis.

Friday, 17th November 2023, 10.30 am - 1.30 pm (followed by lunch at 1.30 pm), location : room 628 ’Olympe de Gouge’ building, place Paul Ricoeur Paris 75013

Philosophy of mathematics in the Middle Ages
**Amos Corbini** (Turin)
*Between ideal and reality. Mathematical disciplines and the theory of proof in the medieval Latin tradition of the Second Analytics (13th-14th century)*

**Sabine Rommevaux-Tani** (Bordeaux-Montaigne)
*Styles of proofs in some texts of mathematics and natural philosophy from the 14th century*

Friday, 8th December 2023, location : Room 002 - Archives Henri Poincaré, 91 av . de la Libération, 3° étage, 54 000 Nancy

**Teaching mathematics in the 18th century**

**Davide Crippa** (Università Ca’ Foscari, Venise)
*Teaching Mathematics in the Eighteenth Century : Giovanni Poleni at the University of Padua*

This contribution falls within the framework of the microhistory of education and focuses on the context of mathematics teaching at the University of Padua during the 18th century. The period between 1700 and 1720 in Northern Italy witnessed a particularly pronounced mathematical renaissance, especially within the universities of Bologna and Padua. This renaissance can largely be attributed to the spread of Leibniz’s new calculus in the peninsula, disseminated through his writings, as well as through the manuals of L’Hopital and the handwritten courses of Johann Bernoulli. The rise of mathematics and Newtonian physics also played a decisive role in this revival in Italy. However, an examination of the printed curriculum programs during most of the 18th century reveals that infinitesimal calculus was not among the disciplines taught at the advanced level, with a few rare exceptions. In the case of the University of Padua, for example, the archives of course programs, or "Rotuli," show that the study of Euclid, enriched with "its applications," constituted the predominant element of university mathematics education until the second half of the 18th century. Nevertheless, what these programs do not reveal – and what will be the subject of my presentation – is that even in the case of a classical subject like Euclid, teaching relied on the use of contemporary sources, thereby incorporating knowledge from more recent literature that circulated among students. My focus will be particularly on the pedagogical activity of Giovanni Poleni, who held the mathematics chair from 1719 until his passing in 1761. However, the scope of my contribution extends beyond specific temporal and geographical boundaries. Considering the teaching of sciences as a means of transmitting knowledge raises several crucial questions that can be generalized to other contexts : What texts were used and disseminated at the time ? How did their selection and adaptation respond to the specific needs of university administrators ? Finally, to what extent were recent developments, including the most contemporary mathematical advances, transmitted through academic courses ?

**Pierre Ageron** (Université de Caen)
*Manuscripts of Courses : Witnesses of Mathematics Education and Circulation in 18th-Century in France*

The manuscript notebooks containing the lessons given by teachers in Latin or French are the most tangible material traces of mathematics teaching in the 18th century. What actual practices do they reveal ? For what purposes and under what conditions were they copied, preserved and disseminated ?

We will focus on two sets of mathematical treatises whose manuscripts we have identified and examined, corresponding to the lessons of Pierre Varignon at the Collège Mazarin from 1688 to 1722 and those of Yves-Marie André at the Jesuit Collège de Caen from 1726 to 1759. We will also look at other manuscripts, often anonymous, held in libraries in Normandy.

We’ll look at what these manuscripts reveal about the tension between Euclidean tradition and new methods, the articulation between pure and mixed mathematics, the pedagogical program and teachers’ expectations. We give examples of how to interpret paratextual elements and detect various forms of intertextuality.

Finally, we’ll offer a few ideas for comparing mathematics teaching manuscripts in eighteenth-century France with those in Arab and Muslim countries at the same time.

Friday, 19th January 2024, 10.30 am - 5.30 pm, location : Room 628 Olympe de Gouges Université Paris Cité 8 Rue Albert Einstein Paris 75013

**Curves and generality in the classical age**

Organisateurs : Sandra Bella (Archives Poincaré) et Simon Gentil (SPHère)

During antiquity, a curve was rarely studied as part of a set ; it was typically examined in isolation. The invention of symbolic notation (by Viète and Descartes) led to an entirely new way of solving geometric problems and an increase in the number of available curves for geometers, particularly for constructing solutions to problems. The proliferation and usefulness of curves led them to become a separate object of study. The primary goal was to establish criteria of accuracy (Bos) that would allow the inclusion of curves within geometry. It is well-known that, in this regard, *La Géométrie* (1637) provides a categorical answer : only curves with an algebraic equation are admitted, and this equation represents their general form. Leibniz broadened the admissible field to include transcendental curves, thus introducing a new form of generality. Analytical methods for determining the properties of curves were devised by geometers in accordance with the concept of generality they had established. It becomes apparent that the concept of a curve emerges inherently linked to the forms of generality induced by the introduction of symbolic notation.

This day seeks to deepen our understanding of how the epistemic value of generality shapes the concept of a curve in the early stages of classical analysis. This exploration is conducted through the examination of mathematical practices of various authors, both minor and major. Our analyses will particularly focus on the different attempts made by these actors to define and implement a general idea of a curve. We will explore how this idea guides the organization and classification of a set of curves or induces the invention of methods, also of a general nature, applicable to a set of curves.

**Program**

- 10.30 am - 12.30 pm
**Olivier Bruneau**(Archives Poincaré)*« Podal curves at MacLaurin : towards a generalization of the curve »*

Abstract

In a short article published in 1718 in the Philosophical Transactions of the Royal Society, Colin Maclaurin (1698-1746) introduced a "new" class of curves, which from the 1840s onwards would be known as podars. Isaac Newton encouraged the young Scotsman to develop his research, and in 1719, Maclaurin published his Geometria Organica, in which a section is dedicated to the description of this type of curve. As far as we know, Roberval was responsible for this construction, but he only applied it to the cycloid. At the same time, podar coordinates were relatively well established and were used for central forces.

In this talk, we’ll present Maclaurin’s results, comparing them with the practice of podar coordinates, and attempt to show how the work of this young scientist remained isolated throughout the 18th century.

**Sandra Bella**(Archives Poincaré)*« The infinitangular polygon : a general concept of Leibnizian calculus ? »* - Lunch
- 2.00 pm - 4.00 pm
**Thierry Joffredo**(Archives Poincaré)*« Gabriel Cramer’s analytical triangle as a representation and tool for studying the general equation of an algebraic curve »*

Abstract

A variation on Newton’s*analytical parallelogram*, borrowed from De Gua de Malves in its triangular form,*the analytical triangle*is a central device in Gabriel Cramer’s*Introduction à l’analyse des lignes courbes algébriques*(published in Geneva in 1750), It is particularly useful in the many examples that populate the pages of the book, when it comes to defining the predominant terms of a particular equation at the origin or at infinity, in order to study the asymptotes, infinite branches or singular points of the associated curve. But this representation of the terms of the general equation of a curve also enabled Gabriel Cramer to state general results about algebraic curves (such as the number of points required to define a curve of a given order), to describe universal methods and procedures for their study and, ultimately, to work on the general classification of third-, fourth- and fifth-order curves. In this presentation, we shall see how Gabriel Cramer uses this device in his work to impose order, universality and generality on this landscape of algebraic curves, barely explored at the beginning of the 18th century, whose "perpetual varieties, constantly recalled to unity, offer the Spirit a spectacle of which it never tires".

**Simon Gentil**(Laboratoire SPHère)*« The General Curve from Descartes to Euler »*

- Friday, 2nd February 2024, location : Room internationale (324) - Archives Henri Poincaré, 91 av . de la Libération, 3° étage, 54 000 Nancy

**Extensions of the notion of number**

**Veronica Gavagna**(Université de Florence, Italie)*... sed quaedam tertia natura abscondita : new numbers and new signs in Italian Renaissance mathematics*

Abstract :

Are the square roots of negative numbers numbers ? And if so, do they obey the usual laws of arithmetic ?

These questions do not arise from trying to solve second-degree equations with negative discriminants, which were thought to be simply impossible, but become inescapable when the solution procedure for third- and fourth-degree equations is found. In fact, in the "solution formula" of Niccolo Tartaglia (1499-1557), square roots of expressions that could be negative also appeared (the so-called "irreducible case"), but in this case it could not be concluded that the equations were impossible, because it was known that they admitted real roots. The irreducible case forced mathematicians of the time to consider the nature of square roots of negative numbers. Girolamo Cardano (1501-1576) studied the irreducible case in the Ars magna (1545), De Regula Aliza (1570), and other writings, but he could not find a way to work with these strange objects, which were neither positive nor negative numbers, but of a different and mysterious nature (tertia natura abscondita), because when raised to the square they produced a negative and not a positive number. Rafael Bombelli (1526-1572) took up the challenge of finding a way to work arithmetically with the roots of negative numbers, the so-called radices sophisticae, and solved it by introducing two new signs, “plus of minus” and “minus of minus”, and by extending the “rule of signs” to include these new signs. In this way, Bombelli had given meaning to the procedure for solving equations of the third and fourth degree, even in the irreducible case, but had he really clarified what the radices sophisticae were ? This is a question that I will try to answer in the course of my talk.

**Catherine Goldstein**(Institut de Mathématiques de Jussieu Paris rive gauche)*Continuous arithmetic in the 17th century*

Abstract :

In the 17th century, geometric representations of integers and the development of an algebraic symbolism unifying all kinds of quantities led to an extension of number problems from integers to other kinds of numbers or quantities. Using a few examples, I’ll look back at the conceptual and mathematical issues raised by these extensions and the reactions they provoked.

**David Rabouin**(CNRS-SPHère UMR 7219 – Université Paris Cité) et**Arilès Remaki**(CNRS-SPHère UMR 7219 – Université Paris Cité)*The origin and use of the term "transcendent" in Leibniz*

Friday, 15th March 2024, 9.30 am - 1.00 pm, location : Room 628 Olympe de Gouges building Université Paris Cité 8 Rue Albert Einstein Paris 75013
**Arabic and Latin mathematics : arithmetic and algebraic questions.**

Program :

- 9.30 am-10.30 am

Marc Moyon (Université de Limoges)*Problem-solving in Liber augmenti et diminutionis (12e s.).* - 10.45 am - 11.45 am

Clelia Crialesi (KU Leuven)*Algebraic Variable and Mathematical Abstraction : An Entanglement of Social Actors and Philosophical Practices in Pre-Modern Italy.* - 12.00 pm - 1.00 pm

Pascal Crozet (Laboratoire SPHère, CNRS)

Numbers and quantities*What connections, what uses, what differences ? Reflections on the gradual attenuation of a distinction, from the advent of algebra to Kamāl al-Dīn al-Fārisī (XIV century).*

Friday, 12th April 2024, 10.30 am - 5.30 pm, location : Room 628 Olympe de Gouges building Université Paris Cité 8 Rue Albert Einstein Paris 75013

Philosophy of mathematics in the Antiquity

**10.30 am-11.30 am**

**Pierre Adam** (Université de Lille)

*From Platon to Euclide, describe what numbers are without adding units.*

Abstract :

Traditionally, there are two main ways of describing a number : ordinal and cardinal. We can then distinguish several subspecies within these two broad categories. The Euclidean definition of number as a plurality made up of units (VII.D2) is one of the subspecies of the cardinal conception.

The aim of this investigation is to determine whether Plato and Euclid can be identified as having any traces of another conception of cardinality, which consists in considering each number as characterized, not by the sum of the units it gathers, but by its divisors. This other conception of cardinality, which can be described as "divisive", gives priority to multiplication over addition, and leads to a privileged role for prime numbers.

**11.45 am-12.45 pm**

**Carole Hofstetter ** (CNRS-SPHère)

*ὑποδιπλάσιoς, ἥμισυς and the description of half in Nicomachus of Gerasa and his readerss.*

**Pause déjeuner**

**2.00 pm-3.00 pm**

Lorenzo Corti (Université de Lorraine)

*Platonists, unity and proof by ecthesis*

**3.00 pm-4.00 pm**

**Chiara Martini** (Université de Cambridge, Corpus Christi College)

*Geometrical Changes. Change and Motion in Aristotle’s Philosophy of Geometry*

Abstract :

It is often said that Aristotle takes geometrical objects to be absolutely unmovable and unchangeable. However, Greek geometrical practice does appeal to motion and change, and geometers seem to consider their objects apt to be manipulated. In this paper, I examine if and how Aristotle’s philosophy of geometry can account for the geometers’ practices and way of talking. First, I illustrate three different ways in which Greek geometry appeals to change. Second, I examine Aristotle’s ontology of geometrical objects and argue that although the truth-makers of geometrical statements are in fact unmovable because they are properties of sensible objects, geometers ‘separate them in thought’ and treat them as substances apt to be modified. Finally, I examine whether allowing for the possibility of manipulating these semi-fictional geometrical individuals creates problems for the applicability of geometry. I find that it does not, insofar as one accepts that geometry is not meant to track physical change but merely to study the instantaneous geometrical configuration of sensible bodies, and is thus only applicable at the instant

Friday, 17th May 2024, location : ROom internationale (324) - Archives Henri Poincaré, 91 av. de la Libération, 3rd floor, 54 000 Nancy

**On the epistemological status of practical mathematics**

program :

- 10.30 am-12.45 pm

Thomas Morel (Bergische Universität Wuppertal)*Quantification practices and the status of mathematics in the modern era*

Abstract

This talk discusses certain developments in the knowledge of quantification in the modern era, with a view to questioning the status of mathematical sciences and practices. While there are now many studies on practical mathematics, as well as on some of its actors (in particular mathematical practitioners), these works most often focus on its impact or reception in scholarly spheres, and in particular on its influence on contemporary natural philosophy.

Another perspective might be to encompass arithmetical and geometrical practices within the technical, economic and even religious activities of their time. It is probably in the field of elementary knowledge, lived and used daily, that the most massive transformations take place - with probable epistemic consequences on the perceptions of mathematical sciences. Seen from this angle, the mathematization of nature in the modern era is a broad social and cultural phenomenon, involving both the scholar and the common man. - Frédéric Métin (Université de Bourgogne)
*Human and mathematics*

- 2.00 pm-3.00 pm
- Dominique Raynaud (Université de Grenoble)
*A few remarks on geometric constructions in practical geometry treatises*

Tuesday, 4th June 2024, Room 628 Olympe de Gouges building (6th floor) Place Paul Ricœur Paris 75013

**Principles of Demonstration in Late Antiquity**

Organized by : Vincenzo De Risi et Ulysse Chaintreuil

You can follow the session via Zoom, for the link please write tosphere-communication@cnrs.fr (objet:AX2024)

Program

- 9:00

Welcoming participants - 9:30-10:45
**Orna Harari**(Tel Aviv University)*Epistemologizing Aristotle’s Theory of Demonstration : The Ancient Commentators on the Principles of Demonstration* - Break
- 11:00-12:15
**Maria Fiorella Privitera**(Université de Genève)

*Philoponus on the hoti esti. In APo. 1. 1, 2, 10* - Lunch break
- 13:45-15:00
**Alvise Lagnerini**(Université de Genève)*Exploring Aristotelian Induction through the Lens of Alexander of Aphrodisias* - Break
- 15:15-16:30
**Maddalena Bonelli**(Università di Bergamo)*Alexandre d’Aphrodise et les axiomes de la philosophie*

. . . . . . . . . . .

Building Condorcet, Université Paris Cité, 4 rue Elsa Morante, 75013 - Paris*.Map

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Metro : lines 14 and RER C, stop : Bibliothèque François Mitterrand or line 6, stop : Quai de la gare. Bus : 62 and 89 (stop : Bibliothèque rue Mann), 325 (stop : Watt), 64 (stop : Tolbiac-Bibliothèque François Mitterrand)