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# Séminaire – Histoire et Philosophie des Mathématiques

## April 22 @ 9h00 - 17h30

**Generality**

Organization: Simon Gentil

**Programme**

**9h30-10h**

Accueil des Participants

**10h-11h15**

**David Waszek** (ENS, ITEM)

*Expressive Means and Generality : Some reflections on the use of algebra in geometry in the early seventeenth century*

Résumé : The use of algebra is often said to make geometrical problem-solving more “general” : problems that, in traditional diagram-based geometry, would have required separate treatments then become susceptible of a unified solution. This talk looks at how such issues actually played out in early-seventeenth-century applications of algebra to geometrical problems, with a special focus on thorny issues regarding problem individuation and the geometrical interpretation of negative results. This talk is partly based on work by, and joint work with, Ken Manders.

**11h15-11h30**

Pause

**11h30-12h45**

**Agathe Keller** (CNRS, SPHERE)

*Commenting on general statements : some working reflections on practices of generality in treatises and commentaries dealing with mathematics in Sanskrit.*

Résumé : What tools can we use to try to describe an actor’s understanding of “generality”, when such an actor does not explicitly state much about this topic, while as historians we are tempted to interpret his or her mathematical statements as reflecting such an aim ? This presentation will look at three sets of early mathematical texts in Sanskrit, investigating how authors and commentators understand the grouping of things that might be considered distinct. The two first case studies- one on Āryabhaṭa’s mathematical sūtras (499) and Bhāskara’s commentary on them (629) on the one hand, and Brahmagupta’s sūtras on mathematics and mathematical astronomy (628) and Pṛthūdhaka’s commentary (ca 850)- deals with mathematical objects and processes. The practices of specification (uddiś-) and variation (udahṛ-) of commentators will be a way to reflect on what then characterizes for them the statements that elicit their mathematical work. The third case is a reflection on what appears as an enduring standard of the representation of mathematical computations on a working surface in manuscripts : I suggest to consider them as representing a generic situation of computation, representing many possible computational media and instruments. In other words, this presentation will describe a variety of practices of generality in Sanskrit mathematical sources, a fact worth noting in itself as it goes against homogenizing conceptions of mathematical practices from South Asia.

**12h45-14h30**

Repas

**14h30-15h45**

**Marco Panza** (Chapman University and IHPST)

*What generality might have been for Euclid ?*

Résumé : In this talk, I’d delineate a conception of generality, which does not depend on any sort of universal generalization of geometrical objects, that seems compatible with Euclid’s conception of geometry and might explain the way he reaches universal results, tough working on particular instanciations.

**15h45-16h00**

Pause

**16h-17h15**

**Filippo Constantini** (CNRS, SPHERE)

*Absolute Generality without a Universal Domain*

Résumé : In this talk, after a brief introduction to the absolute generality debate, I introduce the notion of indefinite extensibility and argue that we can have absolutely general statements about an indefinitely extensible sequence of domains. As a consequence, absolutely general statements do not require the existence of a universal domain of everything.

**Salle : **628 (6è étage, bâtiment Olympes de Gouges)