Séminaire – Histoire et philosophie des mathématiques de l’Antiquité à l’âge classique

Extensions de la notion de nombre

Pascal Crozet (CNRS-SPHère UMR 7219 – Université Paris Cité)
Nombre et grandeur : quels rapports, quels usages, quelles différences ? Réflexion sur l’atténuation progressive d’une distinction, de l’avènement de l’algèbre à Kamāl al-Dīn al-Fārisī (XIVe siècle)

Veronica Gavagna (Université de Florence, Italie)
… sed quaedam tertia natura abscondita : new numbers and new signs in Italian Renaissance mathematics
Résumé :
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)
Les enjeux du continu en arithmétique au XVIIe siècle
Résumé :
Les représentations géométriques des entiers et le développement d’un symbolisme algébrique unifiant toutes sortes de quantités favorisent au XVIIe siècle une extension des problèmes sur les nombres, des entiers à d’autres sortes de nombres ou de quantités. A partir de quelques exemples, je me propose de revenir sur les enjeux tant conceptuels que mathématiques posés par ces extensions et les réactions qu’elles ont suscitées.

David Rabouin (CNRS-SPHère UMR 7219 – Université Paris Cité) et Arilès Remaki (CNRS-SPHère UMR 7219 – Université Paris Cité)
L’origine et l’usage de terme “transcendant” chez Leibniz

lieu : salle internationale (324) – Archives Henri Poincaré, 91 av . de la Libération, 3° étage, 54 000 Nancy