Mathematical Models of Abstract Systems: Knowing abstract geometric forms
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 22 (2013) no. 5, p. 969-1016

Scientists use models to know the world. It is usually assumed that mathematicians doing pure mathematics do not. Mathematicians doing pure mathematics prove theorems about mathematical entities like sets, numbers, geometric figures, spaces, etc., they compute various functions and solve equations. In this paper, I want to exhibit models build by mathematicians to study the fundamental components of spaces and, more generally, of mathematical forms. I focus on one area of mathematics where models occupy a central role, namely homotopy theory. I argue that mathematicians introduce genuine models and I offer a rough classification of these models.

Les scientifiques construisent des modèles pour connaître le monde. On suppose, en général, que les mathématiciens qui font des mathématiques pures n’ont pas recours à de tels modèles. En mathématiques pures, on prouve des théorèmes au sujet d’entités mathématiques comme les ensembles, les nombres, les figures géométriques, etc., on calcule des fonctions et on résout des équations. Dans cet article, je présente certains modèles construits par des mathématiciens qui permettent d’étudier les composantes fondamentales des espaces et, plus généralement, des formes mathématiques. Cet article explore principalement la théorie de l’homotopie, secteur des mathématiques où les modèles occupent une place centrale. Je soutiens que les mathématiciens introduisent des modèles au sens courant du terme et je présente une première classification de ces modèles.

@article{AFST_2013_6_22_5_969_0,
     author = {Marquis, Jean-Pierre},
     title = {Mathematical Models of Abstract Systems: Knowing abstract geometric forms},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 22},
     number = {5},
     year = {2013},
     pages = {969-1016},
     doi = {10.5802/afst.1393},
     mrnumber = {3154584},
     zbl = {1286.00041},
     language = {en},
     url = {http://www.numdam.org/item/AFST_2013_6_22_5_969_0}
}
Marquis, Jean-Pierre. Mathematical Models of Abstract Systems: Knowing abstract geometric forms. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 22 (2013) no. 5, pp. 969-1016. doi : 10.5802/afst.1393. http://www.numdam.org/item/AFST_2013_6_22_5_969_0/

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