Cet article trace le parallèle entre le livre « Homotopy type theory Univalent foundations »:
https://aperiodical.com/2013/06/homotopy-type-theory-a-new-foundation-for-21st-century-mathematics/
Et les « Principia mathematica » de Whitehead et Russell, qui fonde les mathématiques du siècle dernier, et qui est ici :
https://quod.lib.umich.edu/u/umhistmath/aat3201.0001.001/23?view=image&size=100
(En forme très aisée pour la lecture)
https://ncatlab.org/nlab/show/Principia+Mathematica
L’article rappelle qu’il a fallu aux PM 379 pages pour prouver que 1+1=2, tandis que les 250 premières pages du livre HoTT suffisent pour déduire les groupes d’homotopie des sphères à partir des bases de la théorie des types.
Cette page relie HoTT aux théorèmes d’incompletude De Godel, qui datent de 1931, vingt après les Principia mathematica qui sont de 1910:
Le langage formel des Principia mathematica est identique à la théorie des types, inventée par Russell :
https://www.sciencedirect.com/topics/computer-science/principia-mathematica
http://www.nuprl.org/documents/Constable/PrincipiaArticle.pdf
La notation, le langage formel, des Principia peut être étudié ici :
Les espaces sont les objets de l’ ∞-topos Spaces :
Le texte de Nima Rasekh est ici :
http://guests.mpim-bonn.mpg.de/rasekh/talks/PrelimPresentation.pdf
Logics of Worlds is the sequel to Alain Badiou’s much-heralded masterpiece, Being and Event. Tackling the questions that had been left open by Being and Event, and answering many of his critics in the process, Badiou supplements his pioneering treatment of multiple being with a daring and complex theory of the worlds in which truths and subjects make their mark – what he calls a materialist dialectic. The radical recasting of ontology in Being and Event is followed and complemented here by a thoroughgoing transformation in our very understanding of logic, conceived as a theory not of being but of appearing.
Unafraid to resurrect and reinvent the classical themes of philosophy, Badiou gives new meaning to concepts such as object, body and relation, mobilising them in arresting studies that range from the architectural planning of Brasilia to contemporary astronomy, and confronting himself with towering philosophical…
View original post 19 more words
