New foundations : towards tribal unity

L’auteur part de la distinction entre  cinq tribus différentes dans les disciplines du ML (machine learning ) et de l’intelligence artificielle ( AI):

Five Tribes of Machine Learning

«

  1. Symbolists use formal systems. They are influenced by computer science, linguistics, and analytic philosophy.
  2. Connectionists use neural networks. They are influenced by neuroscience.
  3. Bayesians use probabilistic inference. They are influenced by statistics.
  4. Evolutionaries are interested in evolving structure. They are influenced by biology.
  5. Analogizers are interested in mapping to new situations. They are influenced by psychology

«

mais n’en reconnaît pour sa part que quatre, remplaçant les évolutionnistes et les analogistes par les fréquentistes, qui s’opposent aux bayésiens en statistique mathématique :

New Foundations: Towards Tribal Unity

 

Pour unifier ces différentes tribus,  il cite trois développements dans les mathématiques modernes : mathématiques constructives, avec une logique intuitionniste et épistémique qui rejette le tiers exclus, théorie des catégories et topologie algébrique avec les travaux de Grothendieck sur les topoi et les faisceaux :

»

In recent years, category theory has emerged as the lingua franca of theoretical mathematics. It is built on the observation that all mathematical disciplines (algebraic and analytic) fundamentally describe mathematical objects and their relationships. Importantly, category theory allows theorems proved in one category to be translated into entirely novel disciplines.

Finally, since Alexander Grothendieck’s work on sheaf and topos theory, algebraic topology (and algebraic geometry) have come to occupy an increasingly central role in mathematics. This trend has only intensified in the 21st century. »

La théorie homotopique des types #HoTT est aussi citée :

»In metamathematics, researchers investigate whether a single formal language can form the basis of the rest of mathematics. Historically, three candidates have been Zermelo-Frankel (ZF) set theory, and more recently Elementary Theory of the Category of Sets (ETCS). Homotopy type theory (HoTT) is a new entry into the arena, and extends computational trinitarianism by the Univalence Axiom, an entirely new interpretation of logical equality. Under the hood, the univalence axiom relies on a topological interpretation of the equality type. Suffice it to say, this particular theory has recently inspired a torrent of novel research. Time will tell how things develop. »

ainsi que le « computational   trinitarianism »

Computational Trinitarianism

« Computational trinitarianism is built on deep symmetries between proof theory, type theory and category theory. The movement is encapsulated in the slogan “Proofs are Programs” and “Propositions are Types”. This realization led to the development of Martin-Lof dependent type theory, which in turn has led to theorem proving software packages such as Coq.«

Pour unifier la physique , cet article de John Baez est cité :

http://math.ucr.edu/home/baez/rosetta.pdf

C’est cela que je veux dire en parlant d’une « Pensée absolue » dans les mathématiques d’après 1945 (théorie des catégories)  et Grothendieck (topoi) :

https://anthroposophiephilosophieetscience.wordpress.com/2019/08/03/ce-sont-les-mathematiques-de-mclane-lawvere-et-grothendieck-apres-1945-qui-donnent-raison-a-wronski-un-siecle-plus-tot/

avec aussi l’émergence de HoTT en 2006 : l’unification du savoir (scientifique)

This entry was posted in Categorical quantum mechanics, category theory, Grothendieck, Philosophie, Philosophie mathématique, Physique, Science, mathesis, Science-internelle, Théorie des ensembles (set theory), Théorie des topoi (topos theory), topos physics, Wronski, Wronski-Messianisme-séhélianisme-Science-internelle. Bookmark the permalink.