The Grothendieck’s Toposes as the Future Mathematics of AI

Abstract

The aim of this article is to discuss the idea that the notion of topos could become the future mathematics of AI, by giving more emphasis to geometric forms, compared to the currently mainstream approach, which favors numbers through statistical procedures. AI engineers are looking for theories that can formalize the basic elements that shape their daily work, as well as the basic operations that structure how the human brain works. The notion of topos could contribute largely to satisfying these two main aspirations of AI engineering. The article is a part of a questioning on the nature of intelligence and supposedly intelligent machines. In this respect, it would be useful to enrich the approach that assimilates the topos as a kind of bridge that removes all the inessential aspects of things to be connected to keep an common essence that associates mathematical things. One of the most promising avenues from the archeology of the discourse on the topos is the reading by meaning. This art of reading well gives rise to the notion of conceptual strata which shows that the things to be connected have a layered configuration. What seems insignificant –in view of a supposed essence– through one conceptual stratum, might seem important through another, even capital. Rather than calling something ‘insignificant’, a consistent relativism of ‘insignificant for’ would be more appropriate, according to the conceptual stratum considered.