China
Africa
Explore chapters and articles related to this topic
Introduction
Published in James Juniper, The Economic Philosophy of the Internet of Things, 2018
Kato and Nishimura (2013) have published a recent article applying topos theory to cognition. These topos-theoretic techniques that were originally developed for the analysis of quantum gravity as a core element of unified field theory—effectively, by achieving a structural and dynamic integration between quantum mechanics and general relativity theory—have been applied to a specific model of brain activity or thinking. To achieve a formal representation of quantum gravity, relativistic notions of phenomena such as the light cone, gravitational effect by mass, black holes, and the big bang have also been formulated. The main epistemological advantage of this specific application of the t-topos is that it provides a unifying theory of both microcosm and macrocosm based on mathematical notions of a (micro) decomposition of the associated presheaf and a (micro) factorization of the relevant morphism that can be associated with the t-site. It is this aspect of t-topos theory that comes to the fore in the authors’ work on cognition. As Peter Johnstone (1977: 17) has explained: What, then is the topos-theoretic outlook? Briefly, it consists in the rejection of the idea that there is a fixed universe of “constant” sets within which mathematics can and should be developed, and the recognition that the notion of “variable structure” may be more conveniently handled within the universe of continuously variable sets than by the method, traditional since the rise of abstract set theory, of considering separately a domain of variation (i.e. a topological space) and a succession of constant structures attached to a point of this domain. In the words of F. W. Lawvere [1973], “Every notion of constancy is relative, being derived perceptually or conceptually as a limiting case of variation, and the undisputed value of such notions in classifying variation is always limited by that origin. This applies in particular to the notion of constant set, and explains why so much of naïve set theory carries over in some form into the theory of variable sets”. It is this generalization of ideas from constant to variable sets which lies at the heart of topos theory; and the reader who keeps it in mind, as an ultimate objective, whilst reading this book, will gain a great deal of understanding thereby.
Constructing condensed memories in functorial time
Published in Journal of Experimental & Theoretical Artificial Intelligence, 2023
Intuitively, a presheaf assigns some set of data to each point, or neighbourhood of points, on a topological space . Since the presheaf operates on all open sets of , the assignments must be self-consistent, i.e. larger neighbourhoods inherit the assignments of data to their constitutive points. Taking to be a set of events, we can regard typings as data assigned to the events. Hence, we can state: