Entanglement of inner product, topos induced by opposition and transformation of contradiction, and tensor flow

Abstract

The law of unity of opposites, the mechanism of mutual change of quality and quantity, and the rule of dialectical transformation have become the key fundamental problems that need to be addressed in Intelligence Science. It is shown that the spatial-time position x t(u) of object u is the attribute describing where u is existing, the distance d(xt(u),yt(v)) between u and its contradiction v is the expressing relation distinguishing u from v. Based on the mechanism of distance vary with position change ∆xt was controlled by the law of unity of opposites, such that the description of the law can be transformed into a physical problem. By mean of three equivalent definition of distance, some of mathematical construction for describing physical move of u and v, such as Polarization Vector of Inner Product, Entangled Circle, Entangled Coordinates and Clifford Algebra can be induced, such that the Entangled relation both u and v can be transformed into a mathematical problem. The spatial-time position collection {z(xt, yt)} with the collection of time arrows and the displacement arrows (∆xt, ∆yt, ∆t) constructs a category E. A quantity xt belong to a corresponding quality qv(u), during xt varies with time change ∆t in the qualitative criterion [xi, xt), the mechanism that qv(u) is maintaining the same can be abstracted to be a Qualitative Mapping τ(xt, [xi, xT)) from a quantity xt into quality qv(u), and the regulation of different quantity convert into different quality of new quality, can be represented by the Degree Function of Conversion n(xt), a Cartesian Closed Category can be gotten by τ(xt, [xi, xt)) and n(xt). A subobject classifier can be induced by the mechanism of a quality is changing to a simple (or non-essential) quality, so an Attribute Topos can be achieved by them. Tensor Flow and a Fixation Image Operator, and an approach for Image Thought has been presented, and their applications in Noetic Science and Intelligent Science are discussed too.