Related papers: Exploring mathematical objects from custom-tailore…
Topos theory occupies a singular place in contemporary mathematics: born from Grothendieck's algebraic geometry, it has emerged as a unifying language for geometry, topology, algebra, and logic. This book offers a progressive introduction…
The classifying topos of a geometric theory is a topos such that geometric morphisms into it correspond to models of that theory. We study classifying toposes for different infinitary logics: first-order, sub-first-order (i.e. geometric…
Mathematics is a mountain, but students need more than descriptions of the view: they need a trail they can actually walk. This paper presents the Math Teaching Atlas, a framework for mathematical exposition built around route units (single…
We learn mathematics subjectively and must apply it objectively. But sometimes, we apply it subjectively by using wrong intuitions which may be elusive to our eyes. The aim of this note is to disclose the secretes of two kinds of these…
Visual insights into a wide variety of statistical methods, for both didactic and data analytic purposes, can often be achieved through geometric diagrams and geometrically based statistical graphs. This paper extols and illustrates the…
The space-like hypersurface of the Universe at the present cosmological time is a three-dimensional manifold. A non-trivial global topology of this space-like hypersurface would imply that the apparently observable universe (the sphere of…
Abstract axiomatic formulation of mathematical structures are extensively used to describe our physical world. We take here the reverse way. By making basic assumptions as starting point, we reconstruct some features of both geometry and…
This essay considers the special character of mathematical reasoning, and draws on observations from interactive theorem proving and the history of mathematics to clarify the nature of formal and informal mathematical language. It proposes…
The world of mathematics is often considered abstract, with its symbols, concepts, and topics appearing unrelated to physical objects. However, it is important to recognize that the development of mathematics is fundamentally influenced by…
We introduce an abstract topos-theoretic framework for building Galois-type theories in a variety of different mathematical contexts; such theories are obtained from representations of certain atomic two-valued toposes as toposes of…
Some of the basic concepts of topology are explored through known physics problems. This helps us in two ways, one, in motivating the definitions and the concepts, and two, in showing that topological analysis leads to a clearer…
Many different and complementary strategies for translating the basic principle of multiple topological imaging into observational analysis are now available, both for three-dimensional and two-dimensional catalogues.
Normally we judge Topological shapes analytically but they hide significant amount of data in them about coordinate planes and ordered & unordered paris. In this article we will build our intuition and find those datas.
The purpose of this thesis is to study classical combinatorial objects, such as polytopes, polytopal complexes, and subspace arrangements, using tools that have been developed in combinatorial topology, especially those tools developed in…
Being mathematics a natural language to Mankind and to physics, it must be constantly adapted to our necessities and our natural perception. Then, mathematical concepts are not absolute to reality. Although mathematical theories are…
Metric spaces are a fundamental component of mathematics and have a paramount importance as a framework for measuring distance. They can be found in many different branches of mathematics, such as analysis and topology. This paper offers an…
The Deutsch multiverse is collection of parallel universes. In this article a formal theory and a topos-theoretic model of the Deutsch multiverse are given. For this the Lawvere-Kock Synthetic Differential Geometry and topos models for…
We survey indications from different branches of Physics that the fine scale structure of spacetime is not adequately described by a manifold. Based on the hints we accumulate, we propose a new structure, which we call a quantum topos. In…
The topos approach to the formulation of physical theories includes a new form of quantum logic. We present this topos quantum logic, including some new results, and compare it to standard quantum logic, all with an eye to conceptual…
We use the notion of topological data analysis to compare metrics on data sets. We provide two different motivating examples for this. The first of these is a point cloud data set that has $\mathbb{R}^2$ as its ambient space, and is…