Related papers: On proof and progress in mathematics
This is a book on derived foliations, that are a generalisation of classical foliations in the context of derived geometry. The text starts with the basic definitions and constructions, then explore foliated cohomology (with crystal…
We show a possibility to apply certain philosophical concepts to the analysis of concrete mathematical structures. Such application gives a clear justification of topological and geometric properties of considered mathematical objects.
This work concerns a study of the quantum mechanical extension of the work of Horwitz et al. [1] on the stability of classical Hamiltonian systems by geometrical methods. Simulations are carried out for several important examples, these…
For $\Gamma_1$-structures on 3-manifolds, we give a very simple proof of Thurston's regularization theorem, first proved in \cite{thurston}, without using Mather's homology equivalence. Moreover, in the co-orientable case, the resulting…
In this chapter, we propose some future directions of work, potentially beneficial to Mathematics and its foundations, based on the recent import of methodology from the theory of programming languages into proof theory. This scientific…
We discuss a formal system of mathematics. We use it to construct the natural numbers.
We discuss new approaches to fundamental problems of mathematics and mathematical physics such as mathematical foundation of quantum field theory, the Riemann hypothesis, and construction of noncommutative algebraic geometry.
Wigner's "unreasonable effectiveness of mathematics" in physics can be understood as a reflection of a deep and unexpected unity between the fundamental structures of mathematics and of physics. Some of the history of evidence for this is…
One often sees a sharp distinction in mathematics between descriptions from the outside and from the inside. Think of defining a set in the plane through an algebraic equation, or dynamically as the closure of the orbit of some point under…
This paper examines various methods and ideas for humanizing mathematics. The term 'humanizing mathematics' which includes elements of 'aesthetic mathematics' refers to approaches that emphasize the aesthetic, philosophical, and subjective…
Noncommutativity lays hidden in the proofs of classical dynamics. Modern frameworks can be used to bring it to light: *-products, groupoids, q-deformed calculus, etc.
Development of several alternative mathematical models for the biological system in question and discrimination between such models using experimental data is the best way to robust conclusions. Models which challenge existing theories are…
This report on the topics in the title was written for a lecture series at the Southwestern Center for Arithmetic Algebraic Geometry at the University of Arizona.It may serve as an introduction to certain conjectural relations between…
In their account of theory change in logic, Aberdein and Read distinguish 'glorious' from 'inglorious' revolutions--only the former preserves all 'the key components of a theory' [1]. A widespread view, expressed in these terms, is that…
Artificial intelligence assisted mathematical proof has become a highly focused area nowadays. One key problem in this field is to generate formal mathematical proofs from natural language proofs. Due to historical reasons, the formal proof…
This book intends to give the main definitions and theorems in mathematics which could be useful for workers in theoretical physics. It gives an extensive and precise coverage of the subjects which are addressed, in a consistent and…
This extended abstract is about an effort to build a formal description of a triangulation algorithm starting with a naive description of the algorithm where triangles, edges, and triangulations are simply given as sets and the most complex…
The purpose of this paper is to explore the question "to what extent could we produce formal, machine-verifiable, proofs in real algebraic geometry?" The question has been asked before but as yet the leading algorithms for answering such…
This is an overview article. In his Habilitationsvortrag, Riemann described infinite dimensional manifolds parameterizing functions and shapes of solids. This is taken as an excuse to describe convenient calculus in infinite dimensions…
Transverse one dimensional foliations play an important role in the study of codimension one foliations. In \cite{KR2}, the authors introduced the notion of flow box decomposition of a 3-manifold $M$. This is a decomposition of $M$ that…