Related papers: Philosophical Reflections on Intrinsic Differentia…
Some sorts of generalized morphisms are defined from very basic mathematical objects such as sets, functions, and partial functions. A wide range of mathematical notions such as continuous functions between topological spaces, ring…
Human cognition spans perception, memory, intuitive judgment, deliberative reasoning, action selection, and social inference, yet these capacities are often explained through distinct computational theories. Here we present a unified…
Mathematical research is often motivated by the desire to reach a beautiful result or to prove it in an elegant way. Mathematician's work is thus strongly influenced by his aesthetic judgments. However, the criteria these judgments are…
We express the $q$-th Gauss-Bonnet-Chern mass of an immersed submanifold of Euclidean space as a linear combination of two terms: the total $(2q)$-th mean curvature and the integral, over the entire manifold, of the inner product between…
Local inertial frame invariance is taken as the fundamental principle of physical geometry, where a local inertial frame is represented by a verbein. Invariance of the vierbein with respect to local Lorentz transformations then expresses…
In models of emergent gravity the metric arises as the expectation value of some collective field. Usually, many different collective fields with appropriate tensor properties are candidates for a metric. Which collective field describes…
In La G\'eom\'etrie, Descartes proposed a balance between geometric constructions and symbolic manipulation with the introduction of suitable ideal machines. In modern terms, that is a balance between analog and symbolic computation.…
The issues of quintessence and cosmic acceleration can be discussed in the framework of $F(R, {\cal G})$ theories of gravity where $R$ is the Ricci curvature scalar and ${\cal G}$ is the Gauss-Bonnet topological invariant. It is possible to…
Topological invariants such as winding numbers and linking numbers appear as charges of topological solitons in diverse nonlinear physical systems described by a unit vector field defined on two and three dimensional manifolds. While the…
Magnitude is a real-valued invariant of metric spaces, analogous to the Euler characteristic of topological spaces and the cardinality of sets. The definition of magnitude is a special case of a general categorical definition that clarifies…
We consider a theory of modified gravity possessing d extra spatial dimensions with a maximally symmetric metric and a scale factor, whose (4+d)-dimensional gravitational action contains terms proportional to quadratic curvature scalars.…
General Teleparallel theories assume that curvature is vanishing in which case gravity can be solely represented by torsion and/or nonmetricity. Using differential form language, we express the Riemannian Gauss-Bonnet invariant concisely in…
In this paper, we propose that 'embodied mathematics' should be studied not only by reduction to the present individual bodily experience but in an historical context as well, as far as the origins of mathematics are concerned. Some early…
Recently a notion of self-duality for differential equations of maximal cuts was introduced, which states that there should be a basis in which the matrix for an {\epsilon}-factorised differential equation is persymmetric. It was observed…
Differential geometry may be generalized to allow infinitesimals to any order. The purpose of the present contribution is to show that the theory so developed expands received geometrical ideas in an interesting way, rich in potential for…
Geometric algebra is the natural outgrowth of the concept of a vector and the addition of vectors. After reviewing the properties of the addition of vectors, a multiplication of vectors is introduced in such a way that it encodes the famous…
The compactness phenomenon is one of the featured aspects of structuralism in mathematics. In simple and broad words, a compactness property holds in a structure if a related property is satisfied by sufficiently many substructures of that…
The issue of the transformations of units is treated, mainly, in a geometrical context. It is shown that Weyl-integrable geometry is a consistent framework for the formulation of the gravitational laws since the basic law on which this…
In this paper we discuss, from a historical and philosophical point of view, a variation of the meaning of the five postulates in Euclidean Geometry and we make a short reference to D. Hilberts formalism. We examine, throughout the ages,…
In the present manuscript, I examine an intriguing relation at the classical level between general relativity and a theory where matter couples uniquely multiplicatively to geometry in the Lagrangian density. Interestingly, the…