Related papers: The grounding for Continuum
We first give a rigorous mathematical proof that classical mathematics (involving such notions as infinitely small/large, continuity etc.) is a special degenerate case of finite one in the formal limit when the characteristic $p$ of the…
It is a rather universal tacit and unquestioned belief - and even more so among physicists - that there is one and only one set of real scalars, namely, the one given by the usual field $\mathbb{R}$ of real numbers, with its usual linear…
The thesis presents the subject of synthetic topology, especially with relation to metric spaces. A model of synthetic topology is a categorical model in which objects possess an intrinsic topology in a suitable sense, and all morphisms are…
Real numbers in constructive mathematics have always seemed to require compromises of one form or another. Classical proofs of Cauchy completeness require countable choice, Bishop's setoid construction introduces persistent bookkeeping…
We study the set of Dedekind cuts over a linearly ordered Abelian group as a structure over the language (0,<,+,-). Moreover, we obtain a simple set of axioms for the universal part of the theory of such structures. Finally, we prove that…
Classical mathematics (involving such notions as infinitely small/large and continuity) is usually treated as fundamental while finite mathematics is treated as inferior which is used only in special applications. We first argue that the…
Given a trivalent graph in the 3-dimensional Euclidean space, we call it a discrete surface because it has a tangent space at each vertex determined by its neighbor vertices. To abstract a continuum object hidden in the discrete surface, we…
In Mathematical Thought and Its Objects, Charles Parsons argues that our knowledge of the iterability of functions on the natural numbers and of the validity of complete induction is not intuitive knowledge; Brouwer disagrees on both…
We give an overview of the basic definitions of condensed categories, as well as the internal Hom of condensed abelian groups. We give a construction for the internal Hom of condensed sets and apply it to obtain a new proof of a theorem of…
We prove the existence of nowhere continuous bijections that satisfy the Costas property, as well as (countably and uncountably) infinite Golomb rulers. We define and prove the existence of real and rational Costas clouds, namely nowhere…
Cosine similarity, the standard metric for measuring semantic similarity in vector spaces, is mathematically grounded in the Cauchy-Schwarz inequality, which inherently limits it to capturing linear relationships--a constraint that fails to…
Informally speaking, the categoricity of an axiom system means that its non-logical symbols have only one possible interpretation that renders the axioms true. Although non-categoricity has become ubiquitous in the second half of the 20th…
In the setting of constructive pointfree topology, we introduce a notion of continuous operation between pointfree topologies and the corresponding principle of pointfree continuity. An operation between points of pointfree topologies is…
A new integral representation is derived using a definite integral given by Cauchy and used to evaluate a number of integrals containing the finite series of special functions.
String theory is accused by some of its critics to be a purely abstract mathematical discipline, having lost the contact to the simple yet deeply rooted questions which physics provided until the beginning of this century. We argue that, in…
Floating point arithmetic allows us to use a finite machine, the digital computer, to reach conclusions about models based on continuous mathematics. In this article we work in the other direction, that is, we present examples in which…
Computational materials design often profits from the fact that some complicated contributions are not calculated for the real material, but replaced by results of models. We turn this approximation into a very general and in principle…
The real numbers, it is taught at universities, correspond to our idea of a continuum, although the hyperreal numbers are located ``in between'' the real numbers. The number $x + dx$, where $dx$ should be an infinitesimal number and $x$…
We study cross ratios from an axiomatic viewpoint, also known as the study of M\"obius spaces. We characterise cross ratios induced by quasi-metrics in terms of topological properties of their image. Furthermore, we generalise the notions…
Following a line of research initiated in \cite{BBNN}, I describe a general framework for turning reduction concepts of relative computability into diagrams forming an analogy with the Cicho\'n diagram for cardinal characteristics of the…