Related papers: Relative Categoricity and Abstraction Principles
Developing suitable formal semantics can be of great help in the understanding, design and implementation of a programming language, and act as a guide for software development tools like analyzers or partial evaluators. In this sense, full…
Crispin Wright in his 1982 paper argues for strict finitism, a constructive standpoint that is more restrictive than intuitionism. In its appendix, he proposes models of strict finitistic arithmetic. They are tree-like structures, formed in…
Recently, in Axioms 10(2): 119 (2021), a nonclassical first-order theory T of sets and functions has been introduced as the collection of axioms we have to accept if we want a foundational theory for (all of) mathematics that is not weaker…
In a previous publication, we introduced an abstract logic via an abstract notion of quantifier. Drawing upon concepts from categorical logic, this abstract logic interprets formulas from context as subobjects in a specific category, e.g.,…
Fair Machine Learning endeavors to prevent unfairness arising in the context of machine learning applications embedded in society. Despite the variety of definitions of fairness and proposed "fair algorithms", there remain unresolved…
We examine Paul Halmos' comments on category theory, Dedekind cuts, devil worship, logic, and Robinson's infinitesimals. Halmos' scepticism about category theory derives from his philosophical position of naive set-theoretic realism. In the…
Category theory is the language of homological algebra, allowing us to state broadly applicable theorems and results without needing to specify the details for every instance of analogous objects. However, authors often stray from the realm…
Principles of cognitive economy would require that concepts about objects, properties and relations should be introduced only if they simplify the conceptualisation of a domain. Unexpectedly, classic logic conditionals, specifying…
We study two notions of expressiveness, which have appeared in abstraction theory for model checking, and find them incomparable in general. In particular, we show that according to the most widely used notion, the class of Kripke Modal…
Cluster analysis has attracted more and more attention in the field of machine learning and data mining. Numerous clustering algorithms have been proposed and are being developed due to diverse theories and various requirements of emerging…
We introduce a notion of complexity of diagrams (and in particular of objects and morphisms) in an arbitrary category, as well as a notion of complexity of functors between categories equipped with complexity functions. We discuss several…
The concept of category from mathematics happens to be useful to computer programmers in many ways. Unfortunately, all "good" explanations of categories so far have been designed by mathematicians, or at least theoreticians with a strong…
Effective field theories (EFTs) are widely considered by physicists to be explanatory and to be the appropriate frameworks for modelling various phenomena at different scales. At the same time, they are known to be approximate, restricted,…
We apply a framework developed by C. S. Peirce to analyze the concept of clarity, so as to examine a pair of rival mathematical approaches to a typical result in analysis. Namely, we compare an intuitionist and an infinitesimal approaches…
Abstraction is a fundamental tool for reasoning about complex systems. Program abstraction has been utilized to great effect for analyzing deterministic programs. At the heart of program abstraction is the relationship between a concrete…
Shannon's entropy and other entropy-based concepts are derived from the new, more general concept of relative divergence of one "grading' function on a linearly ordered set from another such function. The definition of relative divergence…
This paper deals with different concepts for characterizing the size of mathematical objects. A game theoretic investigation and generalization of two size concepts, which can both be formulated in topological terms, is provided: the so…
We give a self-contained introduction to accessible categories and how they shed light on both model- and set-theoretic questions. We survey for example recent developments on the study of presentability ranks, a notion of cardinality…
This paper presents the contemporary Fundamental Theorem of Asset Pricing as being equivalent to approaches to pricing that emerged before 1700 in the context of Virtue Ethics. This is done by considering the history of science and…
Based on rectangle theory of formal concept and set covering theory, the concept reduction preserving binary relations is investigated in this paper. It is known that there are three types of formal concepts: core concepts, relative…