Related papers: Pure Data Foundation of Mathematics and Computing
In a previous work, "pure data" is proposed as an axiomatic foundation for mathematics and computing, based on "finite sequence" as the foundational concept rather than based on logic or type. Within this framework, objects with…
The foundations of mathematics have long been considered settled by the Zermelo-Fraenkel-Choice axioms. But set theory abounds in models with different truths and even classical questions such as the measurability of projective sets can…
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…
Axiomatizing mathematical structures and theories is an objective of Mathematical Logic. Some axiomatic systems are nowadays mere definitions, such as the axioms of Group Theory; but some systems are much deeper, such as the axioms of…
We consider a set-theoretic version of mereology based on the inclusion relation $\subseteq$ and analyze how well it might serve as a foundation of mathematics. After establishing the non-definability of $\in$ from $\subseteq$, we identify…
Axiomatic set theory is almost universally accepted as the basic theory which provides the foundations of mathematics, and in which the whole of present day mathematics can be developed. As such, it is the most natural framework for…
Discrete mathematics is the foundation of computer science. It focuses on concepts and reasoning methods that are studied using math notations. It has long been argued that discrete math is better taught with programming, which takes…
There is a problem with the foundations of classical mathematics, and potentially even with the foundations of computer science, that mathematicians have by-and-large ignored. This essay is a call for practicing mathematicians who have been…
The recent trend in mathematics is towards a framework of abstract mathematical objects, rather than the more concrete approach of explicitly defining elements which objects were thought to consist of. A natural question to raise is whether…
This paper presents mathematics as a general science of computation in a way different from the tradition. It is based on the radical philosophical standpoint according to which the content, meaning and justification of experience lies in…
While concepts and tools from Theoretical Computer Science are regularly applied to, and significantly support, software development for discrete problems, Numerical Engineering largely employs recipes and methods whose correctness and…
There are different meanings of foundation of mathematics: philosophical, logical, and mathematical. Here foundations are considered as a theory that provides means (concepts, structures, methods etc.) for the development of whole…
This article was motivated by the discovery of a potential new foundation for mainstream mathematics. The goals are to clarify the relationships between primitives, foundations, and deductive practice; to understand how to determine what…
In this article we present an axiomatic definition of sets with individuals and a definition of natural numbers and ordinals. We use the axioms pairs, union, power, regularity and separation. We define the equality of sets and of…
A re-construction of the fundamentals of programming as a small mathematical theory (PRISM) based on elementary set theory. Highlights: $\bullet$ Zero axioms. No properties are assumed, all are proved (from standard set theory). $\bullet$ A…
We introduce a set of eight universal Rules of Inference by which computer programs with known properties (axioms) are transformed into new programs with known properties (theorems). Axioms are presented to formalize a segment of Number…
My purpose is to examine some concepts of mathematical logic, which have been studied by Carlo Cellucci. Today the aim of classical mathematical logic is not to guarantee the certainty of mathematics, but I will argue that logic can help us…
The widespread success of foundation models in natural language processing and computer vision has inspired researchers to extend the concept to scientific machine learning and computational science. However, this position paper argues that…
We present a coherent collection of finite mathematical theorems some of which can only be proved by going well beyond the usual axioms for mathematics. The proofs of these theorems illustrate in clear terms how one uses the well studied…
Abstraction logic is a new logic, serving as a foundation of mathematics. It combines features of both predicate logic and higher-order logic: abstraction logic can be viewed both as higher-order logic minus static types as well as…