Related papers: Foundations
This is a preliminary version of the textbook on integrable systems. The work has been partly supported by Grant Nr.10/2006-RU, Austrian Academic Exchange Service \"OAD and Grant P20164-N18, Austrian Science Fund FWF
This chapter amalgamates some foundational developments and calculations in factorization homology.
In this paper we expound some basic ideas of proof theory for theories of ordinals such that there are many stable ordinals below the ordinals.
This paper has been withdrawn by the author. In this article I review W\"ust's recent handbook on mathematical physics from a philosophical standpoint. It emerges a structural approach to mathematics which evidences the utility of logic in…
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…
This chapter is a short pedagogical introduction to the use of quantum logic for the simulation of complex quantum systems, including a simulation example on actual quantum hardware.
This paper presents a new system of logic, LF, that is intended to be used as the foundation of the formalization of science. That is, deductive validity according to LF is to be used as the criterion for assessing what follows from the…
In this paper I discuss what, according to my long experience, every computer scientist should know from logic. We concentrate on issues of modeling, interpretability and levels of abstraction. We discuss what the minimal toolbox of logic…
We illustrate the concept of mathematical proof.
These are notes on discrete mathematics for computer scientists. The presentation is somewhat unconventional. Indeed I begin with a discussion of the basic rules of mathematical reasoning and of the notion of proof formalized in a natural…
This book is a textbook for the course of foundations of geometry. It is addressed to mathematics students in Universities and to High School students for deeper learning the elementary geometry. It can also be used in mathematics coteries…
In various provers and deductive verification tools, logical transformations are used extensively in order to reduce a proof task into a number of simpler tasks. Logical transformations are often part of the trusted base of such tools. In…
This paper describes a formal proof library, developed using the Coq proof assistant, designed to assist users in writing correct diagrammatic proofs, for 1-categories. This library proposes a deep-embedded, domain-specific formal language,…
This book provides a compact, derivation-oriented introduction to the mathematical foundations of modern generative artificial intelligence. Rather than surveying every recent architecture or implementation detail, it develops a coherent…
Evidence-based reasoning is at the core of many problem-solving and decision-making tasks in a wide variety of domains. Generalizing from the research and development of cognitive agents in several such domains, this paper presents progress…
This is a preprint version of a chapter for Handbook of Algebra.
This essay considers the special character of mathematical reasoning, and draws on observations from interactive theorem proving and the history of mathematics to clarify the nature of formal and informal mathematical language. It proposes…
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 paper describes mathlib, a community-driven effort to build a unified library of mathematics formalized in the Lean proof assistant. Among proof assistant libraries, it is distinguished by its dependently typed foundations, focus on…
Procedural computer languages have long been used in many aspects of mathematics pedagogy. In this work, we examine the use of Prolog, a declarative language for the same purpose. We find the facts+rules aspect of Prolog to be a novel…