Logic in Mathematics and Computer Science

TL;DR

This paper explores the historical development and modern applications of logic in mathematics and computer science, highlighting formal systems, proof theory, and computational methods that underpin foundational and practical advancements.

Contribution

It provides a comprehensive overview of the evolution of logical formalism and its integration into computer science, emphasizing recent developments in proof systems and their applications.

Findings

Development of formal logical systems for mathematics

Advances in proof theory and model theory methods

Application of logic in automated reasoning and verification

Abstract

Logic has pride of place in mathematics and its 20th century offshoot, computer science. Modern symbolic logic was developed, in part, as a way to provide a formal framework for mathematics: Frege, Peano, Whitehead and Russell, as well as Hilbert developed systems of logic to formalize mathematics. These systems were meant to serve either as themselves foundational, or at least as formal analogs of mathematical reasoning amenable to mathematical study, e.g., in Hilbert's consistency program. Similar efforts continue, but have been expanded by the development of sophisticated methods to study the properties of such systems using proof and model theory. In parallel with this evolution of logical formalisms as tools for articulating mathematical theories (broadly speaking), much progress has been made in the quest for a mechanization of logical inference and the investigation of its theoretical limits, culminating recently in the development of new foundational frameworks for mathematics with sophisticated computer-assisted proof systems. In addition, logical formalisms developed by logicians in mathematical and philosophical contexts have proved immensely useful in describing theories and systems of interest to computer scientists, and to some degree, vice versa. Three examples of the influence of logic in computer science are automated reasoning, computer verification, and type systems for programming languages.