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Theories, Sites, Toposes
According to Grothendieck, the notion of topos is "the bed or deep river where come to be married geometry and algebra, topology and arithmetic, mathematical logic and category theory, the world of the continuous and that of discontinuous or discrete structures". It is what he had "conceived of most broad to perceive with finesse, by the same language rich of geometric resonances, an "essence" which is common to situations most distant from each other, coming from one region or another of the vast universe of mathematical things". The aim of this book is to present a theory and a number of techniques which allow to give substance to Grothendieck's vision by building on the notion of classify...
Objects, Structures, and Logics
This edited collection casts light on central issues within contemporary philosophy of mathematics such as the realism/anti-realism dispute; the relationship between logic and metaphysics; and the question of whether mathematics is a science of objects or structures. The discussions offered in the papers involve an in-depth investigation of, among other things, the notions of mathematical truth, proof, and grounding; and, often, a special emphasis is placed on considerations relating to mathematical practice. A distinguishing feature of the book is the multicultural nature of the community that has produced it. Philosophers, logicians, and mathematicians have all contributed high-quality articles which will prove valuable to researchers and students alike.
Theories, Sites, Toposes
According to Grothendieck, the notion of topos is "the bed or deep river where come to be married geometry and algebra, topology and arithmetic, mathematical logic and category theory, the world of the continuous and that of discontinuous or discrete structures". It is what he had "conceived of most broad to perceive with finesse, by the same language rich of geometric resonances, an "essence" which is common to situations most distant from each other, coming from one region or another of the vast universe of mathematical things". The aim of this book is to present a theory and a number of techniques which allow to give substance to Grothendieck's vision by building on the notion of classify...
Cyclic Cohomology at 40: Achievements and Future Prospects
This volume contains the proceedings of the virtual conference on Cyclic Cohomology at 40: Achievements and Future Prospects, held from September 27–October 1, 2021 and hosted by the Fields Institute for Research in Mathematical Sciences, Toronto, ON, Canada. Cyclic cohomology, since its discovery forty years ago in noncommutative differential geometry, has become a fundamental mathematical tool with applications in domains as diverse as analysis, algebraic K-theory, algebraic geometry, arithmetic geometry, solid state physics and quantum field theory. The reader will find survey articles providing a user-friendly introduction to applications of cyclic cohomology in such areas as higher ca...
Mathematics and Computation in Music
This book constitutes the thoroughly refereed proceedings of the 7th International Conference on Mathematics and Computation in Music, MCM 2019, held in Madrid, Spain, in June 2019. The 22 full papers and 10 short papers presented were carefully reviewed and selected from 48 submissions. The papers feature research that combines mathematics or computation with music theory, music analysis, composition, and performance. They are organized in topical sections on algebraic and other abstract mathematical approaches to understanding musical objects; remanaging Riemann: mathematical music theory as “experimental philosophy”?; octave division; computer-based approaches to composition and score structuring; models for music cognition and beat tracking; pedagogy of mathematical music theory. The chapter “Distant Neighbors and Interscalar Contiguities” is available open access under a Creative Commons Attribution 4.0 International License via link.springer.com.
Logic in Question
This contributed volume collects papers related to the Logic in Question workshop, which has taken place annually at Sorbonne University in Paris since 2011. Each year, the workshop brings together historians, philosophers, mathematicians, linguists, and computer scientists to explore questions related to the nature of logic and how it has developed over the years. As a result, chapter authors provide a thorough, interdisciplinary exploration of topics that have been studied in the workshop. Organized into three sections, the first part of the book focuses on historical questions related to logic, the second explores philosophical questions, and the third section is dedicated to mathematical...
Comprehending the Complexity of Countries
This book argues for computer-aided collaborative country research based on the science of complex and dynamic systems. It provides an in-depth discussion of systems and computer science, concluding that proper understanding of a country is only possible if a genuinely interdisciplinary and truly international approach is taken; one that is based on complexity science and supported by computer science. Country studies should be carefully designed and collaboratively carried out, and a new generation of country students should pay more attention to the fast growing potential of digitized and electronically connected libraries. In this frenzied age of globalization, foreign policy makers may – to the benefit of a better world – profit from the radically new country studies pleaded for in the book. Its author emphasizes that reductionism and holism are not antagonistic but complementary, arguing that parts are always parts of a whole and a whole has always parts.
