Hardback
Published: 26 January 2023
304 Pages
234x156mm
ISBN: 9780192867391
Many of the most famous results in mathematics are impossibility theorems stating that something cannot be done. Good examples include the quadrature of the circle by ruler and compass, the solution of the quintic equation by radicals, Fermat's last theorem, and the impossibility of proving the parallel postulate from the other axioms of Euclidean geometry. This book tells the history of these and many other impossibility theorems starting with the ancient Greek proof of the incommensurability of the side and the diagonal in a square.
Lutzen argues that the role of impossibility results have changed over time. At first, they were considered rather unimportant meta-statements concerning mathematics but gradually they obtained the role of important proper mathematical results that can and should be proved. While mathematical impossibility proofs are more rigorous than impossibility arguments in other areas of life, mathematicians have employed great ingenuity to circumvent impossibilities by changing the rules of the game. For example, complex numbers were invented in order to make impossible equations solvable. In this way, impossibilities have been a strong creative force in the development of mathematics, mathematical physics, and social science.
1:Introduction
2:Prehistory: Recorded and Non-Recorded Impossibilities
3:The First Impossibility Proof: Incommensurability
4:The Classical Problems in Antiquity: Constructions and Positive Theorems
5:The Classical Problems: The Impossibility Question
6:Diorisms and Conclusions about the Greeks and the Medieval Arabs
7:Cube Duplication and Angle Trisection in the 17th and 18th Centuries
8:Circle Quadrature in the 17th Century
9:Circle Quadrature in the 18th Century
10:Impossible Equations Made Possible: The Complex Numbers
11:Euler and the Bridges of Konigsberg
12:The Insolvability of the Quintic by Radicals
13:Constructions with Ruler and Compass: The Final Impossibility Proofs
14:Impossible Integrals
15:Impossibility of Proving the Parallel Postulate
16:Hilbert and Impossible Problems
17:Hilbert and Godel on Axiomatization and Incompleteness
18:Fermat's Last Theorem
19:Impossibility in Physics
20:Arrow's Impossibility Theorem
21:Conclusion
Format: Hardback, 676 pages, height x width: 235x155 mm, 35 Illustrations, color;
121 Illustrations, black and white; XXI, 676 p. 156 illus., 35 illus. in color
Series: Applied Mathematical Sciences 213
ISBN-13: 9783031203855
The first edition of this book was published in 1994. Since then considerable progress has been made in both theoretical developments of percolation theory, and in its applications. The 2nd edition of this book is a response to such developments. Not only have all of the chapters of the 1st edition been completely rewritten, reorganized, and updated all the way to 2022, but also 8 new chapters have been added that describe extensive new applications, including biological materials, networks and graphs, directed percolation, earthquakes, geochemical processes, and large-scale real world problems, from spread of technology to ad-hoc mobile networks.
Chapter 1: Macroscopic Connectivity as the Essential Property of Disordered Materials and Media.
Chapter 2: Classical and Poor Man's Percolation Models.
Chapter 3: Variations of the Classical Percolation Model.
Chapter 4: Characterization of Porous Media.
Chapter 5: Percolation Properties of Fracture and Fault Networks.
Chapter 6: Earthquakes and Percolation.
Chapter 7: Conductivity, Diffusivity, and Permeability of Porous Materials.
Chapter 8: Mass Transport, Mixing, and Dispersion in Flow Through Porous Media.
Chapter 9: Multiphase Fluid Flow in Porous Media.-
Chapter 10: Percolation in Evolving Porous Materials: Catalyst Deactivation, Gasification, Fragmentation, and Precipitation.
Chapter 11: Percolation, and Rigidity and Elastic Properties of Materials.
Chapter 12: Morphological and Transport Properties of Composite Materials.
Chapter 13: Rheology and Elastic Properties of Network Glasses, Branched Polymers, and Gels.
Chapter 14: Vibrational Density of States of Heterogeneous Materials.
Chapter 15: Hopping Conductivity of Heterogeneous Materials.
Chapter 16: Applications of Invasion Percolation.
Chapter 17: Percolation in Random Graphs and Complex Network.
Chapter 18: Percolation in Biological Systems.
Chapter 19: Percolation Theory at the Intersection of Ecology, Hydrology, and Geochemistry.
Chapter 20: Explosive Percolation and its Applications.-
Chapter 21: Directed Percolation: From Turbulent Flow to Catalysis and Brain.
Chapter 22: Percolation in Large-Scale Problems.
Format: Hardback, 295 pages, height x width: 235x155 mm, weight: 635 g, 3 Tables, color;
3 Illustrations, color; 7 Illustrations, black and white; XIV, 295 p. 10 illus., 3 illus. in color.
Series: Computer Science Foundations and Applied Logic
Pub. Date: 02-Jan-2023
ISBN-13: 9783031211119
This unique textbook, in contrast to a standard logic text, provides the reader with a logic that actually can be used in practice to express and reason about mathematical ideas.
The book is an introduction to simple type theory, a classical higher-order version of predicate logic that extends first-order logic. It presents a practice-oriented logic called Alonzo that is based on Alonzo Church's formulation of simple type theory known as Church's type theory. Unlike traditional predicate logics, Alonzo admits undefined expressions. The book illustrates, using Alonzo, how simple type theory is suited ideally for reasoning about mathematical structures and constructing libraries of mathematical knowledge.
Preface.- 1 Introduction.- 2 Answers to Readers' Questions.- 3 Preliminary Concepts.- 4 Syntax.- 5 Semantics.- 6 Additional Notation.- 7 Beta-reduction and Substitution.- 8 Proof Systems.- 9 Theories.- 10 Sequences.- 11 Developments.- 12 Real Number Mathematics.- 13 Morphisms 14 Alonzo Variants.- 15 Software Support.- Appendix A: Metatheorems of .- Appendix B: Soundness of .- Appendix C: Henkin's Theorem for .- Bibliography.- List of Figures.- List of Tables.- List of Theorems, Examples, Remarks, and Modules.- Index
Offers the first book-length introduction to simple type theory as a predicate logic
Provides the reader with a logic that is close to mathematical practice
Presents the tools needed to build libraries of mathematical knowledge
Employs two semantics, one for mathematics and one for logic
Emphasizes the model-theoretic view of predicate logic
Includes several important topics, such as definite description and theory morphisms, not usually found in standard logic textbooks
Aimed at students of computing and mathematics at the graduate or upper-undergraduate level, this book is also well-suited for mathematicians, computing professionals, engineers, and scientists who need a practical logic for expressing and reasoning about mathematical ideas.
William M. Farmer is a Professor in the Department of Computing and Software at McMaster University in Hamilton, Ontario, Canada.
Format: Hardback, 300 pages, height x width: 235x155 mm, weight: 641 g, 2 Tables,
color; 1 Illustrations, black and white; XIII, 300 p. 1 illus
Series: Studies in Universal Logic
Pub. Date: 02-Dec-2022
ISBN-13: 9783031223297
How should we think about the meaning of the words that make up our language? How does reference of these terms work, and what is their referent when these are connected to abstract objects rather than to concrete ones? Can logic help to address these questions? This collection of papers aims to unify the questions of syntax and semantics of language, which span across the fields of logic, philosophy and ontology of language. The leading motif of the presented selection is the differentiation between linguistic tokens (material, concrete objects) on the one hand and linguistic types (ideal, abstract objects) on the other. Through a promenade among articles that span over all of the Author’s career, this book addresses the complex philosophical question of the ontology of language by following the crystalline conceptual tools offered by logic. At the core of Wybraniec-Skardowska’s scholarship is the idea that language is an ontological being, characterized in compliance with the logical conception of language proposed by Ajdukiewicz. The application throughout the book of tools of classical logic and set theory results fosters the emergence of a general formal logical theory of syntax, semantics and of the pragmatics of language, which takes into account the duality token-type in the understanding of linguistic expressions. Via a functional approach to language itself, logic appears as ontologically neutral with respect to existential assumptions relating to the nature of linguistic expressions and their extra-linguistic counterparts.
The book is addressed to readers both at the graduate and undergraduate level, but also to a more general audience interested in getting a firmer grip on the interplay between reality and the language we use to describe and understand it.