Format: Paperback / softback, 155 pages, height x width: 235x155 mm, 9 Illustrations, black and white
Series: Mathematics Study Resources
Pub. Date: 20-Nov-2025
ISBN-13: 9783662721780
What kind of objects does mathematics investigate, and in what sense do these objects exist? Why are we justified in considering mathematical statements as part of our knowledge, and how can they be validated? A philosophy of mathematics seeks to answer such questions. In this introduction, we present the major positions in the philosophy of mathematics and formulate their core ideas into clear, accessible theses. Readers will learn which philosophers developed each position and the historical context in which they emerged. Drawing on fundamental intuitions and scientific findings, one can argue for or against these theses such arguments form the second focus of this book. The book aims to encourage readers to reflect on the philosophy of mathematics, to develop their own stance, and to learn how to argue for it.
This book is a translation of the original German 2nd edition. The translation was done with the help of an artificial intelligence machine translation tool. A subsequent human revision was done primarily in terms of content, so that the book may read stylistically differently from a conventional translation.
1 Introduction.- 2 Pythagoreanism.- 3 Platonism.- 4 Rationalism.- 5
Kantianism.- 6 Mathematics in German Idealism.- 7 Logicism.- 8 Intuitionism.-
9 Formalism.- 10 Constructivism.- 11 Structuralism.- 12 Naturalism.- 13
Further Developments.- 14 Appendix: Set Theory.
Format: Paperback / softback, 226 pages, height x width: 235x155 mm, 25 Illustrations, color; 86 Illustrations, black and white
Series: Springer Undergraduate Mathematics Series
Pub. Date: 06-Dec-2025
ISBN-13: 9783032038142
This book offers an introduction to the nature of mathematics, its history, and some of the curious individuals who have shaped it. It explores a range of fundamental ideas that can be appreciated without prior mathematical training.
Highlighting the human side of mathematics and its shared lineage with philosophy, the book embraces the subject’s colorful history, incorporating original source material in the main text. Based on a course taught to non-science majors, it is intended for anyone with an interest in mathematics—including those whose last experience with algebra was less than enjoyable.
1 What is Mathematics?.- 2 Pascals Triangle, Number Magic and the
Apocalypse.- 3 Some Simple Number Mysteries.- 4 Cellular Automata.- 5 Axioms:
the Rules of the Game.- 6 Direct and Indirect Proofs.- 7 Dirichlets Box
Principle.- 8 Adding Infinitely Many Things.- 9 Incorrectly Adding Infinitely
Many Things.- 10 The Other Side of the Story (Plato vs. Wittgenstein).- 11
Different Infinities.- 12 The Monte Carlo Method.- 13 Chaos Theory.- 14
Mathematics as a Tool for Dishonesty.- 15 Able Amateurs and Colorful
Crackpots.- 16 The House Always Wins.- 17 Matching Things.- 18 Conflict and
Cooperation.- 19 The End of Knowledge.
Format: Hardback, 400 pages
Pub. Date: 26-Oct-2025
ISBN-13: 9789819821297
This unique and richly illustrated book explores barycentric calculus, a geometric method grounded in the concept of the center of gravity. Used to elegantly determine triangle centers through weighted points, barycentric coordinates have long revealed deep insights in Euclidean geometry. Now, this book extends those insights to the fascinating realm of hyperbolic geometry, building a powerful bridge between classical and modern mathematical worlds.In Euclidean geometry, over 3,000 triangle centers have been identified using barycentric coordinates. This book introduces readers to their hyperbolic analogs, uncovering remarkable parallels between triangle centers in Bolyai-Lobachevsky geometry and their Euclidean counterparts. The author's innovative use of Cartesian coordinates, trigonometry, and vector algebra adapted for hyperbolic geometry equips readers with familiar yet powerful tools to explore unfamiliar terrain.At the heart of the book is the development of hyperbolic barycentric coordinates, or gyrobarycentric coordinates, within the framework of gyrovector spaces a novel algebraic structure emerging from Einstein's velocity addition and Möbius addition. These gyrovectors underpin the Klein and Poincaré ball models of hyperbolic geometry, just as traditional vectors underlie analytic Euclidean geometry.Whether you are a researcher in geometry, mathematical physics, or relativity, or simply fascinated by the deep structure of space, this book offers a groundbreaking approach to analytic hyperbolic geometry through barycentric and gyrobarycentric coordinates.
Pages: 290
ISBN: 978-981-98-1865-5 (hardcover)
The main purpose of the book is to give a complete presentation of the Theory of a well-known class of special numbers (Eulerian numbers), as well as some generalizations and relatives (Eulerian numbers of the second order, factorial numbers, Euler numbers, etc.), and to give much of their properties, facts and theorems with full proofs.
The "names" of many of the special numbers essential to so many aspects of Number Theory, general Mathematics, and several applied areas are known to every mathematician. Fermat numbers, Mersenne numbers, Fibonacci numbers, etc. Yet actual information on them is often scattered in the actual literature, and those books which do provide this information are often by now out of date. This series is committed to remedying this gap, and this title in particular is dedicated to Eulerian numbers, their relatives, and their generalizations.
Key definitions, collected main properties of considered mathematical objects, the integral main facts of the theory of set-partitions, recurrent equations, and generating functions, the main questions of Eulerian numbers of the first- and second-orders, comprehensive and exclusive collections of practice exercises, a huge Mini Dictionary, and much more is compiled and presented in this text to provide an accessible guide to the field for not only undergraduate students of Mathematics but also for professionals and the general interested audience.
Notations
Preface
Preliminaries:
Combinatorial Configurations
Binomial Theorem and Pascal's triangle
Graphs
Partitions
Recurrent Relations
Generating Functions
Asymptotic Formulas
Permutations:
Definitions and Examples
Different Representations of Permutations
Cycles and Transpositions
Ascents, Descents, Runs and Exceedances
Inversions
Alternating Permutations
Enumeration of Permutations
Eulerian Numbers of the First Order:
Construction of Eulerian Numbers of the First Order
Problems Related to Eulerian Numbers of the First Order
Closed Formula for Eulerian Numbers of the First Order
Generating Function of the Sequence of Eulerian Numbers of the First Order
Properties of Eulerian Numbers of the First Order
Polynomials Related to Eulerian Numbers of the First Order
Eulerian Numbers of the First Order in the Family of Special Numbers
Eulerian Numbers of the Second Order:
Construction of Eulerian Numbers of the Second Order
Problems Related to Eulerian Numbers of the Second Order
Closed Formula for Eulerian Numbers of the Second
Generating Function of the Sequence of Eulerian Numbers of the Second Order
Properties of Eulerian Numbers of the Second Order
Polynomials Related to Eulerian Numbers of the Second Order
Eulerian Numbers of the Second Order in the Family of Special Numbers
Other Numbers Related to Leonard Euler:
Construction of Euler Numbers
Generating Function of the Sequence of Euler Numbers
Closed Formula for Euler Numbers
Properties of Euler Numbers
Polynomials Related to Euler Numbers
Problems Related to Euler Numbers
Other Number Constructions Related to Leonard Euler
Zoo of Numbers
Mini Dictionary
Exercises
Undergraduate students and teachers of Combinatorics, Number Theory, General Algebra, Cryptography and related fields, as well as general audience of amateurs of Mathematics. Professional Mathematicians and postgraduate students can also use the book as a reference book.
Pages: 480
ISBN: 978-981-98-1676-7 (hardcover)
This extensively revised second edition of Inverse Semigroups offers a comprehensive and accessible introduction to the subject, aimed at graduate students and researchers across mathematics. First published in 1998 and still widely read, this seminal text has now been reorganized, updated, and expanded to reflect the many advances in the field over the past two decades.
Inverse semigroup theory continues to play a central role in modern mathematics,
with growing significance in operator algebras — particularly C*-algebras
— as well as in group theory, topos theory, and the study of partial symmetries.
This edition incorporates substantial new material, culminating in two
brand new chapters, which brings the theory fully up to date
Pseudogroups.
Self-similar actions.
Advances in non-commutative Stone duality.
Emerging links with C*-algebras and group theory.
Partial group actions.
Insights into the historical development of the theory.
With no other contemporary text focused solely on inverse semigroups and their broadening web of applications, this volume is an essential resource for both newcomers and seasoned researchers alike.
Introduction to Inverse Semigroups
Abstract Pseudogroups
The Theorem of Ehresmann, Schein and Nambooripad
Basic Concepts
Examples of Inverse Semigroups
E-Unitary Semigroups and E-Unitary Covers
Inverse Semigroups Constructed from Actions
Boolean Inverse Monoids
Non-Commutative Stone Duality
The book is written for anyone with an undergraduate knowledge of mathematics. It is aimed at those with an interest in symmetry in all its forms. Those working in the theory of C*-algebras should find it particularly useful.
Pages: 150
ISBN: 978-981-98-1462-6 (hardcover)
ISBN: 978-981-98-1573-9 (softcover)
BLUF
A Prophetic Remark
The Richness of Nonlinear Equations
In Praise of Linear Equations: Peptide Design
The Surprise in Quadratic and Cubic Maps
IPST: My Past and Future
My Path to the Office of Naval Research
Decision to Stay at ONR
History of ONR
Why Math and Physics
The View from the Other Side of the Desk
My ONR Programs
The Saalfeld and Weisskopf Awards
DARPA
Schroedinger's Dogs and Yellow Pearls
Random Stories and Thoughts: Dumb and Otherwise
Research Professors; Undergraduate and graduate students in science and mathematics; Small business researchers; General readership in physics and mathematics.
March 2026
Pages: 375
ISBN: 978-981-98-1793-1 (hardcover)
This second edition of Linear Operator Equations: Approximation and Regularization provides a modern, accessible, and student-friendly introduction to the theory and numerical treatment of linear operator equations, both well-posed and ill-posed.
With rewritten and expanded proofs, a more consistent structure, and numerous new exercises and examples, the book offers clear pathways for learning approximation theory, regularization techniques, and applications of functional analysis to inverse problems. New topics include truncated singular value decomposition, generalized Tikhonov regularization with stabilizing operators, and a deeper treatment of discrete projection methods for operator equations.
Designed for students and researchers alike, the book balances theoretical rigor with practical insights, making it a valuable resource for courses in numerical functional analysis, inverse problems, and operator theory, or for self-study by applied mathematicians, engineers, and computational scientists.
A newly expanded bibliography, revised notation, and systematic numbering
make this edition even more usable as a reference and teaching tool
Introduction
Basic Results from Functional Analysis
Well-Posed Equations and Their Regularization
Ill-Posed Equations and Their Regularizations
Regularized Approximation Methods
Graduate students at the master's or pre-PhD level in mathematics, applied mathematics, or engineering; Researchers in numerical analysis, regularization theory, and functional analysis; Instructors of courses on operator theory, ill-posed problems, or inverse modeling; It can also serve as a text for an elective course following a second-level functional analysis course at the graduate level.