Mathematics : History & Philosophy

Nonstandard Analysis enhances mathematical reasoning by introducing new ways of expression and deduction. Distinguishing between standard and nonstandard mathematical objects, its inventor, the eminent mathematician Abraham Robinson, settled in 1961 the centuries-old problem of how to use infinitesimals correctly in analysis. Having also worked as an engineer, he saw not only that his method greatly simplified mathematically proving and teaching, but also served as a powerful tool in modelling, analyzing and solving problems in the applied sciences, among others by effective rescaling and by infinitesimal discretizations. This book reflects the progress made in the forty years since the appearance of Robinson??'s revolutionary book Nonstandard Analysis: in the foundations of mathematics and logic, number theory, statistics and probability, in ordinary, partial and stochastic differential equations and in education. The contributions are clear and essentially self-contained. more...

Price: $129.00

What is spacetime? General relativity and quantum field theory answer this question in very different ways. This collection of essays by physicists and philosophers looks at the problem of uniting these two most fundamental theories of our world, focusing on the nature of space and time within this new quantum framework, and the kind of metaphysical picture suggested by recent developments in physics and mathematics. This is a book that will inspire further philosophical reflection. on recent advances in modern physics. - ;Quantum gravity is the name given to a theory that unites general relativity - Einstein's theory of gravitation and spacetime - with quantum field theory, our framework for describing non-gravitational forces. The Structural Foundations of Quantum Gravity brings together philosophers and physicists to discuss a range of conceptual issues that surface in the effort to unite these theories, focusing in particular on the ontological nature of the spacetime that results. Although there has been a great deal written about quantum gravity from the perspective of physicists and mathematicians, very little attention has been paid to the philosophical aspects. This volume closes that gap, with essays written by some of the leading researchers in the field. Individual papers defend or. attack a structuralist perspective on the fundamental ontologies of our physical theories, which offers the possibility of shedding new light on a number of foundational problems. It is a book that will be of interest not only to physicists and philosophers of physics but to anyone concerned with foundational issues and curious to explore new directions in our understanding of spacetime and quantum physics. - more...

Price: $99.00

The hunt for the 'Monster' of symmetry is one of the great mathematical quests, alongside Fermat's Last Theorem, the Riemann Hypothesis, and Poincar--eacute--; Conjecture. The Monster is a giant snowflake in 196,884 dimensions - the largest exception to our neat classifications of symmetry, with a beautiful structure which may turn out to unlock our understanding of symmetry, string theory, and the very fabric of our universe. The story of its discovery became the biggest joint mathematical project of all time - involving determination, luck, and some very extraordinary characters. - ;Mathematics is being driven forward by the quest to solve a small number of major problems - generating excitement in the mathematical world and beyond. Four famous challenges have been Fermat's Last Theorem, the Riemann Hypothesis, Poincar--eacute--;'s Conjecture, and, now, the quest for the 'Monster' of Symmetry. It is this latter that forms the topic of this book. Although its roots go back much further, the quest to understand symmetry really begins with the tragic young genius Evariste Galois, who died at the age of 20 in a duel. He used symmetry to understand algebraic equations, and he discovered that there were building blocks or 'atoms of symmetry'. Most fit into a table, rather like the periodic table of elements, but there are 26 exceptions. The biggest of these was dubbed 'the Monster' - a giant snowflake in 196,884 dimensions. At first the Monster was only dimly seen. Did it really exist, or was it a mirage? Many mathematicians became involved. The Monster became clearer, and it was no longer monstrous but a beautiful form that pointed out deep connections between symmetry, string theory, and the very fabric and form of the universe. The story of the discovery involves some extraordinary characters, and Mark Ronan brings these people to life, and recreates in accessible language the growing excitement of what became the biggest j more...

Price: $18.95

Oscar Zariski's work in mathematics permanently altered the foundations of algebraic geometry. This title is based upon the author's interviews with Zariski's family, colleagues, and students, and on his own memories from a series of tape-recorded interviews made a few years before his death in 1986. more...

Price: $39.95

This is Volume 1 of the first authoritative translation of Archimedes' works into English. Also provided are a scientific edition of the diagrams, a translation of the ancient commentator Eutocius and a commentary, where attention is paid to the cognitive and aesthetic nature of Archimedes' mathematical practice. more...

Price: $124.00