Second edition of a landmark work by one of the world's leading philosophers
Revised, updated, and expanded by the author
Substantial new preface (approximately 16,000 words)
The author provides his current views on the original book and addresses the issues that were raised in its subsequent discussion
Original text revised and corrected throughout
Science Without Numbers caused a stir in philosophy on its original publication in 1980, with its bold nominalist approach to the ontology of mathematics and science. Hartry Field argues that we can explain the utility of mathematics without assuming it true. Part of the argument is that good mathematics has a special feature ("conservativeness") that allows it to be applied to "nominalistic" claims (roughly, those neutral to the existence of mathematical entities) in a way that generates nominalistic consequences more easily without generating any new ones. Field goes on to argue that we can axiomatize physical theories using nominalistic claims only, and that in fact this has advantages over the usual axiomatizations that are independent of nominalism. There has been much debate about the book since it first appeared. It is now reissued in a revised contains a substantial new preface giving the author's current views on the original book and the issues that were raised in the subsequent discussion of it.
1: Why the Utility of Mathematical Entities is Unlike the Utility of Theoretical Entities
Appendix: On Conservativeness
2: First Illustration of Why Mathematical Entities are Useful: Arithmetic
3: Second Illustration of Why Mathematical Entities are Useful: Geometry and Distance
4: Nominalism and the Structure of Physical Space
5: My Strategy for Nominalizing Physics, and its Advantages
6: A Nominalistic Treatment of Newtonian Space-Time
7: A Nominalistic Treatment of Quantities, and a Preview of a Nominalistic Treatment of the Laws Involving them
8: Newtonian Gravitational Theory Nominalized
9: Logic and Ontology
Hardback
Published: 27 October 2016 (Estimated)
288 Pages
234x156mm
ISBN: 9780198790426
Oxford Graduate Texts in Mathematics
Suitable for use in self-study when approaching the subject for the first time.
The universality of the method of outer measure in the construction of measures is emphasized.
A chapter on real functions of one or several real variables studies in detail those relevant properties of functions which are frequently used in analysis proper or in other disciplines. Integral formula in polar coordinates is given a complete proof and is applied to give a brief account of potential integral.
The all-powerful Hahn-Banach theorem in linear analysis is derived from a separation principle which is intuitively more satisfactory.
The book is written with the recognition that aside from its well-known application to mathematical physics, Real Analysis is indispensable for probability theory which is now a very important discipline in modern science.
A glance of measure and integration in Chapter 2 shows quickly the general feature of integration based on measure.
Real Analysis is indispensable for in-depth understanding and effective application of methods of modern analysis. This concise and friendly book is written for early graduate students of mathematics or of related disciplines hoping to learn the basics of Real Analysis with reasonable ease. The essential role of Real Analysis in the construction of basic function spaces necessary for the application of Functional Analysis in many fields of scientific disciplines is demonstrated with due explanations and illuminating examples.
After the introductory chapter, a compact but precise treatment of general measure and integration is taken up so that readers have an overall view of the simple structure of the general theory before delving into special measures. The universality of the method of outer measure in the construction of measures is emphasized because it provides a unified way of looking for useful regularity properties of measures. The chapter on functions of real variables sits at the core of the book; it treats in detail properties of functions that are not only basic for understanding the general feature of functions but also relevant for the study of those function spaces which are important when application of functional analytical methods is in question. This is then followed naturally by an introductory chapter on basic principles of Functional Analysis which reveals, together with the last two chapters on the space of p-integrable functions and Fourier integral, the intimate interplay between Functional Analysis and Real Analysis. Applications of many of the topics discussed are included to motivate the readers for further related studies; these contain explorations towards probability theory and partial differential equations.
1: Introduction and Preliminaries
2: A Glimpse of Measure and Integration
3: Construction of Measures
4: Functions of Real Variables
5: Basic Principles of Linear Analysis
6: Lp Spaces
7: Fourier Integral and Sobolev Space Hs
8: Postscript
Hardback
Published: 12 January 2017 (Estimated)
336 Pages
234x156mm
ISBN: 9780198758914
Whilst being a research monograph presenting new original results, the book is essentially self-contained and contains an introductory chapter reviewing the necessary preliminaries.
The unifying techniques introduced in this book have applications beyond Mathematics.
Good balance and integration between theoretical results and examples/applications.
The book gets the reader to the core aspects of topos theory and the ways in which it can be applied to shed light on a variety of different mathematical subjects.
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 classifying topos educed by categorical logicians. Mathematical theories (formalized within first-order logic) give rise to geometric objects called sites; the passage from sites to their associated toposes embodies the passage from the logical presentation of theories to their mathematical content, i.e. from syntax to semantics.
The essential ambiguity given by the fact that any topos is associated in general with an infinite number of theories or different sites allows to study the relations between different theories, and hence the theories themselves, by using toposes as 'bridges' between these different presentations. The expression or calculation of invariants of toposes in terms of the theories associated with them or their sites of definition generates a great number of results and notions varying according to the different types of presentation, giving rise to a veritable mathematical morphogenesis.
1: Topos-theoretic background
2: Classifying toposes and the 3: A duality theorem
4: Lattices of theories
5: Flat functors and classifying toposes
6: Theories of presheaf type: general criteria
7: Expansions and faithful interpretations
8: Quotients of a theory of presheaf type
9: Examples of theories of presheaf type
10: Some applications
* Standard reference work for researchers in this area
* Second supplementary volume to the revised and enlarged third
edition of General Lattice Theory (Lattice Theory: Foundations)
* Together with Foundations the most comprehensive work on an
explosively growing field
George Gratzer's Lattice Theory: Foundation is his third book on lattice theory (General
Lattice Theory, 1978, second edition, 1998). In 2009, Gratzer considered updating the
second edition to reflect some exciting and deep developments. He soon realized that to
lay the foundation, to survey the contemporary field, to pose research problems, would
require more than one volume and more than one person."
So Lattice Theory: Foundation provided the foundation. Now we complete this project
with Lattice Theory: Special Topics and Applications, in two volumes, written by a
distinguished group of experts, to cover some of the vast areas not in Foundation.
This second volume is divided into ten chapters contributed by K. Adaricheva, N. Caspard,
R. Freese, P. Jipsen, J.B. Nation, N. Reading, H. Rose, L. Santocanale, and F. Wehrung.
Series: Frontiers in Mathematics
* Provides one of the few in the area of Incidence Geometry which
discusses several families of point-line geometries at the same time
* Includes the graph theory necessary for the study of certain pointline
geometries
* Shows the connections between these various point-line geometries
This book gives an introduction to the field of Incidence Geometry by discussing the basic
families of point-line geometries and introducing some of the mathematical techniques
that are essential for their study. The families of geometries covered in this book include
among others the generalized polygons, near polygons, polar spaces, dual polar spaces
and designs. Also the various relationships between these geometries are investigated.
Ovals and ovoids of projective spaces are studied and some applications to particular
geometries will be given. A separate chapter introduces the necessary mathematical tools
and techniques from graph theory. This chapter itself can be regarded as a self-contained
introduction to strongly regular and distance-regular graphs.
This book is essentially self-contained, only assuming the knowledge of basic notions
from (linear) algebra and projective and affine geometry. Almost all theorems are
accompanied with proofs and a list of exercises with full solutions is given at the end
of the book. This book is aimed at graduate students and researchers in the fields of
combinatorics and incidence geometry.
* With worked examples, 55 end of chapter exercises and chapter
summaries
* The equivalence of various definitions of the canonical
transformation is proved explicitly, in contrast to competing books
* Discussion of (global) symmetries and the Noether theorem in the
framework of classical mechanics gives a new approach not covered
by most mechanics textbooks
The revised edition of this advanced text provides the reader with a solid grounding in
the formalism of classical mechanics, underlying a number of powerful mathematical
methods that are widely used in modern theoretical and mathematical physics. It reviews
the fundamentals of Lagrangian and Hamiltonian mechanics, and goes on to cover
related topics such as canonical transformations, integral invariants, potential motion in
geometric setting, symmetries, the Noether theorem and systems with constraints. While
in some cases the formalism is developed beyond the traditional level adopted in the
standard textbooks on classical mechanics, only elementary mathematical methods are
used in the exposition of the material.
New material for the revised edition includes additional sections on the Euler-Lagrange
equation, the Cartan two-form in Lagrangian theory, and Newtonian equations of motion
in context of general relativity. Also new for this edition is the inclusion of problem sets
and solutions to aid in the understanding of the material presented.
The mathematical constructions involved are explicitly described and explained, so the
book is a good starting point for the student new to this field. Where possible, intuitive
motivations are replaced by explicit proofs and direct computations, preserving the level
of rigor that makes the book useful for more advanced students intending to work in
one of the branches of the vast field of theoretical physics. To illustrate how classicalmechanics
formalism works in other branches of theoretical physics, examples related to
electrodynamics, as well as to relativistic and quantum mechanics, are included.