The anthology is a collection of articles contiguous to the
humanities written by renowned mathematicians of the 20th Century.
The articles cover a variety of topics that, for want of a better
name, shall be referred to as ?humanistic.? An important
criterion, thereby limiting the choice, is that the articles
should be accessible to the literate reader who may or may not
have technical knowledge of mathematics.
The articles span roughly a century in time and a wide range in
subject. They are by mathematicians acknowledged by their peers
as outstanding creators whose work has added richly to the
discipline. Each article is preceded by a brief biographical
sketch of the author and a brief indication of the content. In
many articles there is a common theme?the attempt to define the
role of intuition in mathematical activities. Intuition as
contrasted with pure formalism can be briefly, if too simply,
described by asking whether mathematics is invented or discovered.
One object of the anthology is to open to the reader some of the
thoughts and speculations of noted mathematicians often
discoursing on subjects outside the strict confines of
mathematics. It is a way of revealing the humanistic aspects of a
subject that has occupied the attention of humanity from the dawn
of time. The material is accessible to a wide audience, lay as
well as academic.
Contents:
I. Mathematics & the Intellect: Mathematics and Thinking
Mathematically, Mary Cartwright; Mathematical Invention, Henri
Poincare; Thought on the Heuristic Method, Jacques Hadamard;
Mathematical Proof, G. H. Hardy; The Unity of Knowledge, Hermann
Weyl. 2. Mathematics and Human Understanding: Mathematics and the
Arts, Marston Morse; Intuition, Reason, and Faith in Science,
George D. Birkhoff; Logic and the Understanding of Nature, David
Hilbert; the Cultural Basis of Mathematics, Raymond Wilder. 3.
Mathematics and Society: Presidential Address to the British
Association, J. J. Sylvester; The Mathematician, J. von Neumann;
The Community of Scholars, Andre Lichnerowica, History of
Mathematics: Why and How, Andre Weil. 4. Miscellaneous: Does God
Exist? Paul Levy, Goethe and Mathematics, Wilhelm Maak; Leonardo
and Mathematics, Francesco Severi; The Highest Good, Norbert
Wiener.
How should you encode a message to an extraterrestrial? What
do frogs and powers of 2 have in common? How many faces does the
Stella Octangula have? Is a plane figure of constant diameter a
circle, and what has this to do with NASA? 210 = 5 x 6 x 7 = 14 x
15, so just how many numbers can be the product of both two and
of three consecutive integers? Is there any such thing as a truly
correct map? What patterns are possible in juggling?
What do all of these questions have in common? They, and many
others, are answered in this book.
The authors are distinguished mathematicians; some are bright
newcomers while others have been well known in mathematical
circles for decades.
This is a partial record of the Bay Area Math Adventures (BAMA),
a lecture series for high school students (and incidentally their
teachers, parents, and other interested adults) hosted by San
Jose State and Santa Clara Universities in the San Francisco Bay
Area. These lectures are aimed primarily at bright high school
students, the emphasis on bright, and as a result, the
mathematics in some cases is far from what one would expect to
see in talks at this level. There are serious mathematical issues
addressed here.
We hope that this book will capture some of the magic of these
talks that have filled auditoriums at the host schools almost
monthly for several years. Join the students in sharing these
mathematical adventures.
Included in series
North-Holland Series in Applied Mathematics and Mechanics, 46
Description
This book deals with the impact of uncertainty in input data on
the outputs of mathematical models. Uncertain inputs as scalars,
tensors, functions, or domain boundaries are considered. In
practical terms, material parameters or constitutive laws, for
instance, are uncertain, and quantities as local temperature,
local mechanical stress, or local displacement are monitored. The
goal of the worst scenario method is to extremize the quantity
over the set of uncertain input data. A general mathematical
scheme of the worst scenario method, including approximation by
finite element methods, is presented, and then applied to various
state problems modeled by differential equations or variational
inequalities: nonlinear heat flow, Timoshenko beam vibration and
buckling, plate buckling, contact problems in elasticity and
thermoelasticity with and without friction, and various models of
plastic deformation, to list some of the topics. Dozens of
examples, figures, and tables are included. Although the book
concentrates on the mathematical aspects of the subject, a
substantial part is written in an accessible style and is devoted
to various facets of uncertainty in modeling and to the state of
the art techniques proposed to deal with uncertain input data. A
chapter on sensitivity analysis and on functional and convex
analysis is included for the reader's convenience.
Contents
Preface List of Figures List of Tables Introduction
Acknowledgments I Reality, Mathematics, and Computation 1
Modeling, Uncertainty, Verification, and Validation 1.1 Modeling
1.2 1.2 Verification and Validation 1.3 Desirable Features of a
Mathematical Model 2 Various Approaches to Uncertainty 2.1
Coupling the Worst Scenario Method with Fuzzy Sets, Evidence
Theory, and Probability 2.1.1 Worst Scenario and Fuzzy Sets I 2.1.2
Worst Scenario and Evidence Theory 2.1.3 Worst Scenario and
Probabilistic Methods 2.1.4 Worst Scenario and Fuzzy Sets II 2.2
Key Point: Admissible Set 2.3 How to Formulate Worst Scenario
Problems 2.4 On the Origin of Data 2.5 Conclusions II General
Abstract Scheme and the Analysis of the Worst Scenario Method 3
Formulation, Solvability, Approximation, Convergence 3.1 Worst
Scenario Problem 3.2 Approximate Worst Scenario Problem 3.3
Convergence Analysis III Quasilinear Elliptic Boundary Value
Problems 4 Uncertain Thermal Conductivity Problem 4.1 Setting of
the Problem 4.2 Approximate Worst Scenario Problem 4.3
Convergence Analysis 4.4 Sensitivity Analysis 4.5 Numerical
Examples 4.6 Heat Conduction: Special Case 5 Uncertain Nonlinear
Newton Boundary Condition 5.1 Continuous Problem 5.2 Approximate
Problem 5.3 Convergence of Approximate Solutions IV Parabolic
Problems 6 Linear Parabolic Problems 6.1 Stability of Solutions
to Parabolic Problems 6.2 Worst Scenario Problem 6.3 Approximate
Worst Scenario Problem 6.4 Convergence Analysis 7 Parabolic
Problems With a Unilateral Obstacle 7.1 Worst Scenario for a
General Variational Inequality 7.2 Applications to Fourier
Obstacle Problems V Elastic and Thermoelastic Beams 8 Transverse
Vibration of Timoshenko Beams with an Uncertain Shear Correction
Factor 8.1 Eigenvalue Problems 8.2 Worst Scenario Problems,
Sensitivity Analysis 9 Buckling of a Timoshenko Beam on an
Elastic Foundation 9.1 Buckling of a Timoshenko Beam 9.2 Buckling
of a Simply Supported Timoshenko Beam on an Elastic Foundation 9.3
Singular and Negative Values of the Shear Correction Factor 9.4
Summary of the Analysis 9.5 Worst Scenario Problem 10 Bending of
a Thermoelastic Beam with an Uncertain Coupling Coefficient 10.1
Approximations Bibliography and Comments on Chapter V VI Elastic
Plates and Pseudoplates 11 Pseudoplates 11.1 Formulation of a
State Problem 11.2 Stability of the Solution for a Class of
Variational Inequalities 11.3 Application to a Unilateral
Pseudoplate Problem 11.4 Criterion-Functionals and Worst Scenario
Problems 11.5 Approximate State Problem 11.6 Approximate Worst
Scenario Problems 11.7 Convergence of Approximate Solutions 12
Buckling of Elastic Plates 12.1 Buckling of a Rectangular Plate
12.2 Worst Scenario Problem 12.3 Initial Imperfection Combined
from One and Two Halfsinewaves Bibliography and Comments on
Chapter VI VII Contact Problems in Elasticity and
Thermoelasticity 13 Signorini Contact Problem with Friction 13.1
Setting of the Worst Scenario Problems 13.2 Existence of a Worst
Scenario 13.3 Approximate Worst Scenario Problems 13.4
Convergence Analysis 14 Unilateral Frictional Contact of Several
Bodies in Quasi-Coupled Thermoelasticity 14.1 Setting of
Thermoelastic Contact Problems 14.2 Sets of Uncertain Input Data
14.3 Worst Scenario Problems 14.4 Stability of Weak Solutions 14.5
Existence of a Solution 14.6 Comments on Unilateral Contact with
Coulomb Friction Bibliography and Comments on Chapter VII VIII
Hencky's and Deformation Theories of Plasticity 15 Timoshenko
Beam in Hencky's Model with Uncertain Yield Function 15.1 Setting
of the Problem in Terms of Bending Moment and Shear Forces 15.2
Worst Scenario Problems 15.3 Numerical Examples: von Mises Yield
Function 16 Torsion in Hencky's Model with Uncertain Stress-Strain
Law and Uncertain Yield Function 16.1 Problem Setting and
Stability of the Solution 16.2 Worst Scenario Problems 16.3
Approximate Worst Scenario Problems 16.4 Convergence Analysis 17
Deformation Theory of Plasticity 17.1 Setting of the State
Boundary Value Problem 17.2 Admissible Material Functions and the
Unique Solvability of the State Problem 17.3 Continuous
Dependence of the Solution 17.4 Worst Scenario Problems 17.5
Approximate Worst Scenario Problems 17.6 Convergence Analysis
Bibliography and Comments on Chapter VIII IX Flow Theories of
Plasticity 18 Perfect Plasticity 18.1 State Problem 18.2 Worst
Scenario Problems 18.3 Approximate Problems 19 Flow Theory with
Isotropic Hardening 19.1 Formulation of the State Problem 19.2
Uncertain Input Data 19.3 Approximate State Problem 19.4
Approximate Worst Scenario Problems 20 Flow Theory with Isotropic
Hardening in Strain Space 20.1 Variational Formulation of the
State Problem 20.2 Uncertain Input Data 20.3 Regularizations of
Problem P by Kinematic Hardening 20.4 Stability of the Solution
of the Regularized Problem 20.5 Stability of the Stress Tensor 20.6
Worst Scenario Problems 21 Combined Linear Kinematic and
Isotropic Hardening 21.1 Variational Formulation of the State
Problem 21.2 Uncertain Input Data 21.3 Stability of the State
Solution 21.4 Worst Scenario Problems 22 Validation of an
ElastoPlastic Plane Stress Model Bibliography and Comments on
Chapter IX X Domains With Uncertain Boundary 23 Neumann Boundary
Value Problem 23.1 Instability of Solutions 23.2 Reformulated
Newton Boundary Value Problem 23.3 Convergence with Respect to
Sequences of Domains 23.4 Difference Between Two Solutions 23.5
Closing Remarks 24 Dirichlet Boundary Value Problem 24.1
Stability of Solutions 24.2 Difference Between Two Solutions 24.3
Numerical Example XI Essentials of Sensitivity and Functional
Analysis 25 Essentials of Sensitivity Analysis 25.1 Matrix-Based
State Problems 25.2 Weakly Formulated Elliptic State Problems 25.3
General Theorem 26 Essentials of Functional and Convex Analysis
26.1 Functional Analysis 26.2 Function Spaces Appendix V&V in
Computational Engineering Introduction Definitions A View of
V&V Process and Rules for Model Selection Summary
0-19-514877-0, hardback, 856 pages Feb 2005,
Description
Mathematics and logic have been central topics of concern since
the dawn of philosophy. Since logic is the study of correct
reasoning, it is a fundamental branch of epistemology and a
priority in any philosophical system. Philosophers have focused
on mathematics as a case study for general philosophical issues
and for its role in overall knowledge- gathering. Today,
philosophy of mathematics and logic remain central disciplines in
contemporary philosophy, as evidenced by the regular appearance
of articles on these topics in the best mainstream philosophical
journals; in fact, the last decade has seen an explosion of
scholarly work in these areas.
This volume covers these disciplines in a comprehensive and
accessible manner, giving the reader an overview of the major
problems, positions, and battle lines. The 26 contributed
chapters are by established experts in the field, and their
articles contain both exposition and criticism as well as
substantial development of their own positions. The essays, which
are substantially self-contained, serve both to introduce the
reader to the subject and to engage in it at its frontiers.
Certain major positions are represented by two chapters--one
supportive and one critical.
The Oxford Handbook of Philosophy of Math and Logic is a ground-breaking
reference like no other in its field. It is a central resource to
those wishing to learn about the philosophy of mathematics and
the philosophy of logic, or some aspect thereof, and to those who
actively engage in the discipline, from advanced undergraduates
to professional philosophers, mathematicians, and historians.
Product Details
856 pages; 9 line illus.; 6-3/4 x 9-3/4; 0-19-514877-0
About the Author(s)
Stewart Shapiro is O'Donnell Professor of Philosophy at The Ohio
State University and Professorial Fellow at the Arche Centre,
University of St. Andrews.