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With this third edition of Nelson Goodman's The Structure of Appear
ance, we are pleased to make available once more one of the most in
fluential and important works in the philosophy of our times.
Professor Geoffrey Hellman's introduction gives a sustained
analysis and appreciation of the major themes and the thrust of the
book, as well as an account of the ways in which many of Goodman's
problems and projects have been picked up and developed by others.
Hellman also suggests how The Structure of Appearance introduces
issues which Goodman later continues in his essays and in the
Languages of Art. There remains the task of understanding Good
man's project as a whole; to see the deep continuities of his
thought, as it ranges from logic to epistemology, to science and
art; to see it therefore as a complex yet coherent theory of human
cognition and practice. What we can only hope to suggest, in this
note, is the b. road Significance of Goodman's apparently technical
work for philosophers, scientists and humanists. One may say of
Nelson Goodman that his bite is worse than his bark. Behind what
appears as a cool and methodical analysis of the conditions of the
construction of systems, there lurks a radical and disturbing
thesis: that the world is, in itself, no more one way than another,
nor are we. It depends on the ways in which we take it, and on what
we do."
With this third edition of Nelson Goodman's The Structure of Appear
ance, we are pleased to make available once more one of the most in
fluential and important works in the philosophy of our times.
Professor Geoffrey Hellman's introduction gives a sustained
analysis and appreciation of the major themes and the thrust of the
book, as well as an account of the ways in which many of Goodman's
problems and projects have been picked up and developed by others.
Hellman also suggests how The Structure of Appearance introduces
issues which Goodman later continues in his essays and in the
Languages of Art. There remains the task of understanding Good
man's project as a whole; to see the deep continuities of his
thought, as it ranges from logic to epistemology, to science and
art; to see it therefore as a complex yet coherent theory of human
cognition and practice. What we can only hope to suggest, in this
note, is the b. road Significance of Goodman's apparently technical
work for philosophers, scientists and humanists. One may say of
Nelson Goodman that his bite is worse than his bark. Behind what
appears as a cool and methodical analysis of the conditions of the
construction of systems, there lurks a radical and disturbing
thesis: that the world is, in itself, no more one way than another,
nor are we. It depends on the ways in which we take it, and on what
we do."
This book explores the research of Professor Hilary Putnam, a
Harvard professor as well as a leading philosopher, mathematician
and computer scientist. It features the work of distinguished
scholars in the field as well as a selection of young academics who
have studied topics closely connected to Putnam's work. It includes
12 papers that analyze, develop, and constructively criticize this
notable professor's research in mathematical logic, the philosophy
of logic and the philosophy of mathematics. In addition, it
features a short essay presenting reminiscences and anecdotes about
Putnam from his friends and colleagues, and also includes an
extensive bibliography of his work in mathematics and logic. The
book offers readers a comprehensive review of outstanding
contributions in logic and mathematics as well as an engaging
dialogue between prominent scholars and researchers. It provides
those interested in mathematical logic, the philosophy of logic,
and the philosophy of mathematics unique insights into the work of
Hilary Putnam.
In these essays Geoffrey Hellman presents a strong case for a
healthy pluralism in mathematics and its logics, supporting
peaceful coexistence despite what appear to be contradictions
between different systems, and positing different frameworks
serving different legitimate purposes. The essays refine and extend
Hellman's modal-structuralist account of mathematics, developing a
height-potentialist view of higher set theory which recognizes
indefinite extendability of models and stages at which sets occur.
In the first of three new essays written for this volume, Hellman
shows how extendability can be deployed to derive the axiom of
Infinity and that of Replacement, improving on earlier accounts; he
also shows how extendability leads to attractive, novel resolutions
of the set-theoretic paradoxes. Other essays explore advantages and
limitations of restrictive systems - nominalist, predicativist, and
constructivist. Also included are two essays, with Solomon
Feferman, on predicative foundations of arithmetic.
In these essays Geoffrey Hellman presents a strong case for a
healthy pluralism in mathematics and its logics, supporting
peaceful coexistence despite what appear to be contradictions
between different systems, and positing different frameworks
serving different legitimate purposes. The essays refine and extend
Hellman's modal-structuralist account of mathematics, developing a
height-potentialist view of higher set theory which recognizes
indefinite extendability of models and stages at which sets occur.
In the first of three new essays written for this volume, Hellman
shows how extendability can be deployed to derive the axiom of
Infinity and that of Replacement, improving on earlier accounts; he
also shows how extendability leads to attractive, novel resolutions
of the set-theoretic paradoxes. Other essays explore advantages and
limitations of restrictive systems - nominalist, predicativist, and
constructivist. Also included are two essays, with Solomon
Feferman, on predicative foundations of arithmetic.
Mathematical and philosophical thought about continuity has changed
considerably over the ages. Aristotle insisted that continuous
substances are not composed of points, and that they can only be
divided into parts potentially. There is something viscous about
the continuous. It is a unified whole. This is in stark contrast
with the prevailing contemporary account, which takes a continuum
to be composed of an uncountably infinite set of points. This vlume
presents a collective study of key ideas and debates within this
history. The opening chapters focus on the ancient world, covering
the pre-Socratics, Plato, Aristotle, and Alexander. The treatment
of the medieval period focuses on a (relatively) recently
discovered manuscript, by Bradwardine, and its relation to medieval
views before, during, and after Bradwardine's time. In the
so-called early modern period, mathematicians developed the
calculus and, with that, the rise of infinitesimal techniques, thus
transforming the notion of continuity. The main figures treated
here include Galileo, Cavalieri, Leibniz, and Kant. In the early
party of the nineteenth century, Bolzano was one of the first
important mathematicians and philosophers to insist that continua
are composed of points, and he made a heroic attempt to come to
grips with the underlying issues concerning the infinite. The two
figures most responsible for the contemporary orthodoxy regarding
continuity are Cantor and Dedekind. Each is treated in an article,
investigating their precursors and influences in both mathematics
and philosophy. A new chapter then provides a lucid analysis of the
work of the mathematician Paul Du Bois-Reymond, to argue for a
constructive account of continuity, in opposition to the dominant
Dedekind-Cantor account. This leads to consideration of the
contributions of Weyl, Brouwer, and Peirce, who once dubbed the
notion of continuity "the master-key which . . . unlocks the arcana
of philosophy". And we see that later in the twentieth century
Whitehead presented a point-free, or gunky, account of continuity,
showing how to recover points as a kind of "extensive abstraction".
The final four chapters each focus on a more or less contemporary
take on continuity that is outside the Dedekind-Cantor hegemony: a
predicative approach, accounts that do not take continua to be
composed of points, constructive approaches, and non-Archimedean
accounts that make essential use of infinitesimals.
The present work is a systematic study of five frameworks or
perspectives articulating mathematical structuralism, whose core
idea is that mathematics is concerned primarily with interrelations
in abstraction from the nature of objects. The first two,
set-theoretic and category-theoretic, arose within mathematics
itself. After exposing a number of problems, the Element considers
three further perspectives formulated by logicians and philosophers
of mathematics: sui generis, treating structures as abstract
universals, modal, eliminating structures as objects in favor of
freely entertained logical possibilities, and finally,
modal-set-theoretic, a sort of synthesis of the set-theoretic and
modal perspectives.
Varieties of Continua explores the development of the idea of the
continuous. Hellman and Shapiro begin with two historical episodes.
The first is the remarkably rapid transition in the course of the
nineteenth century from the ancient Aristotelian view, that a true
continuum cannot be composed of points, to the now standard,
point-based frameworks for analysis and geometry found in modern
mainstream mathematics (stemming from the work of Bolzano, Cauchy,
Weierstrass, Dedekind, Cantor, et al.). The second is the
mid-tolate-twentieth century revival of pre-limit methods in
analysis and geometry using infinitesimals including non-standard
analysis (due to Abraham Robinson), and the more radical smooth
infinitesimal analysis that uses intuitionistic logic. Hellman and
Shapiro present a systematic comparison of these and related
alternatives (including constructivist and predicative
conceptions), weighing various trade-offs, helping articulate a
modern pluralist perspective, and articulate a modern pluralist
perspective on continuity. The main creative work of the book is
the development of rigorous regions-based theories of classical
continua, including Euclidean and non-Euclidean geometries, that
are mathematically equivalent (inter-reducible) to the currently
standard, point-based accounts in mainstream mathematics.
Hellman here presents a detailed interpretation of mathematics as
the investigation of "structural possibilities," as opposed to
absolute, Platonic objects. After treating the natural numbers and
analysis, he extends the approach to set theory, where he
demonstrates how to dispense with a fixed universe of sets.
Finally, he addresses problems of application to the physical
world.
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