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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.
Hartry Field presents a selection of thirteen of his most important essays on a set of related topics at the foundations of philosophy; one essay is previously unpublished, and eight are accompanied by substantial new postscripts. Five of the essays are primarily about truth, meaning, and propositional attitudes, five are primarily about semantic indeterminacy and other kinds of 'factual defectiveness' in our discourse, and three are primarily about issues concerning objectivity, especially in mathematics and in epistemology. This influential work by a key figure in contemporary philosophy will reward the attention of any philosopher interested in language, epistemology, or mathematics.
Saving Truth from Paradox is an ambitious investigation into
paradoxes of truth and related issues, with occasional forays into
notions such as vagueness, the nature of validity, and the Godel
incompleteness theorems. Hartry Field presents a new approach to
the paradoxes and provides a systematic and detailed account of the
main competing approaches.
Part One examines Tarski's, Kripke's, and Lukasiewicz's theories
of truth, and discusses validity and soundness, and vagueness. Part
Two considers a wide range of attempts to resolve the paradoxes
within classical logic. In Part Three Field turns to non-classical
theories of truth that that restrict excluded middle. He shows that
there are theories of this sort in which the conditionals obey many
of the classical laws, and that all the semantic paradoxes (not
just the simplest ones) can be handled consistently with the naive
theory of truth. In Part Four, these theories are extended to the
property-theoretic paradoxes and to various other paradoxes, and
some issues about the understanding of the notion of validity are
addressed. Extended paradoxes, involving the notion of determinate
truth, are treated very thoroughly, and a number of different
arguments that the theories lead to "revenge problems" are
addressed. Finally, Part Five deals with dialetheic approaches to
the paradoxes: approaches which, instead of restricting excluded
middle, accept certain contradictions but alter classical logic so
as to keep them confined to a relatively remote part of the
language. Advocates of dialetheic theories have argued them to be
better than theories that restrict excluded middle, for instance
over issues related to the incompleteness theoremsand in avoiding
revenge problems. Field argues that dialetheists' claims on behalf
of their theories are quite unfounded, and indeed that on some of
these issues all current versions of dialetheism do substantially
worse than the best theories that restrict excluded middle.
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