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First published in 1990, this book consists of a detailed
exposition of results of the theory of "interpretation" developed
by G. Kreisel - the relative impenetrability of which gives the
elucidation contained here great value for anyone seeking to
understand his work. It contains more complex versions of the
information obtained by Kreisel for number theory and clustering
around the no-counter-example interpretation, for number-theorectic
forumulae provide in ramified analysis. It also proves the
omega-consistency of ramified analysis. The author also presents
proofs of Schutte's cut-elimination theorems which are based on his
consistency proofs and essentially contain them - these went
further than any published work up to that point, helping to
squeeze the maximum amount of information from these proofs.
First published in 1990, this book consists of a detailed
exposition of results of the theory of "interpretation" developed
by G. Kreisel - the relative impenetrability of which gives the
elucidation contained here great value for anyone seeking to
understand his work. It contains more complex versions of the
information obtained by Kreisel for number theory and clustering
around the no-counter-example interpretation, for number-theorectic
forumulae provide in ramified analysis. It also proves the
omega-consistency of ramified analysis. The author also presents
proofs of Schutte's cut-elimination theorems which are based on his
consistency proofs and essentially contain them - these went
further than any published work up to that point, helping to
squeeze the maximum amount of information from these proofs.
In this illuminating collection, Charles Parsons surveys the
contributions of philosophers and mathematicians who shaped the
philosophy of mathematics over the course of the past century.
Parsons begins with a discussion of the Kantian legacy in the work
of L. E. J. Brouwer, David Hilbert, and Paul Bernays, shedding
light on how Bernays revised his philosophy after his collaboration
with Hilbert. He considers Hermann Weyl's idea of a "vicious
circle" in the foundations of mathematics, a radical claim that
elicited many challenges. Turning to Kurt Goedel, whose
incompleteness theorem transformed debate on the foundations of
mathematics and brought mathematical logic to maturity, Parsons
discusses his essay on Bertrand Russell's mathematical
logic--Goedel's first mature philosophical statement and an avowal
of his Platonistic view. Philosophy of Mathematics in the Twentieth
Century insightfully treats the contributions of figures the author
knew personally: W. V. Quine, Hilary Putnam, Hao Wang, and William
Tait. Quine's early work on ontology is explored, as is his
nominalistic view of predication and his use of the genetic method
of explanation in the late work The Roots of Reference. Parsons
attempts to tease out Putnam's views on existence and ontology,
especially in relation to logic and mathematics. Wang's
contributions to subjects ranging from the concept of set, minds,
and machines to the interpretation of Goedel are examined, as are
Tait's axiomatic conception of mathematics, his minimalist realism,
and his thoughts on historical figures.
In From Kant to Husserl, Charles Parsons examines a wide range of
historical opinion on philosophical questions, from mathematics to
phenomenology. Amplifying his early ideas on Kant's philosophy of
arithmetic, Parsons uses Kant's lectures on metaphysics to explore
how his arithmetical concepts relate to the categories. He then
turns to early reactions by two immediate successors of Kant,
Johann Schultz and Bernard Bolzano, to shed light on disputed
questions regarding interpretation of Kant's philosophy of
mathematics. Interested, as well, in what Kant meant by "pure
natural science," Parsons considers the relationship between the
first Critique and the Metaphysical Foundations of Natural Science.
His commentary on Kant's Transcendental Aesthetic departs from
mathematics to engage the vexed question of what it tells about the
meaning of Kant's transcendental idealism. Proceeding on to
phenomenology, Parsons examines Frege's evolving idea of
extensions, his attitude toward set theory, and his correspondence,
particularly exchanges with Russell and Husserl. An essay on
Brentano brings out, in the case of judgment, an alternative to the
now standard Fregean view of negation, and, on truth, alternatives
to the traditional correspondence view that are still discussed
today. Ending with the question of why Husserl did not take the
"linguistic turn," a final essay included here marks the only
article-length discussion of Husserl Parsons has ever written,
despite a long-standing engagement with this philosopher.
This scarce antiquarian book is a selection from Kessinger
Publishing's Legacy Reprint Series. Due to its age, it may contain
imperfections such as marks, notations, marginalia and flawed
pages. Because we believe this work is culturally important, we
have made it available as part of our commitment to protecting,
preserving, and promoting the world's literature. Kessinger
Publishing is the place to find hundreds of thousands of rare and
hard-to-find books with something of interest for everyone
This scarce antiquarian book is a selection from Kessinger
Publishing's Legacy Reprint Series. Due to its age, it may contain
imperfections such as marks, notations, marginalia and flawed
pages. Because we believe this work is culturally important, we
have made it available as part of our commitment to protecting,
preserving, and promoting the world's literature. Kessinger
Publishing is the place to find hundreds of thousands of rare and
hard-to-find books with something of interest for everyone
This scarce antiquarian book is a selection from Kessinger
Publishing's Legacy Reprint Series. Due to its age, it may contain
imperfections such as marks, notations, marginalia and flawed
pages. Because we believe this work is culturally important, we
have made it available as part of our commitment to protecting,
preserving, and promoting the world's literature. Kessinger
Publishing is the place to find hundreds of thousands of rare and
hard-to-find books with something of interest for everyone!
William Parsons (1800 67), third Earl of Rosse, was responsible for
building in 1845 the largest telescope of his time, nicknamed the
'Leviathan'. It enabled the Earl to make unprecedented astronomical
discoveries, including the discovery of the spiral nature of
galaxies. Rosse (then Lord Oxmantown) began publishing scientific
papers on telescopes in 1828, and for the rest of his life made
regular contributions to scientific journals in Ireland, England
and Scotland. He served as President of the British Association for
the Advancement of Science in 1843, and of the Royal Society from
1848 to 1854, and his addresses to those societies are also
included in this collection. Edited by his younger son, the
engineer Sir Charles Parsons (1854 1931) and published in 1926,
these papers show the wide range of the Earl's interests, from
astronomy and telescopes to ancient bronze artefacts and the use of
iron in shipbuilding.
Kurt Goedel (1906-1978) did groundbreaking work that transformed
logic and other important aspects of our understanding of
mathematics, especially his proof of the incompleteness of
formalized arithmetic. This book on different aspects of his work
and on subjects in which his ideas have contemporary resonance
includes papers from a May 2006 symposium celebrating Goedel's
centennial as well as papers from a 2004 symposium. Proof theory,
set theory, philosophy of mathematics, and the editing of Goedel's
writings are among the topics covered. Several chapters discuss his
intellectual development and his relation to predecessors and
contemporaries such as Hilbert, Carnap, and Herbrand. Others
consider his views on justification in set theory in light of more
recent work and contemporary echoes of his incompleteness theorems
and the concept of constructible sets.
Charles Parsons examines the notion of object, with the aim to
navigate between nominalism, denying that distinctively
mathematical objects exist, and forms of Platonism that postulate a
transcendent realm of such objects. He introduces the central
mathematical notion of structure and defends a version of the
structuralist view of mathematical objects, according to which
their existence is relative to a structure and they have no more of
a 'nature' than that confers on them. Parsons also analyzes the
concept of intuition and presents a conception of it distantly
inspired by that of Kant, which describes a basic kind of access to
abstract objects and an element of a first conception of the
infinite.
Charles Parsons examines the notion of object, with the aim to
navigate between nominalism, denying that distinctively
mathematical objects exist, and forms of Platonism that postulate a
transcendent realm of such objects. He introduces the central
mathematical notion of structure and defends a version of the
structuralist view of mathematical objects, according to which
their existence is relative to a structure and they have no more of
a 'nature' than that confers on them. Parsons also analyzes the
concept of intuition and presents a conception of it distantly
inspired by that of Kant, which describes a basic kind of access to
abstract objects and an element of a first conception of the
infinite.
The central project of the Critique of Pure Reason is to answer two
sets of questions: What can we know and how can we know it? and
What can't we know and why can't we know it? The essays in this
collection are intended to help students read the Critique of Pure
Reason with a greater understanding of its central themes and
arguments, and with some awareness of important lines of criticism
of those themes and arguments. Visit our website for sample
chapters!
The central project of the Critique of Pure Reason is to answer two
sets of questions: What can we know and how can we know it? and
What can't we know and why can't we know it? The essays in this
collection are intended to help students read the Critique of Pure
Reason with a greater understanding of its central themes and
arguments, and with some awareness of important lines of criticism
of those themes and arguments.
Kurt Goedel (1906-1978) did groundbreaking work that transformed
logic and other important aspects of our understanding of
mathematics, especially his proof of the incompleteness of
formalized arithmetic. This book on different aspects of his work
and on subjects in which his ideas have contemporary resonance
includes papers from a May 2006 symposium celebrating Goedel's
centennial as well as papers from a 2004 symposium. Proof theory,
set theory, philosophy of mathematics, and the editing of Goedel's
writings are among the topics covered. Several chapters discuss his
intellectual development and his relation to predecessors and
contemporaries such as Hilbert, Carnap, and Herbrand. Others
consider his views on justification in set theory in light of more
recent work and contemporary echoes of his incompleteness theorems
and the concept of constructible sets.
Kurt Gödel was the most outstanding logician of the 20th century and a giant in the field. This book is part of a five volume set that makes available all of Gödels writings. The first three volumes, already published, consist of the papers and essays of Gödel. The final two volumes of the set deal with Gödel's correspondence with his contemporary mathematicians, this fourth volume consists of material from correspondents from A-G.
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