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Contemporary philosophy of mathematics offers us an embarrassment
of riches. Among the major areas of work one could list
developments of the classical foundational programs, analytic
approaches to epistemology and ontology of mathematics, and
developments at the intersection of history and philosophy of
mathematics. But anyone familiar with contemporary philosophy of
mathematics will be aware of the need for new approaches that pay
closer attention to mathematical practice. This book is the first
attempt to give a coherent and unified presentation of this new
wave of work in philosophy of mathematics. The new approach is
innovative at least in two ways. First, it holds that there are
important novel characteristics of contemporary mathematics that
are just as worthy of philosophical attention as the distinction
between constructive and non-constructive mathematics at the time
of the foundational debates. Secondly, it holds that many topics
which escape purely formal logical treatment--such as
visualization, explanation, and understanding--can nonetheless be
subjected to philosophical analysis.
The Philosophy of Mathematical Practice comprises an introduction
by the editor and eight chapters written by some of the leading
scholars in the field. Each chapter consists of a short
introduction to the general topic of the chapter followed by a
longer research article in the area. The eight topics selected
represent a broad spectrum of contemporary philosophical reflection
on different aspects of mathematical practice: diagrammatic
reasoning and representational systems; visualization; mathematical
explanation; purity of methods; mathematical concepts; the
philosophical relevance ofcategory theory; philosophical aspects of
computer science in mathematics; the philosophical impact of recent
developments in mathematical physics.
Paolo Mancosu continues an investigation he began in his 2013 book
Inside the Zhivago Storm, which the New York Book Review of Books
described as "a tour de force of literary detection worthy of a
scholarly Sherlock Holmes". In this book Mancosu extends his
detective work by reconstructing the network of contacts that
helped Pasternak smuggle the typescripts of Doctor Zhivago outside
the Soviet Union and following the vicissitudes of the typescripts
when they arrived in the West. Mancosu draws on a wealth of
firsthand sources to piece together the long-standing mysteries
surrounding the many different typescripts that played a role in
the publication of Doctor Zhivago, thereby solving the problem of
which typescript served as the basis of the first Russian edition:
a pirate publication covertly orchestrated by the Central
Intelligence Agency (CIA). He also offers a new perspective, aided
by the recently declassified CIA documents, by narrowing the focus
as to who might have passed the typescript to the CIA. In the
process, Mancosu reveals details of events that were treated as top
secret by all those involved, vividly recounting the history of the
publication of Pasternak's epic work with all its human and
political ramifications.
The 17th century saw a dramatic development in mathematical theory and practice. With the recovery of many of the classical Greek mathematical texts, new techniques were developed, and within 100 years the rules of modern analytic geometry, geometry of indivisibles, arithmetic of infinites, and calculus had been developed. Although many technical studies have been devoted to these developments, Mancosu provides the first comprehensive account of the foundational issues raised in the relationship between mathematical advances of this period and philosophy of mathematics of the time.
An Introduction to Proof Theory provides an accessible introduction
to the theory of proofs, with details of proofs worked out and
examples and exercises to aid the reader's understanding. It also
serves as a companion to reading the original pathbreaking articles
by Gerhard Gentzen. The first half covers topics in structural
proof theory, including the Goedel-Gentzen translation of classical
into intuitionistic logic (and arithmetic), natural deduction and
the normalization theorems (for both NJ and NK), the sequent
calculus, including cut-elimination and mid-sequent theorems, and
various applications of these results. The second half examines
ordinal proof theory, specifically Gentzen's consistency proof for
first-order Peano Arithmetic. The theory of ordinal notations and
other elements of ordinal theory are developed from scratch, and no
knowledge of set theory is presumed. The proof methods needed to
establish proof-theoretic results, especially proof by induction,
are introduced in stages throughout the text. Mancosu, Galvan, and
Zach's introduction will provide a solid foundation for those
looking to understand this central area of mathematical logic and
the philosophy of mathematics.
The conflict between Soviet Communists and Boris Pasternak over the
publication of Doctor Zhivago did not end when he won the Nobel
Prize, or even when the author died. Paolo Mancosu tells how
Pasternak's expulsion from the Soviet Writers' Union left him in
financial difficulty. Milan publisher Giangiacomo Feltrinelli and
Sergio d'Angelo, who had brought the typescript of Doctor Zhivago
to Feltrinelli, were among those who arranged a smuggling operation
to help him.After Pasternak's death, Olga Ivinskaya, his companion,
literary assistant, and the inspiration for Zhivago's Lara, also
received some of the Zhivago royalties. After the KGB intercepted
Pasternak's will on her behalf, the Soviets arrested and sentenced
her and her daughter, Irina Emelianova, to eight years and three
years of labor camp, respectively. The ensuing international
outrage inspired a secret campaign in the West to win their
freedom.Mancosu's new book-the first to explore the post-Nobel
history of Pasternak and Ivinskaya-provides extraordinary detail on
these events, in a thrilling account that involves KGB
interceptions, fabricated documents, smugglers, and much more.
While a general reader will respond to the dramatic human story,
specialists will be rewarded with a rich assemblage of new archival
material, especially letters of Pasternak, Ivinskaya, Feltrinelli,
and d'Angelo from the Hoover Institution Library and Archives and
the Feltrinelli Archives in Milan.
Paolo Mancosu provides an original investigation of historical and
systematic aspects of the notions of abstraction and infinity and
their interaction. A familiar way of introducing concepts in
mathematics rests on so-called definitions by abstraction. An
example of this is Hume's Principle, which introduces the concept
of number by stating that two concepts have the same number if and
only if the objects falling under each one of them can be put in
one-one correspondence. This principle is at the core of
neo-logicism. In the first two chapters of the book, Mancosu
provides a historical analysis of the mathematical uses and
foundational discussion of definitions by abstraction up to Frege,
Peano, and Russell. Chapter one shows that abstraction principles
were quite widespread in the mathematical practice that preceded
Frege's discussion of them and the second chapter provides the
first contextual analysis of Frege's discussion of abstraction
principles in section 64 of the Grundlagen. In the second part of
the book, Mancosu discusses a novel approach to measuring the size
of infinite sets known as the theory of numerosities and shows how
this new development leads to deep mathematical, historical, and
philosophical problems. The final chapter of the book explore how
this theory of numerosities can be exploited to provide
surprisingly novel perspectives on neo-logicism.
Paolo Mancosu presents a series of innovative studies in the
history and the philosophy of logic and mathematics in the first
half of the twentieth century. The Adventure of Reason is divided
into five main sections: history of logic (from Russell to Tarski);
foundational issues (Hilbert's program, constructivity,
Wittgenstein, Goedel); mathematics and phenomenology (Weyl, Becker,
Mahnke); nominalism (Quine, Tarski); semantics (Tarski, Carnap,
Neurath). Mancosu exploits extensive untapped archival sources to
make available a wealth of new material that deepens in significant
ways our understanding of these fascinating areas of modern
intellectual history. At the same time, the book is a contribution
to recent philosophical debates, in particular on the prospects for
a successful nominalist reconstruction of mathematics, the nature
of finitist intuition, the viability of alternative definitions of
logical consequence, and the extent to which phenomenology can hope
to account for the exact sciences.
Mancosu provides the first comprehensive account of the relationship between mathematical advances of the seventeenth century and the philosophy of mathematics of the period. Starting with the Renaissance debates on the certainty of mathematics, Mancosu leads the reader through the foundational issues raised by the emergence of these new mathematical techniques. He draws a sophisticated picture of the subtle dependencies between technical development and philosophical reflection in seventeenth century mathematics.
Contemporary philosophy of mathematics offers us an embarrassment
of riches. Among the major areas of work one could list
developments of the classical foundational programs, analytic
approaches to epistemology and ontology of mathematics, and
developments at the intersection of history and philosophy of
mathematics. But anyone familiar with contemporary philosophy of
mathematics will be aware of the need for new approaches that pay
closer attention to mathematical practice. This book is the first
attempt to give a coherent and unified presentation of this new
wave of work in philosophy of mathematics. The new approach is
innovative at least in two ways. First, it holds that there are
important novel characteristics of contemporary mathematics that
are just as worthy of philosophical attention as the distinction
between constructive and non-constructive mathematics at the time
of the foundational debates. Secondly, it holds that many topics
which escape purely formal logical treatment--such as
visualization, explanation, and understanding--can nonetheless be
subjected to philosophical analysis.
The Philosophy of Mathematical Practice comprises an introduction
by the editor and eight chapters written by some of the leading
scholars in the field. Each chapter consists of a short
introduction to the general topic of the chapter followed by a
longer research article in the area. The eight topics selected
represent a broad spectrum of contemporary philosophical reflection
on different aspects of mathematical practice: diagrammatic
reasoning and representational systems; visualization; mathematical
explanation; purity of methods; mathematical concepts; the
philosophical relevance of category theory; philosophical aspects
of computer science in mathematics; the philosophical impact of
recent developments in mathematical physics.
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