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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.
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.
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, Godel); 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.
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.
Does syllogistic logic have the resources to capture mathematical
proof? This volume provides the first unified account of the
history of attempts to answer this question, the reasoning behind
the different positions taken, and their far-reaching implications.
Aristotle had claimed that scientific knowledge, which includes
mathematics, is provided by syllogisms of a special sort:
'scientific' ('demonstrative') syllogisms. In ancient Greece and in
the Middle Ages, the claim that Euclid's theorems could be recast
syllogistically was accepted without further scrutiny.
Nevertheless, as early as Galen, the importance of relational
reasoning for mathematics had already been recognized. Further
critical voices emerged in the Renaissance and the question of
whether mathematical proofs could be recast syllogistically
attracted more sustained attention over the following three
centuries. Supported by more detailed analyses of Euclidean
theorems, this led to attempts to extend logical theory to include
relational reasoning, and to arguments purporting to reduce
relational reasoning to a syllogistic form. Philosophical proposals
to the effect that mathematical reasoning is heterogenous with
respect to logical proofs were famously defended by Kant, and the
implications of the debate about the adequacy of syllogistic logic
for mathematics are at the very core of Kant's account of synthetic
a priori judgments. While it is now widely accepted that
syllogistic logic is not sufficient to account for the logic of
mathematical proof, the history and the analysis of this debate,
running from Aristotle to de Morgan and beyond, is a fascinating
and crucial insight into the relationship between philosophy and
mathematics.
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.
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.
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.
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