*Ettore Casari*

- Published in print:
- 2016
- Published Online:
- January 2017
- ISBN:
- 9780198788294
- eISBN:
- 9780191830228
- Item type:
- book

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198788294.001.0001
- Subject:
- Mathematics, Logic / Computer Science / Mathematical Philosophy

A starting point of Bolzano’s logical reflection was the conviction that among truths there is a connection, according to which some truths are grounds of others, and these in turn are consequences ...
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A starting point of Bolzano’s logical reflection was the conviction that among truths there is a connection, according to which some truths are grounds of others, and these in turn are consequences of the former, and that such a connection is objective, i.e. subsisting independently of every cognitive activity of the subject. In the attempt to account for the distinction between subjective and objective levels of knowledge, Bolzano gradually gained the conviction that the reference of the subject to the object is mediated by a realm of entities without existence that, recalling the Stoic lectà, are here called ‘lectological’. Moreover, of the two main ways through which that reference takes place—psychic activity and linguistic activity—Bolzano favoured the first and traced back to it the problems of the second; i.e. he considered those intermediate entities first as possible content of psychic phenomena and only subordinately, on the basis of a complex theory of signs, as meanings of linguistic phenomena. This book follows this schema and treats, in great detail, first, lectological entities (ideas and propositions in themselves), second, cognitive psychic phenomena (subjective ideas and judgements), and, finally, linguistic phenomena. Moreover, it tries to bring to light the extraordinary systematic character of Bolzano’s logical thought and it does this showing that the main logical ideas developed principally in the first three parts of the Theory of Science, published in 1837, can be effortlessly formally presented within the well-known Hilbertian epsilon-calculus.Less

A starting point of Bolzano’s logical reflection was the conviction that among truths there is a connection, according to which some truths are grounds of others, and these in turn are consequences of the former, and that such a connection is objective, i.e. subsisting independently of every cognitive activity of the subject. In the attempt to account for the distinction between subjective and objective levels of knowledge, Bolzano gradually gained the conviction that the reference of the subject to the object is mediated by a realm of entities without existence that, recalling the Stoic lectà, are here called ‘lectological’. Moreover, of the two main ways through which that reference takes place—psychic activity and linguistic activity—Bolzano favoured the first and traced back to it the problems of the second; i.e. he considered those intermediate entities first as possible content of psychic phenomena and only subordinately, on the basis of a complex theory of signs, as meanings of linguistic phenomena. This book follows this schema and treats, in great detail, first, lectological entities (ideas and propositions in themselves), second, cognitive psychic phenomena (subjective ideas and judgements), and, finally, linguistic phenomena. Moreover, it tries to bring to light the extraordinary systematic character of Bolzano’s logical thought and it does this showing that the main logical ideas developed principally in the first three parts of the *Theory of Science*, published in 1837, can be effortlessly formally presented within the well-known Hilbertian epsilon-calculus.

*Leon Horsten and Philip Welch (eds)*

- Published in print:
- 2016
- Published Online:
- November 2016
- ISBN:
- 9780198759591
- eISBN:
- 9780191820373
- Item type:
- book

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198759591.001.0001
- Subject:
- Mathematics, Logic / Computer Science / Mathematical Philosophy

The logician Kurt Gödel in 1951 established a disjunctive thesis about the scope and limits of mathematical knowledge: either the mathematical mind is equivalent to a Turing machine (i.e., a ...
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The logician Kurt Gödel in 1951 established a disjunctive thesis about the scope and limits of mathematical knowledge: either the mathematical mind is equivalent to a Turing machine (i.e., a computer) or there are absolutely undecidable mathematical problems. In the second half of the twentieth century, attempts have been made to arrive at a stronger conclusion. In particular, arguments have been produced by the philosopher J.R. Lucas and by the physicist and mathematician Roger Penrose that intend to show that the mathematicalmind ismore powerful than any computer. These arguments, and counterarguments to them, have not convinced the logical and philosophical community. The reason for this is an insufficiency of rigour in the debate. The contributions in this volume move the debate forward by formulating rigorous frameworks and formally spelling out and evaluating arguments that bear on Gödel’s disjunction in these frameworks. The contributions in this volume have been written by world leading experts in the field.Less

The logician Kurt Gödel in 1951 established a disjunctive thesis about the scope and limits of mathematical knowledge: either the mathematical mind is equivalent to a Turing machine (i.e., a computer) or there are absolutely undecidable mathematical problems. In the second half of the twentieth century, attempts have been made to arrive at a stronger conclusion. In particular, arguments have been produced by the philosopher J.R. Lucas and by the physicist and mathematician Roger Penrose that intend to show that the mathematicalmind ismore powerful than any computer. These arguments, and counterarguments to them, have not convinced the logical and philosophical community. The reason for this is an insufficiency of rigour in the debate. The contributions in this volume move the debate forward by formulating rigorous frameworks and formally spelling out and evaluating arguments that bear on Gödel’s disjunction in these frameworks. The contributions in this volume have been written by world leading experts in the field.

*Selman Akbulut*

- Published in print:
- 2016
- Published Online:
- November 2016
- ISBN:
- 9780198784869
- eISBN:
- 9780191827136
- Item type:
- book

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198784869.001.0001
- Subject:
- Mathematics, Geometry / Topology

This book present the topology of smooth 4-manifolds in an intuitive self-contained way. The handlebody theory, and the seiberg-witten theory of 4-manifolds are presented. Also stein and symplectic ...
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This book present the topology of smooth 4-manifolds in an intuitive self-contained way. The handlebody theory, and the seiberg-witten theory of 4-manifolds are presented. Also stein and symplectic structures on 4-manifolds are discussed, and many recent applications are given.Less

This book present the topology of smooth 4-manifolds in an intuitive self-contained way. The handlebody theory, and the seiberg-witten theory of 4-manifolds are presented. Also stein and symplectic structures on 4-manifolds are discussed, and many recent applications are given.

*Fon-Che Liu*

- Published in print:
- 2016
- Published Online:
- January 2017
- ISBN:
- 9780198790426
- eISBN:
- 9780191831676
- Item type:
- book

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198790426.001.0001
- Subject:
- Mathematics, Analysis

Real analysis in its modern aspect is presented concisely in this text for the beginning graduate student of mathematics and related disciplines to have a solid grounding in the general theory of ...
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Real analysis in its modern aspect is presented concisely in this text for the beginning graduate student of mathematics and related disciplines to have a solid grounding in the general theory of measure and to build helpful insights for effectively applying the general principles of real analysis to concrete problems. After an introductory chapter, a compact but precise treatment of general measure and integration is undertaken to provide the reader with an overall view of the general theory before delving into special measures. The universality of the method of outer measure in the construction of measures is emphasized, because it provides a unified way of looking for useful regularity properties of measures. The chapter on functions of real variables is the core of the book; it treats properties of functions that are not only basic for understanding the general features of functions but also relevant for the study of those function spaces which are important when application of functional analytical methods is in question. The chapter on basic principles of functional analysis and that on the Fourier integral reveal the intimate interplay between functional analysis and real analysis. Applications of many of the topics discussed are included; these contain explorations toward probability theory and partial differential equations.Less

Real analysis in its modern aspect is presented concisely in this text for the beginning graduate student of mathematics and related disciplines to have a solid grounding in the general theory of measure and to build helpful insights for effectively applying the general principles of real analysis to concrete problems. After an introductory chapter, a compact but precise treatment of general measure and integration is undertaken to provide the reader with an overall view of the general theory before delving into special measures. The universality of the method of outer measure in the construction of measures is emphasized, because it provides a unified way of looking for useful regularity properties of measures. The chapter on functions of real variables is the core of the book; it treats properties of functions that are not only basic for understanding the general features of functions but also relevant for the study of those function spaces which are important when application of functional analytical methods is in question. The chapter on basic principles of functional analysis and that on the Fourier integral reveal the intimate interplay between functional analysis and real analysis. Applications of many of the topics discussed are included; these contain explorations toward probability theory and partial differential equations.