Book contents
- Frontmatter
- Contents
- Editor's preface
- Acknowledgments
- Introduction
- Part I The semantic tradition
- 1 Kant, analysis, and pure intuition
- 2 Bolzano and the birth of semantics
- 3 Geometry, pure intuition, and the a priori
- 4 Frege's semantics and the a priori in arithmetic
- 5 Meaning and ontology
- 6 On denoting
- 7 Logic in transition
- 8 A logico-philosophical treatise
- Part II Vienna, 1925–1935
- Notes
- References
- Index
7 - Logic in transition
Published online by Cambridge University Press: 05 March 2012
Book contents
- Frontmatter
- Contents
- Editor's preface
- Acknowledgments
- Introduction
- Part I The semantic tradition
- 1 Kant, analysis, and pure intuition
- 2 Bolzano and the birth of semantics
- 3 Geometry, pure intuition, and the a priori
- 4 Frege's semantics and the a priori in arithmetic
- 5 Meaning and ontology
- 6 On denoting
- 7 Logic in transition
- 8 A logico-philosophical treatise
- Part II Vienna, 1925–1935
- Notes
- References
- Index
Summary
I have long struggled against the admission of ranges of values thereby of classes; but I have found no other possibility to provide a logical foundation for arithmetic. This question is: how are we to conceive of logical objects? And I have found no answer other than this: we conceive of them as extensions of concepts or, more generally, as ranges of values of functions … what other way is there?
Frege to Russell, 28 July 1902The first task in discussing the foundations of (pure) mathematics is to make precise the distinction between it and other sciences, a task which in Principia Mathematica is surprisingly neglected.
Ramsey, Undated manuscript (ASP)Logicism and the foundational crisis
In 1900 Russell underwent the one event in his intellectual life that he was willing to characterize as a “revolution”: He met Peano and was struck by the capacity of Peano's work to shed light on the philosophical nature of mathematics. It was at this time that Russell conceived one of his most fruitful ideas, the logicist project.
Peano had identified a notational system, or a cluster of concepts, that seemed to have enormous expressive power. The hope was that it could be used to express all of mathematics, and Peano's school had been working for years at rewriting different fragments of mathematics in their peculiar notation. Russell suggested that its basic concepts might be reduced to purely “logical” notions, in an as yet undisclosed sense of that word, and that perhaps all the assumptions one needed were those of logic, whatever they might be.
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- The Semantic Tradition from Kant to CarnapTo the Vienna Station, pp. 113 - 140Publisher: Cambridge University PressPrint publication year: 1991
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