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
3 - Geometry, pure intuition, and the a priori
Published online by Cambridge University Press: 05 March 2012
- 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
In philosophy, an intuition can only be an example; in mathematics, on the other hand, an intuition is the essential thing.
Kant, Logik BusoltFor Helmholtz, however, there existed the option: either “necessity of thought” or “empirical origin.” But it is appropriate to add to these: necessity of intuition, and this as pure.
Cohen, Kants Theorie der ErfahrungSo it is entirely implausible that outside the range of pure mathematics we will ever make use of these hypotheses of non-Euclidean spaces.
Riehl, Phil. Krit, vol. 2From the beginning of the nineteenth century, Kant's pure intuition had a rough time in analysis. The rigorization of the calculus banished intuition from the notions of function, continuity, limit, infinitesimal, and all else that had elicited Berkeley's justified complaint. The arithmetization of analysis cornered the pure intuition of time into arithmetic, where Frege would soon deal it a death blow (Chapter 4). Mathematics was not just the theory of abstract magnitudes, numbers, functions, and infinitesimals, however. It was also the science of space, geometry, and here Kantians could rest assured that intuition would never be dethroned. Or so it seemed for a while.
During the nineteenth century, geometry was the battleground of two major epistemological wars. The first, the subject of this chapter, concerned the role of pure intuition in knowledge; the second, surveyed in Chapter 7, took it for granted that that role was nil and questioned the nature of geometric concepts.
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- Information
- The Semantic Tradition from Kant to CarnapTo the Vienna Station, pp. 41 - 61Publisher: Cambridge University PressPrint publication year: 1991