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3 - Geometry, pure intuition, and the a priori

Published online by Cambridge University Press:  05 March 2012

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Summary

In philosophy, an intuition can only be an example; in mathematics, on the other hand, an intuition is the essential thing.

Kant, Logik Busolt

For 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 Erfahrung

So 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. 2

From 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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The Semantic Tradition from Kant to Carnap
To the Vienna Station
, pp. 41 - 61
Publisher: Cambridge University Press
Print publication year: 1991

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