Space, Number, and Geometry from Helmholtz to Cassirer by Francesca Biagioli

By Francesca Biagioli

This ebook deals a reconstruction of the talk on non-Euclidean geometry in neo-Kantianism among the second one 1/2 the 19th century and the 1st many years of the 20 th century. Kant famously characterised area and time as a priori different types of intuitions, which lie on the beginning of mathematical wisdom. The good fortune of his philosophical account of area used to be due now not least to the truth that Euclidean geometry used to be largely thought of to be a version of sure bet at his time. despite the fact that, such later clinical advancements as non-Euclidean geometries and Einstein’s common conception of relativity known as into query the understanding of Euclidean geometry and posed the matter of reconsidering house as an open query for empirical study. The transformation of the idea that of house from a resource of information to an item of study could be traced again to a convention, together with such mathematicians as Carl Friedrich Gauss, Bernhard Riemann, Richard Dedekind, Felix Klein, and Henri Poincaré, and which reveals one in all its clearest expressions in Hermann von Helmholtz’s epistemological works. even supposing Helmholtz formulated compelling objections to Kant, the writer reconsiders diversified concepts for a philosophical account of an identical transformation from a neo-Kantian point of view, and particularly Hermann Cohen’s account of the aprioricity of arithmetic when it comes to applicability and Ernst Cassirer’s reformulation of the a priori of house by way of a method of hypotheses. This publication is perfect for college kids, students and researchers who desire to develop their wisdom of non-Euclidean geometry or neo-Kantianism.

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Kant clearly ruled out logical necessity by considering both sensible and intellectual conditions of knowledge. Arguably, non-Euclidean geometry is consistent with Kant’s view that pure intuition alone can provide mathematical concepts with objective reality. Kant’s argument was directed against Leibniz’s and Wolff’s attempts to infer the existence of mathematical objects from the lack of contradiction in mathematical concepts. Although Kant could not consider the possibility of non-Euclidean geometry, his approach suggests that, while infinitely many geometries can be considered as logical possibilities, only Euclidean geometry is a real possibility according to the form of outer intuition.

This argument sheds some light on Kant’s previous characterization of space as an infinite given magnitude. It should be clear now that the magnitude considered in 4 cannot be of the same kind as those of physical objects: that would contradict the ideal nature of space. , division). Immediateness here indicates that constructions in pure intuition can be accomplished in principle. The discussion of Kant’s claims in the Transcendental Aesthetic emerged from a more general discussion of transcendental idealism about the forms of intuition: namely, the view that these forms are not perceivable themselves, because they provide us with general schemas for the ordering of any perceivable phenomena.

263) Arguably, Helmholtz believed that he should distance himself from Kant insofar as he took an empiricist direction. 230). By contrast, Moritz Schlick, in his comments on the centenary edition of Helmholtz’s Epistemological Writings (1921), connected Helmholtz’s theory of signs with his own project of a scientific empiricism, which, unlike neo-Kantianism, on the one hand, and positivism, on the other, would provide us with a philosophical interpretation of Einstein’s general theory of relativity.

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