By Francesca Biagioli
This ebook bargains a reconstruction of the controversy on non-Euclidean geometry in neo-Kantianism among the second one 1/2 the 19th century and the 1st a long time of the 20 th century. Kant famously characterised area and time as a priori different types of intuitions, which lie on the starting place of mathematical wisdom. The luck of his philosophical account of house used to be due no longer least to the truth that Euclidean geometry used to be largely thought of to be a version of simple task at his time. although, such later clinical advancements as non-Euclidean geometries and Einstein’s common concept 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 concept that of area from a resource of data to an item of study could be traced again to a practice, consisting of such mathematicians as Carl Friedrich Gauss, Bernhard Riemann, Richard Dedekind, Felix Klein, and Henri Poincaré, and which reveals one among its clearest expressions in Hermann von Helmholtz’s epistemological works. even though Helmholtz formulated compelling objections to Kant, the writer reconsiders diversified recommendations for a philosophical account of an identical transformation from a neo-Kantian viewpoint, and particularly Hermann Cohen’s account of the aprioricity of arithmetic by way of applicability and Ernst Cassirer’s reformulation of the a priori of house when it comes to a process of hypotheses. This e-book is perfect for college kids, students and researchers who desire to expand their wisdom of non-Euclidean geometry or neo-Kantianism.
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Additional info for Space, Number, and Geometry from Helmholtz to Cassirer
204–205) Kant maintained that geometry and arithmetic differ both in their methods and in their objects. Therefore, the axioms that for Kant are grounded in the synthesis of the productive imagination concern only magnitudes in general. However, Kant’s cognition in intuition, unlike Helmholtz’s, does not presuppose the speciﬁcation of the magnitude of a quantity. This is the issue of arithmetic. The answer to the question how big something is requires not so much construction in pure intuition, as 18 1 Helmholtz’s Relationship to Kant calculation.
The aprioricity of the axioms of (Euclidean) geometry is ruled out by the possibility of obtaining a more general system of hypotheses by denying supposedly necessary constraints in the form of outer intuition. Although I believe that Helmholtz’s account of spatial intuition can be made compatible with a relativized conception of the a priori, my suggestion is to reconsider the importance of the philosophical debate about the foundations of geometry for the actual development of such a conception.
Friedman reconsiders the Kantian aspect of Helmholtz’s theory for the following reason. The core idea of Helmholtz’s geometrical papers relates to his previous studies in the physiology of vision, because he believed that the distinction between voluntary and external movement, and the capacity to reproduce external changes by moving our own body or the objects around us, lies at the foundation of geometrical knowledge. In particular, Helmholtz pointed out the empirical origin of the notion of a rigid body: solid bodies or even parts of our own body work as standards 12 1 Helmholtz’s Relationship to Kant of measurement according to the observed fact that such bodies do not undergo any remarkable changes in shape and size during displacements.