By José Ferreirós
"José Ferreirós has written a magisterial account of the heritage of set idea that is panoramic, balanced, and interesting. not just does this ebook synthesize a lot prior paintings and supply clean insights and issues of view, however it additionally includes a significant innovation, a full-fledged therapy of the emergence of the set-theoretic strategy in arithmetic from the early 19th century." --Bulletin of Symbolic good judgment (Review of first edition)
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"José Ferreirós has written a magisterial account of the heritage of set thought that is panoramic, balanced, and interesting. not just does this publication synthesize a lot earlier paintings and supply clean insights and issues of view, however it additionally includes a significant innovation, a full-fledged remedy of the emergence of the set-theoretic process in arithmetic from the early 19th century.
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Extra info for Labyrinth of Thought: A History of Set Theory and Its Role in Modern Mathematics
An interesting, short analysis of Grassmann's mathematical method can be found in [Nagel 1939, 215-19]. 14 I. Institutional and Intellectual Contexts truly mathematical propositions are always a priori, not empirical, judgements, since they involve necessity, which cannot be gained from experience, l As we have seen, Kant regarded the world as a representation in the subject, partly determined by the impressions received from the "things in themselves," partly by a priori characteristics of the subject's sensitive and conceptual abilities.
Besides that speculative vision, Hamilton presented very interesting mathematical ideas, including the notion that what is essential in I~ is the presence of a continuous ordering, a detailed and quite rigorous treatment of arithmetic, and the brilliant idea of introducing the complex numbers as pairs of real numbers. It seems that, this time, the philosophical garb was an obstacle for the diffusion of interesting thoughts. Nevertheless, his mathematical and philosophical ideas become well known later, thanks to the preface to his Lectures on Quaternions [Hamilton 1853].
26 I. Institutional and Intellectual Contexts 81 ], although it was the latter who got more involved in physical practices, eventually becoming Weber's teaching assistant around 1854 [Dedekind 1876, 512-13, 515]. Riemann's collaboration with Weber was the background for his many efforts to establish a unified treatment of the laws of nature. A manuscript note that must come from this or a later period, indicating Riemann's research topics, mentions his great research on Abelian and other transcendental functions, and on the integration of partial differential equations, and goes on: My main work deals with a new conception of the known natural laws - expression of them by means of different basic notions - which would make it possible to employ the experimental data on the interactions between heats, light, magnetism and electricity, in order to investigate their interrelation.