Bandini A.'s 3-Selmer groups for curves y^2 = x^3 + a PDF

By Bandini A.

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We thus offer a close analysis of the relevant historical episodes. 49 Chester 2002, 111. 50 Ibid. 51 Chester 2002, 112. 52 Wigner [1959] 1979. 18 1 Introduction enterprise and observe that group theory captures some essential elements of it. ”53 This position was already sketched in the closing years of the 19th century by one of the foremost contributors to group theory, Sophus Lie (1842–1899): Having seen how much the ideas of Galois have little by little been shown to be fruitful in so many branches of analysis, geometry, and even of mechanics, it is surely permitted to hope that their power will become equally manifest in mathematical physics.

83 For the philosopher, van Fraassen, it must have been the case that Gottfried Wilhelm von Leibniz (1646–1716) connected his metaphysical principles of Sufficient Reason and Indifference with the epistemology of symmetry principles; this is, after all, a vivid demonstration of van Fraassen’s claim about the power of symmetry as a philosophical concept. However, for the historian this is running the story in reverse. Granted, van Fraassen is aware that he speculates on the basis of meager data—a brief note; nonetheless, he is confident that invoking symmetry principles in the analysis of Leibniz’s appeal to the two principles is faithful.

The mathematical sense of symmetry, that is, commensurability, did not change in the period from Euclid to Isaac Barrow (1630–1677) and beyond. In contrast to the stability of the mathematical sense, a new aesthetic sense of symmetry was formulated in France and it is found in texts on architecture in the 16th and 17th centuries; these texts drew in turn on Italian sources, going back to the 15th century. In the 18th century the aesthetic sense of symmetry in art and architecture was current, and a few instances of symmetry in scientific contexts can also be identified (see Ch.

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3-Selmer groups for curves y^2 = x^3 + a by Bandini A.


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