ÉVARISTE GALOIS’ GROUP THEORY.  EPISTEMOLOGICAL NOTES ON ITS LOGICAL STRUCTURE.  

RAFFAELE PISANO 

University of Rome “La Sapienza”, Italy    

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http://www.historyofscience.it/ 

Abstract

      Up until the 18^th cent., attempts to produce arguments on infinitely great and infinitely small quantities were numerous and multifarious with the most vital contributions coming mainly from Newton (1642-1727) and Leibniz’ (1646-1716) theories.

Up until the 18th cent., attempts to produce arguments on infinitely great and infinitely small quantities were numerous and multifarious with the most vital contributions coming mainly from Newton (1642-1727) and Leibniz’ (1646-1716) theories. Nevertheless the logical-mathematical and physical meaning of infinitesimal objects was still not specified in an area which itself remained undefined conceptually. As a consequence, this way of conceiving the mathematical sciences and for interpreting physical phenomena (e.g., thermodynamics) produced, in the 19th  cent., well-known speculations on metaphysical objects ( ? ? ). In the tension-filled atmosphere of the era, Galois (1811-1832) played an important role proposing reasoning, as well as, a revolutionary thesis both for his predecessors and contemporaries. Recent historical and educational studies have also confirmed that his thesis seemed more consistent with mathematicians of the mid-20th cent., than rigorous calculus of his era.     Here, I study historical development of the foundations of theory in écrits et mémoirs mathématiques. Regard with his famous demonstration, I analyze logical thought about Permutation Group/Galois’ Group as a property and a measure of symmetry for a given equation. In order, I emphasis his reasoning about: the association of Permutation group (also "substitutions" for Galois) for every equation, logical conditions (for a given group) if an equation is solvable by radicals or not, logical structure on (sufficient condition by Gauss and) necessary condition, his idea to avoid using Lagrange resolvent, the permutation for a group of symmetry also provides a connection between Field theory and Group theory.  This investigation takes me through two categories of historical interpretation: the order of ideas as an element for understanding the historical evolution of scientific thought on one hand, and the use of logics as an element for scanning and controlling the organization of the theory on the other. Obviously the content of this work (in progress) could appear potentially factious, since it cannot be assumed to be the only possible perspective. Key words: Logics, relationship mathematics-physics, foundations, group theory.  

Minimum references

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