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advanced_tools:group_theory [2018/03/28 16:23] jakobadmin [Intuitive] |
advanced_tools:group_theory [2018/03/28 16:24] jakobadmin |
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<tabbox Abstract> | <tabbox Abstract> | ||
- | <blockquote>The case I have chosen to treat in this paper concerns the question as to whether | + | |
- | the group concept should be extended to, or even subsumed by, the [[advanced_tools:category_theory:groupoids|groupoid concept]]. | + | |
- | Over a period stretching from at least as long ago as the early nineteenth | + | |
- | century, the group concept has emerged as the standard way to measure the degree | + | |
- | of invariance of an object under some collection of transformations.4 The informal | + | |
- | ideas codified by the group axioms, an axiomatisation which even Lakatos thought | + | |
- | unlikely to be challenged, relate to the composition of reversible processes | + | |
- | revealing the symmetry of a mathematical entity. Two early manifestations of | + | |
- | groups were as the permutations of the roots of a polynomial, later re-interpreted | + | |
- | as the automorphisms of the algebraic number field containing its roots, in Galois | + | |
- | theory, and as the structure-preserving automorphisms of a geometric space in the | + | |
- | Erlanger Programme. Intriguingly, it now appears that there is a challenger on the | + | |
- | scene. In some situations, it is argued, groupoids are better suited to extracting the | + | |
- | vital symmetries. And yet there has been a perception among their supporters— | + | |
- | who include some very illustrious names—of an unwarranted resistance in some | + | |
- | quarters to their use, which is only now beginning to decline.<cite>[[http://www.sciencedirect.com/science/article/pii/S0039368101000073|The importance of mathematical conceptualisation]] by David Corfield</cite></blockquote> | + | |
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* [[advanced_tools:group_theory:group_contraction]] | * [[advanced_tools:group_theory:group_contraction]] | ||
+ | <tabbox Research> | ||
+ | <blockquote>The case I have chosen to treat in this paper concerns the question as to whether | ||
+ | the group concept should be extended to, or even subsumed by, the [[advanced_tools:category_theory:groupoids|groupoid concept]]. | ||
+ | Over a period stretching from at least as long ago as the early nineteenth | ||
+ | century, the group concept has emerged as the standard way to measure the degree | ||
+ | of invariance of an object under some collection of transformations.4 The informal | ||
+ | ideas codified by the group axioms, an axiomatisation which even Lakatos thought | ||
+ | unlikely to be challenged, relate to the composition of reversible processes | ||
+ | revealing the symmetry of a mathematical entity. Two early manifestations of | ||
+ | groups were as the permutations of the roots of a polynomial, later re-interpreted | ||
+ | as the automorphisms of the algebraic number field containing its roots, in Galois | ||
+ | theory, and as the structure-preserving automorphisms of a geometric space in the | ||
+ | Erlanger Programme. Intriguingly, it now appears that there is a challenger on the | ||
+ | scene. In some situations, it is argued, groupoids are better suited to extracting the | ||
+ | vital symmetries. And yet there has been a perception among their supporters— | ||
+ | who include some very illustrious names—of an unwarranted resistance in some | ||
+ | quarters to their use, which is only now beginning to decline.<cite>[[http://www.sciencedirect.com/science/article/pii/S0039368101000073|The importance of mathematical conceptualisation]] by David Corfield</cite></blockquote> | ||
<tabbox FAQ> | <tabbox FAQ> |