Symmetry and Its Formalisms: Mathematical Aspects

Abstract

This paper explores the relation between the concept of symmetry and its formalisms. The standard view among philosophers and physicists is that symmetry is completely formalized by mathematical groups. For some mathematicians however, the groupoid is a competing and more general formalism. An analysis of symmetry which justifies this extension has not been adequately spelled out. After a brief explication of how groups, equivalence, * Previous versions of this paper were presented at the University of Pittsburgh and at theÉcole Normale Supérieure (Paris). Thanks to the members of those audiences for their comments and questions and especially to John Earman and Gordon Belot. Guay's work was supported by a postdoctoral grant form the Fonds de Recherche sur la Société et la Culture du Québec (Canada). 1 and symmetries classes are related, we show that, while it's true in some instances that groups are too restrictive, there are other instances for which the standard extension to groupoids is too unrestrictive. The connection between groups and equivalence classes, when generalized to groupoids, suggests a middle ground between the two.

The grounding of computational psychoanalysis: A comparative history of culture overview of Matte Blanco bilogic

“…These have also played a certain role in linguistics, starting from Noam Chomsky's formalization attempts up until recent linguistic invariant theory (see [4]). Following almost verbatim [5], the symmetry of the initial state of a given situation implies the complete equivalence between the existing possible alternatives (for example, the left bundle of hay with respect to the right one of Buridan's ass argument).…”

“…Matthews (1919Matthews ( -1987) (see [3,Chapter 10]), according to which the essence of the scattering methods is like to determine the form of an invisible statue (i.e., the mathematical form of the scattering potential) from the angular distribution and intensity of the emerging rays due to a watering of it. 4 For instance, when, in Chapter 1, Section 3 of this work, Freud talks about contact-barriers, we may identify a certain conceptual analogy with the potential wells of elementary quantum mechanics. the physical states.…”

“…On the other hand, following [4], for some time mathematicians have argued for the more general groupoids as more suitable structures for treating symmetries. At the crossroads between groups and groupoids, lies a distinction between local and global symmetries.…”

“…Therefore, groupoids are offering more flexible mathematical structures embracing a wider formal tool box to treat symmetry and related phenomena. Groupoids can be understood not only as generalized groups but we can also see them as generalized equivalence relations: to be precise, it is possible to prove the existence of a certain two-way correspondence between groupoids and equivalence relations: see [4] and [11] for more details. Meant as a generalization of equivalence relations between parts of an object, groupoids open the door to local symmetries.…”

Symmetry and the Metaphysics of Physics

“…(Healey, 2009) Is the group a general enough concept to represent all symmetries? (Guay and Hepburn, 2009) I leave the pursuit of these problems to the interested reader.…”