Symmetry and equivalence

Philosophy of Science, 2009

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). 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.

Commensurability and symmetry have diverged from a common Greek origin. In the Latin of Boethius, commensurable numbers are numbers not prime to one another. With Billingsley's translation of Euclid, commensurable magnitudes, including numbers, have come to be what Euclid himself called symmetric: possessed of a common measure, which for numbers can be unity alone. Symmetry has always had a vaguer sense as well: a quality that contributes to, if it does not constitute, the beauty of an object. The symmetry of a mathematical structure is given by its automorphism group; the size, by its underlying set. We measure the set by counting it, and we may express the result by a particular cardinal number: in Cantor's definition, made precise by von Neumann, this number is a certain set that is equipollent with the original set. For measuring symmetry, strictly speaking, we have no corresponding activity, because we have no simple way to select a representative from each isomorphism class of groups. Noneless, we allude to such representatives, as when we use the definite article to refer to the infinite cyclic group, instead of an arbitrary infinite cyclic group. Equality sometimes means identity, sometimes isomorphism or congruence. In our sign of equality, invented by Recorde, two line segments are depicted that are not the same, but their lengths are the same. It is worthwhile to pay attention to the distinction between equality and sameness, precisely because recognizing the possibility of confusing them has often been a mathematical advance.

2021

We state the defining characteristic of mathematics as a type of symmetry where one can change the connotation of a mathematical statement in a certain way when the statement’s truth value remains the same. This view of mathematics as satisfying such symmetry places mathematics as comparable with modern views of physics and science where, over the past century, symmetry also plays a defining role. We explore the very nature of mathematics and its relationship with natural science from this perspective. This point of view helps clarify some standard problems in the philosophy of mathematics.

Acta Physica Hungarica A) Heavy Ion Physics, 2004

Symmetry played a central role in the works of E.P. Wigner. He made a great contribution to the extension and reinterpretation of the concept of symmetry. He proposed a new classification. The paper investigates in the light of Wigner's works how do we see now the classification of physical symmetries.

Foundations of Science, 2016

Over the past few decades the notion of symmetry has played a major role in physics and in the philosophy of physics. Philosophers have used symmetry to discuss the ontology and seeming objectivity of the laws of physics. We introduce several notions of symmetry in mathematics and explain how they can also be used in resolving different problems in the philosophy of mathematics. We use symmetry to discuss the objectivity of mathematics, the role of mathematical objects, the unreasonable effectiveness of mathematics and the relationship of mathematics to physics.

Philosophy of Science, 2006

Symmetry is commonly perceived as a concept that expresses bilateral or radial relations, which effectively describes spatial arrangements that most people think is in some sense innate to the human mind. So, does the concept have a history? Has it evolved? Was there a revolution? The long history of the concept of symmetry began in classical Antiquity as a single concept with a range of applications, expressing proportionality with a specific constraint. In fact, symmetry was used in two different contexts: in mathematics it had the technical meaning of commensurable, while generally it meant suitable or well proportioned. The latter usage involves an aesthetic judgment arrived at by comparison with an ideal in the relevant domain, in an attempt to establish a certain property of the object, e.g., that it is beautiful or that it functions efficiently. We offer historical evidence that, despite the variety of usages in many different domains, there is a conceptual unity underlying the invocation of symmetry in the period from Antiquity to the 1790s which is distinct from the scientific usages of this term that first emerged in France at the end of the 18th century. We examine the trajectory of the concept in the mathematical and scientific disciplines as well as its trajectory in art and architecture. The changes in the meaning of symmetry from Antiquity to the eighteenth century can be explained by appealing to evolution—nobody in that period claimed to be doing anything new. The philosopher Immanuel Kant is probably the first thinker to indicate that something is fundamentally missing in the traditional account. In 1768 he introduced the concept of incongruent counterparts to indicate a reversal of ordering in entities that are equal and similar but cannot be superposed. However, the key figure in revolutionizing the concept of symmetry was the mathematician Adrien-Marie Legendre who, in 1794, claimed to be doing something new. Indeed, by introducing a principle of ordering he revolutionized the concept, and laid the groundwork for its modern usages.

Archimedes, 2008

Crystallography Reviews, 2017

2016