Background and Motivation
The modern scientific definition of symmetry is that of invariance under transformations. The concept of symmetry has proved remarkably fruitful in recent physical science. Symmetry provides a guide to ontology: the indiscernibility of elementary particles is a form of symmetry. It plays a role in delineating fundamental physical principles: every conservation law is associated with an invariance under a specific transformation. It constitutes a framework for modelling causal processes, by suggesting that effects must possess the symmetries of their causes. It serves as a heuristic principle in theory formulation: symmetry considerations suggest which physical situations must be treated as distinct and which as the same. The attendant notion of symmetry breaking has demonstrated a similar fruitfulness and range of applicability.
Recently, symmetry has come to play important roles in biological science too. The symmetries of plant morphology are regarded as a leading principle in ontogenesis. The bilateral symmetries of animals, and their assessment by potential mates, have been revealed as an important factor in sexual selection.
But foundational questions about the status of symmetry principles in science remain. Do symmetry principles convey empirical information about the world, or are they in some sense trivially true? Are symmetry considerations merely methodologically sound procedures, or do they correspond to the structure of the world? Can the validity of symmetry principles be established a priori by logical means, or must it be learned by experience?
Symmetry also plays important roles in domains other than the sciences. In architecture, for example, symmetry is associated with order, balance, and seemliness. Symmetry principles are similarly prominent in other art forms, though perfect symmetries are frequently regarded in visual art as aesthetically sterile.
The aim of this interdisciplinary Lorentz Center workshop is to review conceptions of symmetry from four different perspectives. The following themes are intended to facilitate interactions across disciplines:
• Order vs. Disorder
• The Epistemological Status of Symmetry Principles in Physics
• Symmetry and Symmetry Breaking in Morphogenesis
• Overarching Perspectives: D’Arcy Thompson and Hermann Weyl
Session 1: Tuesday 11 March 2008
Order vs. Disorder
The opening day is devoted to exploring the general concepts of order and disorder to shed light on the tension between symmetry and asymmetry. On one reading, symmetry is understood as corresponding to a highly ordered state, which requires effort to achieve and maintain, whereas departures from symmetry are understood to correspond to disorder and lack of structure. The concepts of entropy and of the thermodynamic arrow of time encapsulate the idea that isolated systems in nature tend to decay from order to disorder, a conception to which Paul Ehrenfest, a physicist in Leiden, made a signal contribution. On a reading inspired by algorithmic information theory, by contrast, symmetrical states of affairs correspond with low information, whereas increasingly disordered or random outcomes have higher information content. This juxtaposition of perspectives yields some apparently paradoxical conclusions: for example, many physicists would hold that the discovery of a symmetry in phenomena shows nature to be highly structured, whereas one might conclude that such a discovery shows nature to be simpler and less richly structured than if the symmetry failed to hold. The same ambivalence pertains to the role of symmetry in aesthetic discourse: beauty is associated with symmetry in classical theories of art, but many modern onlookers would associate perfect symmetries with lifelessness and lack of aesthetic value. What do these conceptual relations tell us about the relations of order and symmetry?
Session 2: Wednesday 12 March 2008
The Epistemological Status of Symmetry Principles in Physics
Symmetry principles number among the fundamental methodological principles of physics. Physicists often use them as premises in arguments aiming to show that some quantity is conserved, that one state of affairs is observationally or metaphysically indistinguishable from another, or more generally that a certain state of affairs is physically necessary. Symmetry principles appear to arise in a variety of ways, however. Curie’s principle appears to derive from metaphysical considerations concerning the relation of cause and effect. Other symmetry principles are posited on the grounds of high-level considerations, such as transformations and their resultant expression in conservation laws (Noether’s theorem). Yet other symmetry principles are induced from experimental data, such as the approximate symmetries that hold among elementary particles. What different sorts of symmetry principles can be discerned? Can these principles be established by a priori means, or are they based on empirical data? Correspondingly, are these symmetry principles trivially valid or do they convey information about the structure of the world?
Session 3: Thursday 13 March 2008
Symmetry and Symmetry Breaking in Morphogenesis
The concepts of symmetry and symmetry breaking play an important role in accounts of morphogenesis in at least two widely differing contexts: cosmogenesis in cosmology and ontogeny in developmental biology. In both contexts, a symmetrical early state results in an asymmetrical and thus more highly differentiated later state. What is the conceptual basis for treating structure formation as an outcome of symmetry breaking? Can any systematic principles relating symmetry breaking to the evolution of structure be established for these two domains? Is it possible to quantify the degree of symmetry breaking, or construct a measure of how far the outcome lies from the initial symmetry?
Session 4: Friday 14 March 2008
Overarching Perspectives: D’Arcy Thompson and Hermann Weyl
In the final session we revisit two classic texts of the twentieth century: D’Arcy Thompson, On Growth and Form (1917) and Hermann Weyl, Symmetry (1952). Both Thompson and Weyl present mathematical theories of transformations and invariances that are intended as overarching organizing principles in their different domains. In pursuing this goal the two authors offer historical accounts of the concepts at stake. However, while Weyl’s work has remained central to and acknowledged in theoretical physics and science in general, Thompson’s approach largely fell out of fashion in the “modern synthesis” of evolutionary theory, regaining importance only with the rise of evo-devo. What general conclusions can be drawn within a historical framework about the role of theories of transformations, invariances, and symmetries in different branches of science?