These papers resulted from the first workshop in a series exploring concepts that are of central importance to all of the sciences. The series will bring together the perspectives of natural and social sciences and the humanities in intimate, cross-disciplinary dialogue. The intention is to improve our understanding of the concepts discussed by emphasizing their different uses in the many different disciplines, while at the same time focusing on shared concerns and structural commonalities. And this is the peculiarity of the concepts to be examined here – they do not get defined in just one discipline, so that they only form part of the basic concepts of that discipline, as distinguished from the basic concepts of other disciplines. Rather, their unique character derives from the fact that they have different uses in different disciplines, and all of them in the foundations of the respective sciences. The concepts of natural law, causality, of complexity and truth, scientific truth, are of this kind, and so is the concept of symmetry, with which we will start this series.

As we all know, this concept has many different (scientific) meanings, while the basic meaning remains stable. In geometry, for instance, mirroring, rotation and periodicity prove to be symmetry-properties, that is, self-copies (automorphisms) of spaces, which leave the structure of figures and bodies unchanged (invariant). In natural philosophy, symmetry is the property of natural objects to remain invariant under certain operations, namely symmetry-operations. In logic, a two-place relation R is symmetric on a set S, if for any two elements x and y from that set, the following is the case: if the relation R holds between x and y, then it also holds between y and x.

The concept of symmetry is omnipresent, although originally, in Greek antiquity, distinctly different from the modern logical notion. In logic a binary relation R is called symmetric if xRy implies yRx. In Greek, ‘being symmetric’ in general usage is synonymous with ‘being harmonious’, and in technical usage, as in Euclid's Elements, it is synonymous with ‘commensurable’. Due to the second meaning, which is close to the etymology of συ´μμετρoς, ‘with measure’ has likewise to be read as ‘being [in] rational [ratios]’ and displays the origin of the concept of rationality of establishing a proportion. Heraclitus can be read as a master of such connections. Exercising rationality is a case of simultaneously finding and inventing symmetries. On that basis a proposal is made of how to relate the modern logical notion of symmetry, a second-order concept, on the one hand with modern first-order usages of the term symmetric in geometry and other fields, and on the other hand with the notion of balance that derives from the ancient usage of symmetric. It is argued that symmetries as states of balance exist only in theory, in practice they function as norms vis-à-vis broken symmetries.

Our modern representation of elementary processes considers that all physical phenomena result from the interplay of four fundamental interactions: the gravitational attraction, the electromagnetic force, the strong interaction and the weak interaction. Formally, the quantum mechanical description of elementary processes introduces the concept of discrete symmetry, illustrated for instance by space and time inversions. Discrete symmetries play a central role in the elaboration of theories and models, and have profound consequences in the predictions of these theories. For nearly 50 years, it has been observed that, of the four fundamental interactions, only the weak interaction violates mirror symmetry, and all observations so far indicate that it does so in a so-called maximal way. Despite overwhelming evidence of mirror-symmetry breaking, the search for a possibly underlying left–right symmetry has been pursued for many years by dedicated experiments. In this paper we review the context of mirror symmetry breaking in the weak interaction, we describe its interpretation in the framework of the standard model of particle physics and describe current efforts to identify the restoration of the left–right symmetry.

Historically, static symmetric bodies and ornaments are geometric idealizations in the Platonic tradition. Actually, symmetries are locally and globally broken by phase transitions of instability in dynamical systems generating a variety of new order and partial symmetries with increasing complexity. The states of complex dynamical systems can refer to, for example, atomic clusters, crystals, biomolecules, organisms and brains, social and economic systems. The paper discusses dynamical balance as dynamical symmetry in dynamical systems, which can be simulated by computational systems. Its emergence is an interdisciplinary challenge of nonlinear systems science. The philosophy of science analyses the common methodological framework of symmetry and complexity.

It seems to be well known that life cannot arise in a racemic mixture of molecules (L=D). The various physical mechanisms that could influence the homochirality of the molecules responsible for life are reviewed together with a model where these molecules might be produced in the galaxy prior to the formation of the earth. The focus is on the influence of a powerful supernovae explosion to affect the homochirality of these molecules. Possible tests of this concept with meteorites and future space missions are described.

Symmetry and chemistry have been in a fruitful interplay, initially in spectroscopy and crystallography, lately in more traditional domains of chemistry, such as reactivity and conformational analysis. A simple phenomenological approach suffices to get an idea about the symmetries of molecules whereas group theoretical approach greatly facilitates the understanding of molecular vibrations, electronic structure, and the mechanism of chemical reactions. In our discussion, the multi-level relationship between symmetry and chemistry is demonstrated by a sampler of examples, including the variations of symmetry of free molecules and molecular packing in crystals. Symmetry considerations continue to assist chemistry in systematizing and interpreting observations and also in discovering new reactions, molecules, and other materials.

Naturally occurring biological molecules are made of homochiral building blocks. Proteins are composed of L-amino acids (and not D-amino acids); nucleic acids such as DNA have D-ribose sugars (and not L-ribose sugars). It is not clear why nature selected a particular chirality. Selection could have occurred by chance or as a consequence of basic physical chemistry. Possible proposals, including the contribution of the parity violating the weak nuclear force, are discussed together with the mechanisms by which this very small contribution might be amplified. Homochirality of the amino acids has consequences for protein structure. Helices are right handed and beta sheets have a left-hand twist. When incorporated into the tertiary structure of a protein these chiralities limit the topologies of connections between helices and sheets. Polypeptides comprised of D-amino acids can be synthesized chemically and have been shown to adopt stable structures that are the mirror image of the naturally occurring L-amino acid polypeptides. Chirality is important in drug design. Three examples are discussed: penicillin; the CD4 antagonistic peptides; and thalidomide. The absolute hand of a biological structure can only be established by X-ray crystallographic methods using the technique of anomalous scattering.

Symmetry across the midline is present in many animals, together with the left/right asymmetry of several organs, such as the heart in vertebrates. The development of such asymmetries during embryonic development requires first the specification of the midline and then specification of left/right. One model proposes the transfer of molecular asymmetry to the multicellular level. Nodal expression on the left side in mammals and chicks is a key event, and is due to the release of calcium on the left possibly involving an ion pump and the Notch pathway

This review aims to present a speculation about mechanisms that shape the brains of humans and other animals into an asymmetrical organization. To this end, I will proceed in two steps: first, I want to recapitulate evidence from various experiments that show that some but not all asymmetries of the avian brain result from a prehatch light stimulation asymmetry. This should make it clear that avian embryos have a genetic predisposition to turn their head to the right. This results in a higher level of prehatch light stimulation of their right eye. The concomitant left–right difference in sensory input alters the brain circuits of the animal for the entire lifespan in a lateralized way. In the second part of the paper I will present evidence that some of the asymmetries of the human brain take a similar ontogenetic path as those observed in birds. This review provides the evidence that critical ontogenetic processes discovered in animal models could also be involved in the ontogeny of human cerebral asymmetries.

Structural and functional asymmetry in the human brain and nervous system is reviewed in a historical perspective, focusing on the pioneering work of Broca, Wernicke, Sperry, and Geschwind. Structural and functional asymmetry is exemplified from work done in our laboratory on auditory laterality using an empirical procedure called dichotic listening. This also involves different ways of validating the dichotic listening procedure against both invasive and non-invasive techniques, including PET and fMRI blood flow recordings. A major argument is that the human brain shows a substantial interaction between structurally, or ‘bottom-up’ asymmetry and cognitively, or ‘top-down’ modulation, through a focus of attention to the right or left side in auditory space. These results open up a more dynamic and interactive view of functional brain asymmetry than the traditional static view that the brain is lateralized, or asymmetric, only for specific stimuli and stimulus properties.

Asymmetry is a fundamental aspect of the biology of all organisms, and has a deep evolutionary history. The fossil record contains evidence of both morphological and behavioural asymmetries. Morphological asymmetry is most commonly expressed as conspicuous, directional asymmetry (either lateral asymmetry or spiral asymmetry) in body fossils. Few examples of fluctuating asymmetry, a form of subtle asymmetry, have been documented from fossils. Body fossil evidence indicates that morphological asymmetry dates to the time of the appearance of the first life on Earth (Archaean Eon). Behavioural asymmetry can be assumed to have been concomitant with conspicuous morphological asymmetry, but more direct evidence is in the form of trace fossils. Trace fossil evidence suggests that behavioural asymmetry, including nervous system lateralization, was in existence by the beginning of the Palaeozoic Era.

Symmetry is a relevant concept in sociological theories of exchange. It is rooted in the evolutionary old norm of social reciprocity and is particularly important in social contracts. Symmetry breaking through violation of the norm of reciprocity generates strain in micro-social systems and, above all, in victims of non-symmetric exchange. In this contribution, adverse healthconsequences of symmetry breaking in contractual social exchange are analysed, with a main focus on the employment contract. Scientific evidence is derived from prospective epidemiological studies testing the model of effort-reward imbalance at work. Overall, a twofold elevated risk of incident disease is observed in employed men and women who are exposed to non-symmetric exchange. Health risks include coronary heart disease, depression and alcohol dependence, among others. Preliminary results suggest similar effects on health produced by symmetry breaking in other types of social relationships (e.g. partnership, parental roles). These findings underline the importance of symmetry in contractual social exchange for health and well-being.

Symmetry and beauty are often claimed to be linked, particularly by mathematicians and scientists. However philosophers and art historians seem generally agreed that although symmetry is indeed attractive, there is also a somewhat sterile rigidity about it, which can make it less attractive than the more dynamic, less predictable beauty associated with asymmetry. Although a little asymmetry can be beautiful, an excess merely results in chaos. As Adorno suggested, asymmetry probably results most effectively in beauty when the underlying symmetry upon which it is built is still apparent. This paper examines the ways in which asymmetries, particularly left-right asymmetries, were used by painters in the Italian Renaissance. Polyptychs often show occasional asymmetries, which are more likely to involve the substitution of a left cheek for a right cheek, than vice-versa. A hypothesis is developed that the left and right cheeks have symbolic meanings, with the right cheek meaning ‘like self’ and the left cheek meaning ‘unlike self’. This principle is evaluated in pictures such as the Crucifixion, the Annunciation and, the Madonna and Child. The latter is particularly useful because the theological status of the Madonna changed during the Renaissance, and her left–right portrayal also changed at the same time in a comprehensible way. Some brief experimental tests of the hypothesis are also described. Finally the paper ends by considering why it is that the left rather than the right cheek is associated with ‘unlike self’, and puts that result in the context of the universal ‘dual symbolic classification’ of right and left, which was first described by the anthropologist Robert Hertz.