In May 2016 an Academia Europaea symposium with the title ‘Symmetry, Proportion and Seriality: The Semantics of Mirroring and Repetition in Science and the Arts’ was organized at the Freiburg Institute for Advanced Studies (FRIAS). Its major aim was to bring together interdisciplinary perspectives on symmetry and related concepts, comparing humanities approaches with those current in the material sciences.

This article is in two parts: a series of drawings depicting walls, and a series of lyrical or imaginative commentaries on walls. The drawings show walls in multifarious forms, more or less abstract, more or less figurative, constrained by geometrical regularity. The format of presentation elicits the perception of rhythm in the lateral survey of the drawings, which in turn suggests the operation of forms comparable to those encountered in the perusal of poetic language organized over many pages in serial verse. In particular, comprising a visual series, the drawings may be discussed in terms of poetic devices such as echo, verse-line, and meter, and their repeating elements. Besides verbal art, the drawings also function as a schematic representation of walled edifices that embody the ratios and measures of symmetry in physical architecture (a cathedral interior, a city block, a Daedalian labyrinth, for example). Mindful of both analogies – with poetic language and physical architecture – the commentaries attempt reflection on the experience of symmetry in the drawings.

In the last two decades, mirroring systems have been detected in the monkey and in the human brain. The mirroring mechanisms have been considered as the neural basis for social cognition and interpersonal reactivity, and they have been assumed to support imitation, sharing of emotional states and empathy. Here I would like to compare ‘mirroring phenomena’ to ‘symmetrization phenomena’. In psychoanalytic literature, the construct of symmetrization has been proposed in the context of the Bi-Logic theory by Matte Blanco in 1975, on the basis of clinical evidence obtained in the psychoanalytical setting and following a theoretical analysis derived from the Freudian distinction between Primary and Secondary Processes. I will consider two different types of behaviours, empathic social interactions and the creation of transitional objects and spaces as defined by Winnicott in order to argue that symmetrization, in Matte Blanco’s terms, cannot be reduced to mirroring. I will then sketch a hypothesis on the interplay between the symmetric aspects of the mind and external reality in the development of higher relational functions of humans, also taking playing, arts and creativity into account. Finally I will describe the paradigmatic shift in neuro-imaging studies that was introduced with the discovery of the ‘Default Mode Network’ and its potential relevance in the research on the symmetric and asymmetric aspects of the human mind.

Hierarchy has properties of symmetry and scale. All that is related to hierarchy provides important perspectives on many other domains. The primary focus here is on renowned psychoanalyst Ignacio Matte Blanco’s Bi-Logic. Bi-Logic relates to the two modes of being, respectively symmetry and asymmetry in thought and reasoning and brain processes, in unconscious and in conscious modes of being. Some further consequences and implications are noted. These include relevance and potential in social sciences; in qualitative as well as quantitative literary analysis, and content analytics of text, speech, visual recording, and so on; in security and forensics; and in the contemporary theme of research and development relating to Big Data analytics.

The word ‘symmetry’ expresses in everyday language the correspondence in form, size, and arrangement of parts on opposite sides of a plane, line, or point. Such a concept of correspondence is the main idea of most definitions of this term in the various humanities disciplines discussed in this volume of European Review. But if we go back to the Greek origins of this term, we discover that the understanding of symmetry was completely different, that it signified mathematical commensurability of all important elements of a three-dimensional figure. This article discusses the termʼs initial meaning and its importance for the design and appreciation of beautiful architectonic and human bodies. It goes on to provide an outline of the main phases of the step-by-step development from the original concept towards its modern meaning established in the sixteenth and seventeenth centuries.

During the Renaissance, the language of proportion became a unified theory capable of encompassing the understanding of the world within a coherent theological, philosophical and artistic framework. Music, with its harmonic paradigm, plays a key role in this construction. From the fifteenth century through to the end of the sixteenth century, architects and architectural theorists made reference, both in new treatises and commentaries to Vitruvius, to musical matters, transforming architecture into the summa of knowledge. The affinity to music was grounded on both a common mathematical and rhetoric gnosiology. Formerly conceived of as ideal, numbers became eloquent, reinforcing the quantitative paradigm of proportion with its qualitative one. The language of proportion as a compositional tool reveals the shift between the Middle Ages and the Renaissance: while the Medieval tèchne based on modular thinking provides beauty and universal truth using the technique of repetition, the Humanist paradigm of variety produces pleasure and individual truth – a condition typical of the premodern.

The general concept of symmetry is realized in manifold ways in different realms of reality, such as plants, animals, minerals, mathematical objects or human artefacts in literature, fine arts and society. In order to arrive at a common ground for this variedness a very general conceptualization of symmetry is proposed: the existence of substitutions, which, in the given context, do not lead to an essential change. This simple definition has multiple consequences. (i) The context dependence of the notion of symmetry is evident in the humanities but is by no means irrelevant, yet is often neglected, in science. By taking this context dependence into account, the subtle problematics of concept formation and of the ontological status of ‘similarities’ become evident. In general, the substitutions underlying the concept of symmetry are not really performed but remain in a state of virtuality. Counterfactuality, freedom and creativity come into focus. The detection of previously hidden symmetries may provide deep and surprising insights. (ii) Related to this, due attention is devoted to the aesthetic dimension of symmetry and the breaking of it. (iii) Finally, we point out to what extent life is based on the interplay between order and freedom, between full and broken symmetry.

By looking at three significant examples in analysis, geometry and dynamical systems, I propose the possibility of having two levels of realism in mathematics: the upper one, the one of entities; and a subordinated ground one, the one of objects. The upper level (entities) is more the one of ‘operations’, of mathematics in action, of the dynamics of mathematics, whereas the ground floor (objects) is more dedicated to culturally well-defined objects inherited from our perception of the physical or real world. I will show that the upper level is wider than the ground level, therefore foregrounding the possibility of having in mathematics entities without underlying objects. In the three examples treated in this article, this splitting of levels of reality is created directly by the willingness to preserve different symmetries, which take the form of identities or equivalences. Finally, it is proposed that mathematical Platonism is – in fine – a true branch of mathematics in order for mathematicians to avoid the temptation of falling into the Platonist alternative ‘everything is real’/‘nothing is real’.

Roman Jakobson famously defined poetry as pivoting on the metaphorical axis with parallelism as a major feature, and James Kugel argues that parallelism is the defining feature of biblical poetry. The parallel structure – including its variations of symmetry and antithesis – is crucial for classical Chinese poetry. In drawing on both Chinese and Western critical views on the symmetrical structure of parallelism and antithesis, this paper will explore the relationship between the cognitive and linguistic correlation in the formation of parallel structure in literary language, particularly poetry, and argue for the basis of parallelism as deeply embedded in the mind and manifested in literary expressions.

Starting from a distinction between different historical and philosophical concepts of ‘form’ as they have been discussed by Władysław Tatarkiewicz, this article argues that visual poetry is constituted and concretely shaped by its implicit, sometimes even explicit, reflection upon ‘form’. In three paragraphs different concepts of ‘form’ are briefly discussed with regard to selected examples of visual poems: (1) form as ‘proportion’; (2) form as the counter-concept of ‘content’; and (3) form in the sense of ‘contour’. The first part focuses on examples of twentieth-century visual poetry playing with the sonnet form and exposes its rigidly proportional visual structure. In addition, the strategy of turning form to serial account is illustrated. There is a long tradition of engaging with the sonnetʼs history and generic features via the very sonnet form, either in order to defend this highly artificial poetic genre (as an example by August Wilhelm Schlegel illustrates), or in order to criticize or parody it. Visual poetry sonnets reduce this poetic genre to its proportions, thus questioning under which preconditions the reader identifies a ‘sonnet’ at all. The second section of the article presents two examples of concrete visual poetry (by Eugen Gomringer and Mathias Goeritz) that play with the notion of ‘content’ by foregrounding this ludic element and the poetic processes that are represented indirectly by the respective poems’ visual structure. The third section is dedicated to the complementary concepts of ‘outline’ and ‘dissolved contours’, focusing on the contrast between traditional instances of the contour poem and more recent examples (Carlfriedrich Claus, Eugen Gomringer) which expose the tension between form and its dissolution.

Classical compositions, such as those of Claude Lorrain (1600–1680), show balanced situations where the depicted objects seem to represent beauty – they appear to exist in an ideal order; they seem right, once and for all. They allow their viewers, who live in a contingent world, believe that unity and beauty are attainable. English landscape gardens in the eighteenth century offered viewers the experience of moving in real settings that seem to reproduce the canvasses of classical landscape painting. Contemporary visitors know this is an imitation; they appreciate that this is not everyday reality but a kind of realized utopia. Artworks such as the early ones by Rembrandt Harmenszoon van Rijn (1606–1669) and those from the nineteenth century onwards (Paul Cézanne, Claude Monet), make us understand that a final rightness is not possible; that it is only possible to try again and again to attain the real – without ever arriving at the one right solution. In their endeavour to depict reality, these artists engage in a serial elaboration of the same subjects and objects of their art. These series of nearly the same subjects invoke the impossibility of getting it right once and for all. Piet Mondrian illustrates the development from seriality, as repeated attempts to depict the same with a difference, to the practice of seriality as an experiment in symmetry and balance that both goes back to Claude Lorrain and modernizes the concept of the ideal for application to abstract art. In contemporary art after the Second World War serial structures in single artworks were common, thus superseding compositions with their discredited promise of ideality. Series occurred not only in the contemporary visual arts, but also in literature, music and in the theatre. The repetitive structures of Minimal Art offer to the beholder the possibility of experiencing ‘reality’ in an age of media and fakes. My example will be the repetitive structures of Christian Boltanskiʼs oeuvre which provides an opportunity to reflect on the similarity and difference of people in contemporary democracies. My main argument is therefore that the change from composition to seriality corresponds to historical change.