According to Pierre Francastel, “Art and mathematics are the two poles of any logic thinking—the major thinking modes of humanity.”1 In fact, since Greek Antiquity, art and mathematics have always nurtured a natural dialogue, forming together
According to Pierre Francastel, “Art and mathematics are the two poles of any logic thinking—the major thinking modes of humanity.”1 In fact, since Greek Antiquity, art and mathematics have always nurtured a natural dialogue, forming together a tissue of opposed polarities, fruitful tensions and partial osmoses. Concepts, symbols and materials often separate the artist and the mathematician, but a common strategy—to think and imagine through dialectic, relational webs—brings them together. For Francastel, “As the mathematician … combines schemes of representation and prediction where the real is associated with the imaginary, so the artist confronts elements of representation with other imagination issues. In both cases, the dynamism of a thought that becomes aware of itself by expressing itself and materializing itself in ‘signes-relais’ [“relay-signs”] extends and encompasses elements of experience and those of the logic of the mind.”2 The transits between reality and imagination, the Many and the One, the particular and the universal, the local and the global, constitute the very core of mathematical and artistic creativity. If their signes-relais are often different (and, in some cases, the specificity of mathematical signes-relais preclude the possibilities the artist has of entering into the realm of mathematics), the underlying relational protogeometry which governs the intertwining of the different signs can be extremely close.
In what follows, fragments of that common relational protogeometry will be revealed, and will be applied to an understanding of some strong weaving dynamics in contemporary art and mathematics. Referring to The Glass Bead Game, Hesse mentions in a 1943 letter how his game and Castalia were meant “not as eternally valid ideals, but as potential worlds, conscious of their own relativity.”3 Relationality, potentiality, and relativity, are in fact key twentieth-century forces. Contrary to popular interpretation, entanglement and intersection, built over a non-absolute ground, do not necessarily lead to postmodern skepticism. As Knecht writes at the end of his Waldzell encounter with Designori, “The entire life, whether physical or spiritual, is a dynamic phenomenon, of which the Glass Bead Game contemplates its esthetic side, especially through the framework of rhythmic phenomena.”4 Following Castalia’s Magister Ludi, the mesh of materiality and spirituality, the crossings and echoes along reason and imagination, the intertwined rhythmic spectrum of all senses, the dynamic blend of polarities, are not only still possible, but become really potentiated in our transmodern world, as Glass Bead‘s research platform and journal are demonstrating.
The roots of a unified, non-foundational vision that encompasses art and mathematics as fragments of a multidimensional protogeometry of culture can be traced back at least to Novalis (end of the eighteenth century). Riemann surfaces (mid-nineteenth century) partially solve the tensions and ramifications between the Many and the One, in the mathematical domain of complex variables. Peirce’s pragmaticism (end of the nineteenth century) offers a general architectonics of knowledge, where phenomenological obstructions and transits help to explain the emergence of creativity. If we consider fragments and ramifications as residues, or strata, and we focus our vision on the history of art, Warburg’s Mnemosyne Atlas (early twentieth century) provides another main reference to unveil the complex protogeometry of cultural exchanges. Profiting from dual definitions of art and mathematics proposed by Focillon and Lautman (mid-twentieth century), we can then set up a faithful panorama of the strong dialectics in play. Finally, through the invention of mathematical category theory and its developments in the work of Grothendieck and followers (end of the twentieth century), deep non-classical logics and non-separated geometries (sites and sheaves in Grothendieck toposes) can be used to further clarify the oppositions and conciliations between art and mathematics.
The first section of this paper presents diverse forms of dialectics between art and mathematics, following Novalis, Warburg, Focillon and Lautman. The second section explores some tensions between the local and the global, both in art and mathematics, profiting from Peirce’s architectonics. The third and final section proposes embedding the local/global dialectics in art and mathematics along multilayered surfaces (coming from Riemann surfaces) and multidimensional sites (coming from Grothendieck toposes), governed by continuous, intuitionistic logics.
1.Dialectics between Art and Mathematics
Due to the unavoidable opacity of mathematical signes-relais, mathematical creativity is seldom understood outside its practitioners. Nevertheless, a sustained French tradition has delved into the question,5 revealing the complex weaving of inventiveness in mathematics. From the concrete (multitude of careful examples) to the abstract (general representation patterns), from a precise reckoning of difference (existence and uniqueness theorems in local frames) to all-encompassing integral views (universal structures and mathematical categories), from syntactical restraint (proof theory) to semantic freedom (model theory), from full control (deduction, rational clearness) to 