The Topological Imagination:
Spheres, Edges, and Islands1
by Angus Fletcher
WILLIAM J. URBAN
In his latest work, Angus Fletcher does not so much double-down on the Romantic bet that humanistic concerns always-already trump enlightened ‘progress’ as suggest something much more paradoxical: it is only by returning to science that art can be shown to take precedence. But this is not the science of Galileo, Newton, or Descartes. The science of these men is deemed negative on net, having led, for instance, to the ecological catastrophe said to be awaiting our Earth. Rather, it is the science which Euler effectively inaugurated in 1735 when he solved Königsberg’s Seven Bridges problem, viz. the mathematical field of topology. The Topological Imagination is to be lauded for championing the transformative power that topology has to fundamentally reshape our thinking on everything from language and literature, to reality and the world at large.
Far from anything like a formal treatise, Fletcher nevertheless begins by introducing topological principles that are, regrettably, only elaborated slowly and haphazardly throughout the remainder of the book. In these pages you learn how topology distinguishes itself from geometry by dampening concerns for the quantitative. As a logic of positioning, site or place, topology avoids measurements of size and similar magnitudes to instead focus on the perception of shape, sequence and order, as well as disparity and other deviations from ideal unity. In this way “topology insists on the human primacy of the qualitative” (169) and should therefore be a welcome addition to our interpretive arsenal, as it provides a suitable guide to our aesthetic sensibilities.
Key for Fletcher is the principle of homeomorphism. Topology considers any two objects (e.g., a sphere and pyramid, or a coffee mug and doughnut) as equivalent if one can be stretched, twisted, or molded into the other without cutting into their continuous form. Such transformations prove analogous to the complex changes taking place in our environment and in the mimetic arts. By studying the abstract field of topology, you gain insights into these and other concrete fields. The Topological Imagination is a work which lays out many of these insights, with the help of Einstein, Bohm, Shakespeare and Coleridge, to name just a few enlisted to the cause.
This methodological approach also raises the question of "invariance amidst change – perhaps the chief property of topological thought" (139). With Fletcher’s tendency to define topology strictly in homeomorphic terms (cf. also 15, 79, 172-73), his accent on this question is coherent enough. Hence, when our imaginations mold and reshape materials into a more flexible union, we do so against the backdrop of “previous fixities” (58). Enter the topological figure of the book: the sphere. Of the dozens of transformations explored in this book, each is invariably discussed against an “esemplastic unifying vision of things” (57) of which the sphere is the ideal model. Such a model has no edges, so its smooth surface can hold out a promise of unity – at least in some future time. For Fletcher repeatedly points out how no actually existing sphere, or its topological equivalent, is completely devoid of edges. Our oblong Earth, for instance, possesses the natural edges of mountain ranges, shorelines and cellular membranes, while we continuously populate our immediate environs with the artificial edges of national borders, buildings, tables, and CDs. All of these objects equally rise up from Earth’s two-dimensional surface. But in doing so, they do not substantially alter the underlying sphericity of our place in the cosmos. If Donne wrote that “no man is an island” in a phrase used to entitle the final chapter, Fletcher goes one better in fine Heideggerian fashion: islands themselves are never truly isolated from the “larger continent,” but are instead to be “imagined to incorporate the single personhood into a vaster scale of being” (184).
Conflating topology with homeomorphism, however, has a downside. It overlooks the radical potential this science has when coupled with our human concerns. By only considering “bounded transformations” (9) of the type which leaves the underlying topological shape intact, we remain forever caught in the spherical universe of meaning. Such transformations cannot therefore account for the fact that non-meaning and nonsense can and do exist in our universe.
Topologists, however, equally consider transformations of the non-homeomorphic type. They have established operations which effect certain cuts and sutures into spherical surfaces to arrive at aspherical surfaces like the Möbius strip and cross-cap. While Fletcher does acknowledge that the “cut creates but also threatens the continuity of a surface manifold” (166), this insight, into how cuts can effectively double as sutures, remains undeveloped. This is unfortunate since it would greatly assist his project. Thinking with the cross-cap, for instance, could free us from having to “lean upon the sphere” (194) in our “quest for unity” (21), since this paradoxical figure harbors within itself a singular point which, nonsensically, embodies or collapses its horizon. Directly assuming such a point would suspend the need to offer up “a prayer to the horizon” for an imagined “emergent occasion” (194) to instead strategically place us within striking distance of our supposedly ailing biosphere. Doing so would not achieve the “balance of temperament” that Fletcher is looking for when praising Obama as its exemplary model (171); arguably, it is precisely the latter’s reticence to act on the global stage with the full force of his office that has put our globe in grave peril, threatening to upset the delicate balance in play since 1945. Rather, the crosscap would call us, paradoxically, to double-down on our “conquests of nature” (194) if we wish to enact the type of cuts – environmental, political and otherwise – that are necessary to sustain our Earth for current and future generations.
Despite not cutting deep enough into its spherical contours to come to pointed conclusion, The Topological Imagination is nevertheless enthusiastically recommended. The sheer scope of Fletcher’s literary excursions will be enough to entice the humanities scholar, while any scientist looking to expand his field will find it of equal value. It is a fine addition to the growing literature on how topology allows science and art to be thought together.
1 Fletcher, Angus. The Topological Imagination: Spheres, Edges, and Islands (Cambridge, MA: Harvard UP, 2016)