Topology,again: From crumpled handkerchiefs to stretched Möbius strips

Abstract

In response to the article by Unnithan Kumar et al., I ask whether there is an audience for a more mathematically focussed utilisation of topology amongst post-structural thinkers writing on space and spatiality in human geography. While the authors suggest that concepts from topology can be useful for approaching spatial relations in less precise and more multiple ways, human geographers are much more likely to engage with conceptual tools from philosophy and social theory.

When I was undertaking my PhD in the late 1990s, ‘actor-network theory’ was reaching into many corners of the discipline. Part of the appeal, it seems, was that spaces were not posited as inert settings for social action, but were products of processes of association, interaction, relating and network-building. Geographers, like many social theorists, were searching for theories which could effectively describe highly mediated and distanciated socio-spatial relations and non-linear spatial patterns, enabling scholars to abandon the idea that we all live in a uniform, pre-existing, dimensioned, containing space which follows Euclidean or Cartesian geometric principles. The concept of topology – and particularly the idea of deformed or stretched topological spaces – appeared attractive for thinkers such as Kevin Hetherington, John Law and Annemarie Mol who were tracing the spatialities of different aspects of material culture, from museum objects (Hetherington, 1997a, 1997b) to technologies such as the bush-pump (Law and Mol, 2001; see also Mol and Law, 1994).

The concept of ‘topological space’ had – and still has – something of a shock factor, challenging seemingly commonsense concepts of space that are grounded in Western philosophical and mathematical traditions (see Merriman, 2022). For me, I found the use of analogies – a crumpled handkerchief or piece of paper – helpful for illustrating how social and cultural proximity do not always correlate with spatial distance (see Hetherington, 1997a). In using ideas or figures from topology, many of these social scientists were revisiting theories which have repeatedly surfaced in both the discipline of geography and in wider society. One of the most common of these was to consider how modern technologies of communication and transport produced a more complex and variegated geography of distance and proximity, whether experienced as the ‘annihilation of space by time’, ‘time-space convergence’, ‘time-space distanciation’, or ‘time-space compression’ (see Merriman, 2022). It was, however, Marcus Doel's (1996) Deleuzian concept of a ‘scrumpled geography’ which resonated most with me when I was conducting my doctoral research – something I considered alongside the writings of Hetherington, Law and Mol. A focus on movement and change necessitates a rethinking of what Doel terms a ‘pointillist’ geography, focusing on processes of spacing and enfolding rather than a geography of points and lines (Doel, 1999). It was these various writings which led me to invoke the concept of topology to figure ‘the continual arrangement and placing of the socio-material spaces of the motorway’ (Merriman, 2004: 159) and explore how the distinctive ‘corridoring’ and ‘tunnelling’ effects of the M1 produced a series of ‘linear, nodal, material, imaginative and discursive geographies’ (Merriman, 2011: 211). It was not the complex mathematical theories, calculations and conceptual figures of topology which appealed to me, but the simplified analogies used by social scientists to objectify and ground the relations I was exploring.

More recent research has deepened social science and humanities engagements with topology, from writings focussing on what has been termed ‘cultural topology’ (Lury et al., 2012; Shields, 2013) to studies of power relations (Allen, 2011, 2016). Writing in this journal in 2011, John Allen argued that certain kinds of power relation could usefully be understood as ‘power topologies’, sparking a number of responses. These different ‘post-mathematical’ engagements with topology are very effectively discussed by Martin and Secor (2014), who outline the long-standing fascination of post-structuralist spatial thinkers with topology ‘as a metaphor, a heuristic device, an analytical approach, a figure, and an ontological relationship’ (p.421). They outline early engagements with topology by social science and humanities thinkers, including philosophers Martin Heidegger, psychologist Kurt Lewin, Lacanian psychoanalysts, and political theorist Georgio Agamben (who himself influenced political geographers such as Derek Gregory, Alison Mountz, Paolo Giaccaria, and Claudio Minca). Martin and Secor (2014) ask whether topology is simply the latest in a long line of concepts being imported into the discipline to ‘lend a scientific veneer to humanist and social scientific work’ (p.433). They also highlight how many writings simply use topology as ‘a synonym for relationality’, saying ‘nothing about the rules of connection, disconnection, and transformation’ (p.435).

Many of the same geographic writings discussed by Martin and Secor (2014) are revisited in this new article by Siddharth Unnithan Kumar, Andy Stirling, David Abram, and Sanjiv Kumar, titled ‘Topology beyond application: Drawing social and mathematical worlds into rhythm’. In the paper's overview, they outline their disciplinary backgrounds, for these are not post-structuralist geographers importing ideas from mathematics, but a team whose credentials span mathematics and different strands of the social sciences:

'Siddharth is a mathematician and geographer, working across disciplines to collaboratively develop mathematics as an ecological practice… Andy is a transdisciplinary sociologist engaging with power in science and technology policy. David is a cultural ecologist, geophilosopher, and sleight-of-hand magician. Sanjiv is a retired mathematician and astronomer’ (Unnithan Kumar et al., 2026: 5).

It is pleasing to not only see scholars from these different fields working together, but also to be engaging with geographic writings and publishing in a human geography journal. I always enjoy seeing how human geography travels into other fields, even if this example possesses a certain circularity, as these mathematicians interpret geographers’ use of mathematical ideas.

I started reading with a positive outlook, although I felt a little out of touch with current debates in human geography on topology, having not written about the concept for 4 or 5 years (since Merriman, 2022). Unnithan Kumar et al. claim to ‘detail, with several diagrams, key concepts in mathematical topology which critically engage and augment geographers’ theories of space’ (p.1). Given the long-standing interest of geographers in topology, I had assumed that the authors might find fruitful common ground in the topological explorations of notable spatial scientists such as Bill Bunge, William Garrison, Peter Haggett, Waldo Tobler, and William Warntz, or perhaps even the contemporary work of scholars in spatial analysis and GIS. It is in these different traditions of spatial science, mathematical geography, and data visualisation that the fields of geography and mathematics are often most closely entangled and aligned, and I wondered if a discussion of this work might have led them to unpick the mathematics/geography binary which seems to reoccur at various points in the article – concluding perhaps that these are two sets which overlap. To their credit, the authors do discuss how mathematical topology traverses and cuts across ‘the qualitative/quantitative binary’, but this will come as no surprise to many human geographers, with a number having previously emphasised the non-positivist assumptions and qualitative judgements and categorisations underpinning fields like set theory (see Merriman, 2022).

Instead of engaging with spatial science writings, the authors focus on post-structuralist accounts of (topological) space. Unnithan Kumar et al. review key writings by sociologists, philosophers, political theorists, and anthropologists such as Bruno Latour, Marilyn Strathern, John Law, Annemarie Mol, Rob Shields, Isabelle Stengers, Georgio Agamben, and others. I was left, however, with little sense of how their reading of these key thinkers diverged from that of Martin and Secor (2014) over 10 years ago, and I was not entirely convinced that their quite descriptive review of ‘recent topologies in geography’ served as more than an aide memoire.

The article then proceeds to explore how certain key concepts from topological mathematics can add to geographic theories of space. Unnithan Kumar et al. announce that they ‘refrain from using symbolic notation from set theory’ on the grounds that ‘this can serve to gatekeep mathematicians’ knowledge’, but their illustration of a series of complex topological concepts using sketch-diagrams reproduces a similar kind of ‘gating’ effect. Few human geographers will be familiar with the quite technical concepts discussed, unless, of course, they have a background in advanced mathematics. There are, indeed, fairly lengthy discussions of concepts that will be unfamiliar to – and probably quite impenetrable to – many human geographers, from quotient spaces and three spheres, to indirect topology and homeomorphism.

The suggestion seems to be that mathematics, and particularly topology, shows the way, and points to answers – whether in describing, or explaining. This is not an invocation of topological figures as useful analogies for visualising relations or disrupting simplistic accounts of spatial containment, but mathematics as a powerful set of conceptual tools for enacting and explaining. This is not an article proposing playful, experimental, or propositional post-structural musings on mathematical figurations. Rather, concepts like boundaries, sets, connectedness, unions, continuity, subject, and object seem to be deployed in quite specific and tightly defined ways, taking a conventional approach to mathematics which is, as the authors state, ‘in accordance with standard mathematical textbooks’ (p.21). If this is conventional (though advanced) mathematical topology, then is this deployed in some radical or revolutionary new approach to space and spatiality? Well, yes and no. On the one hand, towards the end of the article, the authors do ask whether conceptions of ‘rigour, precision, and messiness’ may need ‘rethinking’, and whether ‘a degree of descriptive indeterminacy’ is required in order to render social worlds visible (p.21). On the other hand, I’m not sure we need concepts from topology in order to highlight multiplicity, mobility, plasticity, and change in human geography.

Given that this team of mathematicians, geophilosophers, and social scientists aim to use mathematical topology to explore a range of essentially philosophical questions, I am surprised that they do not engage with the thinking of French philosopher Alain Badiou. Across a large number of books, Badiou has, famously, explored the philosophy of mathematics and numbers, asserting that ‘mathematics is ontology’ (Badiou, 2005 [1988]: 15) and ‘number is being’ (Badiou, 2008 [1990]). Badiou focuses on ‘set theory’, ‘category theory’, and what we can, in the broadest sense, refer to as number theory, alongside a detailed engagement with philosophical conceptions of being. Unlike Unnithan Kumar et al., Badiou does not shy away from using mathematical symbols and specifically notational symbols from ‘set theory’. But neither does he refrain from tackling foundational ideas in Western philosophy.

In conclusion, I am unclear what the authors might hope to achieve with this article. While their emphasis on key concepts in topology for a geographic audience is not itself a problem, I’m not convinced that many post-structuralist geographers are looking for a more precise mathematical language for understanding spatiality and spatial relations. And if topology can provide an alternative language that is imprecise, then why might human geographers look to mathematics rather than philosophy or social theory for its conceptual tools? The authors question those who may seek to ‘apply’ concepts of topology to their empirical studies, but while preferring to call for an ‘attunement’ of topological theories to particular social situations, and challenging various binaries, Unnithan Kumar et al. could well be criticised on semantic grounds, for ‘attunements’ and ‘applications’ are not always dissimilar.