Art, Ethnomathematics and the Digital Era

A bit about different histories of math & representational methods


I'm a person both in love for theory as for image making. Often I feel scattered between such passions, but most of the time they are complementary. They're different ways of thinking, of trying to understand something.

My interest for worldwide cultures is directly connected to my experience as a Brazilian woman, raised in Rio de Janeiro, where each year we make offerings for the goddess of the ocean, Yemanjá, but our academic curriculum is mostly a failed (terribly underfunded) attempt to follow outdated versions of European and American standards. Also, as it's common to Brazilians, my personal experience means a family background of continuous ethnical and cultural mix, often in pretty violent ways. Loving history and philosophy, analysing how things came to be, is a deeply inspiring and liberating activity that became my path to understand my possible places and what is meaningful in the world.

The following text is mostly based in academic research, I still need to refine my thoughts on my own experiences with different cultural groups, from media artists, indigenous groups, governmental officials, art dealers, quilombos and etc etc.

The concepts here presented were inspirations for several art pieces, specially the project "Ethnocomputing Experiments", done with Wenqi Li in 2018.

This text has not been reviewed by an english speaking native. Therefore, it's full of weird phrase constructions and other mistakes.

I decided I'd publish anyway. I’ll be glad to hear constructive feedback on this.

The Cokwe people, originally from present-day Angola, have a tradition of gathering around a bonfire or in the shade of leafy trees to enjoy good stories while illustrating them with drawings in the sand. One of these tales chronicles the encounter between a hunter and a wild chicken in the bush; the chicken sees the man approaching and begins to run.

In the story, the bird is a symbol of cleverness and intelligence, so although it is considerably near the entrance to its house when the encounter happens, the chicken decides to run in wide zigzags, because if it ran straight it would be caught or the man would know the location of its hiding place. However, zigzagging is not enough, the hunter would eventually understand the tactic and then catch the bird. So the chicken changes direction, runs in a narrower zigzag, changes again, speeding up the race in a straight line. It comes close to home and finds that the hunter is still around, so it resumes the zigzag, parallel to the first moment of the race. Once again it changes direction and accelerates, alternating between zigzags and accelerated straight lines, until the bird can return home while the hunter is tired and confused.

This tale is part of Paulus Gerdes’s book “From Ethnomathematics to Art-Design and Circulant Matrices” (2010, p. 34–38), but also appears in several publications by the same author. In the mentioned book, the researcher recounts his experience of telling such a story to various audiences, while accompanying them with the execution of a geometric algorithm that illustrates the chicken saga.

Many of Paulus Gerdes books are available for free as ebooks in the website

Gerdes was part of the first generation of Ethnomathematics, a field study that started with the Brazilian researcher Ubiratan D’Ambrosio in the 70’s. One of the main influences on Ethnomathematics was the pedagogy of the oppressed, developed by the also Brazilian Paulo Freire. Another key influence was Oswald Spengler early 20th century cultural history. The originality of Spengler’s work came from the understanding that science history would be too limited if it focused on formally written concepts. His most famous example was ancient architecture, how its construction applied many concepts that hypothetically were only formalised centuries later. Gerdes made a similar argument, devoting several papers to prove that Pythagoream Theorem was implicit in traditional building and drawing techniques.

Paulus Gerdes was a dutch mathematician, but he spent most of his life in Mozambique, where he was naturalised. According to the researcher, ethnomathematics was a necessity once the country had finally achieved independence. Though it was a much needed profession, students had little interest in becoming math teachers, mostly because they understood such subject as a foreign one, part of the coloniser’s culture but not their own. Ethnomathematics became a way to inspire people in the art of finding mathematical practices in the local markets, in the way small business operated, but also in the intricate weaving patterns found in baskets, traditional drawings, all sort of objects seen as crafts. Nevertheless, in comparison to other researches in the field, the outstanding part of Gerdes work is how he connected the study of traditional crafts with contemporary science.

When we read his books on mono linear patterns known as ‘sona’, that were once common in Angola, we discover how they are geometrical algorithms, how we can read the path done by the line as a matrix system. By the end of his explanation, he uses the same logic of the traditional patterns to create circulant matrices and we get to a much deeper level of mathematical discussion. Furthermore, another interesting point about the 'sona' is how it’s similar to several traditional practices spread around the globe, from the celts, to ancient Egypt, but specially to the Kolams in India.

Another of the best research examples of Gerdes work is his comparison of weaving techniques, such as the Malasyan “Sepak Raga” ball, and carbon molecules known as buckerminsterfulleres (a homage to the visionary architect Buckmister Fuller), that were only studied formally in the eighties. In the researcher’s view, the similarity is not a coincidence, but an issue of optimal design.

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Sepak Raga traditional ball, Buckminster Fuller Carbon molecules, weaving technique used to make the balls…

In the US one of the leading researchers in a similar field is University of Michigan professor Ron Eglash, who went from ethnomathematics to what he calls ethnocomputing. He develops it specially as a pedagogical approach, through the usages of the Culturally Situated Design Tools. Eglash was once a grad student of Donna Haraway, also studying with Angela Davies, and got international attention for his 2007 TedGlobal Talk on African Fractals. The talk is a summary of the research he’s done for a homonymous book, published in 1999, in which he defends that fractal design is historically far more present in the African continent than in other areas. In his vision, mathematical ideas are necessarily connected to a cultural understanding of the world, then the presence of fractals in several African societies would be related to religious beliefs, to particular ways of thinking about time and life cycles. These societies would have different world views from cultures where the Euclidian geometry played a much deeper role and fractal geometry is a new phenomenon, such as the western world.

One should remember how shocking and difficult were the formalisation and development of non Euclidian Geometry during 19th century Europe, even if now they could look like friendly complementaries. According to philosopher of science Michael Friedman, despite the widespread recognition of Kant’s role as the founder of modern philosophy and history of science, the developments of mathematics during the 19th century, specially geometry, made the original Kantian approach obsolete, as it was directly based in Newtonian physics. Kant is particularly revered for separating science from religion, if Newton was a highly christian man, for Kant what matters is Absolute Reason. Geometry had a special role in Kant's epistemology and once his premisses were questioned, once Euclidian's axioms were not an incontestable logical truth anymore, philosophers and mathematicians became obsessed with the search for solid ground over which science could flourish (noteworthy examples are Bertrand Russell and Ernst Cassirer). A great reference to understand what the new geometries represented is the short novel ‘Flatland’, in which a square tries to describe to his peers in a bidimensional world how a tridimensional reality is, and how in theory many more dimensions could exist : the inhabitants of two dimensions are terribly outraged and the square is imprisoned for life. Such questions were essential to Einstein's relativity theory.

Eglash goes deep into intellectual history, tracing the reception of Ifa and Bamana binary and recursive divination system into Europe through the alchemists, whose practices were studied by Leibniz, possibly influencing his mathematical research. Or how George Cantor’s famous ternary set looks a lot like an Egyptian architecture capital, and knowing that a Cantor’s relative was an Egyptologist, it is possible that Cantor could have had access to such images. Recreating the history of intellectual influence is a very tricky work, but one thing should be clear: as the world knowledge became accessible to empires, it meant certain groups of people, such as scholars in leading universities, could drink inspiration from all sorts of different sources, while the same does not happen with the colonies. As Claudia Zaslavsky explains in her seminal book “Africa Counts: Number and Pattern in African Culture”, from 1973, understanding local mathematical systems became key for the metropolis to employ and charge taxes on Africans in their colonial administrations. Therefore, a lot of technical material on such systems became available for 19th century Europe.

Reading Gerdes’ work, Eglash’s and few other researchers, we are led to the conclusion that the study of such traditional techniques could be relevant for nowadays scientific debate. One should just remember that origami was only recently taken seriously for advance technological development. Gerdes’ approach connects traditional crafts with contemporary science, he was interested in good design solutions and tried to understand the whole process of making something, from the raw material to the final product. This is not so distant from several other current trends in design, art and technology. The maker movement for example, which is a tech oriented DIY movement, combines hacker and open-source culture with traditional arts and crafts, from metal and woodworking to sewing and whatever technique fits better for a project. One of the people doing innovative work on such topic is Harshit Agrawal and his Traditional Cad Tools project:

These days, we are seeing a rise in the digital maker movement, propelled by digital fabrication machines like 3D printers. These are democratizing manufacturing, however, a point to note is that all of the CAD (computer-aided-design) software tools to design objects for digital fabrication have been developed in an industrial context. This inherently means that these tools support operations like extrude, revolve etc., but not traditional operations like weaving techniques (coiling, twining etc.) for basketry for example. Therefore, the digital design language lacks a representation of the diverse making traditions and these are not accessible to people to design with. — Harshit Agrawal

Digital fabrication evangelists (like FabLab’s Neil Gershenfeld or historian Mario Carpo) common cultural reference is not the African fractals nor patterns to be found throughout the world, but the Italian Renaissance. Renaissance has been the most prolific case study for art historians since the formalisation of this academic field and now it seems to maintain a similar attraction to the world of new media. Gershenfeld defends the Renaissance as the role model of a social organisation in which collaboration between different knowledge fields is truly incentivised.

Linear perspective, the Renaissance preferred method for visual representation, was a game changer. One of art history’s most famous essays, “Perspective as a Symbolic Form” published in 1927 by Erwin Panofsky, considers this method the clearest expression of a deep transformation towards the understanding about space and human vision, therefore on how we perceive reality and the place of the human being in the world. He argued that linear perspective was based on an understanding of physical space as an objectified, a measurable form, within a notion of continuous and homogeneous space. A painting using such method of representation would present all of its objects according to precise proportions within a space receding towards one (in a few cases more than one) defined vanishing point. Therefore, it became possible to make accurate reconstructions of a scene or buildings using two dimensional images as references.

Linear perspective was related to a rationalist world view, in which the individual, the observer, is at the center. This argument is important because it highlights how artistic/representational/mathematical methods and concepts can be shaping agents of major changes in the way we deal with reality.

To understand this better, the research led by Philipp H. Lepenies in his book “Art, Politics and Development. How linear perspective shaped policies in the western world” is particularly insightful, constantly making a parallel of linear perspective’s vanishing point with the idea of one directional understanding of history and progress, as if there was only one goal (and mostly one path) to pursue for the whole of humanity. Lepenies highlights the influence of linear perspective in the development of Illuminism and even for nowadays international politics, specially ideas of foreign aid and economic development. Another relevant argument is Mario Carpo’s , who defends that after linear perspective, only computer aided design and the digital revolution managed to bring a really original contribution to our representation methods and design possibilities; even the modernists were still designing with square and compasses, bidimensional technical drawing had a continuous improvement since XV century but no change of paradigm.

If during the Renaissance mathematics and art were deeply interconnected, as the scientific revolution unfolded in the West, math became a much more abstract field with no easy method of representation. Specialisation also played a role and the arts were eager to defend its autonomy and specificity as a knowledge field. Painters and architects also became more distant to each other. Mario Carpo argues that Newton’s calculus created beautiful curves, however, making solid three dimensional curves was mostly still an artisanal process, relying on tools like the spline. When Viollet-Le-Duc, a prominent 19th century architect, looked for the theoretical foundations of his field, he was far more interested in the sciences of life than by the new calculus based mathematics of his time. The history of the mathematical representational methods that changed such situation is a bit too long for this text, but it’s worth to highlight the development of vectors during 19th century, Bézier curves around the 1960s, Computer Aided Design and the beginning of drawing softwares in 20th century, and finally Digital Fabrication, which is capable of making extremely precise 3 dimensional curves.

As Panofsky tried to make clear, a change in representational methods has a much wider impact than one would initially suppose. Mario Carpo makes a long explanation on the importance of the Albertian Paradigm, which states that the real artwork is not the building, but the project, the drawing. According to the researcher, such idea was hard to really implement during the Renaissance era, as workers had much more autonomy and would creatively contribute for the overall construction, but it settled the trend for architecture, the arts, and the industrial revolution labor system: if the original artwork is all embedded in the project drawing, then the people who execute it are completely detached from the act of creation. The same project can be recreated anywhere else, the context matters little. Authorship is revered, but restricted to a few outstanding individuals. Design becomes an activity on its own.

Even as painting and architecture became distant from mathematical studies, the trend of highlighting the individual creator instead of the collective only augmented since the Renaissance. One of the biggest references to study such phenomenon is Jakob Buckhardt’s idea of the birth of individualism during the Italian Renaissance, part of his “The civilization of Renaissance in Italy”.

A very different approach to creation and geometry is described by Ron Eglash. In the cover of his book, as showed in the image above, there’s a weaved fabric, created by an artist/craftsman in one sitting. People continuously serve him coffee so that he can keep his task, the shapes become smaller as the artist energy becomes more intense but also exhausted. The geometry is directly connected to a state of being and the maker is the artist, though one must always remember that the concept ‘artist’ has a very different connotation, market and social expectations in each cultural context. Another curious example is the Nankani home, in which the space a person is allowed to occupy increases according to life phases.

The homes of the Nankani are fractal series of cylinders. Here we see just one compound, a cylinder of cylinders that get smaller as you rotate counter-clockwise around the central courtyard. In this village, your life follows the architecture. Your first first rite of passage is from mother’s womb to the birthing room. Your next is to crawl into the courtyard. Your next is from the courtyard to the village as a whole, and finally from the village to the world. — CSDT website

Anthropology is particularly important to understand our own cultural symbols. It can seem easier to look at groups who are not incorporated into industrial and financial capital production logic, groups that we can identify as 'the other’, and see the correlations between symbols and social organisation. Often a relevant exercise is to do a similar analyses on ourselves, trying to understand what system of values are we following, where do they come from, and if they’re compatible to a future we would desire to live in (how could we have a better social organisation in our digital era?). Or trying to grasp the complexity of living in multicultural societies, where ethnicity becomes terribly tricky to define. Understanding the non linearity and the multiple networks of influences in History is essential for us to understand how symbols, concepts, are created and transformed. The more recent research on linear perspective's debt to arabic optical science is one of the innumerable examples of the surprising paths of history, of what the encounter of different cultural world-views can create. Also, having broad cultural references can give us insight into what alternatives paths we could create.

What is universal and what should be local, in terms of shared knowledge and practices? With the current development of the global economy, places become more similar, though not always in an advantageous way to most who don't concentrate wealth. Ole Skovsmose and Renuka Vithal 1997 article “The end of innocence: a critique of ‘ethnomathematics” explains Ethnomathematics as a response from former colonies to Eurocentrism in mathematical education while european/american educational methods were becoming mainstream all over the world after World War 2, following a massive industrialisation movement and its need of specialised labor. Similar to what Paulus Gerdes argued, the authors point out that such methods were too often not adapted to local cultural contexts and former educational practices. However, though they recognise the importance of the research field, Skovsmose and Vithal criticise Ethnomathematics using as example the case of South Africa, explaining that too much focus on local cultures or ethnicity could lead to a kind of educational ‘apartheid’, as schools would teach students differently according to their race. One can easily see how problematic that can become. Instead of fighting structural inequality problems, a naive ethnomathematics or identity politics approach could end up reinforcing racial stereotypes.

One direction view, one vanishing point we all follow, not even western mathematics accepts such notions anymore. Relativity and quantum theory portray a much more complex an ambivalent notion of space and time than the science behind linear perspective. If art and mathematics have not always been deeply intertwined, in the age of computer's ubiquity, to understand the machine’s logic is each day more important. Algorithmic art nowadays comprehends a wide variety of practices, from generative patterns, to computer aided design, machine learning, to any image editing software. Similar to the process described in the last paragraph about mathematics educational methods, the newer image creation and sharing tools also tend to contribute to a homogenisation trend. The history of linear perspective is a very clear example of the relations between representational methods, social organisation and power structures. So are the history of the Nankani people and the 'sona' drawings, if we look deeper into them. Happily, we do have access to diverse cultural references, even if it's still necessary to make the effort of building more digital cultural databases and of democratising such concepts and tools. How should cultural databases be created and shared? How can more people actively participate in this process?

Looking at world history can always be a source of inspiration for imagining new utopias. I'm a weird combination of extremely rational with extremely intense/impulsive. Sometimes I manage to find balance.

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