Every now and again I go back to mathematics as a visual discipline. It is fundamental to our understanding of the world and its visualisation processes can be used across all disciplines. It is usually geometry that people like myself begin with when working with mathematical ideas and in the visual arts geometry is often linked with perspective and technical drawing systems, especially as a measuring tool. For instance we use right angled verticals to assess other angles, we assess the rotation of our looking through angles of degrees, find heights of buildings by using trigonometric relationships; the rise of computer aided design cementing these relationships. However there are other aspects of visual thinking that come directly out of thinking with numbers and the one I've recently begun to appreciate is 'category theory'. Like so many things in this world, my interest began from something said to me in conversation. I was having a coffee with an ex-student of fine art, who is also a university lecturer in computing and he was helping me think though some ideas about how one thing represents another, or is like something but is not the same as something. I was trying to think around some of the work I had done representing things we don't see but can feel, such as loneliness or pain. In trying to make an equivalence I was worried that in translation I was losing particular associations that only worked in verbal languages. My associate began drawing and asked me if I had ever heard of Category Theory*, an area of mathematics that I was soon to realise had all sorts of resonance in relation to my own thought processes, especially when I found out that as a theory it could be used to get over the idea of one thing being equal to another. The escape from equality being something I decided was similar to my own worries about whether or not what I was doing represented 'reality' in the way I needed it to; I needed to escape from the problem of the noumenon or the thing-in-itself.
But what is category theory? It seems to me that it's to do with how you group things together. For instance, is there a set with 2.5 elements? As sets of things are collections of equal units, the answer would be no. But there's a "groupoid" with 2.5 elements. To get it, just take a set with 5 elements and fold it in half. The point in the middle gets folded over, and becomes half a point.
The physical and visual image here is a powerful drawing thought idea and it allows us to think in 'groupoids' as well as 'sets'. This 'trick' allows us to see that there are other possibilities of arraigning things, and in that thought lies the possibility of creating an interesting metaphor, one that suggests that things are not always as they appear to be. For instance 1 + 2 = 3 could be questioned. Only 3 can equal 3 in the sense of being exactly the same as 3. 1 + 2 being something else, it is a single thing that is being added to a double thing, not a treble thing that has always been a treble thing. In it's history a treble thing might point to a time when it was a double thing that was added to a single thing, but that is a very different history to that of a treble thing that has always been a treble thing. This is of course opening another door, because there will also be things that were first of all double and then they were added to a single, thus forming a treble thing.
Category theory is all about possibilities and it is driven by diagrams. So if we look at the three dots in set 'G', at first sight it looks as if these could be equal to the three dots of set 'H', but then we realise that the dots could be matched up in several ways.
Category theory diagrams
In category theory the dots are objects. The arrows as seen in the top diagram are also important because they don't just indicate possible movement or connection, they indicate a real change in the identity of the object. What has gone on before is an object's history and this history will change its nature. A thousand pounds saved up by a poor person over a lifetime might look the same as a thousand pounds used by a millionaire to pay for a night's stay in a hotel, but in reality they are totally different sums of money.
Category theory questions the use of the equal sign, which states that things are exactly the same. There are important complexities in the way quantities are related which suggest that there is a need to reformulate mathematics in the looser language of equivalence and into this looser language we might be able to slip in the language of art, in that it is constantly been used to find equivalences for other things.
It is interesting to look at the relationships set out in the diagram below. The slippage between individual instances of diverse forms and universal principles is one whereby it is impossible to define where the dividing line is. In particular it is impossible to determine the exact edge between the microscopic and the macro world, all the scales from Angstroms to meters are found in a biological entity like myself, and the complex hierarchical assemblies of various building blocks that we are made of seem to follow principles of form that can be found in other systems as well. In this way we can begin to discover new metaphors, ones that rely on deep structural echoing and that Giesa, Wood, Spivak and Buehler (2011) have termed “concept webs” or “semantic networks”.
A concept web
This diagram shows how you can think about the ways that the building blocks of protein can be connected and where possible weaknesses might be.
Giesa et al. describe how category theory can be used to link the fundamental structural principles behind biological protein materials with entities such as social networks, by comparing both of their underlying structural principles. To do this they have constructed the term 'ologs'. An olog follows a rigorous mathematical formulation based on category theory, therefore it is suitable for sharing concepts with other ologs. They show that an olog for the protein and an olog for a certain social network feature have identical category-theoretic representations and that there is an isomorphism between them; thus demonstrating that the relationship between structure and function at different hierarchical levels, can be effectively represented by ologs. Because metaphors have always been developed by artists looking for similarities between things, I also think that although Giesa et al. indicate that this type of thinking could help engineering, life sciences, and medicine, I see no reason to not add art into the mix. It would seem to me that if biological materials evolved to perform specific biological functions, then the higher-level structures that biological entities like ourselves also evolved such as societies, may well have deep down core similarities, as certain mathematical patterns are no doubt more useful than others in the creation of structures that can survive in a hostile world. It would seem therefore that in order to reflect upon the workings of a natural system, we do not need to understand everything about it, only the principles out of which we believe the functions arise. Therefore a drawing can represent an equivalence by showing how patterns from different models of thinking can find themselves grouped together. In doing so it can also be a tool that refers to and evaluates differences in historical backgrounds, so that when rendering two similar images of a hundred pesos it does not simply make them the same, but renders them different by nature. The problem of the noumenon or the thing-in-itself is also overcome because of the concept of representing an equivalence, in this way there is never any need to reproduce the thing in itself, only equivalences and they can be many and varied and always subject to interpretation.
Two examples of 100 peso notes
*Category theory formalises mathematical structure and its concepts in terms of a labeled directed graph called a category, whose nodes are called objects, and whose labelled directed edges are called arrows.
References
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