An analysis of the forms that appear in mathematical discourse and the structures through which mathematical spaces are shared. A presentation of work from a new research project centred around the blog infiltratemathematics.wordpress.com. Spatial and physical language allude to the conceptual spaces of mathematical thought, captured by notation with iconic and symbolic properties.
The language developed to express mathematics has made it possible to talk about strange and complex subjects using hyper-refined symbolic expressions. It is very difficult to access mathematics without knowledge of the conventions that give its notations their meaning. As an artist with a sincere fascination with mathematics and the ontology of mathematical objects I am investigating ways for outsiders to gain access to some of the ideas and culture of the discipline, and embodied and extended mind theories suggest that observable, physical aspects of mathematics can be understood as integral to the discipline.
I have been carrying out observations of the ways in which experts communicate
through writing, drawing, speech and gesture, in order to learn about mathematics from
the outside. As Latour and Woolgar aimed to do in their study of scientific culture(
Mathematics is given a physical form in various aspects of a presentation, and
careful analysis of the gestures and metaphorical language used by experts to guide one
another through mathematical spaces can help the outsider to learn something about the
shape of those spaces. Here I present some observations and responses from a lecture
given by André Neves on Min-max theory and its applications(
Lakoff and Nuñez' Where Mathematics Comes From describes mathematics as arising
from cognitive metaphorical mappings that extend the structures of bodily experience
into conceptual domains(
To an outsider, some of the gestures used in a mathematics presentation seem very
surprising; theoretical, high-dimensional 'objects' are described with movements that
trace out a physical shape or manipulate them in space.
“...a hypersurface or the same hypersurface with different orientation, they are the
same object.”(
As Neves speaks this sentence, he accompanies it with a flipping of the hand from palm up to down and back. This suggests flipping around a particular axis, the hand either becoming a representation of the hypersurface or supposed to be manipulating it in space, drawing out the interesting assertion that “they are the same object”, rather than two similar descriptions of theoretical surfaces. This description of the two hypersurfaces as being the same object is not reflected in the formal notation of this piece of mathematics; rather it gives descriptions for two manifolds which have certain characteristics in common but others that differ. As with much of Lakoff's Cognitive Linguistics, Lakoff and Núñez' model rests on cognitive metaphorical mappings that set up a statement that connects two domains; in this instance the cognitive metaphor might be something like HYPERSURFACES ARE PHYSICAL OBJECTS IN SPACE. What is missing, though, is an account of how certain characteristics are chosen to be mapped and others are not; we might understand the hypersurfaces as maintaining aspects of their form like objects when flipped, but we aren't given the impression that it would be possible to pick one up and put it in one's pocket.
Attempts have been made to reconcile Sperber and Wilson's Relevance Theory(
It is useful to think about sculpture as an ostensive act made with physical material. Which aspects of a physical form or a material promise the greatest cognitive effect in return for attention might depend on the way they are shaped, how they are combined, or how they have been experienced previously in the wider world. Taking an approach to the physical communication of mathematics that focuses on these processes of selection presents one framework through which mathematics and sculpture might relate.
Among the many alternatives to Lakoff and Núñez' book, De Freitas and Sinclair's
2014 book Mathematics and the Body(
Neves focuses on the word 'embedded' on the blackboard, drawing a line around the area on which it is written as he asserts that this is the reason that the theorem is false. By highlighting a particular part of the written statement the speaker alters the audience's perception of the problem, drawing additional attention to an element that may not have been prominent in their initial reading. The location of that problem, though, is an interesting question. What is sure is that the problem is not a purely visual thing; it isn't something that the mathematician would normally carry around written down, to look at; the speaker doesn't copy the words or the shapes from a visual representation up on to the board, and the audience doesn't look to the shapes of the letters to provide the explanation.
Why, then, does Neves emphasise the writing on the board using a drawn shape, though the 'embedded' condition could just as well be emphasised using speech alone? The expectation must be that the speaker has chosen this as the optimal, most efficient way to highlight the appropriate idea. The problem was written at the top of the board, fixing initial conditions, their spatial position emphasising their priority. By returning to this writing, the speaker emphasises that the reason why the theorem in this particular case is false is something that requires nothing more than the conditions set out in the original problem to deduce; the audience doesn't have far to go. Writing is also static and visible. It is present on the board while the speaker is writing something else, and it can be referred back to in a concrete way. For people unfamiliar with the problem this property allows them to 'see' and have easy access to the elements of the problem in a way that perhaps the speaker can have without reference to written forms after a long period of familiarity. Material aspects of mathematics offer certain affordances that are invaluable to its communication.
“...we want to make sure that we capture these k-projective planes...”(
Brian Rotman has written brilliantly about the different entities invoked in a
mathematical paper: the person reading it is echoed in a Subject implied by the
language of direction who must surely carry these commands out, and a yet more
abstracted Agent is supposed to execute the more unrealistic commands, such as taking
the sum of an infinite series(
The story of the original discovery by the mathematician, often full of happenstance, collaboration and mistakes, is transformed into an explanatory narrative through a highly selective retelling that gives particular positions and relationships to the author, audience and mathematical community. Though the work presented is already proved and published, presentations are littered with the language of collective endeavour: we want to do something, we'll try this. The mathematician becomes a performer, a storyteller, showing the audience the footholds to create the mathematical meaning in their own minds. The dynamic of the retelling shows where the important connections are made, and which developments should be given the most attention.
The use of the collective “we” also places this as part of the work of the 'entity' that is the mathematical community, which must, because of extreme specialism within fields, be relied upon to validate it. Once accepted, a proof shifts from a story to an object and is tagged with a name and a date, and used as a point from which to build. Frequent references to names and dates pepper mathematical speech – Cauchy data, Riesz potential, “by Khan-Markovic”. This reference points to a certain person at a point in history, their life's work and history turned into an object to be moved around like a building block in a proof. It also obscures the complex socio-cultural factors that have decided which name is attached to a concept, particularly given that mathematics is subject to Stigler's law: the law that no scientific discovery is named after its original discoverer.
Figure 3 shows an example from the studio practice that will run alongside these
written reflections, using physical elements to explore themes of interpretation,
representation and abstraction. This is a model of these three threads running through
mathematical discourse, referencing the shapes seen in the storyteller ceramics made
by the Pueblo people in New Mexico(
Creative practice in this study will be used as a demonstrative exploratory medium, making use of non-verbal experimentation concerned with methods of representation and abstraction. Using it in combination with observation can engage with mathematics on different terms to those commonly used in art-mathematics work, which tends to focus on artefacts from mathematics more than the processes that create them. Rather than creating an attractive illustration I want to present insights into the culture of the discipline, and use a visual, spatial language to make them accessible to an audience whose knowledge of the mathematical world is normally severely restricted.
This paper is a sketch for an ongoing exploratory research project whose aim is to bring back interesting observations from the world of mathematics and present them in a way that is tangible and engaging even to the non-expert. The first result of this research is a blog on which sculpture made in response to these observations is being published. Though the difficulties are significant, I believe it important to investigate ways to access and discuss cultural and conceptual structures in this fascinating realm.
The incompleteness of this experience is key. It calls to mind Susan Gerofsky's
desire to find a place in mathematics education to “dwell with ambiguity”(
In this paper I have discussed some of the ways that shape and our experience of physical reality are manifested in mathematics, and how this can be fed into sculptural work. Going forward I will be carrying out more observations at conferences and will be exploring the role of movement and gesture and what it can communicate about the shape of mathematical reality. I will also be investigating the metaphors that are evoked when a mathematician is asked to explain their work to a layperson and how their form expresses aspects of the work done, as well as their inherent limitations, in a set of humorous sculptural diagrams. Though inherently challenging, I believe that this interdisciplinary project will have many surprising perspectives to offer, to mathematicians and outsiders alike.