The Bridge Between Worlds: Relating Position and Disposition in the Mathematical Field by Lorenzo Lane

This is an excellent ethnography and well worth reading.

Lane discusses use of

1. Common and private space
2. Formal and informal spaces
3. Productive and relaxation space p. 143

By “socialising ideas” I simply mean that concepts become part of assemblages of people and things. Ideas become grounded in day to day life through their attachment to people. At the institute people speak of Sarah’s technique, Penelope’s data, Luke’s theorem, and use these names to act as handles for searching MathSciNet, Arxiv.org, or google scholar, for related material by which to trace the origins of the ideas. p. 178

Knowledge in mathematics is embodied. Individuals act as repositories of knowledge, and many researchers prefer to ask a question directly to an expert, in order to find out more about a problem or idea, rather than consulting written material. The oral transmission of knowledge in mathematics is a means through which information contained in publications can be expanded and summarised, so as to provide the key points, motivations and overall structures of proofs. Interviewees highlight the importance of asking experts about a problem, paper or approach, which can serve a number of purposes:
1. As a time-saving device.
2. As a means of acquiring interesting techniques and problems.
3. As a way of getting up to speed in an area.
4. As a means to socialise and share interests in mathematics.p. 179

N: So I’m not fast in oral communication. So I like to discuss maths with others, but at the level that is of motivation and general interests, questions. But when I want to think really deep, I need a pen, and I need some loneliness at some point to think through a question. When someone is watching me I am not so good at thinking in front of somebody. But I need communication and exchange about questions, in fact to be interested about a problem I need to understand why we should all care, as a community, about this question, not just “Oh it’s a hot problem – you should try this”.
IN: So you use people to get a problem and interrogate them to find out why the problem is interesting?
N: Yes I use people in that way, and also I see it as a cultural thing also. So I like to exchange, and see what the vision of mathematics we have…Doing maths with others is all about what questions we can work on between ourselves. It’s not just “Oh lets go to the blackboard and solve a problem”, it’s rather: “Oh have you heard of that, and look I can do this, describe some simple examples and so on”. At some point exchanging around this, and maybe having a small idea or small example, and then afterwards have more quiet and just think about the natural places where these ideas should sit. p. 179-80

Adrian notes that reading the formalised, published papers is sometimes not enough. Rather a “fluid” approach is necessary, by which an individual can question a person who is a “knowledge-repository” on specific questions related to a concept or proof, in order to “get the bare-bones” or the “gist” of an argument. This is an important task to undertake before a researcher decides to fully commit to investing time and mental energy in reading a given paper. The informal back-channels of questioning knowledge-repositories serves as a means of scoping out potential problems or useful concepts, narrowing down the search parameters needed for later literature reviews and google-scholar/ arxiv.org searches. p. 181

Before any presentation, such preparatory work takes place. Individuals will ask what they call their “stupid” or “silly” questions before face-to-face encounters, utilising close friends or colleagues, or else will use the question and answer site MathOverflow.com. Then when it comes to asking the expert, or giving an exposition, one does so effortlessly.
Such displays of effortlessness are important in showing that a researcher possesses “natural” ability within their field, and thus they are important in developing their reputation as a competent, rigorous, trustworthy researcher. In public, much of the hard work of learning is made invisible. Through downplaying the effort involved in understanding concepts, so individuals create distance between themselves and their audience. Some individuals, I am told, are perceived as “naturals”, “geniuses”, “gods” because of their effortless ability to perform and demonstrate their understanding of mathematics. When many such “naturals” talk in private between themselves, or when being interviewed, the struggles involved in grasping concepts are revealed. p. 183

This mathematical corpus forms part of the habitus: the system of scheme-resource sets which produce orientations and dispositions towards phenomena. The conceptual machinery that a mathematician acquires thus serves to produce a certain way of perceiving mathematical objects and spaces, and also generates dispositions by which they are able to confront problems within certain mathematical domains. Through becoming familiarised with certain techniques, objects, languages or papers, mathematicians furnish their habitus, and this, in turn, re-shapes the perceptual lenses by which they are able to perceive a problem. p. 209

J: So yes he told me to read this one article and said I would understand things. But it was not well written, it was something like 30 pages. He said “OK, so if you read it in the first year it will be good for you”. I said “30 pages in one year?…yes I can do that [laughs]”. But this was really hard. For every word you have to understand a lot of things. You can read that, that, that, that and you just get used to the different notions.
He had a lot of time for me, so I could just discuss about things, and he explained a lot of things to me. It’s easier to read when someone explains the story before. I mean you don’t have the details but you know where you’re going, and you just fill in the details when you read. p. 201

In order to “keep alive” within the mathematical landscape, a researcher has to adapt to their changing environments, and learn new languages by which to communicate with their colleagues. Over time, they need to learn new vocabularies and gain access to new ways of inhabiting and exploring the mathematical landscape. This process of developing a shared working vocabulary is not confined to the office, the institute, or the seminar room, but as the following conversation with Han and Bernie indicates, the process of building a shared knowledge base is continuous, happening throughout the working day:
H: This is very important: the word vocabulary. Because this vocabulary has built up not only through working one day a week, but this vocabulary has been created when going to take a drink, or when walking, or when talking about another subject. This vocabulary pops up in math, so it’s difficult to say. Every day there is new input into this vocabulary. It took us really one month to have a common vocabulary, but this vocabulary has become richer and richer with time.
B: But it was about one month, and we could speak to one another, and we knew what the other guy was talking about. But we did other things. During this month we were explaining and asking and saying “I do not understand could you explain in more detail?”
H: This is like what we said at lunch, he was speaking German and I was speaking in French… p/ 215

We shall see in the next section how, through the mastery of technique, the tools and language of mathematics are moved to the background of consciousness. The perception of the mathematical landscape becomes second nature, and the researcher learns to use their intuition. We shall explore the processes by which the language of mathematics becomes naturalised and backgrounded, so that working memory is freed up, in order to focus on the process of discovery, creativity, and assembly. Mathematics, we shall see, becomes a craft, and the mathematician is transformed into the bricoleur. p. 216

This role of the body is played out through the practices involved in constructing mathematical proofs. It is through manipulating material representations by assembling, selecting, sorting, ordering, tinkering and relating that the mathematician builds up the machinery of proof. p. 217

G: In the last 40 years I simply never spent a week without thinking about mathematics, a day yes, but not a week. It’s like ruminating you know, we’re like cows but what we eat is mathematics. It’s a long rumination. Some people work late at night, and usually the best are like that and I’m not one of the best. For me it’s like a long rumination, OK, mathematics is with you whatever you do. You can go and something else, of course, but it is always escorting you around. I was mentioning these notebooks, they are with me essentially always, and sometimes I don’t touch them for a week because nothing happens but they are with me anyhow, where ever I am. p. 218

N: I have mathematical notes but it’s still vague, when I write down, it’s still at the early stage and it’s really badly handwritten. It’s more like a collection of “Oh this can work like that” or else just sentences to myself. But I can show you this. So currently [flicks through notebooks] it’s a mixture of papers that I have printed, and notes like this where I’m working on some random walks in some spaces, and these are the random walks. These are the kinds of stuff that I write. This is really not that precise. And when it’s crystallised into being something special like this, you have a statement labelled lemma and you can see that there is no proof behind, because when I’m at the level of proof then I will start writing things down the paper. And you can see stuff like this, where we can relate these two theories and these are notes from a discussion with a colleague. Not very precise, but quite helpful. At some stage it’s written in the computer and at some stage I will just chuck these. Maybe I will take this same paper and write down something else on it. At some point when I’m thinking I just need to write something down to help me think. I don’t keep these notes. The notes I keep are already type-set.
In fact seeing it like this – only a very few people will it be meaningful for. I think what is interesting is to see that even if the paper has no picture, but I’ve wrote a picture for myself here… So one important thing in writing is to help the mind to concentrate. Sometimes it’s not important if it’s right. I explained this to a collaborator once: that I just need to write. It’s not strictly useful to write down, but it helps to think. And also the mathematician does not want to be naked in public. You don’t want to show that you were really dumb at some point. p. 220-1

Writing, as Nemo tells us, helps him concentrate, to reformulate and to re-figure concepts. Such thoughts start off as vague, but over time they are shaped into something much more concrete, “crystallising” into definitions or lemmas. Mathematical writing thus provides a physical scaffold for crafting ideas. For many researchers this scaffolding is a way of offloading certain processes from working memory, distributing memory onto an external storage device. p. 222

Gordon explains that the picture sketched out on the blackboard is more than just a depiction of an object, like in a photograph; rather it is an interpretation of an abstract concept which is given expression within a painting. The painting, depending on the skill of the artist, can encode within it a number of concepts (Gombrich 1977: 44). Understanding what information is encoded within an artwork depends on one’s understanding of the style and language of the artist. The act of sketching out objects thus is a means of condensing mathematical concepts into a visual short-hand, which provides a further scaffold for constructing complex arguments. Through condensing information into a visual format, arguments or concepts become subject to visual inspection and interrogation, which facilitates the process of pattern discovery and the synthesis of information, as p. 234

I’m not sure about all this talk of encoding and art, but there are some interesting things here.

J: I first draw pictures and then go and try to explain it in a mathematical way and I ask “what is this picture really trying to say”. That’s the really hard problem. It’s like: “OK I have to write every details…ahaha” How do I do this? SO it has to be a real proof so everybody says “yes this is true”…For me I like speaking with my hands. But then we have to write rigorously… But if you look at my drafts it’s like picture, picture, picture.

Oftentimes mathematicians begin by working with pictures, building up intuitions about objects through manipulating such pictures. It is only later on in the proof construction process that these physical intuitions are erased, as the pictures are abstracted away, in the formal process of “writing up”. In writing up much of the informal processes of experimentation, as well as the physical intuitions on which arguments are based, get written out of the end publication. However, in erasing pictures and examples from the published proofs, it can often become difficult for other mathematicians to follow a proof. p. 235-6

These modes of visualising mathematical reality are referred to as gaining “intuition”. This “intuition” is what we refer to here as a perceptual frame. This perceptual frame is built up over the course of one’s career as a mathematician, and comprises of the vocabulary of concepts that an individual has internalised within long-term memory. p. 240

I’m not sure I understand this part – what is being pictured? Sometimes it’s a thing seen as spatial. I guess I can understand ‘picturing’ a sort of perceptual or mental state, perhaps. But I’m not sure I’m clear that that’s what people are referring to.

But perhaps the most important aspect of visual representations is the fact that they simplify complex constructions into visual short-hands. They are able to condense complex information into symbols, signs and graphics, which represent complex processes and phenomena. Through simplification and condensation into two and three dimensional drawings or algebraic symbols, so the background structure of the construction becomes more apparent. p. 242

I think there might be more going on here. There’s a lot of structure added in when we write something out in notation and so on – I’m not sure it’s just simplification, I think we actually must add something in. There’s been some research that shows experts reading mathematics in structured, language-like ways: http://www.tandfonline.com/doi/abs/10.1080/09541440600709955 . Might turn out to be relevant.

Creating visual analogues to abstract mathematical constructions transforms concepts into perceptible objects. The visual representation approximates the structure of the mathematical construct, sharing with it certain characteristics. Visual representations in mathematics are used in similar ways to how metaphors are used in natural language. The visual representation bears certain resemblance to, or shares certain features with, the mathematical construct that it refers to. The visual representation thus acts as a schema for organising the abstract mathematical construct. Modifying the visual representation becomes analogous to transforming the mathematical construct in some way.

Visual representations serve as frames for viewing and organising the abstract, mathematical landscape. The visualisations are not the mathematical objects themselves, but merely approximations of these objects, projected into our physical reality. p. 242-3

This gets tricky…

Such ways of thinking about or translating mathematical concepts are called “tools” by mathematicians, and they provide a way of educating their perceptions into being able to perceive the mathematical landscape. p. 243

This is interesting! I love all the ‘tools’ stuff, but not the idea of them allowing you to perceive a pre-existing mathematical world.

I explore the field as a competitive field, in which individuals are motivated by gaining social capital, through making their knowledge visible and recognisable. Ideas, in order to be assimilated within the wider field, and constituted as capital, must be structured so as to fit certain socially valued characteristics. Knowledge therefore must be coordinated and related to existing bodies of knowledge, in order to be accepted and assimilated within the wider field. I explore this process of coordinating local frames of reference to global reference frames, and show how individuals shape their ideas so as to conform to certain standards and exemplars. p. 250

He discusses authority in the next chapter.

Advertisements

Artist as ethnographer

http://www.corner-college.com/udb/cpro2ZgGKfArtist_As_Ethnographer.pdf

http://csmt.uchicago.edu/annotations/fosterartist.htm

Hal Foster wrote something important about the artist as ethnographer.

  • the subject is othered, and the researcher still has the power
  • the ethnographer envies the artist’s reflexivity, while the artist envies the ethnographer’s access to an ‘other’ that has various properties attributed to it

“…this setup can promote a presumption of ethnographic authority as much as a questioning of it, an evasion of institutional critique as often as an elaboration of it.” p. 306

“…the artist, critic or historian projects his or her practice onto the field of another, where it is read not only as authentically indigenous but as innovatively political!” p. 307

As I’m a white western academic studying other white western academics, I suppose my subject is pretty close to where I am. I’m also endeavouring to de-other them, to dismantle this overly romanticised image that we have that takes mathematics to be so mysterious.

http://www.tandfonline.com/doi/pdf/10.1080/02560046.2013.855513

Arnd Schneider and Chris Wright (2006: 4) assert that ‘[a]nthropology’s iconophobia and self-imposed restriction of visual expression to text-based models needs to be overcome by a critical engagement with a range of material and sensual practices in the contemporary arts’. p. 460
Based on Hal Foster (1995):
Does this artist consider his/her site of artistic transformation as a site of political
transformation?
Does this artist locate the site of artistic transformation elsewhere, in the field
of the other (with the cultural other, the oppressed postcolonial, subaltern or
subcultural)?
Does this artist use ‘alterity’ as a primary point of subversion of dominant
culture?
Is this artist perceived as socially/culturally other and has s/he thus limited or
automatic access to transformative alterity?
Can we accuse the artist of ‘ideological patronage’?
Does this artist use ‘alterity’ as a primary point of subversion of dominant
culture?
Does the artist work with sited communities with the motives of political
engagement and institutional transgression, only in part to have this work recoded
by its sponsors as social outreach, economic development, public relations?
Is this artist constructing outsiderness, detracted from a politics of here and now?
Is this work a pseudo-ethnographic report, a disguised travelogue from the world
art market?
Is this artist othering the self or selving the other?
Based on Andrew Irving (2006: 14):
Can this artist be criticised for underlying assumptions of misplaced
temporalisation
whereby non-Western practices, be they artistic or otherwise,
are seen as some throwback to earlier, more primitive forms of humanity?
Based on Lucy Lippard:
Is the artist wanted there and by whom? Every artist (and anthropologist) should
be required to answer this question in depth before launching what threatens to
be intrusive or invasive projects (often called ‘interventions’) (Lippard 2010:
32).

p. 463-4

Less concerned with the possibilities of
accurately representing the ‘other’ and his/her culture, the ethnographer nowadays
aims to comparatively relate his/her own cultural frame to that of the ‘other’, in
view of establishing an interactive relation. Ethnographers furthermore look at
cultural practices in which attention is paid to inter-subjectivity, where one relates
engagement with a particular situation (experience) and the assessment of its
meaning and significance to a broader context (interpretation) (Kwon 2000: 75). The
idea that one actually can ‘go native’ and ‘blend in’, so as to completely integrate and
participate in a particular culture, has been criticised as exoticism. Yet the stress on
ethnography as an interactive encounter is of crucial importance, as ‘the informant
and the ethnographer are producing some sort of common construct together, as a
result of painstaking conversation with continuous mutual control’ (Pinxten 1997:
31, see also Rutten and van. Dienderen 2013). p. 465
Ingold (ibid: 10) proposes to shift anthropology and the study
of culture in particular ‘away from the fixation with objects and images, and towards
a better appreciation of the material flows and currents of sensory awareness within
which both ideas and things reciprocally take shape’. p. 465

Practice

After reading some of Graeme Sullivan’s work on art practice as research (Sullivan, 2004) I have started seeing the relationships between some of his tripartite schema and aspects of my research. In particular, he outlines three approaches to practice as research:

  • thinking in a medium (working with media through making)
  • thinking in a language (responding to discourse, interpretive work)
  • thinking in a context (art that engages with the wider world, in a critical, generative way)

I was struck by the parallel between these and the structure I’ve been working on for understanding mathematical communication and how it is successful, comprising statements within a problem, the cognitive environment that gives those statements meaning, and the institutional structures that make it possible to build those cognitive environments.

In a similar way, I am beginning to think about my practice in terms of similar trios.

From one perspective, I am working creatively with the media of mathematical communication (the paper, the chalkboard), using moments from observed material to recontextualise and talk back to (working with, for example, sketched diagrams that change valence when placed in a frame), and introducing my practice in the institutions I am visiting (such as making animations made on chalkboards, and photographing work spaces).

As well as using this as a structure for particular lines of enquiry going forward, I am using this as a means to reconsider some of the wider practice experiments that I have been engaged in so far, on my practice page (https://infiltratemathematics.wordpress.com/practice/)

Hushed Tones: the modern art gallery as intensifier

A colleague and I are giving a paper at Beyond Meaning conference in Athens on the 12th September. The abstract follows.

 

Art is a complex and difficult field for analysis from other disciplines, and any analysis that straightforwardly identifies art with communication is bound to be rejected. It seems possible, however, to accept that art is intentionally produced and that meaning generation takes place in the knowledge of that attributed intentionality. In that case, it can be considered a good candidate to be addressed using theories of communication that focus on complex inferential processes in the mind of the viewer (as opposed to resting on a simple encoding/decoding model). The attribution of intentionality gives art a meaning beyond the kind of pleasure evoked in experiences of natural beauty. Institutional theories of art such as that of Danto (1983) suggest that institutional sanctioning is the primary condition for the identification of an object as art, and we put forward a charitable justification of such ideas grounded in a cognitivist theory of communication. We intend not to attempt novel interpretations of our own, but to offer an explanation of the mechanics of interpretations as they tend to exist, as Wilson (2011: 74) does for literature.

A challenge posed by modern art, following the shifts in attitudes toward craftsmanship and appropriation that came with the birth of the readymade and conceptual art, is how it is possible to discern art from non-art; a further question is how it is that objects are imbued with the kind of weighty significance that justifies the far-ranging interpretations reached in its scholarship, not to mention its price. Using a cognitivist framework based on relevance theory (Sperber & Wilson, 1996), we propose that gallery spaces function analogously to a linguistic intensifier, such as ‘really’ or ‘very’, and that the art-world framing of an object is instrumental in attributing the intentions that will bring forth appropriate interpretations in a viewer. An attribution of artistic intention prompts the investment of more effort, and so the bringing forth of more complex, effortful and multidimensional interpretations. Galleries, we suggest, justify the effort required to make such an interpretation and further decrease the effort required to produce some interpretations. From this perspective we consider how such framing might be achieved and thus how the profound, ambiguous networks of unresolved possibilities that make up artistic appreciation might be constructed.

This approach deals capably with the use of ritual in art and the repurposing of readymades by Duchamp and Warhol, and offers insight into difficult questions such as the artworld appropriation of outsider making, the problems of migrating Banksy’s street art into the gallery and the fetishisation of the gallery space. In each of these we consider the cultural ramifications of each act and the means by which this is realised in the mind of an individual at a gallery space. We also consider, therefore, how the cultural is grounded in the cognitive needs and tendencies of individual and, conversely, how the cognitive environment of the individual is grounded in their interactions with culture.

References

Danto, A. C. (1983). The Transfiguration of the Commonplace: A Philosophy of Art (Reprint edition). Cambridge, Mass.: Harvard University Press.

Sperber, D. and Wilson, D. (1996). Relevance: Communication and Cognition. Wiley.

Wilson, D. (2011). Relevance and the interpretation of literary works. In: Observing linguistic phenomena: A festschrift for Seiji Uchida. pp.3–19.

 

Relevance and rationality by NICHOLAS ALLOTT

Abstract
Subjects’ poor performance relative to normative standards on reasoning tasks has
been supposed to have ‘bleak implications for rationality’ (Nisbett & Borgida, 1975).
More recent experimental work suggests that considerations of relevance underlie
performance in at least some reasoning paradigms (Sperber et al., 1995; Girotto et al.,
2001; Van der Henst et al., 2002). It is argued here that this finding has positive
implications for human rationality since the relevance theoretic comprehension
procedure is computationally efficient and well-adapted to the ostensive
communicative environment: it is a good example of bounded and adaptive rationality
in Gigerenzer’s terms (Gigerenzer and Todd, 1999), and, uniquely, it is a fast and
frugal satisficing heuristic which seeks optimal solutions.

A singularity: where actor network theory breaks down an actor network becomes visible by Héle`ne Mialet

Abstract In this article I propose we rethink the nature of the individual human
subject in a landscape where cognition has been distributed (Hutchins, Clark),
individuality has been transformed into associations between heterogeneous actors,
and human and non-human agency has been reconceived as a product of attribution
(Actor Network Theory). Reengaging with the material developed in my book
Hawking Incorporated, where I did an ethnographic study of Stephen Hawking, the
man and the persona, I will extend my original analysis to extract and map the
processes through which the individual human subject is constituted. Turning upside
down all the notions of which the ‘‘subject’’ is supposed to be made by exteriorizing,
materializing, collectivizing, and distributing, mind, body, and identity, I will
make visible the ramifications that constitute the subject through processes of distribution
and singularization. To the powerful myth of the ‘‘disincorporated brain,’’
I propose an antidote—the concept of the distributed-centered subject.

This is interesting – a highly descriptive ethnography of Stephen Hawking and his team in Actor Network Theory terms.