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.

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