Chalk: Materials and Concepts in Mathematics Research by Michael J. Barany & Donald MacKenzie

There is, we contend, an essential relationship between the supposedly abstract concepts and methods of advanced mathematics and the material substituents and practices that constitute them. This process operates even in the rarefied realm of mathematical research, where the pretense of dealing purely in abstract, ideal, logical entities does not liberate mathematicians from their dependence on materially circumscribed forms of representation. p. 2

Observations that would be “old news” about other sciences or unsurprising to those acquainted with mathematical practice are nevertheless significant in a context where so few investigations of the sort we report here have been undertaken. p.2

Due to the obfuscations of temporal distance and conceptual difficulty, however, historians and sociologists of mathematics have struggled to account for the ongoing achievement of original knowledge in a research context, such as has been ventured for laboratory sciences. P. 3

Lurking in the seminar’s subtext and between the lines of multiple interviews was the open secret that mathematicians, even those in the same field, working on the same topics, or veterans of multiple mutual collaborations tend to have comparatively little idea of what each other does.14 Mathematics is a staggeringly fragmented discipline whose practitioners must master the art of communicating without co-understanding. Indeed, mathematicians seem persistently preoccupied with sharing their work with each other, boldly blinding themselves to the petty incommensurabilities of their studies in order to join, on scales ranging from meetings with collaborators to international congresses, in mutual mathematical activity. P. 4

They describe the limited preparation for a seminar and minimal notes normally taken.

Seminars are thus conditioned on a great deal of shared training in discursive and conceptual norms. P. 7

These references to people and concepts work to dissolve temporal as well as professional boundaries. In an interview, one junior researcher spoke undifferentiatedly of insights from a senior colleague gleaned, respectively, from a conversation the previous week and a body of that colleague’s work more than two decades old. So, too, do old and new theorems and approaches coexist in a seamless technical matrix on the seminar blackboard, thereby enacting an epistemology of mathematics that actively looks past concepts’ context-specifcities. P. 8

There are some notes about how collaborations and projects tend to work.

Every researcher interviewed had stories about conclusions that had either come apart in the attempt to formalize them or been found in error even after the paper had been drafted, submitted, or accepted. Most see writing-up as a process of verification as much as of presentation, even though the mathematical effort of writing-up is viewed as predominantly “technical,” and thus implicitly not an obstacle to the result’s ultimate correctness or insightfulness. P. 8-9

For presenters, presentations can drive the writing-up process by forcing the speaker to cast recent results in a narrative that can be used in both talks and papers, one that mobilizes both program and project to construct an intelligible account of their work (cf. Ochs and Jacoby 1997). Preparing a piece of work for public consumption requires the impartition of an explanatory public logic where ideas develop according to concrete and recognizable methods. Seminars force researchers to articulate their thinking in terms of a series of significant steps, unavoidably changing the thinking in the process by forcing it to conform to a publicly viable model or heuristic. Finally, the seminar’s audience joins in the constitution of a shared public logic that frames their own projects in turn.

Thus, the “following” that takes place in the seminar and extends to other areas of mathematical communication consists of more than the mere sequential comprehension of inscriptions and allusions. \Following” structures the production and intelligibility of entire programs of mathematical research, as well as of the communities that engage in those programs. P. 9

  1. 12 they discuss the features of blackboards and how these guide discourse

Thus spatialized, statements can be mobilized or demobilized by emphatic or obfuscatory gestures. P. 13

Not every trouble has a work-around. Similar to a ball-point pen or pencil on paper, chalk must be dragged along the board’s surface to leave a trace. Entrenched mathematical conventions from the era of fountain pens, such as “dotting” a letter to indicate a function’s derivative, stymie even experienced lecturers by forcing them to choose between a recognizable dotting gesture and the comparatively cumbersome strokes necessary to leave a visible dot on the board. P. 14

As Livingston (1986, 171) observes, mathematical proofs are not reducible to their stable records. Arguments are enacted and validated through their performative unfolding, an unfolding as absent from circulable mathematical texts as it is essential to the production and intelligibility of their arguments. P. 14

On such a medium, the fact that the once-written text does not tell the final story about a mathematical concept allows a potentially infinite variety of descriptions simultaneously to apply to an object or situation under consideration. Where Suchman (1990, 315) and Suchman and Trigg (1993, 160) depict the board as the medium for making objects concrete, we would stress the board’s corresponding ability to make those concepts mutable without threatening their persistence as Platonic entities. Where Suchman (1990, 315) and Suchman and Trigg (1993, 160) depict the board as the medium for making objects concrete, we would stress the board’s corresponding ability to make those concepts mutable without threatening their persistence as Platonic entities. Thus, when a speaker returns later to add a necessary condition to a definition or theorem-statement, it can be seen as an omission rather than an error in the speaker’s argument|the condition can be made to have been there all along at any such point as that anteriorized conceptual investment is required for the lecture to go forward. P. 15

On the other hand, blackboard writing seems supremely open to annotation, adaptation, and reconfiguration. Symbols and images can be erased, redrawn, layered, counterposed, and “worked out” on the board’s surface. Such “immobilized mutables” form a constitutive matrix for mathematical creativity. P. 18

Scrap paper writing shares many characteristics with chalk writing. Both rely on augmentations, annotations, and elisions as concepts are developed through iterated inscriptions designed to disrupt the formal stability of mathematical objects. P. 18-19

The time and thought required to understand and verify each such detail makes mathematical papers subject to similar issues of trust, credibility, and reproducibility as have been described for the natural sciences, but the presentation of mathematical texts as (in principle) self-contained means that their circulation and deployment can have a decidedly different character. P.22

They establish lemmas and theorems that can be invoked as settled relations between specific  mathematical phenomena, and they present methods and manipulations that can be used by others to establish different results. … The mathematicians with whom we spoke almost never read papers in their entirety-and certainly not with the goal of total comprehension. P. 22

This process of reformulating official papers into research instruments can span several media. A single page of one researcher’s notepad, shared during an interview, visibly manifested a series of translations from an article to penned equations to an email to spatial gestures and further writings. P. 23

Particular and ideosyncratic inscriptions and realizations are utterly central to the practice of mathematics. Paradoxically, mathematical inscriptions (especially on blackboards) work in ways that speci fically (and, as we have argued, misleadingly) assert the opposite|that ideas somehow do not depend on the ways in which they are mobilized. … Mathematical work rests on self-effacing technologies of representation that seem to succeed in removing themselves entirely from the picture at the decisive junctures of mathematical understanding. P. 23

In most people’s experience, mathematics is a static body of knowledge consisting of concepts and techniques that are the same now as they were when they were developed hundreds or thousands of years ago and are the same everywhere for their users and non-users alike. Little would suggest that there are corners of mathematics that are changing all the time, where as-yet unthinkable entities interact in a primordial soup of practices that constantly struggle to assert their intelligibility. Such is the realm and such are the objects of mathematical research. P. 24

In contrast to well-worn accounts of representation in the natural sciences, the story of mathematics is less about the hidden work of taming a natural phenomenon according to ideals than about the very public work of crafting those putatively independent ideals from their always-already-dispensable material manifestations. P. 24-5


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