Math II (Несколько текстов для зачёта), страница 5

I want here to pose the question differently and, specifically, to challenge the search for intrinsic markers of scientific drawing that would make it in some instances "private" and in others "public." As we learn from Jacques de Caso's essay on Theophile Bra, Bra's drawings surely cannot be understood as the expression of a purely interior or subjective sensibility. For example, at least one of Bra's cosmological sketches was clearly tied to his views of public discussion about changes in the structure of Saturn's rings; Bra even wrote to the French astronomer and optician, Dominique-Francois-Jean Arago, about the problem.(n12) Nor does the geometry of Dirac issue from an isolated form of reasoning. Dirac's fascination with projective geometry is anything but a private language in Ludwig Wittgenstein's sense--as we will see momentarily (fig. 3).

In both instances (Bra's cosmologies, Dirac's geometry) the drawings neither issue entirely from the public domain nor are they sourceless fountains from a reservoir of pure subjectivity. Tracking Bra's worldly iconological sources or Dirac's public sources in geometry would surely prove both possible and profitable. And yet there is something important in the circumstance that both Dirac and Bra constructed a domain of interiority around these practices. It is not that Dirac's geometric drawing or Bra's cosmogenic images were intrinsically interior or psychological--there is no separate logic here that could provide a universal demarcation criterion splitting the public from the private. Rather, both Dirac and Bra drew a line (so to speak) around their drawings. Both assiduously hid their pictures from the public gaze, and refused (in the case of Dirac) even to admit them into his published arguments. One suggestive concept helpful in capturing this delineation of the private might be Gilles Deleuze's notion of the fold. For Deleuze the "content" of what is infolded is not intrinsically separate from the exterior; there is no metaphysical otherness dividing inside from outside. Instead, interiority is itself the product of an outside pulled in, a process that Michel Foucault called subjectivation because it makes contingent, not inevitable, the formation of what is understood as self.(n13)

I want to push this notion of infolding or subjectivation in two directions. First, my concern here is with an aspect of the private that bears on the epistemic, rather than one that posits lines of individuation that separate a self from others and the world. That is, what interests me is the historical production of a kind of reason that comes to count as private (rather than, for example, the production of the psychological sense of self more generally).(n14) Second, building on this epistemic form of subjectivation, my concern is to explore the historical process by which this takes place. On such a view, the question shifts: How does a form of public inquiry and argument (geometry) come to count as private, cordoned-off.reason?

Public Geometry, Private Geometry

The issue, therefore, is not what makes the interior or the private metaphysically distinct from the exterior and public, but rather how this inbound folding occurs over time. How, in our instance, did projective geometry pass from the status of a state religion at the time of the French Revolution to become, for Dirac, a repressed form of knowledge production that must remain consummately private--that is, how was geometry infolded to become, for Dirac, quintessentially an interior form of reasoning? What are the conditions of visibility that govern its place (or suppression) in demonstration?

So a new set of questions displaces those with which I began. Not the philosophical-psychological question: How do interior rules of combination differ from exterior rules of combination? But rather: What are the specific conditions that govern the separation of certain practices from the public domain? Not: How, linguistically or psychologically, does public science get created by successive transformations of the private domain? But rather the inverse: How do the "private" structures of visibility (specifically in drawing) get pulled in from the public arena to form a domain aimed, in the first instance, at the inward regulation of thought (rather than outward communication)? Consequently what we have is not quite the Deleuzian question either--not the transhistorical elucidation of what he calls the topology of the fold, but rather the historical process of the folding itself. What happens, over time and across places, such that features of public demonstration become private forms of reasoning?

During the late eighteenth century, descriptive geometry (later known as projective geometry) was first heralded by Gaspard Monge, as preeminent mathematician, as political revolutionary; and as director of the Parisian Ecole polytechnique. As Lorraine Daston and Ken Alder have shown, Monge's texts and the Polytechnique curriculum more generally were all oriented toward the school's mission to train engineers.(n15) Descriptive geometry, the science of a mathematical characterization of three-dimensional objects in two-dimensional projections, was supposed to serve not only mathematicians and engineers but also the Polytechniciens who would become the nation's future high-level carpenters, stonecutters, architects, and military engineers.(n16) For a generation of Monge's successors--Polytechnicien engineers including Charles Dupin, Michel Chasles, and Jean-Victor Poncelet--descriptive geometry became much more than a useful tool. Geometry, they contended, would hold together reason and the world.

For Monge and his school, physical processes including projection, section, duality, and deformation became means of discovery, proof, and generalization. This physicalized geometry defined a new role for the engineer as an intermediary lodged between the state and the artisan. Geometrical, technical drawing, "the geometry of the workshop" became at one and the same time a way of organizing the component parts of complex machines and a scheme for structuring a social and workplace order.(n17) Geometry became a way of being as well as the proper way of founding a basis for mathematics. Indeed, at the Ecole polytechnique, geometry became an empirical science. Auguste Comte came to speak of an empirical mathematics, Lazare Carnot exploited physically motivated transformations in geometry and identified correlates between mathematical entities and their geometrical twins.

Geometry was practical and more than practical. Certainly for Dupin, Chasles, Poncelet, and their students, geometry towered above all other forms of knowledge as the paragon of well-grounded argumentation, better grounded, in particular, than algebra. Projective geometry came to stand at that particular place where engineering and reason crossed paths, and so provided a perfect site for pedagogy. As Monge insisted, projective geometry could play a central role in the "improvement" of the French working class--"Every Frenchman of sufficient intelligence" should learn it, and, more specifically, geometry would be of great value to "all workmen whose aim is to give bodies certain forms.(n18) Enthusiastically Henri Saint-Simon and his followers adopted the cause in their utopian planning. Descriptive geometers established classes across Paris, joined the geometrical cause to republicanism, and launched a wider commitment to worker education. In 1825, Dupin proclaimed in his textbook that geometry "is to develop, in industrials of all classes, and even in simple workers, the most precious faculties of intelligence, comparison, memory, reflection, judgment, and imagination.... It is to render their conduct more moral while impressing upon their minds the habits of reason and order that are the surest foundations of public peace and general happiness."(n19) Both before and after the French Revolution, geometry, as Alder notes, became the foundational skill in the training of workers--several thousand passed through the various popular art training programs. Geometry would teach both transferable skills crossing the trades and at the same time stabilize society by locking workers into the social roles previously occupied by fathers.(n20)

Geometry did not, however, survive with the elevated status it had held in France at the highwater mark of the Polytechniciens' dominance. Analysts displaced the geometers. Among their successors was Pierre Laplace, for whom pictures were anathema and algebra was dogma. It was not in France, therefore, but rather in Britain and Germany that educators, scientists, and even politicians took up the cause of descriptive geometry with the conjoint promise of epistemic and pedagogical improvement. So although the French mathematical establishment had turned decisively to analysis in the last third of the nineteenth century, the British did not. Euclid had long reigned over British education as an exemplar of good sense and a pillar of mental training. By 1870, however, there was a widespread and disquieting sense that the British were losing to the Continent in the race for science-based industry. Geometry was no exception. In January 1871, leading mathematicians of the British Association for the Advancement of Science joined a committee known as The Association for the Improvement of Geometrical Teaching. Their goal was to produce a reform geometry better suited to technical and scientific education, in a form less rigid than that demanded by the purer mathematicians and enforced on schools. New methods of geometrical argument were introduced, and teachers began to step away from the definitions, forms of argument, and order of theorems dictated by the historical Euclidean texts. Such a loosening of Euclid's hold over the schoolchild's mind did not go undisputed. By 1901 the reformers (aiming to join geometry to the practical arts) and conservatives (hoping to preserve its purity) had settled into such powerfully opposed camps that separation seemed inevitable.(n21)

These, then, were some of the nineteenth century's territories of geometry: Up until the 1860s or so, the French celebrated projective geometry as joining high reason with practical engagement of the working class; then this physicalized geometry faded from the scene. In Britain, accompanying the rapid expansion of industrial, technical education, Victorian descriptive geometry became the symbol and means of socio-educational uplift, improving the lot of young workers, including those of the working class. For the mathematician-logician Augustus De Morgan, for example, geometry was a route to knowledge in general--as he argued in 1868: "Geometry is intended, in education,... to [unmask] the tricks which reason plays on all but the cautious, plus the dangers arising out of caution itself."(n22)

Over the last decades of the nineteenth century, the teaching of geometry in Britain gradually moved away from a rigid Euclid-based textual tradition toward a more expansive interpretation of geometry's basis. In part this shift issued from the marketplace. No longer would it be adequate for the teaching of geometry to exemplify sound reasoning as an end utterly unto itself. Instead, geometry came to have a practical significance as well--crucial for the upbringing of engineers, the upper tier of tradesmen, and scientists. One widely distributed encyclopedia of technical education put it bluntly: "It is impossible to overstate the importance of a knowledge of Geometry forming as it does the basis of all mechanical and decorative arts, constituting, in fact, the grand highway from which the various branches of drawing diverge."(n23) At the same time, part of the freeing of geometry from its purely descriptive roots was an increasing emphasis by reformers on "modern" methods including, prominently, non-Euclidean and projective geometry of higher dimensions. Pressured by both practical and research exigencies, geometry came to illustrate sound reasoning not by being purely descriptive of an ideal world, but rather by instantiating a reason best captured by a multiplicity of approaches.(n24)

So much for the general historical condition of geometry as a very public epistemic ideal and educational method: as a defining feature first of republican and then working-class French pedagogy it continued into the 1870s and beyond in Germany, and re-emerged within the technical education movement of Victorian England. What, then, are the specific historical conditions under which drawing came to count for Dirac both as a reliable home of reason and as a "private" science, judged by him variously as too hard to print, too arcane for physicists to understand, insufficiently persuasive, or insufficiently concise to merit publication?

Dirac's trajectory in mathematical physics took him across several of geometry's territories, temporal-spatial regions where geometrical drawing was laid out differently from one to the next. The goal in following that arc is to see how it came to pass that what had been the most public of mathematical regimes could become, for Dirac as he moved across this shifting map of geometry's fortune, a most private refuge of thought. Here is an account that begins not with an assumed intrinsic dynamics of interior (psychological) style, but rather with the historical creation of a kind of science judged private: the epistemic subjectivation of the geometrical. This is, therefore, not so much an attempt to follow Dirac's biography, but rather to observe Dirac as a kind of movable marker in order to track the conditions under which reasoning through drawing came to be classed as something to be, in his word, "suppressed," interiorized, made to constitute the private scientific subject.

Zero in on Dirac as we turn from the generic Victorian British trade school to Dirac's secondary school, the Merchant Venturers' Technical College, in Bristol. This was where Dirac's father, Charles Dirac, taught, and where Dirac himself received his primary and secondary scientific-engineering education. Created out of various mergers of the Free Grammar and Writing School, the Merchant Venturers' Navigation School, and various forms of the Bristol Diocesan Trade and Mining School, Dirac's school had stabilized both its structure and name in 1894.(n25) Charles Dirac took his degree at the University of Geneva and then, in 1896, came to Merchant Venturers' where he pursued a long career teaching French. A feared figure on the faculty ("a scourge and a terror" according to some of the students), Charles Dirac clearly reveled in the disciplined teaching of language--especially French, but others too, including Esperanto.(n26) Dirac the younger often claimed that he simply stopped speaking to avoid having to perform at home in perfect, grammatically correct French. Dirac's wife put it this way: "His domineering father made it a rule to be spoken to only in French. Often he had to stay silent, because he was unable to express his needs in French. Having been forced to remain silent may have been the traumatic experience that made him a very silent man for life."(n27)