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Kjeldsen on Rheinberger via Epple April 28, 2009

Posted by Will Thomas in Methods.
Tags: , , , , ,

Continuing on last week’s discussion about the sufficiency of current methodology, I’d like to take a look at Tinne Hoff Kjeldsen’s piece, “Egg-Forms and Measure-Bodies: Different Mathematical Practices in the Early History of the Modern Theory of Convexity,” from the latest Science in Context (free issue!), and particularly the function of her invocation of Hans-Jörg Rheinberger’s array of “epistemic things”.

For those not familiar with the modern theory of convexity, fear not.  I’m mainly interested in the topic because it is central to the mathematical theory of linear programming, which is an important part of the canon of operations research techniques, and Kjeldsen, a historian of mathematics, is the top expert on the subject.  She has a long line of papers explaining how the rather discontinuous history of convexity theory can be understood in terms of its development as parts of mathematicians’ varying projects—what she has previously referred to as different “tasks”.  Her work is extremely useful to people like me who need to figure out what any of this has to do with military doctrine-building and radical British scientists—you’d be surprised—and are reluctant to spend too much time on the nitty-gritty details on the history of things like convexity theory.

The history of mathematics is a nice place to address this issue, because this history is relatively coherent from antiquity to the present in comparison to other fields of study.  As a consequence, historians of mathematics have found it to be more legitimate to address transhistorical mathematical problems as addressed across large gaps of time.  In venues such as the Archive for History of Exact Sciences, history maintains a sort of unusual purity where clearly internalist mathematical developments remain of interest in and of themselves.

Nevertheless, as many historians of mathematics have long noted, even mathematics does not proceed according to any sort of predetermined logic, and attention to context matters.  Back in 1980, for example, Henk Bos pointed out in the Roy Porter and G. S. Rousseau volume Ferment of Knowledge that readings of the history of 18th-century rational mechanics had depended crucially on what one took the history of rational mechanics to be about, emphasizing different achievements and individuals as they suited the project one was interested in.  One’s story could change considerably depending on whether one was interested in foundations or analysis, or, for that matter, whether one was interested in theories or styles.

In the piece at hand, Kjeldsen returns to an analysis of convexity in the hands of Hermann Minkowski and Karl Hermann Brunn.  For reasons Kjeldsen explains, Brunn’s work entailed a definition and non-analytical sytematic exploration of the properties of geometrical objects he called “egg-forms”.  While Brunn was interested in the relationship between these objects and ovals, his main concern was arriving at a more generalized understanding of the egg-form itself.  Minkowski, on the other hand, used convex geometries to analyze algebraic problems, and then later turned to analyze some of the more peculiar properties of his “measure-bodies”.  Kjeldsen’s work is far more detailed, of course, but our concerns here are methodological.

Kjeldsen discusses the relationship between Brunn’s project and Minkowski’s project by drawing on her fellow historian of mathematics Moritz Epple’s translation of Hans-Jörg Rheinberger’s discussions of “epistemic” objects (illustrated as physical apparatus) into mathematical “epistemic techniques”.  Rheinberger’s ideas relate to the ability of experimental appartus to generate enticing scientific problems of their own, which as Epple’s work and Kjeldsen’s discussion of Minkowski show, can also apply to more abstract tools.

I think Epple and Kjeldsen are correct in their adoption of Rheinberger’s terminology. You can do it; the question in my mind is: why would you want to?  Knowing Kjeldsen’s mastery of the history of convexity theory, I don’t think she needed the theory to understand her history.  Rather, I think it is a sort of sugar coating that makes a mightily abstruse article—which even I don’t really understand, and I have a small familiarity with the topic—palatable for publication in a general history journal.  Finding a recent general subject of methodological discussion to which the history can be fitted provides a simple way of mainstreaming discussions in the subdiscipline of the history of mathematics.  I can’t say if this is what Kjeldsen had in mind, but I know it’s done (heck, I’m currently trying to do it in a non-history-of-science journal, a point not unnoticed by one of my referees!  NB. I have a separate article coming out in SiC this summer of which I’m rather more proud).

My concern over this kind of practice is that it creates a false historiographical problematic, which promotes a fractured historiography.  The shifts between tool of research and problem of research that Kjeldsen describes do not require an unusually strenuous conceptual foundation—it is simply a matter of identifying, characterizing, and charting historical projects and problems.  While relating this to a seemingly fresher epistemological problematic might spotlight the work to a larger audience who might otherwise be inclined to take a pass, it also creates the impression that broad historiographical methodologies remain tentative and fragile, thereby promoting the continued accumulation of case study literature that tests and modifies a perpetually tentative and amorphous methodology.

In fact our methodologies are very robust, and their full potential has been realizable for some time.  When journal articles spotlight methodological significance or reference one of a handful of broad historiographical reference points, and downplay more directly relevant historiography, it masks the synthetic potential of a very rich and longstanding body of research.  Yet, methodological development preceding broad synthesis is, as I always argue, more rule than exception in the priorities of contemporary historiographical practice.



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