RePoSS #11: The Mathematics of Niels Henrik Abel: Continuation ...

1.4. Reflections on methodology 13
description of counter examples and the role that they play in mathematical change
elaborates the role played by anomalies in the Kuhnian model and suggests a more
refined view on the status of a mathematical theory in crisis.
The Lakatosian theory of mathematical evolution is present as background through-
out; it will surface sporadically in parts II–IV and become important again in the final,
more analytical part V.
M. EPPLE’S epistemic configurations. Quite recently, EPPLE has suggested the no-
tion of epistemic configurations in order to be able to discuss change in mathematics
in another context. 6 In EPPLE’S analysis, epistemic configurations consist of epistemic
objects and epistemic techniques and are manipulated in mathematical workshops. The
concept of epistemic objects encompasses the immaterial objects with which mathe-
matics deals. These are manipulated and investigated by a number of methods of
producing (or obtaining) mathematical knowledge; these methods are the epistemic
techniques. The precise applicability and range of EPPLE’S concepts and their use-
fulness in historical analysis is not the primary objective here. Instead, as with the
inspirations of KUHN and LAKATOS, I have taken the liberty of using EPPLE’S terms
to ease the analysis and discussion of what I believe to be a fundamental change in
mathematics in the early nineteenth century: the change from formula based to concept
based mathematics which is addressed in chapter 21.
1.4.3 Existing literature
Being one of the important mathematicians of the nineteenth century, ABEL’S person
and his mathematics have been subjected to study for a multitude of different rea-
sons. A few general trends of the literature on ABEL can profitably be identified at
this point. 7 At the relevant places in the subsequent parts, references are given to the
secondary literature which is listed in the bibliography.
ABEL in the history of mathematics literature. In the professional literature in the
history of mathematics, ABEL is often mentioned in order to illustrate one or more of
the following aspects:
1. ABEL’S life story is invoked to illustrate the conditions of young mathematicians
two centuries ago. This aspect is closely related to the biographies treated below.
2. ABEL’S letters from Paris are used to illuminate how the confrontation with
CAUCHY’S new rigor brought about a radical change. For instance, U. BOTTAZZ-
INI (⋆1947) quotes in extenso from these letters in his comprehensive account of
the evolution of analysis in the nineteenth century. 8
6 (Epple, 2000).
7 For a thematic listing of the ABEL literature, see also (Sørensen, 2002).
8 (Bottazzini, 1986).