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

Chapter 10
Toward rigorization of analysis
From the time the calculus emerged in the 17 th century until the end of the 18 th cen-
tury, mathematicians and philosophers were wary when confronted with questions
concerning its foundations. To some extent ignoring foundational questions, mathe-
maticians focused on creating new results which could be useful in answering inter-
esting questions, for instance in the field of mathematical physics. To some mathe-
maticians working toward the end of the 18 th century, rigorously founding the calcu-
lus remained one of the few open problems; but one of relatively lesser importance
than the development of new analytical results. 1 To others, primarily J. L. LAGRANGE
(1736–1813), the foundations of the calculus became a prestigious mathematical research
problem. 2
The transformation of concepts, theorems, and proofs in the process of rigorization
in analysis have been subject to a variety of historical enquiries; in the following, em-
phasis is given to establishing and illustrating certain ideas and developments which
are of importance in subsequent chapters. 3
10.1 EULER’s vision of analysis
To understand the revision and the contents of the refocus on rigor, some aspects of
eighteenth century analysis are of key importance. In particular, the results and tech-
niques of L. EULER (1707–1783) dominated the way mathematicians worked in the
field for half a century.
Focus on functions and formal equality. Beginning with his influential monograph
Introductio in analysin infinitorum, 4 EULER promoted functions to become the basic ob-
1 See for instance the quotations in section 3.3 frequently invoked to document a belief in the stagnation
of the mathematical sciences.
2 For LAGRANGE’S algebraic approach to the calculus, see e.g. (Grabiner, 1990); for its influence on
CAUCHY, see (Grabiner, 1981b). The best general presentation of the development of analysis in the
nineteenth century is, I think, (Bottazzini, 1986).
3 For the evolution of rigorization in analysis, see e.g. (Bottazzini, 1986; Jahnke, 1999; Lützen, 1999).
4 (L. Euler, 1748).
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