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

Chapter 14
Reception of ABEL’s contribution to
rigorization
In the course of the nineteenth century, analysis underwent an elaborate program of
rigorization which effected both the techniques, results, and questions of the dis-
cipline. Basic notions such as real numbers, continuous functions, integrals, and
trigonometric functions were revised and deeply changed as reflections of a funda-
mental transition in the ways mathematicians thought about their subject. The chang-
ing concepts and attitudes have been studied intensively by historians. 1 In the twen-
tieth century, N. H. ABEL’S (1802–1829) critical attitude and his part in the revision of
analysis have received some interest but in the first decades after his death, these were
not issues which attracted the most interest to his mathematics. The following section
briefly discusses the reception of ABEL’S work on rigorization and certain aspects of
the subsequent development.
14.1 Reception of ABEL’s rigorization
Because of the overwhelming development of analysis in the nineteenth century and
ABEL’S apparently limited direct impact, the reception of ABEL’S contribution to the
rigorization movement is only briefly described from two different perspectives. 2
14.1.1 Binomial theorem
The most immediate reaction to ABEL’S binomial paper was actually a non-reaction.
In 1829 and 1830, A. L. CRELLE (1780–1855) published two papers on the binomial theorem
demonstrating that the subject had not been closed by ABEL’S paper of 1826. 3
CRELLE’S proofs were based on his previous research on the so-called analytical facul-
1 See e.g. (Bottazzini, 1986; Hawkins, 1970).
2 I hope to subsequently substantiate the analysis of the reception of this part of ABEL’S research
through detailed studies of the works of selected, later mathematicians.
3 (A. L. Crelle, 1829a; A. L. Crelle, 1830).
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