# The habit of definition

G. H. Hardy wrote (apropos of the task of assigning values to divergent series):

It is plain that the first step towards such an interpretation must be some definition, or definitions, of the 'sum' of an infinite series, more widely applicable than the classical definition of Cauchy. This remark is trivial now: it was not a triviality even to the greatest mathematicians of the 18th century. They had not the habit of definition: it was not natural to them to say, in so many words, `by $$X$$ we mean $$Y$$'. There are reservations to be made ... but it is broadly true to say that mathematicians before Cauchy asked not 'How shall we define 1-1+1-…?' but 'What is 1-1+1-…?', and that this habit of mind led them into unnecessary perplexities and controversies which were often really verbal.

Are there good accounts of the process by which the mathematical community came to adopt the habit of definition? Also, did Cauchy have a fully modern attitude? For instance, did Cauchy ever apply his definition to conclude that such-and-such a series does not converge?

• Cauchy was quite fierce in his criticism of earlier authors including Lagrange for using what Cauchy referred to as the "generality of algebra", which includes (but is not limited to) what we would call analytic continuation today. Commented Dec 21, 2023 at 15:13
• Thanks, Mikhail! I’d forgotten about “generality of algebra”. It gets mentioned in en.m.wikipedia.org/wiki/Principle_of_permanence Commented Dec 21, 2023 at 15:51
• As regards the last sentence of my question, Katz provides the answer: he quotes Cauchy as saying (after Cauchy defines “convergent”) “… On the contrary, if, as n increases indefinitely, the sum … does not approach any fixed limit, the series will be divergent and will not have a sum” [Katz, p. 712]. No agnosticism there. Commented Dec 21, 2023 at 16:01
• Note that this is a different Katz. Commented Dec 21, 2023 at 16:05
• did Cauchy ever apply his definition to conclude that such-and-such a series does not converge? --- Definitely YES. See [1] (Cauchy reference and annotation for it) in this HS & M answer. Commented Dec 21, 2023 at 21:14