The excellent answers by Carlo Beenakker and Padraig Ó Catháin have inspired me to do some reading, and I have come to the understanding that the contrast between English and Continental mathematics alluded to in Trunk's quote was not solely one of "rigor," at least not in the sense in which I understand the word "rigor." To me, rigor has to do with the *precision* of one's definitions and statements and the *logical correctness* of one's proofs. On the other hand, it seems that the debates during the time period in question had at least as much to do with what I would call the role of *abstraction* in mathematics as with rigor per se.

Joan Richards, in the paper cited by Beenakker, gives as an example Lagrange's approach to developing calculus on the basis of Taylor series rather than on infinitesimals. (Since Lagrange was French and Taylor was English, this already poses some challenges to any attempt to draw a sharp line between English mathematics and French mathematics, but never mind that for now.) According to Richards, Lagrange was motivated in part by an attempt to avoid the perceived lack of rigor associated with infinitesimals. On the other hand, if you take a Taylor series approach, then there arises the question of rigorously proving the existence and uniqueness of a Taylor series. From a modern point of view, one could argue that if one is simply developing a theory of *formal* power series, then the manipulation of Taylor series is perfectly *rigorous*; but then that raises the question of whether mathematics should be concerned with studying pure abstractions or whether it needs to stay connected to concrete examples and applications to physics.

Richards tries to argue that French mathematicians were more comfortable with working with formal abstractions independently of any connection to concrete examples, whereas English mathematicians were more concerned with having their feet firmly planted on the ground of concrete examples. Another example she cites is De Morgan's account of Cauchy's theory of limits. It is well known that Cauchy's definition of limits was a major step toward making calculus rigorous by modern standards. De Morgan, who was British, clearly had high regard for Cauchy's work and was trying to encourage others to study Cauchy, but Richards emphasizes that De Morgan took great pains to elaborate Cauchy's terse abstractions with plenty of concrete examples. Once again this seems to me a difference not in rigor, but in one's attitude toward "abstract" versus "concrete." Similarly, Babbage (who was English) criticized the obscurity of infinitesimals, and so does not support a narrative that English mathematicians were unconcerned with rigor (again, as I understand the term "rigor").

Another interesting example cited by Richards is George Peacock. Peacock promoted something he called the "principle of the permanence of equivalent forms." According to Richards, Peacock "baldly asserted the legitimacy of
generalizing from the truth of specific arithmetic forms, through the truth of their
symbolical counterparts, to the truth of their algebraic forms. So, for example, it
allowed one to pass from
$$5^2 - 3^2 = (5 + 3)(5 - 3)$$
and other similar cases, through the generalization that
$$a^2 - b^2 = (a + b)(a - b), \quad\text{when $a$ is greater than $b$}$$
to the even more general statement that
$$a^2 - b^2 = (a + b)(a - b), \quad\text{whatever the values of $a$ and $b$.}$$
The truth of this final statement is meaningless in the arithmetic of counting numbers
from whence it was generalized, because negative numbers do not exist there. Its
validity, therefore, rested squarely on Peacock's quasi-inductive principle. Peacock's
principle is the epitome of the inductive approach that Cauchy derided in his 1821
*Cours*." At first glance, this does seem to support the claim that English mathematicians such as Peacock were not so concerned with rigor, and perhaps goes some way toward explaining the examples from Sylvester and Cayley that Padraig Ó Catháin cited.
The following passage from George Peacock and the British origins of symbolical algebra, by Helena M. Pycior, also seems to suggest, at first glance, that Peacock himself regarded his work as lacking the rigor of arithmetic and Euclidean geometry:

Arithmetic and geometry, he argued, were based upon axioms which were "necessary and self-evident consequences
of the definitions"; in symbolical algebra, however, there were
"properly speaking, no axioms, since the propositions, immediately deducible from the definitions and assumptions, must be
considered rather as the necessary and immediate consequences
of defined operations, than the necessary and self-evident results of reasoning." … He explained, "We are supposed to be in possession of a science
of arithmetical algebra … whose laws of combination are capable of strict demonstration, without the aid of any principle
which is not furnished by our knowledge of common arithmetic." Thus, according to Peacock, arithmetic and
geometry were based upon self-evident, necessary axioms from
which were derived the laws of these two sciences. Symbolical
algebra, on the other hand, was not based on self-evident axioms,
but on defined operations.

However, I think there is an alternative reading possible, which (in modern language) would be that Peacock was interested in pursuing the formal consequences of the axioms for a ring or a field, and distinguishing the nature of these "axioms" from the axioms of Euclidean geometry or Peano arithmetic. On this reading, Peacock was not being unrigorous (and ironically, he would be an English mathematician who was comfortable with reasoning in the abstract without worrying too much about the connection with the concrete).

To sum up, it seems to me that although there seem to have been some differences in philosophy between English and Continental mathematics, it doesn't seem that it can be neatly summarized in terms of differing attitudes toward "rigor" as we understand that term today. This is not to say that Hardy did not have any influence on the philosophy of English mathematics, but I suspect it was more nuanced than simply "bringing rigour" to England from the Continent.

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