Unrigorous British Mathematics Before G.H. Hardy

Question

I was looking at a bio-movie of Ramanujan last night. Very poignant.
Also impressed by Jeremy Irons' portrayal of G.H. Hardy.

In G.H. Hardy's wiki page, we read:

. . . "Hardy cited as his most important influence his independent study of Cours d'analyse de l'École Polytechnique by the French mathematician Camille Jordan, through which he became acquainted with the more precise mathematics tradition in continental Europe."

and

. . . "Hardy is credited with reforming British mathematics by bringing rigour into it, which was previously a characteristic of French, Swiss and German mathematics. British mathematicians had remained largely in the tradition of applied mathematics, in thrall to the reputation of Isaac Newton (see Cambridge Mathematical Tripos). Hardy was more in tune with the cours d'analyse methods dominant in France, and aggressively promoted his conception of pure mathematics, in particular against the hydrodynamics that was an important part of Cambridge mathematics."

Are we to understand from this that up to the late 1800s, British mathematics used only partial or inductive proofs or what ?

On the face of it, this would have been quite a state of affairs.

What exactly – in general or by a specific example – did Hardy bring to mathematics by way of rigour that had previously been absent ?

If someone introduced a new and sketchily proven theorem in the days of Hardy's childhood – and we are talking about Victorian times here (…) – then surely all the mean old men of the profession would have been disapproving of it and would obstruct its publication ?

Answer 1

Rigor and Clarity: Foundations of Mathematics in France and England, 1800-1840 explains in some detail how British mathematicians in the early 19th century viewed the role of rigor in the formulation and proof of mathematical theorems.

Rigor is now accepted as a universal good in mathematics. The differences between the French and the English at the turn of the century indicate that this was not always the case. [...] For Cauchy mathematical rigor was achieved when mathematical terms were defined unambiguously, so that they could be confidently used in subsequent proofs. The English did not agree that the essence of mathematics was captured in the abstract notion of rigor advocated by Cauchy and his school.

For the nineteenth-century English, mathematical theorems, no matter how beautifully proved, did not stand alone. Their validity lay in the concepts they illuminated; these concepts existed independently of the systems describing them. In this view mathematics was not created, it was discovered, and the value of the discovery lay in the understandings it generated rather than in the mathematical structure itself.

The English constructed for the subject a conceptual foundation that they found both strong and appropriate. Rigor as Cauchy and his followers understood it failed to capture the true spirit of legitimate mathematical development. The English would have agreed with the French that mathematics must be exact, but for them exactitude concerned the fit of mathematical definition to underlying concept, rather than precision in use. This way of seeing the issue supported an English style, just as Cauchy's notions of rigor came to support a French style, throughout the century.