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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 ?

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    $\begingroup$ I think Hardy himself gave some (really sad-looking) examples of English work on Fourier analysis in the 19th century. Unfortunately I can't remember the reference, but, going by memory, the examples suggested that, if anything, mentioning examples involving Cayley and Sylvester amounts to steelmanning 19th century English mathematics a little (whereas I suppose Hardy was doing the opposite). Also, textbooks were a mess; Littlewood gives an example in his miscellany on how it took a page not to define what a complex analytic function was. $\endgroup$ May 12 at 19:51
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    $\begingroup$ Is there anything helpful at HSM.SE? $\endgroup$
    – shoover
    May 13 at 15:17
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    $\begingroup$ Regarding German rigor, the NAS Memoir of R. Courant tells how K. Friedrichs "described the excitement he felt when as a young student he read Courant’s presentation of geometric function theory in Hurwitz-Courant: It is true that there were some passages in which matters of rigor were taken somewhat lightly, but the essence came through marvelously. I was reminded of this effect much later, when I heard Courant play some Beethoven piano sonata.... $\endgroup$
    – Firestone
    May 13 at 15:35
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    $\begingroup$ Could you explain what does your last phrase mean? Specifically, what does it have to do with rigour? $\endgroup$ May 14 at 17:01
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    $\begingroup$ Strange Hilbert's name has not been mentioned by anyone although he was born some 15 years before Hardy and would be the embodiment of what many would see as German rigor. $\endgroup$
    – Trunk
    May 17 at 21:41

6 Answers 6

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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.

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    $\begingroup$ Some points on this: 1) Even in the 1800s, many diverse applications could be found for a single math concept. To me, this would more indicate a primacy of a math concept over any physical phenomena manifesting that same concept than vice-versa. 2) Validation of a theorem via vindication of its predications in areas of application would surely demand a thorough proof, lest exceptions make predication/design (e.g. engineering) based on it open to failure. 3) The pace of math 'discovery' in the physical world is slower than the pace of application discovery after its math concept development. $\endgroup$
    – Trunk
    May 11 at 14:25
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    $\begingroup$ Carlo Beenakker, are there examples in this paper of unrigorous English mathematics? Based on your quote, I see at least two possible interpretations. 1. English mathematicians would give what we would consider to be incomplete proofs, but not regard them as being incomplete, and would make statements that we would consider to be imprecise but would regard them as being precise. 2. English mathematicians had the same understanding as we do of what constitutes a precise statement and a complete proof, but did not think it was always necessary to make precise statements and give complete proofs. $\endgroup$ May 11 at 23:25
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    $\begingroup$ @TimothyChow I think the second proposal is closer to the truth. In the examples I've given Sylvester and Cayley are aware that they've proved the smallest interesting case of a general phenomenon. They consider it clear that the pattern they observed will generalise, and make this claim. I presume they would have tested further cases before writing. But, in my reading, they know that their proof is not complete. $\endgroup$ May 12 at 7:58
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    $\begingroup$ @TimothyChow --- one example of "unrigorous English mathematics" which is discussed in this article is the law of the permanence of equivalent forms. I guess I can email the pdf of the article upon request (that should be "fair use"). $\endgroup$ May 12 at 12:49
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    $\begingroup$ Very interesting. Some of it reminds me of what Rota said [in Indiscrete Thoughts] about Alfred Young (of Young tableaux fame): "Alfred Young's style of mathematical writing has unfortunately gone out of fashion: it is based on the assumption that the reader is to be treated as a gentleman with a sound mathematical education, and gentlemen need not be told the lowly details of proofs. As a consequence, we have to figure out certain inferences for which Young omits any explanation out of respect for his readers." [Of course, those papers are notorious nowadays for their apparent opacity.] $\endgroup$
    – Todd Trimble
    May 12 at 13:46
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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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    $\begingroup$ It occurs to me that even today, people sometimes talk about "formal" manipulations (e.g., of infinite series or products) that are heuristic and not "rigorous." Given that the OP mentioned Ramanujan, perhaps this is the real question: Did Hardy complain about the lack of rigor in Ramanujan's arguments in a way that Hardy's predecessors might not have? And if so, was that because Hardy was influenced by Continental mathematicians? $\endgroup$ May 12 at 13:28
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    $\begingroup$ (+1) for the distinction between abstraction and rigor (in the sense of precision and logical correctness) as probably being especially relevant in the difference between mathematics in Britain and "the continent" during the 1800s. British mathematics during this time also seems to me a lot more focused on issues arising out of formal manipulations and topics more directly relevant for mathematical physics (of that time). $\endgroup$ May 12 at 17:22
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    $\begingroup$ Well, but Babbage was swimming against the current, wasn't he? $\endgroup$ May 12 at 19:59
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    $\begingroup$ @HAHelfgott He wasn't the only critic of infinitesimals. Perhaps the most famous critic, Berkeley, was Irish. See Florian Cajori's article Discussion of Fluxions: From Berkeley to Woodhouse for more historical information. $\endgroup$ May 12 at 22:01
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    $\begingroup$ Minor quibble: Lagrange was not French, but Italian. (His nationality put him in a precarious position during the French Revolution, when all foreign nationals were expelled from the country. However, the expulsion decree granted him a personal exemption. He later acquired the French citizenship in 1802.) $\endgroup$
    – Leo Moos
    May 12 at 22:11
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These examples relate to algebra rather than analysis, but might in any case be useful.

The papers of JJ Sylvester are full of deep insight and entertaining prose, but are also full of unsupported claims and partial proofs. He often writes that he is certain something is true but hasn't the time to demonstrate it. As a specific example in his paper on "Thoughts on Inverse Orthogonal matrices, ..." from 1856 he constructs the character tables of abelian groups and claims (incorrectly) that all Hadamard matrices are equivalent to one such.

As another example, the Cayley-Hamilton theorem was proved by Cayley for 2x2 matrices and by Hamilton for quaternions. At least in the case of Cayley, there was a (correct) statement that the result would generalize. If memory serves, Cayley's proof is a direct computation for 2x2 matrices; something which doesn't generalize particularly well.

Even into the twentieth century, this style of thinking persisted among older mathematicians. Thomas Muir gave a proof of Hadamard's inequality, which considered only minors of order 2 in a 4x4 matrix. Again, he asserted that this would generalise readily, though one of the reasons that he disapproved of Hadamard's original proof was that it relied on induction. So it's unclear what a satisfactory generalization would look like to him; though in fact he had proved the result to his own satisfaction.

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  • $\begingroup$ You paint a picture of a myriad of limited and low-order-only proofs and unsupported assertions available to mathematicians of the period. A very sparsely-runged ladder upon which to climb with surety to any major conclusion. Maybe the industrial revolution and resulting competition between UK, France and Germany impacted on academic as well as technological exactitude as each nation reverted to core cultural values in their drive for industrial supremacy. We don't know to what extent national imperatives affected academic outlook in those times, e.g. research funds from government. $\endgroup$
    – Trunk
    May 11 at 14:36
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    $\begingroup$ You seem to be approaching this discussion from a decidedly anachronistic viewpoint: aside from the fact that "research funds from government" simply didn't exist at the time, you are criticising 19th-century mathematics because it does not fit 21st-century criteria. $\endgroup$ May 11 at 15:12
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    $\begingroup$ Having asked for examples of how 19th-century mathematicians understood "proof" and "correctness", you're now arguing that they were wrong to hold those views. There's not much point in arguing with people who have been dead for 150 years. $\endgroup$ May 11 at 15:13
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    $\begingroup$ @Trunk I think that's an uncharitable interpretation of what I've written. These mathematicians were working before the development of modern techniques (and in fact the modern techniques were developed in part to refine their results and place them on a solid foundation). They were sophisticated mathematical thinkers with superb intuition for finding interesting and fundamental results. Their mathematical language was inadequate to describe what they perceived in some cases, and their intended audience could be expected to fill in an generalise arguments in others. $\endgroup$ May 12 at 8:01
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    $\begingroup$ "full of unsupported claims and partial proofs" — I think that this is quite general, not only for English mathematicians, but for pioneers in general. I guess that Poincaré's Analysis Situs and Italian school of algebraic geometry fall into this as well. $\endgroup$
    – Z. M
    May 12 at 18:53
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An example of Hardy's opinion may be found in his excoriating review of a textbook by Edwards on integral calculus:

Mr Edward's book may serve to remind us that the early nineteenth century is not yet dead. He directs our attention to "the admirable and exhaustive works of Legendre, Laplace, Lacroix, Jacobi, Serret, Bertrand, Todhunter, etc."; from which he has learnt, for example, that "a limit may be of finite, infinite, or indeterminate value," that "the processes of integration are necessarily of a tentative nature," and that any convergent series may be integrated term by term. Two proofs are offered of the last proposition. In the first it is stated to be valid "provided the series V itself, and the series V formed by the integrations of the separate terms, are both absolutely convergent." Mr. Edwards italicises the last condition, but we have no idea why it is inserted, for there is no pretence of making any use of it, nor is its meaning explained.

It is difficult for a reviewer to know what to say about such a book, except that it cannot be treated as a serious contribution to analysis. Twenty years ago it might have been necessary to establish the point in detail; it would be a waste of time now, when the battle for accuracy has been won. There is always the danger, however, that a student who reads a textbook may suppose that the statements which it contains are true. We should therefore state explicitly that the "general theorems" asserted in this book are often false, and that, even when they are true, the arguments by which they are supported are generally invalid.

One ought, of course, to judge a book by a different standard, as a storehouse of formulae useful for instructional purposes. Of such there is an abundance, including a good many which are seldom found in other books, and often entertaining or even important. We may mention Catalan's formula for the surface of an ellipsoid, results concerning roulettes and glissettes, the theorems of Fagnano, Burstall, Graves, MacCullagh, Schulz, and others. The book, in short, may be useful to a sufficiently sophisticated teacher, provided he is careful not to allow it to pass into his pupil's hands.

In defence of Edwards, any lack of rigour didn't seem to have been an obstacle to users such as Dirac.

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    $\begingroup$ Off-topic for the original question, but I really like the final sentence of the quote. I have said something similar to undergraduates when asked "why do we need to study e.g. $\epsilon-\delta$ proofs". My answer is that the modern way of thinking about analysis allows mere mortals like me to reason about complicated mathematics, and prove new things etc. Hardy's point is that the alternative seems to be to rely upon intuition, or physical reasoning, which is okay if you are Dirac, but doesn't exactly scale... $\endgroup$ May 14 at 11:06
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The following subsection of Hardy's Divergent Series (zbMath link, freely available on Archive.org) occupies pages 18-20, and gives his thoughts on the matter and some citations. In particular he gives the example of Transactions of the Cambridge Philosophical Society, Vol. 8 (Archive.org link) (see bolded text below). I've taken some text out for brevity, but the whole section (and every part of the book I have read) is a joy to read.

1.7. A note on the British analysts of the early nineteenth century. We end this chapter with a few remarks about British work on these subjects during the years 1840-50, which has been analysed very carefully by Burkhardt in the article from which we have quoted. It was a long time before the writings of the great continental analysts were understood in England, and these British writings show a singular and often entertaining mixture of occasional shrewdness and fundamental incompetence.

(1) The dominant school was that of the Cambridge 'symbolists', Woodhouse, Peacock, D. F. Gregory, and others. They represented what may be described as the ' $f(D)$ ' school of analysis. They started from 'algebra', and had something of the spirit, though nothing of the accuracy, of tho modern abstract algebraists. They dealt in 'general symbols', on which operations were to be performed in accordance with certain laws: 'the symbols are unlimited, both in value and in representation; the operations upon them, whatever they may be, are possible in all cases; ...' But the foundations of their symbolism were both inelastic and inaccurate. 'They insisted on a parallelism between 'arithmetical' and 'general' algebra so rigid that, if it could be maintained, it would effectively destroy the generality; and they never seem to have realized fully that a formula true with one interpretation of its symbols is quite likely to be false with another. They were also very much at the mercy of catchwords like 'what is true up to the limit...', and it is not surprising that their permanent contribution to analysis should have been negligible.

[...]

(2) There is one volume of the Transactions of the Cambridge Philosophical Society (vol. 8, published in 1849 and covering the period 1844-9) which contains a very singular mixture of analytical papers and gives a particularly good picture of the British analysis of the time. It contains Stokes's famous paper ' On the critical values of the sums of periodic series', in which 'uniform convergence' appears first in print; papers by S. Earnshaw and J. R. Young which are little more than nonsense; and a long and interesting paper by de Morgan on divergent series, a remarkable mixture of acuteness and confusion.

[...]

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    $\begingroup$ Thanks for this excellent citation. Perhaps the main reason that "Hardy is credited with reforming British mathematics by bringing rigour into it" is that Hardy criticized his predecessors so harshly. Reading more of this section of Divergent Series, I see that one of his main criticisms is that earlier mathematicians had a "disinclination to give formal definitions" and that "mathematicians before Cauchy asked not 'How shall we define $1-1+1-\cdots$?' but 'What is $1-1+1-\cdots$?'" Hardy's account of divergent series does indeed seem to be much more rigorous than previous accounts. $\endgroup$ May 13 at 18:05
  • $\begingroup$ You're welcome; I didn't think to connect it with Cauchy; I didn't realise this was roughly the same time period! $\endgroup$ May 14 at 12:01
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The quote in question contrasts Hardy’s rigor with “the hydrodynamics that was an important part of Cambridge mathematics.”

So to understand Hardy’s role, it makes sense to look at that hydrodynamics, e.g. Alfred Basset’s “Treatise on Hydrodynamics, with Numerous Examples”, conveniently available online at the Internet Archive. (Since Basset studied math at Cambridge, did his math afterwards without any professorship, and published multiple editions of this book with Cambridge University Press, I think it’s fair to call this “Cambridge math”; one could probably find this or similar books listed in Cambridge math syllabi too.)

Browsing through the book, I do not see a lot of proofs; I see calculations from mathematical hypotheses deemed appropriate for various physical problems.

So Hardy brought his rigor to pure mathematics; what came before him in England was less about pure mathematics done unrigorously, and more about applied mathematics done under different constraints.

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  • $\begingroup$ Thank you for your succinct response. $\endgroup$
    – Trunk
    May 13 at 18:42

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