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Hardy’s famous correspondence with Ramanujan highlights an interesting aspect of the mathematics discipline; the appreciation of a mathematical expression prior to knowing its validity.

Why/how was Hardy impressed with Ramanujan’s early work on continued fractions if he didn’t even know they were true?

There is the proclamation by Hardy:

Must be true, because, if they were not true, no one would have the imagination to invent them

— G. H. Hardy, 1913

...which suggests there is some aesthetic to the expression that is noticed prior to any proof. What is this aesthetic? Is it the apparent symmetry? Some immediate familiarity to other known expressions?

Or is it simply plugging in numbers into the RHS and being able to immediately validate that it “must/should” be true?

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  • $\begingroup$ Your question presupposes that to know a proof of vailidity of something is the same as to know it is true, which is not generally the case. $\endgroup$
    – Lee Mosher
    Mar 23, 2021 at 14:15
  • $\begingroup$ Correct, but then what does “to know it is true” mean? $\endgroup$
    – Cybernetic
    Mar 23, 2021 at 14:16
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    $\begingroup$ Analogy: sometimes you know that a movie will be great (not necessarily a hit though) even before watching it. Watching it only confirms your belief. $\endgroup$
    – Paramanand Singh
    Mar 23, 2021 at 14:33
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    $\begingroup$ You should also study the foreword of A Mathematician's Apology which discusses som of Hardy's thought process when he received Ramanujan's letter. $\endgroup$
    – Paramanand Singh
    Mar 23, 2021 at 15:22
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    $\begingroup$ There is no need of continued fraction and Ramanujan summation tag. Please remove them. I have added soft-question tag. $\endgroup$
    – Paramanand Singh
    Mar 24, 2021 at 3:08

2 Answers 2

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I quote a few excerpts from the foreword (by C. P. Snow) to A Mathematician's Apology (by G. H. Hardy) (emphasis in italics mine) :

One morning early in 1913, he found among the letters on his breakfast table, a large untidy envelope decorated with Indian stamps. When he opened it, he found sheets of paper by no means fresh, on which, in a non-English holograph, were line after line of symbols. Hardy glanced at them without enthusiasm.

So Hardy felt, more than anything, bored. He glanced at the letter, written in halting English, signed by an unknown Indian, asking him to give an opinion of these mathematical discoveries. The script appeared to consist of theorems, most of them wild or fantastic looking, one or two already well-known, laid out as though they were original. There were no proofs of any kind. Hardy was not only bored, but irritated. It seemed like a curious kind of fraud.

That particular day, though, while the timetable wasn’t altered, internally the things were not going according to plan. At the back of his mind, getting in the way of his complete pleasure in his game, the Indian manuscript nagged away. Wild theorems. Theorems such as he had never seen before, nor imagined. A fraud of genius? A question was forming in his mind. As it was Hardy’s mind, the question was forming itself with epigrammatic clarity: is a fraud of genius more probable than an unknown mathematician of genius? Clearly the answer was no. Back in his rooms in Trinity, he had another look at the script. He sent word to Littlewood that they must have a discussion after hall.

Anyway, by nine o’clock or so they were in one of Hardy’s rooms, with the manuscript stretched out in front of them. Apparently it did not take them long. Before midnight they knew, and knew for certain. The writer of these manuscripts was a man of genius. That was as much as they could judge, that night.

Fron the above it should be clear that at first Hardy got really bored and irritated with Ramanujan's letter. But somehow the theorems of the letter got registered in his mind and this was probably due to extreme nature of the theorems (wild, not seen before, couldn't be imagined by Hardy).

When Hardy and Littlewood met together they probably tried to verify some of the theorems and concluded that there was some genuine mathematics involved. In particular the theorems on Rogers-Ramanujan continued fraction were the most difficult. Hardy remarked that "they defeated him completely". The quote by Hardy in your question is also related to the same Rogers-Ramanujan continued fraction formulas.

I have shared some of my thoughts on Ramanujan in a blog post and I have mentioned there something about the nature of Ramanujan's formulas:

  • meaning of the formula could be understood by anyone with basic knowledge of algebra and calculus
  • focus was on special cases of general formulas with actual numbers rather than general formula itself
  • minimal use of symbolism and wherever possible indicate a pattern by exhibiting it numerically or by writing about the pattern in English rather than describing pattern via a formula
  • each formula had a certain unexpectedness providing a shock treatment to the reader
  • most of the formulas had deep theories behind them
  • avoiding use of $\sum$ and $\prod$ symbols to represent infinite series and products

I think these qualities are reasonable enough to impress some people. It is also well known that Ramanujan sent the same letter to H. F. Baker and E. W. Hobson and they did not give any feedback. So probably the formulas couldn't impress everyone.

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See Robert Kanigel's book, The Man Who Knew Infinity, in particular the chapter "I Beg to Introduce Myself...". There is a decent amount of detail there about Hardy and Littlewood's process of investigation. Here are some excerpts:

The first wasn't new to Hardy, who recognized it as going back to a mathematician named Bauer. The second seemed a little different... Hardy and others would show how these series were derived from a class of functions called hypergeometric series...

... they struck [Hardy] as "much more intriguing, and it soon became obivous that Ramanujan must possess much more general theorems and was keeping a great deal up his sleeve." Some theorems in Ramanujan's letter, of course, did look comfortably familiar... Hardy has proved theorems like it, had even offered a similar one as a mathematical question in the Eudcation Times fourteen years before. Some of Ramanujan's formulas actually went back to the days of Laplace and Jacobi a century before.

[Hardy and Littlewood] began to reach a judgement... "There is always more in one of Ramanujans formulae than meets the eye... In some the interest lies very deep... but there is not one which is nor curious and entertaining."

So we can chalk up the decision to not classify Ramanujan as one of the many cranks that used to send letters as part familiarity with some of the results, part entertainment value of the results, and partially that there seemed to be more general results, leading to curiosity. You can find more details in the book I mentioned.

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