Srinivasa Ramanujan (1887-1920) is the case that tests the chapter's argument at its hardest edge, because his discovery specialty arrived from outside the entire Western credentialing apparatus, in a form the Western apparatus did not know how to recognize, and because his death at thirty-two cut short a career whose completed output already constitutes one of the most extraordinary contributions to pure mathematics in the historical record. Ramanujan was born in 1887 in a Tamil Brahmin family in the town of Erode in southern India. His family was poor. His formal mathematical education ended at roughly the secondary-school level — not because he ran out of talent, but because he became so obsessed with mathematics that he failed every other subject at the Government Arts College at Kumbakonam and lost his scholarship. He spent his late teens and early twenties working through the mathematics textbooks he could find — most notably G. S. Carr's 1886 A Synopsis of Elementary Results in Pure and Applied Mathematics, a British cram book for undergraduates preparing for the Cambridge Tripos — and filling notebooks with theorems he derived independently, without the formal proofs that European mathematical practice required, and often without any way of knowing whether the theorems had already been proved by other mathematicians or were entirely new to the literature.
The notebooks Ramanujan filled between approximately 1903 and 1914 contain somewhere between 3,900 and 4,000 individual mathematical results. Some had been proved before by European mathematicians, and Ramanujan had independently derived them via routes nobody else had taken. Some were new and correct. Some were new and partially correct — conjectures that could be sharpened by later mathematicians into fully proved theorems. Some were wrong. The proportion of new and correct results in the notebooks is extraordinary, and the specific character of those results is distinctive enough that subsequent mathematicians have spent a century mining the notebooks for material and are still finding things in them. Ramanujan's work in modular forms, partition theory, infinite series, continued fractions, and what would later be called mock theta functions has shaped branches of contemporary mathematics that did not exist during his lifetime and that developed in part because his notebook entries pointed to territory nobody else had seen.
In January 1913, Ramanujan wrote a letter to G. H. Hardy at Trinity College, Cambridge, enclosing a sample of his theorems and asking whether Hardy would be willing to look at the work. Hardy received the letter, looked at the theorems, and initially assumed they were either plagiarized from existing results or produced by a crank. He showed them to his Trinity colleague J. E. Littlewood. Hardy and Littlewood spent an evening working through the sample and concluded, as Hardy later wrote, that the theorems "must be true, because if they were not true, no one would have the imagination to invent them." Hardy wrote back, and then began the two-year effort required to bring Ramanujan from Madras to Cambridge — an effort that involved overcoming Ramanujan's Brahmin religious objections to crossing the ocean, persuading the Indian colonial administration to support the journey, and building the institutional arrangements that would let Ramanujan work inside the Cambridge mathematical community as a legitimate Fellow rather than as an exotic visitor.
What followed is one of the most lopsided and productive mathematical collaborations in the historical record. Between 1914 and 1919, Hardy and Ramanujan produced a series of joint papers on partition theory, number theory, and analytic methods that established Ramanujan as a first-rank mathematician in the Western tradition and that changed the landscape of several subfields. Hardy provided the integration work: the formal proof apparatus, the translation of Ramanujan's intuitive results into the notational and argumentative form the Cambridge mathematical community could receive, the institutional standing, the election of Ramanujan to the Royal Society (he was the second Indian ever elected, and at thirty-one was one of the youngest Fellows in the Society's history), and the access to journals and colleagues who could evaluate and extend the work. Ramanujan provided the discovery work: the torrent of original results whose structure Hardy himself could not have produced but which Hardy had the specialized mathematical equipment to formalize once Ramanujan had extracted them.
The collaboration ended when Ramanujan's health collapsed — probably from the combination of tuberculosis, severe dietary problems during wartime Cambridge (he was a strict vegetarian, the wartime rationing was inadequate to his specific needs, and he ate too little for years), and the specific stress of operating inside an institutional culture that was often racially hostile to him and whose food, climate, and social norms he never fully adapted to. He returned to India in 1919 and died the following year at thirty-two. The Lost Notebook, discovered by George Andrews in the Trinity College library in 1976, contains additional results Ramanujan wrote in the final year of his life, many of which anticipated mathematical developments that would not appear elsewhere until the late twentieth century. Bruce Berndt and several generations of subsequent mathematicians have spent decades editing and proving the contents of both the main notebooks and the Lost Notebook, and the editing project is not finished. The discovery specialty produced more than an integration community could absorb in a single human lifetime.