Paul Erdős (1913-1996) is the case where the discovery specialty produced integration infrastructure for itself, because no institutional form of its era could contain it. Erdős was born in Budapest to Jewish mathematician parents who home-schooled him through his early years. (The family had lost two older daughters to scarlet fever shortly before Erdős's birth, and his parents kept him home in consequence.) He was doing original mathematics in his teens. He earned his doctorate from the University of Budapest at twenty-one. He fled Hungary during the rise of fascism in the 1930s, spent time at Princeton and at a series of temporary positions across the United States and Europe, and by the 1950s had begun the specific pattern of existence that would define the second half of his life: he owned no permanent home, kept almost no possessions, lived out of two suitcases, and traveled continuously between the homes of fellow mathematicians around the world, arriving unannounced or on minimal notice, working furiously on collaborative papers for days or weeks at a time, and then moving on.
Erdős published somewhere in the neighborhood of 1,500 mathematical papers during his career, more than any other mathematician in history. He collaborated with approximately 500 other mathematicians, which is why the concept of an Erdős number — the collaborative distance between a mathematician and Erdős, calculated along the graph of who had co-authored papers with whom — became a cultural feature of the late twentieth-century mathematical community. The Erdős number of two is the standard informal badge of mathematical standing in most subfields. The Erdős number of three is almost universal. What Erdős had built, without intending to build it and without writing any formal description of what he was doing, was a distributed research institution that had no headquarters and no funding source and no hierarchy. The institution was Erdős's itinerant presence. He showed up at your house. You had to feed him — he expected three meals a day of simple food and large amounts of coffee, invoking the famous remark (originally attributed to his friend Alfréd Rényi) that "a mathematician is a machine for turning coffee into theorems." You had to give him a desk. You worked on whatever problem he wanted to work on while he was there, and when he left, you had one or more joint papers in various stages of completion. Erdős's solution to the integration problem was distributed integration. He found five hundred partners and maintained rolling relationships with all of them. The network was the institution.