[Warning: non-mathematician here, apologies if I'm trespassing, but this seemed like the right place to ask the question.]
I've heard it referred to many times (although I don't know if there's a single specific source) that in the nineteenth century, a single able mathematician could understand and engage in the totality of the subject, all sub-fields included (and if that was, perhaps, untrue by the end of that century, it was true at some point earlier). Clearly even well before the end of the twentieth century this was no longer possible. The scope, number and depth of sub-specialties that emerged in the twentieth century had no precedent in the history of math.
What caused the tree of mathematics to grow such a huge number of new branches in the twentieth century and at such speed? What I'm try to get at more specifically is whether it "just happened" or are there certain identifiable preconditions that were met by the end of the nineteenth century which enabled the rapid subsequent advances? Did Gauss, Riemann, Galois, Abel, Cauchy, to name but a few of the luminaries from the time, create a critical mass of discovery, lines of enquiry and tools which which made the twentieth century 'explosion' possible?