[[Geometry]] has expanded to accommodate [[topology]]. The study of [[number]], called [[algebra]] at the beginning undergraduate level, extends to [[abstract algebra]] at a more advanced level; and the study of [[function (mathematics)|function]]s, called [[calculus]] at the Freshmancollege freshman level becomes [[mathematical analysis]] and [[functional analysis]] at a more advanced level. Each of these branches of more ''abstract'' mathematics have many sub-specialties, and there are in fact many connections between pure mathematics and applied mathematics disciplines. Undeniably, though, a steep rise in [[abstraction]] was seen mid-century.
In practice, however, these developments led to a sharp divergence from [[physics]], particularparticularly from 1950 to 1980. Later this was criticised, for example by [[Vladimir Arnold]], as too much [[Hilbert]], not enough [[Henri Poincaré|Poincaré]]. The point does not yet seem to be settled (unlike the foundational controversies over [[set theory]]), in that [[string theory]] pulls one way, while [[discrete mathematics]] pulls back towards proof as central.
