The first word which came to my mind is fission. It's perhaps most commonly used in the context of nuclear physics, but it doesn't need to be restricted to that context. It could go directly into your first example sentence:
After the fission of Western philosophy into these major directions, new thinkers began to emerge.
Looking through Google Books, I get the impression that it's a moderately popular word in the social sciences. E.g. in Encounters and Transformations: The Archaeology of Iberia in Transition we find sentences like
... incipient complexity led more often to social fragmentation or fission than to pristine state formation. I argue that too little attention has been devoted to fragmentation and fission and that more sophisticated models be developed to account for these alternate trajectories of social evolution.
Google Books also turned up the word in a number of titles or subtitles, often in opposition to fusion, as in the white paper Fission Or Fusion: What Kind of Commercial Culture Will Emerge in Southeast Asia?, but not always. An example without fusion is the book Household Strategies for Survival 1600-2000: Fission, Faction and Cooperation.
The British National Corpus turns up a couple of nice examples in natural science literature. From The Pacific by Simon Winchester:
The slow fission of Gondwanaland produced two oceans -- the Indian, where Africa, India and Antarctica were hauled away from each other; and the southern portion of the Atlantic, where South America and its clearly closely-fitting neighbour Africa (which possessed an uncanny coastal match first noted by Francis Bacon) sprang apart.
And from Exploring the night sky with binoculars, by the legendary Patrick Moore:
It used to be thought that a binary [star] resulted from the fission or breaking-up of a formerly single star which was spinning rapidly, and became unstable.