While there are many factors as to the resurgence of Paris, the main focus is that of its resurgence is due to the only other person I've endlessly written about, Louis XIV.

As a result of the poor leadership of his father, Louis XIII, the French nobility had effectively rebelled in an event called The Fronde. This was a somewhat civil war resulting from issues with old "feudal" rights that were being trampled by the king that resulted from the centralization during and after the Hundred Years War. This war would have a major impact on the young Louis XIV, causing him to have an interest in culling and twisting the nobility towards his will.

As a result, he looked to pull the French Crown away from Paris and to his newly created palace, Versailles. While this should be a problem for Paris, it still helps since it requires the nobility to not only have a residence in Versailles or Paris, it would require the nobility to have access to the comforts they might want from home. As a result there is a massive shift of money towards Paris and Versailles (which is a small distance away). Thus painters, scuplters, arists, and more came to Paris but also philosophers and wannabes.

As a result of the nobility being concentrated in a certain area, the famed Salon culture spring from the nobility's interest in philosophy, culture, debates, and casual partying. With the Salon culture, the Enlightenment was expanded and effectively exported from the popularity of coffee and discussion. Further the French came to dominate the last part of the Enlightenment through people like Rousseau and Voltaire.

As such, Paris became a cultural center not due to education but rather because of politics against the nobility.

I am dubious of any claim that a place (Paris or anywhere else) is the cultural capital of the world.