I am curious to learn in more detail why Rene Descartes was not knighted like Isaac Newton and many other great mathematicians.
The French word for "knight" is "chevalier". Descartes did not need to be knighted; he inherited the title "chevalier" and was frequently called "Chevalier du Peron" or "Chevalier Seigneur du Peron", as in this portrait: http://www.britishmuseum.org/research/collection_online/collection_object_details.aspx?objectId=3610101&partId=1&people=88521&peoA=88521-1-6&page=1
This answer was based on an earlier version of the question which asked why Descartes was not made a Sir
The primary reason is that "Sir" denotes a knighthood in the British honour system, and is normally open only to British subjects(*).
René Descartes was a French subject and therefore not eligible.
As noted by Bernard Massé in his comment, the equivalent in 16th-century France would have been to make him a count, marquis or similar title. There are at least two factors which may have contributed to this not happening: firstly he lived a good part of his life outside France (in the Netherlands and Sweden), and secondly his philosophy was in direct opposition to that of the Scholastics who still ruled the Sorbonne.
(*) According to the official website of the UK monarchy, honorary knighthoods are occasionally bestowed on foreign citizens, but they are not entitled to call themselves "Sir".
Another point is that Isaac Newton, as well as other important mathematicians in England and France, was also a civil servant (Master of the Mint and other offices). He was president of the Royal Society and Member of Parliament.
For instance, Viete was privy councillor to two Kings of France and a high-level civil servant. Brouncker was already noble by birth.
Descartes went to the Netherlands for freedom of thought and study as well as to learn from Dutch scientists and was not member of the french establishment, although most members of his family were distinguished civil servants (member of provincial parlements), with land properties, mostly in Britany and Poitou.