Here is a summary of what D. G. Van Curen [DGVC, pp. 2-4] says about Mathijs Jansz Van Ceulen, the progenitor of the American Van Keurens. (In his 1998 book, he talks about the Mathijs van Ceulen who was a Director of the DWIC--I will summarize that in another note.)
* First known record, about 1639, concerns a parcel of land on Manhattan known as Van Keulen's Hook [see Riker]. The land patent identified Conraet Van Keulen, relationship unknown, as the original claimant.
* Mathijs purchased land known in 1646 as "van Ceulen's Bouwerie" [Riker] in Haarlem. This parcel was listed in a probate document in 1667. The document also mentioned land in the Esopus area of now-Ulster county.
* Because his patronymic was Jansz or Jansen, we know that his father's name was almost certainly Jan. He had daughters named Annetje and Catryn, suggesting that his mother's name was one of those. There is a record of a Jan Jansen van Ceulen and wife Annetje Janz, baptizing a daughter, Annetje at St. Niclaaus DRC in Amsterdam, 12 Oct 1606. Considering that these are all common names, the connection is far from sure.
* Disproved: Some have cited a baptism of a Mathijs at Austin Friars DRC in London, 1602. However, an examination of the record shows that the father's name was Mathijs, not Jan.
* There's no record that he ever lived in Manhattan or Esopus. Instead he appears to have lived at Fort Orange.
* Killaen van Renssalaer's journal says that Mathijs died 13 Oct 1648. Court records on 15 and 22 identify his widow as Margriet Hendrickse. Two months later she is identified as the wife of Thomas Chambers although she is pregnant with Mathijs's child, Matthys.
* Baptism records at the Fort Orange (Albany) DRC are missing for that period. A subsequent rector destroyed the old records. However, other records allow us to infer the following children and approximate dates of birth: Catryn about 1642 (so as to be 18 at time of marriage, 3 October 1660); Annetje about 1644 or 1645 (so as to be more than 21 at a 1667 estate division); Jan abt 1646-47 (as he was noted to be under 21 at the estate division); and Matthys soon after his father's death).
* Renssalaer's journal mentions that Mathijs had been paid for baking bread.
* Margriet's second husband, Thomas Chambers, himself childless, granted his family coat-of-arms to stepchildren, Jan and Matthys.
[DGVC] Van Curen, D. G. The Descendants of Mathijs Jansen van Ceulen: A History of the Van Steenberghs, Peersens, Jansens, Van Keurens and Related Families, from Their Beginnings in Kingston, Ulster County, New York. 2nd edition of "Van Keulen/Van Keuren, Van Kuren/Van Curen". Chesterfield, Missouri: Mira Digital Publishing, 2016.
[Riker] Riker, James, Henry Pennington Toler, and Sterling Potter. Revised History of Harlem (City of New York.): Its Origin and Early Annals Prefaced by Home Scenes in the Fatherlands; or Notices of Its Founders before Emigration. Also Sketches of Numerous Families and the Recovered History of the Land-Titles. With Illustrations and Maps. Revised and enlarged. New York, NY: New Harlem Publishing Company, 1904. https://www.myheritage.com/research/record-90100-97234527/revised-history-of-harlem-ny?s=273564081