Hi Alex,
Humble immigrants who immigrated to New Amsterdam were used to using patronyms. They actually didn't want to have anything to do with last names. But when the English came closer and closer to them they were forced to chose last names. The nice thing about patonyms is that you can be sure that Mathys Cornelissen / Mathijs Cornelisz had a father whose name was Cornelis.
You have to remember when working in the New Amsterdam area that there is a transition period between the Dutch and the English.
I don't have any idea why his marriage record which is in Dutch states that he came from Jutland. if he came 'from Hoorn'. His brothers and sisters had very Dutch first names. Van Horne is also a Dutch name.
This entry states about the same thing: http://archiver.rootsweb.ancestry.com/th/read/VANHORNE/1999-01/0916...
It wasn't until 1811-1812 that people in the Netherlands that it was mandatory to choose last names if the weren't already going by a certain name.
Getting a little bit away from the subject I read just a day or two ago that even now in the 21st century there are no last names in Iceland !
(Some chose patronyms, others decided to go by their occupations. Some people were so innocent that they chose ridiculous names like pannekoek (pancake), mierenhoop (ant hill) etc. They sometimes even chose dirty names! They didn't realize that a last name was important for tax purposes.)