There seems to be two ways of spelling for ‘van Gogh’: with small v or big V. It may have some reason I do not know having small van in the middle of Vincent van Gogh, but why do they still spell small van in the first part of his shortened name Van Gogh? I hope the answer to be added why they use small van in the middle of three words?
There's a long discussion of this point in the Talk section of Wikipedia. The upshot seems to be that the official Dutch convention (the painter was a native of the Netherlands) requires a separate prefix ("tussenvoegsel") on a surname, such as van or de, to be lowercased when it is preceded by the forename or initial but capitalized when it is not.
These prefixes are mostly prepositions or articles and are regarded as secondary; in alphabetical lists the name is entered under the first letter of the 'primary' name, not the prefix.
Such rules vary from country to country, and in 'ungoverned' languages like English are left to individual writers and institutions to work out on their own. Here, for instance, are the house rules for alphabetizing at the Yale University Music Library, and here are the rules for capitlization.
Perhaps input from a Dutch Native is useful here. I will not vouch that this is the way it should be according to American formal guidelines, although my sense is that it should follow the rules from the native language.
In Dutch 'van' is a 'tussenvoegsel' when used within a full name, as StoneyB mentioned. It's written in lower case when writing 'Vincent van Gogh'. However, it's written in upper case when it is the start of the name, such as the sentence:
"Ik houd van de schilderijen van Van Gogh" -> 'I love Van Gogh's paintings'
As you can see, when mentioning the name exclusively by last name the 'Van' is capitalized. It is never interpreted as a first name, as there is a limited set of such 'tussenvoegsels'. The first 'van' you see in this sentence translates to 'of/from', making the sentence literally 'I love the paintings from Van Gogh'.
For names like 'Erik van der Beek' the lower case form is more legible than 'Erik Van Der Beek' would have been. The last name is often represented as 'Beek, van der' in management systems, since so many last names start with these typical 'tussenvoegsels'.
Here are more examples of complicated last names, and how they should be written in dutch:
Anke de Graaf B.G. van Vooren dhr. dr. mr. G.H.V. van den Herik Henny Dijkstra-van der Haar dhr. Den Braber-ten Haaf professor Ten Cate dominee Van den Berg bakker De Vries het verzameld werk van Van het Reve
Note: In Belgium the convention of turning the prefix into lowercase when adding the first name is not followed.
"van" is a preposition in the Dutch and Afrikaans languages, meaning "of" or "from". It is also a common prefix in Dutch language surnames. Ludwig van Beethoven is another famous 'van', though he is usually known as 'Beethoven'. 'da' in Italian (eg Leonardo da Vinci) works in the same way.