Full Idea
I classify conceptualists according to what they say about properties or concepts. If someone classified properties as existing independent of language I would classify her as a realist in ontology of mathematics. Or they may be idealists or nominalists.
Gist of Idea
Conceptualist are just realists or idealist or nominalists, depending on their view of concepts
Source
Stewart Shapiro (Thinking About Mathematics [2000], 2.2.1)
Book Reference
Shapiro,Stewart: 'Thinking About Mathematics' [OUP 2000], p.26
A Reaction
In other words, Shapiro wants to eliminate 'conceptualist' as a useful label in philosophy of mathematics. He's probably right. All thought involves concepts, but that doesn't produce a conceptualist theory of, say, football.