Full Idea
There is one and only one serious argument for the existence of mathematical entities, and that is the Indispensability Argument of Putnam and Quine.
Clarification
The argument is that science requires mathematics
Gist of Idea
The Indispensability Argument is the only serious ground for the existence of mathematical entities
Source
Hartry Field (Science without Numbers [1980], p.5), quoted by Stewart Shapiro - Thinking About Mathematics 9.1
Book Reference
Shapiro,Stewart: 'Thinking About Mathematics' [OUP 2000], p.227
A Reaction
Personally I don't believe (and nor does Field) that this gives a good enough reason to believe in such things. Quine (who likes 'desert landscapes' in ontology) ends up believing that sets are real because of his argument. Not for me.