Sentence:  I suspect that in their hearts most working mathematicians are <WSD>Platonists</WSD>, in that they take it as a matter of unassailable if unprovable fact that mathematical reality stands apart from the world, and is at least as real as the world, and possibly gives shape to the world, as Plato suggested. 
Meaning of the word marked with <WSD> tags:  One who holds similar beliefs, particularly that abstract ideas are real.