Full Idea
There is no more to understanding the real-number structure than knowing how to use the language of analysis. .. One learns the axioms of the implicit definition. ...These determine the realtionships between real numbers.
Gist of Idea
Understanding the real-number structure is knowing usage of the axiomatic language of analysis
Source
Stewart Shapiro (Philosophy of Mathematics [1997], 4.9)
Book Reference
Shapiro,Stewart: 'Philosophy of Mathematics:structure and ontology' [OUP 1997], p.138
A Reaction
This, of course, is the structuralist view of such things, which isn't really interested in the intrinsic nature of anything, but only in its relations. The slogan that 'meaning is use' seems to be in the background.