That's my understanding as well (although I have no idea what his opinion on "external reality" would be - probably that we can't test any theories about it as opposed to the world of human perception, so it isn't a useful idea).

I think the way some philosophers tend to phrase it is that we develop mathematics as a common language which we agree to use to communicate our theories, which we can attempt to verify via perception. Our choice of axioms etc is pretty free, but we tend to pick the ones which are both minimal and most useful for communication and making verifiable predictions.

I think that where the positivists and Quine differ is that the positivists say that once you have chosen axioms, you can make purely analytic statements, whose truth (within that system), only depends on reasoning according to those axioms, as opposed to statements whose truth does depend on our perceptions. Quine disagrees either that this is a useful distinction or that there is any difference at all, I'm not sure.