The kinds of mathematical quasi-empiricism

The received view concerning mathematics is the one, that mathematics is a priori, and that mathematical knowledge develops via intelektuelle Anschauung rather than by analyzing empirical data. Mathematical proofs seems to be immune to empirical refutation, and in particular the development of mathematics does not in any way resemble the development of e.g. physics.
On the other hand, it is quite clear, that mathematics play a fundamental role in science, and it is often considered to be rather just a useful tool, which provides a language and a conceptual system allowing to express statements concerning empirical world. Such views stress the dependence of mathematics upon physics. In the article, I present two quite different aspects of this problem: the ontological and the methodological aspects.
According to Quine, our argumentation in favor of mathematical realism should be based on the analysis of ontological commitment of empirical theories. There is no other compelling argument for mathematical realism. According to Lakatos, mathematical knowledge develops in a way similar to empirical science: it is fallible, and the proper model to describe it is the model of proofs and refutations. In the article I describe and contrast these two points of view.