I don't want to put words in hazov's mouth here but Gödel is a total nonsequitor here. This thread isn't about formal set theory. The question is not whether mathematics can be axiomatized in a satisfactory way in first-order logic, but what it is mathematicians study. Since not everything in mathematics is obviously related to geometry or numbers, it is hard to write a satisfactory definition; just look at the difficulty that, say, Wikipedians have had trying to cook up a canonical one. But saying that mathematics is just about rigor or abstract structures makes just about everything mathematical. Not all scholars are mathematicians. And if mathematicians just study abstract structures, what's so special about, say, elliptic PDE, or Fourier analysis, or commutative algebra?