Mathematics is generally & popularly judged a science in the basic duality: science - humanities. As enemies and collaborationists. The border heavily & fiercely policed.
However, it seems to me that Poppers theory, which entitles science-hood by falsification doesn't apply to mathematics at all.
What can it mean that the Number Theory is falsifiable? Certainly a tightly-focused question will either be true or false. More general conjectures & ideas will be true when enunciated as the mathematical landscape is seen and a new shape formed. For example the Langlands programme (higher dimensional representation theory). Significance through aesthetics & ethics seem the key theme. The serious intent (ethic) towards the good & beautiful (aesthetics) towards the reverance & delight of contemplation. Platonism in essence.
Badiou characterises knowledge as four domains (conditions) - love, science, art & politics.
Is it then love - Number Theory being the material incarnation of a mathematicians embrace and adoration of Number?
If not then is it art - Number Theory being the glorification of Number through steady & inspired craft. As a cathedral to God, so Number Theory to the One?
If neither then could it be politics - creating harmony amongst bickering wilful abstract entities intent on having it their way? Number theory being a nation of number systems.
If none of these, then does it lie with Philosophy, the place from which these four conditions converge (in Badious system)? One is reminded that Badiou states that mathematics is the very ontology of philosophy. He may give it space there, but does anyone else?