Abstract
The purpose of this paper is to raise two ontological problems in mathematical activities through a consideration of the difference between epistemological and socio-cultural approaches. The former approach has discussed how mathematical objects should be constructed, while the latter has asserted the sociocultural relativity of the nature of mathematical objects. We compare them from Quine’s ontological point of view. Our resulting conclusion is, ontology in mathematics education should distinguish between ontological problems within domains and of domains themselves. The former problem is:“ What objects exist within the current target domain?” The latter problem is: “Do the current target domain exist in the first place?” If students agree that there exists the current target domain, then what mathematical objects exist in the domain is determined independently from human minds. If students in classrooms doubt the existence of the current target domain or some of them consider different target domains, then what mathematical objects exist cannot be uniquely determined in principle. On the existence of mathematical objects, we can only reach consensuses relative to target domains. The practical implication from this paper depends on which ontological problems students face in classrooms. Each ontological problem requires mathematics teachers to make students reflect the following question at the end of the lesson: 1) in the former case, “What are mathematically correct answers?” 2) in the latter case, “What answers can be derived from different ontological standpoints emerging from classrooms?”
