One of the most important open problems in physics is to reconcile quantum mechanics with our classical intuition. In this talk we look at quantum foundations through the lens of mathematical foundations and uncover a deep connection between the two fields. We show that Cantorian set theory is based on classical concepts incompatible with quantum experiments. Specifically, we prove that Zermelo-Fraenkel axioms of set theory (and the background classical logic) imply a Bell-type inequality. Consequently, quantum experiments violating Bell inequality cannot be described in the framework of classical set theory. This suggests that a non-Cantorian set theory could be a better framework to capture the elusive nature of quantum world. Finally, we discuss several possible options for a future logico-mathematical framework compatible with quantum experiments.