The metaphysical exploration - an attempt to rectify the foundation of Kant's "transcendental philosophy" - led us to the findings that while "appearances in general are nothing outside our representations," appearances themselves are things that exist outside our representations. In other words, appearances themselves, i.e., categories, while serving "only for the possibility of empirical cognition," would last forever in virtue of the "transcendental truth" or the "transcendental ideality of appearances," irrespective of the existence of humans on this planet. This insight opens the way to whole new realms of mathematics and philosophy, enabling us to find a solution to the conundrums in mathematics, such as inductive numbers, Cantor's continuum hypothesis, Zermelo's axiom, and the infinite or transfinite cardinal numbers.