When infinitesimal calculus was first established, the limit of a function was defined in terms of infinitesimal or infinity. As the naïve way at that time contradicted itself, new methods such as the epsilon-delta definition and non-standard analysis were devised to avoid the contradiction. Although mathematicians are not aware of it, it is Kant’s transcendental idealism that lays the foundation for these methods. That is to say, these methods succeed in solving the paradox of infiniteness because they abandon the cognition of “infiniteness in itself” and accomplish the Copernican Revolution that converts the cognition of infiniteness into the infiniteness of cognition.

An ontological argument for the existence of God defines God as the greatest perfect being and states that He must exist because He would not be perfect or the greatest, if He remained only in thought. Anselm and Descartes proposed it, and Kant pointed out that this argument was wrong in deducing a synthetic judgment from an analytic one, but the real problem of this argument is that the greatest perfect being whose existence it proves is quite different from what Christians regard as God, namely the omniscient and omnipotent supreme being.

Philosophers often use the adjective transcendental, for example, when they speak of the transcendental consciousness, the transcendental intersubjectivity and so on. I will explain this seemingly difficult technical term as plainly as possible.