How does Kant argue for the immortality of the soul?

In The Critique of Pure Reason, Kant says this about the immortality of the soul:

“PURE practical reason postulates the immortality of the soul, for reason in the pure and practical sense aims at the perfect good (summum bonum), and this perfect good is only possible on the supposition of the soul's immortality. It is the moral law which determines the will, and in his will the perfect harmony of the mind with the moral law is the supreme condition of the summum bonum.”

• ​As usual, you need to understand the language Kant uses before you can get to the kernel of his ideas. You need to know what the perfect good consists of from reading his other writings. But in this passage, he is saying that pure reason aims at moral perfection. To him, perfection is progress towards the perfect good. It is not the actual attainment of this state of perfection, it is the process.

“The principle of the moral destination of our nature--that only by endless progress can we come into full harmony with the moral law--is of the greatest use, not only for fortifying the speculative reason, but also with respect to religion. In default of this, either the moral law is degraded from its holiness, being represented as indulging our convenience, or else men strain after an unattainable aim, hoping to gain absolute holiness of will, thus losing themselves in fanatical theosophic dreams utterly contradicting self-knowledge.”

• He’s saying that pure reason is naturally drawn to the progress needed to attain the perfect good, which is in harmony with moral law. He feels that moral law is holy and our immortal soul is drawn to its holiness. At the same time, he blasts people who just lie back and let religions make their decisions for them, in the hope that they will somehow be saved without doing the soul-searching moral behavior requires.

“For a rational, but finite, being the only possibility is an endless progression from the lower to the higher degrees of perfection. The Infinite Being, to whom the time condition is nothing, sees in this endless succession the perfect harmony with the moral law.”

• The Infinite Being, the Immortal Soul is the seat of pure reason. And to the infinite soul, time has no meaning, by virtue of the fact that infinity mean timelessly forever. Also, the progression of the soul towards perfect harmony with the moral law is a fixed and immutable path that is not impacted by the activities of our lower selves.