Irony of Theistic Skepticism
Descartes's philosophical position is almost postmodern in its all-encompassing skepticism; he even worries that he himself might not exist. Ironically, since he is a Christian, he never really doubts that God exists, which is completely out of character for a postmodern skeptic (although he predates them by centuries); he doubts the existence of everything in the world except for God, which is precisely the opposite of most people today.
Irony of Deception
Descartes is worried that he is being deceived by some higher power, probably a demon, into believing in his own existence. Interestingly, he also mentions the possibility that it is God himself who is deceiving him, preventing him from seeing the truth, which is an odd thought experiment considering Descartes's obviously strong faith. The irony is that there really is no way to tell if God is deceiving him, despite God's claims of ultimate goodness.
Irony of Dualism
Descartes proposes a sharp dualism between material and immaterial substances, clearly preferring the latter. The mind is almost the only true essence of a person, as he believes, and the body is untrustworthy, especially considering the ambiguous nature of the senses. He is baffled, however, by the ironic lodging of the soul inside the body, which he doesn't quite understand. In the end, he must simply decide to trust God or risk eternal skepticism.
Irony of Postmodernism
The course Descartes adopts in order to find truth (systematically destroying reality and then building it back up again from the ground level) is a very postmodern one, something that might happen in the later 20th century. Ironically, Descartes lived centuries before postmodernism came about, and his religious views would have prevented him from being a part of the intellectual movement anyway.
Irony of Cogito Ergo Sum
Descartes is famous for his maxim cogito ergo sum (I think, therefore I am), which is articulated in different words in Meditations on First Philosophy. This argument is inherently ironic, although the somewhat humorous irony in no way detracts from the effect of the argument: Descartes says that he thinks, therefore he exists; that means he thinks that he thinks, and therefore he exists; and if he realizes that irony, then he thinks that he thinks that he thinks, and therefore he exists.
