Galileo is right (Dramatic Irony)
A rather large instance of dramatic irony that hangs over the entire play is the fact that, by the time the play was written and performed, audiences and readers would know that Galileo was correct in his calculations; his view has decisively triumphed over the previous, earth-centered model. Science has taken his studies seriously and his proposed structure of the solar system has been canonized in physics and astronomy. Thus, there is a discrepancy between what the reader/audience member knows and the fact that many of the characters fail to believe Galileo's authority as a scientist.
Ludovico does not want to learn what Galileo has to teach (Situational Irony)
Early in the play, Ludovico visits Galileo and wants to be his pupil, largely for social reasons. He tells Galileo, "Nobody can drink a glass of wine without science these days, you know.” As he sees it, science is on trend, a popular topic of conversation that private tutelage would help him to master. He is not that interested in science itself, and ironically enough, he ends up disagreeing with Galileo's teachings, in spite of having no more scientific expertise. Ludovico breaks his engagement with Galileo's daughter when Galileo's theories prove too controversial. It is ironic that after seeking out Galileo's expertise for social reasons, Ludovico does not actually want to learn what Galileo has to teach, again for social reasons.
Science and mathematics are not considered as important as philosophy (Dramatic Irony)
Throughout the play, characters speak dismissively about the sciences, favoring philosophy and theology. Additionally, science and math are seen as less economically valuable than philosophy and theology, and important figures like the monk dismiss Galileo's work as a bad career decision. This is ironic to a modern audience because we know that work like Galileo's will in fact become a cornerstone of the modern economy, while philosophy and theology are now perhaps the epitome of "useless" careers. While these forms of study were valuable when the church was dominant, we know that it's precisely scientific research like Galileo's that will bring science to the forefront and make it the driver of modern industry and capitalism.
Andrea takes Galileo's work to Holland (Situational Irony)
When Galileo recants his claims about the Copernican system, Andrea is exceedingly disappointed in him, and cuts ties with him, deeming him a traitor the scientific cause. At the end, he goes to visit Galileo not because he wants to, but because he has been tasked with checking in on the old scientist. In the process of reconnecting with his mentor, he learns that Galileo has not given up working on innovative scientific theories, and Galileo hands over his Discorsi for Andrea to bring along with him to Holland. Thus, Andrea's expectation and disappointment are turned around by his mentor's gift of new work, and he becomes the one to carry the writing to a place where science is more accepted. This is ironic, since Andrea was the most disappointed by Galileo's recanting, and it completely subverts his expectations about his reunion with his mentor.
