We might be able to relate to Dom Paulo's problems more than any other character in A Canticle. The poor guy opens his house to a bunch of near-strangers, and before he knows it, they're eating his food, rummaging through his library, and fighting with each other. Then war comes a-knocking at his door.
It could only be worse if this was Christmas at the McCallisters'. So why does Dom Paulo open his home to Thon Taddeo, if he knows the man's going to be trouble? Because knowledge is power, Shmoopers. And Dom Paulo wants to spread that power around.
Dom Paulo fears a communication breakdown has occurred between the church and the institutes of education and knowledge. And he's right. In Francis's day, the Church was the only place to seek out an education—K-8, high school, college, you name it (5.6).
By Dom Paulo's time, the collegium has sprung up, and the scholars there no longer trust the monks. The whole reason Taddeo wants the Memorabilia sent to the collegium in the first place is so his colleagues won't feel the "monks had tricked him" into believing anything unfounded (12.97). In effect, the collegium doesn't trust the abbey is on the up-and-up.
So Dom Paulo wants Taddeo to come to the abbey to learn and even to teach. The reason invites the scholar to give a lecture on his studies because:
The esoteric gulf between Christian monk and secular investigator of Nature [will surely be] narrowed by a free exchange of ideas, he felt. […] More communication, not less, [is] probably the best therapy for easing any tension. (19.65)
And that's Dom Paulo's goal in one concise quote.
And his goal seems to be met—on the monks' side of things at least. Nothing Thon Taddeo and his colleagues study, from refracted light (20.103) to the origins of man (20.114), becomes a point of contention between the monks and the thon. The study of knowledge is their common ground.
So what is the problem, then? What causes this communication breakdown? Politics, of course.
Once Dom Paulo gets to know Taddeo, he sees a man stuck within Hannegan's political system. Personally, Taddeo condemns Hannegan's actions, but he cannot openly oppose the mayor. So he has the choice of either supporting the collegium or literally murdering it (21.8). Yikes.
Dom Paulo and his monks, on the other hand, stand on the periphery of Hannegan's political system. They can, and do, oppose his war efforts. The abbot tries to guide the thon away from being Hannegan's accomplice. He asks the Taddeo:
"But you promise to begin restoring Man's control over Nature. But who will govern the use of the power to control natural forces? Who will use it? To what end? How will you hold him in check? […] Mankind will profit, you say. By whose sufferance?" (21.64)
The unstated answers there are: Hannegan will profit, and just about everyone else will suffer. Dom Paulo further pushes Taddeo, telling him he can "serve God first, or to serve Hannegan first" (21.73). But even if Taddeo does not want to serve Hannegan, he cannot bring himself to serve God.
The tension between Dom Paulo and Taddeo (Church and Science) is not one regarding the search for knowledge in the novel. Both seek knowledge and help the other seek their knowledge. It's a political rift.
Dom Paulo sees his master (God) as serving the people for the people's sake. He sees Taddeo's master (Hannegan) as serving himself at the people's expense. And that's just not cool in Dom Paulo's book. Can you blame him?