A classic example of the limits of science, and an example of a conflict between the Church and science is the idea of geocentrism.
The Church has taught that the earth does not move, and the sun and stars move around it. The ancient Greeks proposed a heliocentric theory, which the early Church Fathers contested based on Scriptural interpretation.
Centuries later, Corpenicus proposed a heliocentric theory, which he felt would simplify the cosmology of Ptolemy. He sold it to a Pope as a more efficient way to describe the motions of planets as part of a program to improve the calendar. In fact, Corpenicus' heliocentric model required 48 epicycles (including "epicyclets") to Ptolemies 40 to gain the same accuracy! [Kepler's elliptical orbits simplified things, but still was not more accurate than Ptolemy's model until modern times when Fourier Transforms (a fancy form of many, many, epicycles, effectively) were applied to curve fit observed deviations of planetary motion from the ideal Keplerian motion.)]
Galileo picked up Corpenicus' model and proclaimed basically that Scripture was wrong, the earth orbited the sun, and the sun was the center of the universe. Three Popes ended up condeming the writings of Corpenicus and Galileo. If the Corpenicus' writings and Galileo treated their theories as a theory, and not as proven reality, the Church would have stayed out of it. Finally, after the editors of Corpenicus' works removed the few sentences that proclaimed heliocentrism to be more than a theory from his works (as demanded in the inquisition), the works were removed from the index. The proclamations were never rescinded.
Today, heliocentrism is rejected, as is geocentrism, all in favor of acentrism. No one knows if there is a center to the universe, scientifically. Equally factual, No one has proven to date that the earth rotates or translates! I.e., geocentrism has by no means been disproven. Acentrism is more philosophical than scientific. The theory of General Relativity operates under the assertion that one can pick any center to describe the universe. A postulate of the theory is that there are no preferred reference frames (i.e., any center will do). This is how General Relativity is formulated.
All observation we make in the universe is of relative motion. We see the sun and stars rotating around us, but mathematically, equally probable is that the earth turns (the current OPINION). Foucalt's pendulum, Sagnac effect, light gyroscopes, etc. can detect rotation, but cannot distinguish between rotation of the earth and rotation of the universe.
The interesting predicament is that though science strongly disagrees with geocentrism, it has not been able to disprove it.
Some interesting quotes:
Cosmologist George Ellis in Scientific American:
"People need to be aware that there is a range of models that could explain the observations," Ellis argues. "For instance, I can construct you a spherically symmetrical universe with Earth at its center, and you cannot disprove it based on observations." Ellis has published a paper on this. "You can only exclude it on philosophical grounds. In my view there is absolutely nothing wrong in that. What I want to bring into the open is the fact that we are using philosophical criteria in choosing our models. A lot of cosmology tries to hide that."
As stated by Max Born in his famous book,"Einstein's Theory of Relativity",Dover Publications,1962, pgs 344 & 345:
:"...Thus we may return to Ptolemy's point of view of a 'motionless earth'...One has to show that the transformed metric can be regarded as produced according to Einstein's field equations, by distant rotating masses. This has been done by Thirring. He calculated a field due to a rotating, hollow, thick-walled sphere and proved that inside the cavity it behaved as though there were centrifugal and other inertial forces usually attributed to absolute space.
Thus from Einstein's point of view, Ptolemy and Corpenicus are equally right."