An embryonic form of calculus was first proposed by Bonaventura Cavalieri, a 17th century mathematician, Catholic priest, and mentee of Galileo Galilei, in 1641. In modern calculus, shapes are divided into infinitely many (infinitesimally) small chunks. The total area of a shape is then computed as the sum of the areas of each tiny chunk. It's quite useful, and underlies all modern Physics! Cavalieri's contribution forms the basis of this framework: he postulated that geometrical objects are comprised of an infinite number of tiny, 'indivisible' parts.
Cavalieri was taken to task by a fellow mathematician and Catholic priest, but of a different order (the Jesuit order): Paul Guldin . Each mathematician held deep-seated philosophical and spiritual worldviews which shaped their approach to mathematics. It goes to show how much variation there is in the Catholic Church..!
Scientists today continue to be shaped by fundamental ideologies; by 'unprovable' intuition. For example, it is a great leap of faith to think that the Universe can be comprehended at all by the human mind! As Einstein famously said, "the most incomprehensible thing about the Universe is that it is comprehensible." The mingling of science and philosophy can be fruitful, enlightening, and beautiful, as in the example of Georges Lemaitre, father of the 'Big Bang' theory, who was motivated to study the Universe by his faith in God. It can also be dangerous, however: like Paul Guldin - who refuted calculus because of his belief in a rigid Euclidean Universe - or Einstein - who refused to accept the postulates of Quantum Mechanics because "God does not play dice" - scientists can become wilfully blind to ideology-challenging evidence.
A balance can of course be struck between philosophy-guided intuition and rigorous deductive logic; to quote Einstein yet again, "science without religion is lame; religion without science is blind."
...the men argued for more than purely mathematical reasons. They were members of two religious orders with similar spellings but very different philosophies: Guldin was a Jesuit and Cavalieri a Jesuat. The former believed in using mathematics to impose a rigid logical structure on a chaotic universe, whereas the latter was more interested in following his intuitions to understand the world in all its complexity.