Galileo's work tended to be more precise than that of Aristotle because he used mathematics. Mathematics may be employed in connection with a philosophy and the two together may work well as science. There is no need to maintain mathematics as a strictly separate discipline in order to accurately perceive and explain the world around us. It takes a peculiar philosophy to engage mathematics in total separation from other factors attending to human reason.
Galileo was not mor precise than Ptolemy. Actually the opposite. But Copernicus and Galileo had the imagination to infer the best physical model for the data.
In the long run it is not mathematical logic that makes for consensus in science. It is the ability to move research forward.