Platonists suffered from a bad epistemology and metaphysics, so they believed that ideas they made up in their heads were true regardless of what experience dictated. They did not understand where concepts came from, how we formed them, or what they were; they did not understand where generalizations came from, how they were formed, or what they were; they did not understand reasoning or logic or understanding itself; they did not understand the connection between the conceptual level and the perceptual level — so they made some big, fundamental mistakes. Cognitive failure.
So they thought experience and the evidence of the senses were misleading, illusory, deceptive, and full of lies.
So they played a game of “saving the appearances” (i.e., experience and the facts):
“My basic contention is that, like Greek mathematical astronomy, Greek geometrical optics was subject to a particular methodological paradigm that falls under the rubric of ‘saving the appearances.’ According to this paradigm, the sensible world consists of appearances or illusions, primary among which are the appearances of irregularity or disorder. Called anomalies, witch irregularities are irrational and therefore unreal in Platonic terms, and if they are to be rationalized, or ‘saved,’ they must be reduced to perfect regularity so that their underlying reality can be discovered. This end is achieved by applying a fundamental and perfectly simple principle of order.” — From the book Ptolemy’s Theory of Visual Perception: An English Translation of the Optics By Ptolemy by A. Mark Smith
In the book Galileo Galilei – When the World Stood Still, Atle Naess wrote:
“Galileo’s radical renewal sprang, nevertheless, from the Aristotelian mind set, as it was taught at the Jesuits’ Collegio Romano: human reason has a basic ability to recognize and understand the objects registered by the senses. The objects are real. They have properties that can be perceived, and then ‘further processed’ according to logical rules. These logical concepts are also real (if not in exactly the same way as the physical objects).”
The “radical renewal” means Galileo fundamental changed how science was done and thought of. He rejected the Platonic foundation that was so deeply in error and that had steered people wrong. They had failed. They were not even doing science! They were prescientific or antiscientific.
The “radical renewal” was kicking out Platonism and instead being Aristotelian (with improvements).
Galileo himself said:
”I should even think that in making the celestial material alterable, I contradict the doctrine of Aristotle much less than do those people who still want to keep the sky inalterable; for I am sure that he never took its inalterability to be as certain as the fact that all human reasoning must be placed second to direct experience.”
From the Second Letter of Galileo Galilei to Mark Welser on Sunspots, p. 118 of Discoveries and Opinions of Galileo, translated by Stillman Drake, (c) 1957 by Stillman Drake, published by Doubleday Anchor Books, Doubleday & Co., Garden City, New York