Intuition always leads us, analysis follows its lead.
When einstein conceived of the speed of light and Gravity, they were both understood intuitively first and the analysis was abstracted to make sense of his visions.
The more important the discovery, the more difficult it is for the analytical scientific method to confirm it. Einstein was in a rush to prove his general theory of relativity mathematically, after he understood it intuitively.
David Hilbert, perhaps the leading mathematician in the world was in a race with him to provide the equations to help prove his theory scientifically. Einstein won the race, though Hilbert’s contribution to Relativity was enormous.
Hilbert, through the use of set theory as devised by Georg Cantor, was able to conceive of a space that allowed for Einsteins space-time fabric. Einstein was not truly a specialist like Cantor and Hilbert were, he was a generalist, he saw the big picture and knew analysis well enough to imagine the universe as no other could before him.
Einstein said, if the equations don’t work, change the equations to account for what is (paraphrase), and by this he meant that intuition leads analysis, not that analysis itself was not important. Hilbert and other Formalists, had to reach back to Plato and his philosophy to retrieve a construct that allowed for the study of the tiny constituents of relativity and quantum mechanics.
What had been considered abstract objects, became real objects by employing Platos Forms, and we found equations that allowed for what we could never otherwise possibly study.