I would think that not only could you advance pretty far in math and science with intuition, but a high degree of intuition is absolutely essential!
Otherwise, what would you do when you came to the part of the problem where the instructor says, "and it should be intuitively obvious that the three and a half foot equation on the left side simplifies to 2x ......"
Of course, Pierre LaPlace is the patron saint of mathematics professors. When Nathaniel Bowditch was translating LaPlace's work into English, he noted, "I never came across one of Laplace's 'Thus it plainly appears' without feeling sure that I have hours of hard work before me to fill up the chasm and find and show how it plainly appears".
Edit: Added smilie. Of course, it's really only half kidding, since it certainly helps if the problem seems intuitively obvious in hindsight, at least. But that's going on the assumption that intuition is really just an accumulation of past experience.