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I would argue that culture does not play that much of a role in mathematics anymore. Gone are the days where some mathematical issue could mean a revolution in the way we think (like the addition of zero to the West, or the square root of two to the Ancient Greeks).
The contentions come within the field, or from philosophers. I'm thinking of things like nonstandard analysis, set theory axioms, fractional calculus, and the battles of different philosophical ideas about mathematics, like intuitionism versus formalism. This is all academic, though they can be very important to scientists and engineers in some cases.
As far as greater culture, it plays no sigificant role in mathematics---people in China are doing similar things to people in the USA now---unlike thousands of years ago, where the way you live played a role in what you studied. For example: Chinese focusing on solving various algebraic problems (writing some nice textbooks) instead of trying to prove abstract theorems.
I believe the public sees true mathematics as something of a black box, and they are happy about that. People tend to not care (seeing it as mundane), or tend to see it as some sort of magic or game. See the large pile of books about numerology, puzzle books for enthusiasts, and study guides for algebra. This is not so much the public engaging in mathematics as it is coping with mathematics in various ways.
There is one exception I think, and that is statistics. Though very few people really understand even the basics, everyone reads newspaper polls and likes to hear about the latest pharmaceutical study. The utility of mathematics is something everyone can appreciate now. Mathematics for its own sake is a different story.
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