Just curious, I majored in Mathematics in college and I had professors with varying views and philosophies of mathematics all the way from an almost platonic view (the perfect ideal equilater triangle exists "somewhere") to the more critial view that mathematics has not bearing on reality but is only in our minds. Where do you fall?

The best mathematics Profs are (neo-)Pythagoreans.

They're the ones to tend to see and appreciate the poetry in mathematics, it's not mere a means to support the sciences but equally and, perhaps more importantly so, a form of art. The sciences can eat the scraps that fall from the table of mathematics, we should do mathematics for its own sake.

'Mathematics, rightly viewed, possesses not only truth, but supreme beauty — a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show. The true spirit of delight, the exaltation, the sense of being more than Man, which is the touchstone of the highest excellence, is to be found in mathematics as surely as poetry.' - Bertrand Russell

How do you feel about the paradoxes created in certain fields of mathematics such as in set theory?

No, if there's a 'Paradox', it's just a contradiction which means the theory is wrong and it's time to go back to the drawing board. That's how mathematics work, mathematicians don't always get it right on their first try, sometimes it takes centuries. Contradictions were found in Naive Set Theory, so ZFC was presented, but it doesn't include the universal set or proper classes, so it was found to be incomplete, New Foundations Set Theory was then presented, it solves these completeness issues, but has yet to be proven consistent (though it hasn't been disproven either, as no contradictions have been found), much of it has been proven consistent, but not the theory as a whole.

Set theory's a work in process, that we haven't seen the final product of a theory begun in the late 19th century shouldn't surprise us. It took over 2000 years to settle questions about Euclid's fifth postulate. And more rigorous formulations of the theory have expanded the number of axioms required (to 8, I believe, but it's been a while since I've taken geometry). There's no problem with Mathematics, only with our current understanding of the same; just like there's no problem with the laws of physics, just with our understanding of them at the current time.