Nowadays, a scientist is considered someone who is well-versed in all the sciences and maths, and not merely a specialized field.
I do not agree either.
In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.
I don't mean it that literally such that a mathematician should have knowledge of physics, biology, chemistry, medicine, psychology etc. I mean a basic amount of math, biology, chemistry and biology. Most institute have the requirement of taking courses from the 4 aforementioned categories for graduation. And certain disciplines require a high degree of knowledge from another subject (i.e. physicists need a strong knowledge of analysis).
First of all, in my opinion mathematics is not a science.
What kinds of institutes?
Here lies the reader who will never open this book. He is forever dead.
Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and Punishment
As far as I know, MIT, Harvard and several others have General Institute Requirements. That means even non-math/science majors/minors have to take such courses. As for your first statement, I disagree.