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.
I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.
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.