Mathematicians SHOULD be able to program, first, because they have to to do their work, second because thinking analytically, handling all circumstances, etc, should come naturally to trained mathematicians

I'd have to say that while I agree with the second half, very few of the mathematicians that I dealt with while I was in graduate school had any need to program to do their work (and at that time in my life, nor did I).

Of course there are some (depending on field of interest) that do.