More and more I am becoming convinced that one should know at least one programming language very well as a mathematician of this century. Is my conviction justified, or not applicable?
All of the answers so far seem to pass over this initial question. While it's true that a much larger percentage of mathematicians nowadays do calculations on computers etc, I do not agree with the sentiment that every mathematician should learn a programming language. (There's also the question of what this means: e.g., do I know Python if I can use it as a calculator, or do I know Python if I can write my own web browser in it?)
Certainly being able to program has many uses, and also is a marketable skill outside of academia, but just as there are many kinds of math there are many kinds of mathematicians.
Here are some of my reasons for disagreeing:
Learning how to program (and program properly) takes time. For many mathematicians, that time may be better spent thinking about mathematics proper.
If one ever needs to do some simple calculations, there are many mathematical software packages mentioned in other answers that can be effectively used without learning how to properly program in the relevant language (e.g., based on tutorials one can do simple calculations with elliptic curves in Sage without knowing anything about Python).
Even if one is interested in serious calculations where based on the mathematical software available some serious programming is needed, it may be more efficient (and beneficial) to strike up a collaboration with someone who works on computational mathematics.
I know many successful mathematicians who do not program.