On the other hand mathematicians are generally thought to peak early with their best work in their 20s and 30s (although like programmers there are certainly many cases that go against this "rule"). I think the idea is that mathematics and programming are supposed to benefit from creative vision uninfluenced by previous works whereas science and medicine are supposed to benefit more from experience.
If you don't see the massive structural difference between the practice of pure mathematics and the practice of software, that means you misunderstand both software development and mathematics.
Programming and mathematics are many things. Sure, it might be hard to see how maintaining a web site has has do with mathematics, but if you need to design a novel algorithm it is worth actually working out the proofs to show that it works (and if you want to publish it in a journal that is often essential).
Computer science is a branch of math, but programming is not. People frequently confuse the two. The difference between using mathematical constructs to inform your engineering and discovering new mathematical constructs is fundamental and the practices do not intersect much at all.
Among mathematicians this is widely regarded as an old wives’ tale, there are ample examples of older mathematicians doing their best work outside that age range.
