Some might be too bored to be good programmers, and some might have issues with corporate nonsense and workplace politics. But purely in terms of the ability to manipulate symbolic systems to useful effect, the base level is more than high enough for most programming jobs.
Evidence: for a long time, math BSc/PhD quals were highly valued by software houses. This continues to be true to an extent, especially at the high end with FP/ML.
The percentage of programmers who can become math PhDs is... lower.
First of all what does "can become" mean? If it means with sufficient training and supervision they can learn to be programmers, then that's also true for the inverse.
Furthermore, we need only one counterexample to make it false, and I happen to know a few math PhDs that are not that great at programming even though one of them actually works as a programmer.
> If it means with sufficient training and supervision they can learn to be programmers, then that's also true for the inverse.
No way. Most programmers are genetically unfit to do research mathematics. Many programmers can't take in an idea and expel it back out without corruption because they've got some cosmic ray simulation device in their brain stem. Or they just don't have the creativity. For example, think of all the people that complain about interview warm-up questions or think they're something you'd memorize.
This largely ignores the fact that research is 90% reading papers (or doing lab work in less pure fields) and trying to come up with something. Fighting for money, writing papers, producing graphs/charts, etc.
Pure maths research is undeniably simpler, but not that much. Look at HoTT (homotopy type theory), or reverse maths (https://github.com/ericastor/rmzoo/). These are sufficiently close to programming - because they are largely composed of programming tasks.
Furthermore, researchers usually don't do work alone, they are usually enrolled in some kind of a program, with a supervisor, mentor, guide, or at least a program/faculty chair. And even if they are totally on their own, they can start doing work on unsolved problems. Usually people new to research start by doing a survey paper for a certain field, to get an overview of recent and past progress and problems, solutions and techniques.
Yes, 99.9% of programmers would never become the next Tao, cranking out blog posts, books, polymath papers, lectures and otherwise results every few days/months, but that doesn't mean they couldn't do pure maths research. But luckily they don't have to. Because it's a very different realm than programming. (Or even protocol design, IETF work, low level microcode work, or run of the mill mobile apps.)
Some might be too bored to be good programmers, and some might have issues with corporate nonsense and workplace politics. But purely in terms of the ability to manipulate symbolic systems to useful effect, the base level is more than high enough for most programming jobs.
Evidence: for a long time, math BSc/PhD quals were highly valued by software houses. This continues to be true to an extent, especially at the high end with FP/ML.
The percentage of programmers who can become math PhDs is... lower.