Programmers are less intelligent, skillful and diligent on average than research mathematicians? How do you figure that? This statement appears to rely on a circular definition of intelligence - maths is "hard" therefore people who do it must be "intelligent".
Given the examples in the linked discussion of cases where not only did mathematicians write buggy proofs but it took years to figure out a mistake existed at all, let alone what it was, I find it hard to believe that. To the extent it may appear superficially to be true it's likely the result of the academic setting in which there is no connection to real world relevance and the only pressure that exists to get things done comes from peers and the need to publish papers - but if those papers achieve very little and your peers papers also achieve little, that's totally OK.
Compare to the world of the working programmer who is judged not by how fast he writes code but by the real world positive impact of that code, and it is easy to see how the mathematician may come across as more diligent or intelligent. But it's just an artifact of the pressure-free environment.
I was also surprised to learn that the idea of checking mathematical proofs using Coq is considered just as exotic or impractical as checking programs. I had thought that Coq and other proof assistants were in wide usage for this use case, as surely pure mathematics is simpler to reason about formally than entire programs ... but apparently not. The fact that maths proofs are still mostly checked by hand, whereas machine-checked proofs (like static type systems) are widely used by even novice programmers, is hardly reassuring.
"Programmers are less intelligent, skillful and diligent on average than research mathematicians?"
Keep in mind that there are a lot more programmers than professional mathematicians, and programmers are more likely to be well paid than mathematicians. As a result, mathematicians are self-selected for doing math while most programmers are only in it for the paycheck, on top of the fact that a mediocre mathematician probably isn't going to be able to make a living in math.
"Compare to the world of the working programmer who is judged not by how fast he writes code but by the real world positive impact of that code,..."
I suspect that you have an excessively positive opinion of the work of most programmers. I myself know far too many who are both slow and detrimental to the projects they are assigned.
As for formal mathematical proofs, I bet you will find that most mathematicians don't do them for the same reason that essentially no coffee is verified: it adds a lot of complexity (the same complexity, I suspect) without any obvious benefit.
Research mathematicians are certainly more highly trained than non-research programmers, at least for the first few years of their career. A PhD, at least in the USA, is really intensive.
What’s the connection between being highly trained and being “intelligent, skillful and diligent”? Circus animals are definitely highly trained after all.
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 contemporary mathematical theories such as "Inter-universal Teichmüller theory"[1] are so complicated that only one or two dozen people in the world can understand them. Proof assistants like Coq do not help with those kind of theories at all. In a nutshell, research mathematics has become fairly complicated, to say the least. (I'm saying that as a layman, from what I've gathered about it. I'm not a mathematician.)
> Programmers are less intelligent, skillful and diligent on average than research mathematicians? How do you figure that?
Well, the selection process is much harder. It's an empirical claim that could be falsified, but the claim seems reasonable in lack of any counter-evidence.
> This statement appears to rely on a circular definition of intelligence
The statement is a metaphor. Mathematicians are commonly deemed intelligent, with high variability of course, as university graduates generally are. On the meta level, mathematicians provided a lot of the basis which computers and programs are build from. Now it's poetic freedom to include programmers who venture into mathematics in this fuzzy term.
I think it's largely a product of resources. There are fewer positions and more people who want to become research mathematicians so only those who are extremely well qualified even get to the post. The reason for the lack of resources is that the research mathematician's work isn't usually immediately profitable (hence their pay is often relatively low relative to their software engineering counterpart). By contrast, many companies operate their software development process in a way that allows them to recruit developers with minimal training and even deploy them in a way which they find useful, even if in fact that work might be completed more quickly by a smaller group of more experienced/capable programmers.
I think you're right about the pressure-free environment bit, too - some places developers are so bogged down with fire-fighting that planned development with real foresight rarely gets developed.
Moreover I think there are just more software developers, with much more variance in their abilities than research mathematicians.
Given the examples in the linked discussion of cases where not only did mathematicians write buggy proofs but it took years to figure out a mistake existed at all, let alone what it was, I find it hard to believe that. To the extent it may appear superficially to be true it's likely the result of the academic setting in which there is no connection to real world relevance and the only pressure that exists to get things done comes from peers and the need to publish papers - but if those papers achieve very little and your peers papers also achieve little, that's totally OK.
Compare to the world of the working programmer who is judged not by how fast he writes code but by the real world positive impact of that code, and it is easy to see how the mathematician may come across as more diligent or intelligent. But it's just an artifact of the pressure-free environment.
I was also surprised to learn that the idea of checking mathematical proofs using Coq is considered just as exotic or impractical as checking programs. I had thought that Coq and other proof assistants were in wide usage for this use case, as surely pure mathematics is simpler to reason about formally than entire programs ... but apparently not. The fact that maths proofs are still mostly checked by hand, whereas machine-checked proofs (like static type systems) are widely used by even novice programmers, is hardly reassuring.