>How come we jump to CORRECT conclusions? Even if the guess was not quite correct, it usually was a good hunch that, properly adjusted, will open up new territory.
How much does survivor bias play into this idea that all hunches are at least on the right track? For example, I once wanted to study mathematics in grad school and spent my free time looking for interesting patterns. I ended up only doing one not very interesting presentation as an undergrad. Because of that and a middling score on the GMAT, I decided a math Phd wasn't a good choice for me.
But shouldn't people with a similar experience still included when looking at how "mathematicians" guess? Are mathematicians just more likely to have been people that had a good guess and therefore decided to keep studying mathematics?
So what I want to know is: how good are their guesses beyond their first big win? Maybe even in a topic unrelated to their previous work? That'd be the more useful question if you want to find productive habits or traits.
And examining famous outliers isn't very helpful except for a biography.
Disclaimer: I haven't finished the article. It's long and rambling, so I might not finish it.
I'm an applied mathematician. I don't do much theorem proving in my day to day work, though I have worked closely with people who do. In my experience, the "guessing" is just half of what makes really good mathematicians really good. The other part is verification: taking intuition and turning it into proof. Most of the time the proof works out exactly as one would expect, occasionally one discovers something -- maybe a technical wrinkle that can be overcome with a bit more effort but requires a new idea, maybe a major conceptual gap overlooked before. I'm not a professional programmer or an electrical engineer, but I've been around people who were very good at those things, and being able to think simultaneously at a conceptual level and dig down to the nitty gritty details are both important.
In short, good mathematicians make correct guesses because they've been down the wrong path many times, and know where their intuition may lead them astray.
The thing about math (and most other subjects) is that at the undergrad level, we teach mostly the nitty gritty stuff. In PhD courses we gradually start to teach how to think at a more conceptual level. But the part that I find both hardest and most rewarding -- taking vaguely understood ideas and giving them precise mathematical formulation -- isn't done much in math departments.
Edit: I know I'm not addressing your point directly, which I think is a good one. More clarifying some of the points raised in the article from my perspective.
I think the quality of guesses is affected by prior experience. This seems obvious, but it means that hunches won't help a new grad find interesting problems. It also means that a lifelong mathematician has a lot of failed and successful attack vectors for problems of a certain class. Not only by virtue of experience, but also because the field and competition to publish and produce results makes it necessary to cultivate a sharp instinct for viable solutions to avoid wasting significant time ... and it weeds out those that do not cultivate this skill. (Note, weeding out includes making 7 figures in industry so dont consider that a dig).
You can call this survivorship bias, and it may be that successful mathematicians are being weeded out by factors other than their ability to quickly solve problems.
Think about chess players: 95% of the time Magnus Carlsen immediately spots the right move to play (by his own numbers). The work is then backing it up by analysis or search.
If we didn't have a "hunch" in mathematics guiding us towards proofs or interesting results, we would be doing a completely random walk, or be brute forcing the search space, which would never lead anywhere interesting.
Sure, this hunch may be stronger with some people than others, but it mostly just comes from experience.
I don't understand your objection. Did you have bad hunches? Did you check that your bad hunches were actually meaningless?
If you failed at becoming a mathematician because you have usually bad hunches, why should you be included as a counterexample to a claim that mathematicians have usually good hunches? P=>Q is fully compatible with (not-P AND not-Q).
Something I found useful and related was George Polya's book series "Mathematics and Plausible Reasoning" [1]. It goes into how we make "good" guesses in mathematics and how to improve the way we recognize patterns. If you are curious about this subject and want to improve your "mathematical intuition", this series helped me tremendously.
He's hanging a lot on the taxicab story, but it turns out that he had done some investigation of that class of numbers some years before, and had written down 1729 in one of his notebooks in that context. It's just as plausible that he just remembered it.
How much does survivor bias play into this idea that all hunches are at least on the right track? For example, I once wanted to study mathematics in grad school and spent my free time looking for interesting patterns. I ended up only doing one not very interesting presentation as an undergrad. Because of that and a middling score on the GMAT, I decided a math Phd wasn't a good choice for me.
But shouldn't people with a similar experience still included when looking at how "mathematicians" guess? Are mathematicians just more likely to have been people that had a good guess and therefore decided to keep studying mathematics?
So what I want to know is: how good are their guesses beyond their first big win? Maybe even in a topic unrelated to their previous work? That'd be the more useful question if you want to find productive habits or traits.
And examining famous outliers isn't very helpful except for a biography.
Disclaimer: I haven't finished the article. It's long and rambling, so I might not finish it.