
Mathematical Concepts: How Do Mathematicians Jump to Conclusions? (1998) - rfreytag
https://www.edge.org/conversation/verena_huber_dyson-on-the-nature-of-mathematical-concepts-why-and-how-do-mathematicians
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vharuck
>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.

~~~
kkylin
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.

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focus2x
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.

[1] [https://www.amazon.com/Mathematics-Plausible-Reasoning-
Two-V...](https://www.amazon.com/Mathematics-Plausible-Reasoning-Two-
Volumes/dp/1614275572)

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empath75
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.

~~~
gowld
> Now return to Ramanujan and see how the first thing that springs to the
> naive eye beholding the number 729 is that adding 81 = 9^2 turns it into 810

The article is bewildering.

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gnatman
This subject is discussed (although from a different angle) at length in
Douglas Hofstadter's book Fluid Concepts and Creative Analogies. [0]

There's also good reading to be done around this subject as it pertains to the
problem of induction. [1]

[0]
[https://en.wikipedia.org/wiki/Fluid_Concepts_and_Creative_An...](https://en.wikipedia.org/wiki/Fluid_Concepts_and_Creative_Analogies)
[1]
[https://en.wikipedia.org/wiki/Problem_of_induction](https://en.wikipedia.org/wiki/Problem_of_induction)

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adamnemecek
This is the idea behind intuitionism.

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dang
Verena was my teacher, one the biggest influences in my life, and a good
friend.

