

"Problem Solvers" vs "Theory Developers": The Two Cultures of Mathematics. [pdf] - asciilifeform
http://www.dpmms.cam.ac.uk/~wtg10/2cultures.pdf

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Spyckie
This observed pattern can be described by personality typing. I like to think
of it as 2 classifications of how our brain explores - depth-based vs breadth-
based search. Problem solving vs random walk. Directed vs discovered.
Convergent vs divergent, etc. Essentially, one type enjoys digger deeper
(defined by getting closer towards a goal) into a topic, while another enjoys
expanding outward (exploring a topic, no goal required).

The majority of people enjoy doing one type of thinking and hate doing the
other, which causes this culture difference. Depending on the class of
problems that a field has to tackle, the community may be dominant in one or
the other. Mathematics is a field in which a random walk style of exploration
can yield fascinating results, while an engineering field is much more likely
to benefit from a focused approach. Usually, the type of thinking that yields
more results becomes dominant in the community and gets the right to snub the
other.

The difference and contention, I think, arises purely out of the dislike of
doing the type of thinking that you're not naturally inclined to. This dislike
is rooted in unfamiliarity and a strong sense of worth that parallels our
inclinations. The contention is rarely settled by arguments of logic,
efficiency or practicality - it is not that people don't believe in logic, but
rather because the arguments don't address the core issue. The core issue is
our natural inclinations, which at its roots, are innate or at least solidly
ingrained in us by the end of childhood.

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RiderOfGiraffes
I know Tim. In additional to being a phenomenally good mathematician, his
clarity and elegance of expression spills over into his writing and his
lectures. Some people take a dislike to his style, but I can't see why.

He's also a really, really nice guy, and it's been a privilege on occasion to
work with him. My contributions were miniscule, but he was fantastic to work
with. I hope to do so again.

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amichail
Isn't this more of an age thing? Mathematicians start out as problem solvers
and later become theory developers?

As you age, it becomes harder to solve difficult problems but easier to
develop theories due to greater experience.

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slackenerny
I don't think so, based on historical evidence. At least it doesn't seem to be
like before- and after-45 or other midlife age. From all great theory
builders, we rather see like this age being closer to late twenties. They
start out by solving some minor problems, often ambitiously taking the
labourous technical ones, and then boom, out of nowhere new stage-setting
theory, not necessarily much related to previous work. These frameworks open
new hard problems which then are becoming mainstream, while frameworks that
emerge by accretion of facts and experience hardly make such impact. I give
Alain Connes as an example; man steadily, for most of his academic life i.e.
last 30 years, outlines theories at a rate of one per two or three years.

