
Take It to the Limit (2010) - drjohnson
http://opinionator.blogs.nytimes.com/2010/04/04/take-it-to-the-limit
======
quotient
I'm a mathematician, and I actually hadn't seen the proof that $A = \pi r^2$
via showing that the shape formed by that particular arrangement of the slices
of the circle becomes a square as the number of slices is taken to infinity.
That's a very cool proof.

~~~
anon4
He didn't prove the wiggly-sided paralellogram -> rectangle transition when
the angle of each slice goes to 0. He also did not prove that he is allowed to
take the angle to 0. I can grant him that any angle > 0 results in a less and
less squigglier paralellogram with more and more vertical sides with an
obvious proof, but that doesn't prove either that the limit is a rectangle, or
that reasoning at the limit is transitive to reasoning about the circle.

I don't doubt the result or the method - they both seem intuitively correct,
they just aren't proven.

~~~
judk
Good luck seeking proof.

[http://en.m.wikipedia.org/wiki/Regress_argument](http://en.m.wikipedia.org/wiki/Regress_argument)

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omegant
I really love how mathematics takes much more sense when explained this way,
with historical background. It gives the sense of a story, of people
struggling with concepts and generations needed to solve them, not only the
area but also how they didn't have calculus, all this parts they make easier
for me to understand concepts and remember concepts. When you put 2000 years
of mathematic development in front of a person and explain it like it was
obvious, something brakes inside you. I understand that from a certain point
you have to study mathematics the hard way. But for kids and people starting
with mathematics, it´s better a "history-story" approach.

~~~
nhaehnle
I don't know whether the actual history of mathematics is useful for
understanding mathematics.

However, I do believe that a plausible story of how somebody came up with
concepts is very important. It can be tempting to structure a mathematics
lecture in a very sober/bland style of definitions followed by theorems with
proof. However, definitions usually arose because somebody was trying to solve
a particular problem, and this should be part of mathematics teaching.

It is usually said that "mathematics is not a spectator sport". To fully learn
it, you have to put yourself into the shoes of somebody who invents certain
definitions and seeks out certain theorems because they have some ulterior
question that motivates them.

I would go beyond what you wrote by saying that this applies to advanced
mathematics as well. The truly great papers tend to lucidly lay out the
thought processes that explain _why_ definitions and proof steps are chosen to
be of a certain form.

~~~
omegant
For me it is. For example, I read "Fermat´s last theorem". The great thing is
that the author tells in a vivid way, how different mathematicians attacked
different parts of the theorem, starting with Fermat, up to the present . He
also explains how the tools developed along the time found real world
applications.

I can not say that I understand the concepts explained at the book, but now I
feel that I could even start learning Number theory (well maybe this is a
stretch, but you get the idea)

On a side note please for give my parent comment. I was writing from the
Iphone seated at a plane, so I had to hurry and didn't have time for
corrections.

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Jun8
If you're interested in pi, the book _A History of Pi_ by Petr Beckman is a
must read, not only for the good information but also for his nonconventional
approach.

Note that Archimedes used a 96-sided polygon to estimate bounds on the value
of pi ([http://en.wikipedia.org/wiki/Pi](http://en.wikipedia.org/wiki/Pi))!
This was quite a feat since arithmetic calculations and algebraic was one area
that ancient Greeks were weak at.

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mturmon
Strogatz has written a series of such pieces for the NYT. He's very good. The
series is at [http://opinionator.blogs.nytimes.com/category/steven-
strogat...](http://opinionator.blogs.nytimes.com/category/steven-strogatz/)
and one nice piece from it is
[http://opinionator.blogs.nytimes.com/2010/03/14/square-
danci...](http://opinionator.blogs.nytimes.com/2010/03/14/square-dancing/)

Revised to add: there is at least one subsequent series as well, at
[http://opinionator.blogs.nytimes.com/category/me-myself-
and-...](http://opinionator.blogs.nytimes.com/category/me-myself-and-math/)

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SanderMak
If you're fascinated by infinity, you should check out 'Infinity and the mind'
by Rudy Rucker. It really puts the concept of infinity in it's historical
perspective and even contains a unique conversation between Rucker and Godel.
(shameless plug: I wrote a review on my blog
[http://branchandbound.net/blog/bookreview/2013/01/bookreview...](http://branchandbound.net/blog/bookreview/2013/01/bookreview-
infinity-and-the-mind/))

~~~
gala8y
Also, there is a very good, popular movie by BBC Four 'Dangerous Knowledge'
[0].

"Beneath the surface of the world, are the rules of science. But beneath them,
there is a far deeper set of rules – a matrix of pure mathematics which
explains the nature of the rules of science and how it is way we can
understand them in the first place. In this one-off documentary, David Malone
looks at four brilliant mathematicians – Georg Cantor, Ludwig Boltzmann, Kurt
Gödel and Alan Turing – whose genius has profoundly affected us, but which
tragically drove them insane and eventually led to them all committing
suicide. The film begins with Georg Cantor, the great mathematician whose work
proved to be the foundation for much of the 20th-century mathematics. He
believed he was God’s messenger and was eventually driven insane trying to
prove his theories of infinity."

[0] [http://watchdocumentary.org/watch/dangerous-knowledge-
video_...](http://watchdocumentary.org/watch/dangerous-knowledge-
video_417b79d43.html)

~~~
ColinWright
I beg to differ - this "documentary" is complete crap.

Some of the math is acceptable, some of it is atrocious, and the assertions
that thinking about infinity is what drove these men insane is just completely
bonkers.

Tabloid drivel at its absolute worst.

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downer73
I can't help but think of the movie Scarface when I see this title.

