

Mathematics is full of wonderful but relatively unknown theorems - joeyespo
http://1ucasvb.tumblr.com/post/90160448163/mathematics-is-full-of-wonderful-but-relatively

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mneary
My first thought when I read this title was that there should be a site like
Hoogle for mathematical proofs, where one could search by type signature for
existing proofs. I'm not sure if people often think of proofs in terms of
their type signature, but by Curry-Howard it would be doable.

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infruset
This is a complicated problem. I know some Coq developers are trying to build
a search tool, but the problem is there are many ways to state a theorem and
you want to not only find statements that exactly correspond to your request,
but also those that are convertible to it. And there it gets ugly.

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curiousHacker
I completely agree about the interconnectedness of different branches of
mathematics and think it is difficult to visualize and intuit because we don't
yet have the language.

My guess: Eventually, as we begin to understand mathematics better, we will
elucidate the underlying properties of theorems and there will be better
"tagging"; the underlying language must also change. This will not happen for
a minimum of 30+ years as new advances must be made and institutional faculty
will object lest their life be rendered irrelevant.

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juliangamble
I saw a talk that said the branch of Mathematics that dealt with the
interconnectedness of Mathematics was called 'Category Theory'. You can see it
here: [http://vimeo.com/17207564](http://vimeo.com/17207564) and Part 2 here:
[https://www.youtube.com/watch?v=yilkBvVDB_w](https://www.youtube.com/watch?v=yilkBvVDB_w)

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gosub
Category theory by Tom LaGatta[1] is also a nice introduction. He suggests
taking a look at the nLab website[2], a wiki of Maths and Physics from the
categorical point of view.

[1]
[https://www.youtube.com/watch?v=o6L6XeNdd_k](https://www.youtube.com/watch?v=o6L6XeNdd_k)
[2] [http://ncatlab.org/](http://ncatlab.org/)

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jonsen
Related:
[http://en.wikipedia.org/wiki/Visual_calculus](http://en.wikipedia.org/wiki/Visual_calculus)

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j2kun
How can a theorem be "poorly" used?

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mdlthree
[http://en.wikipedia.org/wiki/Central_limit_theorem](http://en.wikipedia.org/wiki/Central_limit_theorem)

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j2kun
If you use the central limit theorem truly as a theorem, then you're using it
correctly. Either a theorem applies or it does not. That's the point.

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drvortex
The way this is presented in the blog post is as if this is a really
insightful but now under-appreciated gem of mathematics.

But all that description is saying is, when a 2D shape is made by rotation,
its area is the multiple of the 1D generator and the 1D path it takes? You
don't say! And then volumes! when a 3D volume is made, its a 2D shape going
through 1D path ...wow.

Forgive me if I am not impressed. This is not an unknown theorem but a trivial
mathematical fact that one learns somewhere around Grade 7 in school. Of
course, Pappus deserves credit for discovering in 300 AD, and the paper by the
Goodmans is nice to have a general formal proof of. But even that paper
concedes that they are simply proving a general proof for completeness.

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lucasvb
But it isn't that simple. The insight is in the importance of using the
centroid of the shape and the path it traces. It can't be any other point for
the simpler form to work.

That's the cleverness, and the reason the centroid has that property is
interesting enough, and not immediately obvious. But by understanding why it
works, we can see we can apply it for much more general cases than surfaces of
revolution, which is usually the only treatment the theorem gets out there.
The purpose of the post was to illustrate the generality of the theorem.

~~~
drvortex
The centroid is the average of all points of a shape. It is therefore the only
point that contains information about the entire shape. Any other point
doesn't. Naturally, since paths are distances and can also be averaged in the
same way, it is actually quite obvious that the path travelled by the centroid
of a shape is also the average path of all the points on that shape. In fact,
it directly follows from the definition of a centroid consider that the path
of a shape is the addition of one more dimensional coordinate to each of the
points on that shape.

Sorry,I still do not see what makes this so special.

