10 Awesome Theorems & Result | Hacker News Reader

10 Awesome Theorems & Result

algorithm.co.il

30 pointsadmp9 comments

So glad to see "Stokes' theorem"! Back in college, I devoured the books "A Course in Mathematics for Students of Physics" by Paul Bamberg and Schlomo Sternberg that covered exterior calculus. It was awesome to see kirchoff's laws in circuits, flux and loop integral laws in electromagnetism all take the same form.

My favorites:

- Diffie-Hellman Key Exchange - I really do find it wonderfully counter intuitive that you can have two systems agree on a secret value while communicating completely "in the clear" - http://en.wikipedia.org/wiki/Diffie%E2%80%93Hellman_key_exch...

- That you can implement recursion in the untyped lambda calculus - the best known being the Y Combinator (!) http://en.wikipedia.org/wiki/Fixed_point_combinator

There was a thread on Quora recently asking what the most beautiful equation was http://www.quora.com/Mathematics/What-is-the-most-beautiful-..., which while it is obviously the Einstein field equations was nice to read through in a similar vein.

My favorite was always Fubini's theorem in 3D calculus the result of which is basically "meh, it doesn't matter which order you integrate"

I'd count among my favourites:

* the Schroeder-Bernstein theorem in set theory

* the fact that the axiom of choice is equivalent to the statement that every set is well-ordered: how can something so intuitively correct and something so intuitively wrong be equivalent?

* the general notion of dualism in category theory. Two for the price of one!

* the ways fixpoints can be used in TCS: the Y combinator (heh), and the Kleene and Knaster-Tarski theorems. The best part is that they're all intuitively the same.

Excellent choices. (The aforementioned sentence does not require use of AoC ;)

In fact, the moments he describes in #3 are partly why I'm currently pursuing a Math and not a CS degree.

As a guy pursuing a CS degree, I'm compelled to be a constructivist (what use is a proof if you can't get an algorithm out of it?) and reject both AC and the well-ordering theorem. Find me a well-ordering of the reals and then we'll talk. :)

edit: speaking of proofs and algorithms, how could I forget the Curry-Howard correspondence. It just makes so much sense.

eh eh :)

About an hour after taking a set theory exam it dawned on me that a proof I gave is correct iff the continuum hypothesis is true. The distance between a Fields medal and a F was never so small :)

edit: you made me wiki it. Funny, I don't seem to know/remember it. Thanks, seems interesting :)

Regarding theoretical proofs... What's the point of an algorithm if you don't propel its use through a viable product? I'm currently (concurrently with the studies) in a CS research position and it is a difficulty for me... Don't get me wrong, I love my job, but for me if there is any downside then this is it.

«The Axiom of Choice is obviously true; the Well Ordering Principle is obviously false; and who can tell about Zorn's Lemma?»

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