Insanely Long Proofs


There are theorems whose shortest proof is insanely long. In 1936 Kurt Gödel published an abstract called “On the length of proofs”, which makes essentially this claim.

But what does ‘insanely long’ mean?

To get warmed up, let’s talk about some long proofs.

Long proofs

You’ve surely heard of the quadratic formula, which lets you solve

$latex a x^2 + b x + c = 0 $

with the help of a square root:

$latex displaystyle{ x = frac{-b pm sqrt{b^2 – 4 a c}}{2 a} }$

There’s a similar but more complicated ‘cubic formula’ that lets you solve cubic equations, like this:

$latex a x^3 + b x^2 + c x + d = 0 $

The cubic formula involves both square roots and cube roots. There’s also a ‘quartic formula’ for equations of degree 4, like this:

$latex a x^4 + b x^3 + c x^2 + d…

View original post 2,644 more words


Leave a Reply

Fill in your details below or click an icon to log in: Logo

You are commenting using your account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s