As I have been told, the normal distribution is special because of 4 criteria:

1) The errors do not depend on the coordinate system.

2) Errors in perpendicular directions are independent.

3) Large errors are less likely than small errors.

4) The sum of the distribution the (CDF) over the infinite domain equals 1.

My question is: Aren't there a lot of bell-shaped curves out there that meet these criteria?

For example $\displaystyle \frac{1}{\pi(1+x^2)}$ meets the three first assumptions plus

$\displaystyle \int_{-\infty}^{\infty} \frac{1}{\pi(1+x^2)}=1$

This one is also a lot more tractable. So why is the normal distribution considered so unique and special?