What does Hamming mean by "we live in $L_2$"? In 1980, Hamming wrote [0]: 

We live in what the mathematicians call $L_2$ — the sum of the squares of the two sides of a right triangle gives the square of the hypotenuse.

What does $L_2$ means in this context? Note: 2 is written as a subscript for L.
Obviously, the part following the dash refers to the Pythagorean theorem.

[0] Richard W. Hamming, The Unreasonable Effectiveness of Mathematics, The American Mathematical Monthly, Volume 87, Number 2, February 1980.
 A: $L_2$ here means the so-called "$L_2$ norm". A norm is a way to give a length to a vector. If we have vector $v=(v_1, ..., v_n)$, its length in the $L_p$ norm is $||v||_p=(|v_1|^p + |v_2|^p + ... + |v_n|^p)^{1/p}$. For $p=2$ we get the length as given by the Pythagorean theorem. In particular, this is the way we see distance in our world.
A: $L_2$ is a norm on vector spaces. Very loosely speaking, given two orthogonal vectors $u$ and $v$ in a vector space, a norm on that space is a notion of length in that before space, and the the difference between the $L_a$ norms is the relationship between the lengths of $u, v$ and $u+v$.
$L_2$ is the norm where the relationship is given by the Pythagorean theorem. $L_1$ is the norm where the length of the sum is equal to the sum of the lengths. And $L_\infty$ is the norm where the length of the sum is the length of the longest of the two.
A: It means that distance in our world is $L^{2}$ metric, i.e. Euclidean norm, which satisfies the Pythagorean theorem. For example, if a right triangle with base length 3 and height 4, then we know that its hypotenuse has length 5, by the Pythagorean theorem. 
More generally, if we have two point $(x_{1}, y_{1}, z_{1})$ and $(x_{2}, y_{2}, z_{2})$ are given in our real world (you may have to set a coordinate for this), then the distance between two points is given by 
$$
((x_{1}-x_{2})^{2} + (y_{1}-y_{2})^{2} + (z_{1}-z_{2})^{2})^{1/2}
$$
by the Pythagorean theorem. 
In general, we can define $L^{p}$ metric for $1\leq p \leq \infty$, which is given by 
$$
d_{p}(P_{1}, P_{2}) = ((x_{1}-x_{2})^{p} + (y_{1}-y_{2})^{p}+(z_{1}-z_{2})^{p})^{1/p}
$$
for $1\leq p<\infty$ and 
$$
d_{\infty}(P_{1}, P_{2}) = \max\{|x_{1}-x_{2}|, |y_{1}-y_{2}|, |z_{1}-z_{2}|).
$$
In honest, the Hamming's claim is not true for some situations. For example, we can draw a triangle on the earth where all the length of the sides are equal but the angles are all 90 degrees. This situation occures because earth is not flat, which is a special case of non-euclidean geometry. 
