TITLE: Why is Milnor K-theory not ad hoc?
QUESTION [37 upvotes]: When Milnor introduced in "Algebraic K-Theory and Quadratic Forms" the Milnor K-groups he said that his definition is motivated by Matsumoto's presentation of algebraic $K_2(k)$ for a field $k$ but is in the end purely ad hoc for $n \geq 3$. My questions are:

What exactly could Milnor prove with these $K$-groups? What was his motivation except for Matsumoto's theorem?
Why did this ad hoc definition become so important? Why is it so natural?

REPLY [19 votes]: Also, about the motivations of Milnor, it is quite natural to try to understand the Witt ring of a field, classifying quadratic forms over this field (in char not 2). This ring has a natural filtration by the fundamental ideal, and it is natural to try to understand the associated graded ring, which is simpler than the Witt ring. One approach is to understand it by generators and relations. The relations defining Milnor's K-theory are elementary ones obviously satisfied in the graded Witt ring, and there are very few of them. Milnor's conjecture (now a theorem) says that Milnor K-theory mod 2 is isomorphic to that graded Witt ring. It is equivalent to the formulation with étale cohomology, but probably, an important part of Milnor's original motivation was about quadratic forms, as one can see in his original paper.
It is quite surprising that, a posteriori, this simple K-theory appears as such a fundamental object in intersection theory. There is a nice (and seminal) paper by Totaro explaining in rather elementary and geometric terms the simplest case of the connexion between Milnor K-theory and higher Chow groups, mentioned by Denis-Charles Cisinski. 
Milnor K-theory is the Simplest Part of Algebraic K-Theory, K-theory 6, 177-189, 1992