High-level overview of Razborov's method of approximations What is Razborov's method of approximations? Can someone give a high-level overview and the intuition behind it?
Let $f$ be a Boolean function on $n$-bits. Let $Z = f^{-1}(0) \subseteq 2^n$. Let $C$ be circuit on n bits and of size $m$ and gates $g_1, \ldots, g_m$. $g_i$ also denotes the function on $n$-bits computed by subcircuit with $g_i$ as the last gate. The first $n$ gates are for the input $x_1, \ldots, x_n$. The goal is to show that $C$ of size $m$ cannot compute $f$. Consider all computations of $C$ on inputs from $Z$. A computation assigns values to outputs of gates. Let $B$ be the Boolean algebra of $P(Z)$.

The idea is to consider for any function  $g$ on $n$-bits how well it approximates $f$ on $Z$. Let $||g|| = \{w \in Z \mid g(w) \neq 0 \}$.

For an ultrafilter $F \subseteq B$ we can define a new computation by ultraproduct from it: $c(g_i) = 0$ iff $||g_i|| \not\in F$. Because an ultrafilter is essentially a set of consistent computations for 0 values the resulting $c$ is a valid computation. It would follow that $f(c_1, \ldots, c_n) = 0$. We created a new computation from existing ones. Since all ultrafilters on finite sets are principal $c_1,\ldots,c_n \in Z$. This works for any circuit, we have not exploited the fact that the circuit is of size $m$. 

The next idea is now to exploit the finiteness of the circuit to construct a new input that is outside $Z$ and $f(w) \neq 0$ but the circuit does not notice because of its limited size and therefore still outputs 0. So it does not compute $f$.

We need to relax the definition of ultrafilter so that we can get an input outside $Z$. In place of ultrafilters we use upward-closed subsets of $B$ ($a \in F$ and $a \subseteq b$ implies $b \in F$) that preserve meets($a,b \in F$ implies $a \cap b \in F$).

Let $W_F = \{w \in 2^n \mid w_i = 0 \to ||\lnot x_i|| \in F, w_i \neq 0 \to ||x_i|| \in F\}$. $W_F$ is the set of inputs consistent with $F$. If $F$ is prime ($a \cup b \in F$ implies $a \in F$ or $b \in F$) and nonfull ($\emptyset \not\in F$) then for each $i$, $F$ contains either  $||x_i||$ or $||\lnot x_i||$ and $W_F$ contains only a single input.

We are going to relax preservation of meets. In place of all meets in the Boolean algebra we will preserve a small number of them. Let $|f|$ be the smallest number $k$ of meets $M = (a_1 \cap b_1, \ldots, a_k \cap b_k)$ such that for all upward-closed, nonfull, $M$-preserving $F$, $W_F \subseteq Z$. 

Let $m$ be the circuit complexity of $f$. Razborov proved that $\frac{1}{2}|f| \leq m \leq O(|f|^3 + n^3)$.

Note that this inequality holds for all functions. To prove a circuit size lower bound $m$ show that for all $m$-meets $M$, there is a $F$ that satisfies the conditions but its $W_F$ is not contained in $Z$. Moreover any strong circuit lower bound can be proven by this method because of the second inequality.

The actual part of a circuit lower bound proof is to show that for given $m$, for any $m$-meets there is such an $F$. In the case of monotone circuits the condition about $W_F$ simplifies to $w_i \neq 0 \to ||x_i|| \in F$ so coming up with $F$ is easier. 

Alexander Razborov, On the Method of Approximation, 1989.
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Mauricio Karchmer, On Proving Lower Bounds for Circuit Size, 1995.

Tim Gowers, Razborov's method of approximation, 2009. pdf