Let $p,q$ be two points. On pg 671 of "Road to Reality", Penrose says that integrating the amplitudes of all paths between $p$ and $q$ would be infinite. Hence, we need the concept of a Feynman propagator.
In principle. we are to form the sum (integral) of all $e^{iS/\hbar}$ for paths originating at $p$ and terminating at $q$, but this is certainly wildly divergent as it stands. On the other hand, we can suppose that the sum $K(p,q)$ has some kind of mathematical existence, and we ask what formal algebraic and differential properties….including an appropriate positive frequency condition..fix the form of $K(p,q)$ uniquely…and this gives us the Feynman propagator we seek.

From what I understand, $K(p,q)$ is an integral, and hence should be a number (in this case, $\infty$). What kinds of algebraic properties can such a number possess? What does it having positive frequency mean?

What is the Feynman propagator? Is it a path between $p$ and $q$, or is it the number $K(p,q)$?
EDIT:
3. Penrose then says
"in the case of a Dirac particle, the momentumspace propagator turns out to take the form $i(\displaystyle{\not} P – M+i\epsilon)^{1}$, where $\displaystyle{\not}P=\gamma^aP_a$, the quantity $P_a$ being the $4$momentum that the particle happens to have, for the chosen path under consideration."
What chosen path is being talked about? It seems the propagator is a function of $p,q$, and not a chosen path between $p,q$.
Best Answer
Given two points $p$ and $q$, the propagator $K$ spits out a (complex) number $K(p, q)$. As a function of $p$ and $q$, yes $K$ might take infinite values. But there is nothing weird about functions that take infinite values satisfying differential or algebraic equations. For example the function $f$ mapping $x$ to $f(x) = 1/x$ satisfies the equation \begin{equation} f'(x) = \frac{1}{x}f(x) \end{equation} even though $f(x)$ is infinite at $0$.
The positive frequency condition Penrose mentions (he introduces it earlier in the book in Section 24.3, although he doesn't really make it clear how it applies to the propagator) basically says that we should consider paths between the points $p$ and $q$ from the earlier point to the later point. (Things should go in time order.)
Edit: I suggest you take a look at the Wikipedia section on the Feynman propagator. Notice that the (real space) propagator is written as as an integral over momentum $\mathsf{p}$. Roughly speaking this is an integral over the different momenta the particle can have as it propagates between the two real space points.
Notice that in the following section, the momentum space propagator is the integrand of the real space propagator (up to an exponential factor), without the integral sign. The momentum space propagator is defined at a definite value of momentum $\mathsf{p}$ the same way the real space propagator is defined at definite initial and final real space points. We could write the momentum space propagator as an integral as well, but it would be an integral over real space instead of momentum space.