Questions:

1. What is the probability amplitude ("proclivity", a complex number) associated with each path?

2. How can we sum over an infinite number of paths?

3. Is it correct?  (Equivalent to Schrodinger's time development equation.)

4. How can this scheme possibly have a classical limit?
   (What if we played baseball and the ball took every path from bat to fielder?)

Answers:

1. a) For each path, calculate the classical action S [time integral of (KE - PE)]
   b) Amplitude is proportional to exp{i S/\hbar}

2. Interesting technical question about path integration:
   a) Consider representative paths, find the proportionality constant for each consideration.
   b) For considerations where the number of paths approach infinity, the proportionality constant vanishes to zero.
   c) Only one scheme produces a finite sum amplitude.

3. It's not obvious, but it is correct ... see Feynman and Hibbs, "Quantum Mechanics and Path Integrals".

4. It makes sense that paths near the classical path would have bigger magnitude amplitudes ...
   but no!  Every path has the same magnitude amplitude.  Even the crazy loop-de-loops!

   Instead, the classical limit comes through interference.

   We use the time-honored strategy of "divide and conquer": consider a pencil of paths clustered around
   any particular path.  We sum over paths within pencils, then sum over pencils.

   For most pencils:
      the variation in action from one path to another means that
      action/\hbar varies wildly from one path to another meaning that
      the amplitude phases vary all over the map meaning
      destructive interference!

   But if you have a pencil where
      the variation in action from one path to another is minute, meaning that
      action/\hbar varies little from one path to another meaning that
      the amplitude phases are all about the same meaning
      constructive interference!

   In the classical limit only that one constructive pencil will be relevant.
   It falls where the action doesn't vary from one path to its neighbor, that is
   it falls where the action is a maximum. 

   We have explained the classical "principle of least action".