How does Fermat's principle explain the shape of a lens in Feynman's derivation?

[neworder1]

I have a problem in understanding Feynman's derivation of a lens' shape using Fermat's principle. Feynman writes that we have to choose such a surface of the lens that all optical ways from the source S to the focal point F will have the same length (so all will be taken by the traveling light). However, how do we know that the light chooses its path so as to minimize the distance from S to F and not to another point G, or any other point? When Feynman derives Snell's law while source and target points are fized, it's OK, bu here tHe light doesn't "know" that there is a focal point, it could go anywhere (i.e. destination point is not fixed in any way). So I don't see how we can use Fermat principle here.

 

[mgb_phys]

The same way water 'knows' to flow down hill or electricty 'knows' to flow through the lowest resistance.
Fermat's principle is either the greatest idea in physics - in a way it pre-invented both relativity and quantum mechanics, or it's just a convenient short cut to get the right answers.
It depends on your philosphical veiw of the nature of physics really.