Re: Ch. 1 - The Mystery of Being
Though I haven't read the book, my Ph.D. in physics makes me inclined to join the discussion anyway.
The point is probably that traditional philosophy, which largely ignores physics theories like quantum mechanics, can't be taken seriously as a way of understand the nature of reality.
Actually, that sounds a bit poetic for Feynman. That statement is more a way of translating the path-integral formulation into words.
Actually, there are (at least) three different ways of expressing the equations of quantum mechanics: matrices / Hilbert spaces, operators / differential equations, and path integrals / sum-over-histories. However, they are all mathematically equivalent, which implies that the correctness of one implies the correctness of the others.
However, there's definite value in having multiple mathematical representations. From a technical standpoint, different formulations are more helpful in modeling different physical situations. Philosophically, each mathematical structure provides a different perspective of the physical world.