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BTW, my given name is Robert Lewis; people call me Bob Lewis. That's "B" as in Banach, "o" as in "operator", "L" as in "Labochevsky", an "ew" diphthong pronounced generally as the "u" in "grand U-nified theory, "i" as in "integral" and "s" as in "simplex". Any questions?
@Mitch: The OP asked, "By this I really mean the science of how computers work." This is not a purely theoretical issue, as evidenced by our participation in MO!
@Timothy Chow--can you give a reference, hopefully available online, for undecidability of evaluation of integrals? I've heard of such things before and would like to know more.
To do integrals like Feynman did, you have to practice a lot. You have to just do integral after integral by this method until you get a gut feel for it. You have to love evaluating integrals--grungy calculations involving calculus and algebra. I'm pretty sure Feynman started doing this before he went to MIT. So if you want to catch up, have at it.
In the first paragraph do you mean to say $M(f,t):=\langle \bar f \cdot(f \circ T_{t}) \rangle$? because that's how you use it in the second paragraph. In other words, what is A$?
In the second paragraph of the question, you define $d(p,q)$ as the infimum of$L(\gamma)$ for $\gamma$ lying in $R^{n}$. This appears to trivialize your question, since then $d(p,q) = |p - q|$. Are you sure, as Deane Yang suggested, you don't want $\gamma$ to be a curve in $M$?
