You're wrong about LC being (necessarily) Lisp-y. Church's original notation for the lambda calculus was not based on S-expressions. (Those were invented 30+ years later by John McCarthy.) I only used s-expression notation for the factorial example to drive home the point that you can actually run this code.
Ah, I did not mean to imply that LC is Lisp-y. I just meant that the notation you used is valid lisp (as far as I can tell) and can therefore be read, learned about, executed and deconstructed rather easily.
I was downvoted a few times, but I actually think the line "it's just lisp" is a better response to the commenters question than any of the other comments that attempt to explain what LC is. Why explain an algorithmic construct to someone in mathematical notation, if there's an executable notation that's easier on the eyes (at least for a programmer)?
I certainly used Lisp notation, but if you think it's "just Lisp" then you're kind of missing the point. It's not just "just Lisp", it's "just lambda". There is nothing else in there, no primitives, no numbers, no CAR, CDR, CONS, COND or QUOTE. And, to drive home the point that the Y combinator is in there, no DEFUN, LABELS or LETREC. (Well, OK, there's an IF in there, but even that could be gotten rid of with a lazy lambda.)
Basically it looks so complicated because it lacks as much structure as a language can possibly lack while still being turing complete. This makes it easy to map to lambda-calculus, something mathematicians use for reasoning about programming.
edit: am I wrong?