I don't know if these ramblings will help, but here are some thoughts :) First of all, the Y-combinator is a function of a function of a function (of a number)

I don't know if these ramblings will help, but here are some thoughts :) First of all, the Y-combinator is a function of a function of a function (of a number), so we're three levels deep, which is part of what causes the head-spinning, I think. So ignore the Y-combinator for the moment. Simply consider the function of a function (of a number) F: F = function (f) { return function(n) { return (n == 0) ? 1 : (n*f(n - 1)); }; }; Here the argument f is a function number -> number, and the return value from F is itself also a function number -> number. We can pass whatever functions from numbers to numbers we want to F to make new functions from numbers to numbers. E.g., we could pass in exf1 = function(n) { return n }; and we'd have F(exf1) = function(n) { return (n == 0) ? 1 : (n*(n - 1)) }; Or we could pass in something like exf2 = function(n) { return 1; } in which case we'd get F(exf2) = function(n) { (n == 0) 1 : n - 1 }; So now imagine god gives us the factorial function: fact: number -> number fact(n) = n*(n - 1)*...*1 Then if we pass this in to F we get F(fact) = function(n) { return n == 0 ? 1 : n*fact(n - 1) } But the above is simply a recursive definition of fact! So F(fact) = fact, i.e., fact is a fixed point of F. Now consider the Y combinator. We know the Y combinator gives us a fixed point of a functional, i.e., F(Y(F)) = Y(F) (Note Y(F) is simply a function, even though the argument to Y is a functional, i.e., Y: functional -> function or Y: (function -> function) -> function or Y: ((number -> number) -> (number -> number)) -> (number -> number)) So we know that one possible value for Y(F) is fact. The article makes the claim that there is only one fixed point of F, which means Y(f) must actually be fact, but to be honest, I'm not sure why we know there can't be other fixed points of F which Y(F) might give us (though maybe this is obvious and I'm being stupid :). But anyway we can tell by tracing through the definition of Y that it we really do get Y(F) = fact.