[genezeta]

The name, fixed point, is a more generic or general concept in mathematics.

Given a function f, we say that an element c in the domain of f is a fixed point for f if it satisfies that f(c) = c. In a different nomenclature, if you consider f a "transformation", then c satisfies that it "remains fixed through the transformation".

The fixed-point combinator is called so, because it "produces fixed-points for a function". That is, given a function f, then FPC(f) is a fixed-point for f. So, if we call c = FPC(f), then f(c) = c. Or, more classically f(FPC(f)) = FPC(f) or f(Y(f)) = Y(f).

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Note that when talking about fixed points in general, the domain of the f function is whatever -but usually you may have studied it with numbers-. But when talking about the fixed-point combinator the domain is functions themselves.

[youzicha]

Sure I understand that, but nevertheless I have seen at least 3 people who were confused in the same way. (c.f. https://youzicha.tumblr.com/post/728810860450136065). I think the issue is that if you see an equation f(c) = c you think about trying to evaluate the function f at the point c, since that's what you do in almost all other situations involving a function call.