Vladimir_Nesov comments on A Sketch of an Anti-Realist Metaethics - Less Wrong
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Just to clarify, I wanted to point out that sentences are not the same category as beliefs (which in local parlance are anticipations of observations). There can be gramatically correct sentences which don't constrain anticipations at all, and not only the self-referential cases. All mathematical statements somehow fall in this category, just imagine, what observations one anticipates because believing "the empty set is an empty set". (The thing is a little complicated with mathematical statements because, at least for the more complicated theorems, believing in them causes the anticipation of being able to derive them using valid inference rules.) Mathematical statements are sometimes (often) useful for deriving propositions about the external world, but themselves don't refer to it. Without further analysing morality, it seems plausible that morality defined as system of propositions works similarly to math (whatever standards of morality are chosen).
The question is, whether this should be included into the ideal map. To peruse the analogy with customary geographic maps, mathematical statements would refer to descriptions of regularities about the map, such as "if three contour lines make nested closed circles, the middle one corresponds to height between the heights of the outermost one and the innermost one". Such facts aren't needed to read the map and are not written there.
If my remark seemed snarky, I apologise.
What's the distinction between the two? (Useful for deriving propositions about smth vs. referring.)
The derived "propositions about" are distinct from the mathematical statements per se. For example:
Mathematical statement: "2+2 = 4" (nothing more than a theorem in a formal system; no inherent reference to the external world).
Statement about the world: "by the correspondence between mathematical statements and statements about the world given by the particular model we are using, the mathematical statement '2+2=4' predicts that combining two apples with two apples will yield four apples".