I understand why people might think this was a snarky and downvote worthy comment with an obvious answer, but I greatly appreciated this comment and upvoted it. That is to say, it fits a pattern for questions the answers of which are obvious to others, though the answer was not obvious to me.
What’s worse, at first thought, within five seconds of thinking about it, the answer seemed obvious to me until I thought about it a bit more. Even though I have tentatively settled upon an answer basically the same as the one I thought up in the first five seconds, I believe that that first thought was insufficiently founded, grounded, and justified until I thought about it.
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.
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”.
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”.
If you build an inference system that outputs statements it proves, or lights up a green (red) light when it proves (disproves) some statement, then your anticipations about what happens should be controlled by the mathematical facts that the inference system reasons about. (More easily, you may find that mathematicians agree with correct statements and disagree with incorrect ones, and you can predict agreement/disagreement from knowledge about correctness.)
If you build an inference system that outputs statements it proves, or lights up a green (red) light when it proves (disproves) some statement, then your anticipations about what happens should be controlled by the mathematical facts that the inference system reasons about.
That’s why I have said “[t]he 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”.
What’s the distinction between the two? (Useful for deriving propositions about smth vs. referring.)
The latter are sentences which directly mention the object (“the planet moves along an elliptic trajectory”) while the former are statements that don’t (“an ellipse is a closed curve”). Perhaps a better distinction would be based on the amount of processing between the statement and sensory inputs; on the lowest level we’ll find sentences which directly speak about concrete anticipations (“if I push the switch, I will see light”), the higher level statements would contain abstract words defined in terms of more primitive notions. Such statements could be unpacked to gain a lower-level description by writing out the definition explicitly (“the crystal has O_h symmetry” into “if I turn the crystal 90 degress, it will look the same and if I turn it 180 degrees...”). If a statement can be unpacked in finite number of recursions down to the lowest level containing no abstractions, I would say it refers to the external world.
“[t]he 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”.
This doesn’t look to me like a special condition to be excused, but as a clear demonstration that mathematical truths can and do constrain anticipation.
“Directly mentioning” passes the buck of “referring”, you can’t mention a planet directly, the planet itself is not part of the sentence. I don’t see how to make sense of a statement being “unpacked in finite number of recursions down to the lowest level containing no abstractions” (what’s “no abstractions”, what’s “unpacking”, “recursions”?).
(I understand the distinction between how the phrases are commonly used, but there doesn’t appear to be any fundamental or qualitative distinction.)
There has to be a definition of base terms standing for primitive actions, observations and grammatical words (perhaps by a list, to determine what to put on the list would ideally need some experimental research of human cognition). An “abstraction” is then a word not belonging to the base language defined to be identical to some phrase (possibly infinitely long) and used as an abbreviation thereof. By “unpacking” I mean replacing all abstractions by their definitions.
Now I’m confused. Beliefs that don’t correspond to the territory are what we call “wrong”.
Is “this sentence is true” a wrong belief?
Its a classic problem case. I think its semantic function calls itself and so it is meaningless. See here.
I understand why people might think this was a snarky and downvote worthy comment with an obvious answer, but I greatly appreciated this comment and upvoted it. That is to say, it fits a pattern for questions the answers of which are obvious to others, though the answer was not obvious to me.
What’s worse, at first thought, within five seconds of thinking about it, the answer seemed obvious to me until I thought about it a bit more. Even though I have tentatively settled upon an answer basically the same as the one I thought up in the first five seconds, I believe that that first thought was insufficiently founded, grounded, and justified until I thought about it.
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”.
If you build an inference system that outputs statements it proves, or lights up a green (red) light when it proves (disproves) some statement, then your anticipations about what happens should be controlled by the mathematical facts that the inference system reasons about. (More easily, you may find that mathematicians agree with correct statements and disagree with incorrect ones, and you can predict agreement/disagreement from knowledge about correctness.)
That’s why I have said “[t]he 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”.
The latter are sentences which directly mention the object (“the planet moves along an elliptic trajectory”) while the former are statements that don’t (“an ellipse is a closed curve”). Perhaps a better distinction would be based on the amount of processing between the statement and sensory inputs; on the lowest level we’ll find sentences which directly speak about concrete anticipations (“if I push the switch, I will see light”), the higher level statements would contain abstract words defined in terms of more primitive notions. Such statements could be unpacked to gain a lower-level description by writing out the definition explicitly (“the crystal has O_h symmetry” into “if I turn the crystal 90 degress, it will look the same and if I turn it 180 degrees...”). If a statement can be unpacked in finite number of recursions down to the lowest level containing no abstractions, I would say it refers to the external world.
This doesn’t look to me like a special condition to be excused, but as a clear demonstration that mathematical truths can and do constrain anticipation.
“Directly mentioning” passes the buck of “referring”, you can’t mention a planet directly, the planet itself is not part of the sentence. I don’t see how to make sense of a statement being “unpacked in finite number of recursions down to the lowest level containing no abstractions” (what’s “no abstractions”, what’s “unpacking”, “recursions”?).
(I understand the distinction between how the phrases are commonly used, but there doesn’t appear to be any fundamental or qualitative distinction.)
There has to be a definition of base terms standing for primitive actions, observations and grammatical words (perhaps by a list, to determine what to put on the list would ideally need some experimental research of human cognition). An “abstraction” is then a word not belonging to the base language defined to be identical to some phrase (possibly infinitely long) and used as an abbreviation thereof. By “unpacking” I mean replacing all abstractions by their definitions.