The post inspired me to think about the differences between abstraction types. This is relevant to my own research too, so I got stuck on (re)writing this comment for more than a week. The main idea I converged to: we should analyze abstractions through things which make them useful. The first section below is the most relevant to the OP, it contains 4 potential distinctions between physical and mathematical abstractions.
***
Distinction 1A. A physical abstraction predicts only a specific aspect of a situation or a trivial sum of specific aspects. A mathematical abstraction is useful for proving non-trivial facts about entire situations. Basically the same abstraction can be both physical or mathematical, depending on how it’s used. Examples:
“Ants” is a physical abstraction. “Moving points” is a mathematical abstraction when used to prove a theorem about ants on a stick. After you’ve proved the theorem, you can easily predict how an infinite number of situations would unfold.
“Number of sheep” is a physical abstraction when used to track the number of sheep.[1] “Numbers” is a mathematical abstraction when used to show that you never know when you can split your sheep into equal groups with more than 1 sheep in each (because some numbers are prime and there’s an infinity of primes).
“Forces” is a physical abstraction when used to predict the movement of a pendulum. “Vectors” is a mathematical abstraction when used to prove the hairy ball theorem.
“Programs” is a physical abstraction when used to predict physical computers. “Programs” is a mathematical abstraction when used to prove the insolubility of the halting problem.
“King safety” is a physical abstraction when used to predict one aspect of the chess position (whether the king is likely to get checkmated soon). “King safety” is not a mathematical abstraction when used as a part of an evaluation function, predicting the development of the entire game. Because you can’t use king safety to prove what the evaluation function should be.
“Elastic ball” is a physical abstraction. “Elastic sphere” is a mathematical abstraction when used to prove the Borsuk–Ulam theorem.
Distinction 1B. Physical abstractions refer to groups of objects existing on roughly the same scale and behaving similarly, groups conceptualizable as “singular entities”. Physical abstractions capture properties of those objects which are directly valuable/impactful. Mathematical abstractions do this much more rarely. Their main purpose is to reveal or transfer non-trivial facts about objects.
Referents of “animals”, “minerals”, “cars” exist on roughly the same scale and behave similarly. “Animalness” of an object makes it valuable and impactful. “Animals” can be thought of as a sort of singular entity. Referents of “vectors”, “functions”, “metric spaces” exist all over the place. “Vectorness” of an object doesn’t make it valuable or impactful. Both gravity and noses can be “vectors”, but you can’t think of “gravity + noses” as a coherent entity. Mathematical abstractions are too abstract, they bundle too much dissimilar stuff.[2]
This is a reframing of your Take 4 and suffers from the same problem (e.g. “animals” also exist in books and “real animals + book animals” is not a coherent entity). Though maybe it makes a stronger case that the problem is merely an edge case.[3]
Distinction 1C. Physical abstractions are inside the same causal system, they shape each other by adding causally linked details to each other. Physical abstractions are implicitly reflected in each other, like jigsaw puzzle pieces or the Indra’s net. For example, trees shape animals and animals shape trees. Mathematical abstractions can be combined into a single system too, but they won’t have the same interconnectedness. (This is a variation on Takes 1 & 3 and feels related to the well-abstractibility prior.)
From a drop of water a logician could infer the possibility of an Atlantic or a Niagara without having seen or heard of one or the other. (c.) A Study in Scarlet[4]
Distinction 1D. Physical abstractions wouldn’t be useful if there were too many of them and they were too difficult to recognize. For example, if you wanted to get some berries in a forest, but there were too many types (with wildly different properties) which were too hard to recognize, then knowing the types would be practically useless. Therefore, there should be something limiting the diversity and hardness-of-recognition of important physical abstractions. Meanwhile, in math you can create infinities of unrecognizable objects (e.g. uncomputable numbers and unpredictable programs). 1D is related to 1B and 1C.
***
Adjectives wouldn’t be useful if the contents of your visual field (and other perceptual fields) didn’t tend to follow certain “composition rules”, such as “different color patterns tend to correspond to different physical abstractions”. Moreover, it’s probable that composition rules themselves wouldn’t be useful if we didn’t have “meta composition rules” to decide which composition rules to apply. This is related to truesight. Some of my previous comments on the topic: one, two. “Adjectives” and “(meta) composition rules” are different types of abstractions, the same way “mathematical abstractions” and “proofs & situations-where-proofs-are-applicable” are different. Technically independent, but the former wouldn’t be useful without the latter.
Distinction/definition 2A. An adjective doesn’t reveal non-trivial facts and is far less predictive than a physical (noun) abstraction. An adjective doesn’t reference a coherent group of objects (e.g. consider the set of all “red” objects or the set of all “big” objects). A composition rule reveals non-trivial statistical facts and references a group of arbitrary situations (which can be split into subgroups of arbitrary objects close in spacetime). Ditto a meta composition rule.
I guess we can keep defining abstraction types. For example, why are verbs useful? Partially for the same reason adjectives are useful (certain actions (transformations of the world) make physical abstractions easier to recognize). And partially because certain actions are important regardless of what does them. For example, if something is moving towards you, it’s very important regardless of whether it’s an animal, a rock, liquid or gas.
A big inconvenience is that the definition of an abstraction type might depend on arbitrary other abstractions. Maybe the most complex philosophical reasoning is about discovering new abstraction types, customized for a specific problem you’re solving / a specific level of reality you’re analyzing.
If that’s correct and we can’t directly classify all abstraction types, what can we do? I hope we can study how mismatch between abstraction types leads to misalignment. Studying what makes abstractions entangled with our universe (OP’s theme) is a special case of that. In the pessimistic case, even the special case would require finding a set of abstraction types which can express any abstraction type (akin to Turing completeness).[5]
***
In the last section I’ll try to explain the most ambitious hope about all this. Those are the most unfinished thoughts.
One topic I’m interested in is something like “emergence (of meaning) in abstractions”. When an abstraction feels qualitatively different from the things it’s abstracting over or the things it’s composed of. Or when an instance of a general abstraction feels qualitatively different from all other instances. Put differently, sometimes it feels like the definition of an abstraction feels much more important than the aggregation of things fitting the definition—as if we’re making a creative leap from examples to the general theme. This might be related to semantic frames and to holism vs. molecularism in general (“can a word be defined independently from the entirety of the language?”).
I can’t give a good example, so here’s a bunch of average ones: “aggression” (more than “dangerous directed actions”), “happiness” (more than a type of pleasure), “kindness” (more than a type of helpfulness), “corrigibility” (ditto), “shit” (more than unpleasant/useless matter), “running” (more than a “fast sequence of actions in a direction”), “murder” (more than a type of destruction), “general intelligence” or “sapience” (more than a sum of specific skills and feelings), “scary” (doesn’t feel like a generalization of particular scary things or scary features), “kafkaesque” or “inconvenient” in general, “cheesy” or “cheap” in general, “harm” or “trauma” (more than a type of lasting disutility), “fragility” (more than a type of instability), “irony” (more than a type of contradiction), “massive” (more than “very big”). Feels like lots of words in natural language have an extra informal oomph. Metaphors (“love is war”, “job is a jail”, “price is rising”) might be one of the main drives behind it. Even if all this is unnatural, there’s gotta be some general psychological mechanism by which concepts get wrapped in extra layers of meaning which feel so intuitive, right?
Maybe my main idea (“analyze abstractions through things which make them useful”) could help to analyze this. For example, “massive” means more than “very big” because “massive” is useful in more ways than a generic adjective. It helps to quickly recognize objects (like generic adjectives do) but also helps to quickly predict important outcomes (like generic verbs do). If you meet a massive bear, you’re unlikely to beat it. If you encounter a massive mountain, you’re unlikely to cross it. If you have a massive debt, you’re unlikely to pay it off.
If we’re lucky, this type of analysis generalizes really well and we can interpret (almost) any thought like this. If we’re even more lucky, this type of analysis naturally leads to a more specific version of ARC-style correlation analysis (it should be plausible because types of abstractions correspond to types of correlations).
If you have N sheep, you can make a prediction that you’ll most likely have N sheep at a later time. But it’s a trivial combination of each sheep’s object permanence.
Mathematical abstractions are made useful by proofs and situations-where-proofs-are-applicable. Proofs don’t refer to any physical objects at all. Situations-where-proofs-are-applicable, like mathematical abstractions themselves, refer to very incoherent groups of physical objects.
Also, maybe “computations about animals + computations about book animals” is a much more coherent entity. But then we need to define the notion of “scale” for computations.
Suppose that this symbolic-to-us language would be suboptimal for compactly representing the universe. The compression process would want to use some other, more “natural” language. It would spend some bits of complexity defining it, then write the world-model in it. That language may turn out to be as alien to us as the encodings NNs use. The cheapest way to define that natural language, however, would be via the definitions that are the simplest in terms of the symbolic-to-us language used by our complexity-estimator.
The post inspired me to think about the differences between abstraction types. This is relevant to my own research too, so I got stuck on (re)writing this comment for more than a week. The main idea I converged to: we should analyze abstractions through things which make them useful. The first section below is the most relevant to the OP, it contains 4 potential distinctions between physical and mathematical abstractions.
***
Distinction 1A. A physical abstraction predicts only a specific aspect of a situation or a trivial sum of specific aspects. A mathematical abstraction is useful for proving non-trivial facts about entire situations. Basically the same abstraction can be both physical or mathematical, depending on how it’s used. Examples:
“Ants” is a physical abstraction. “Moving points” is a mathematical abstraction when used to prove a theorem about ants on a stick. After you’ve proved the theorem, you can easily predict how an infinite number of situations would unfold.
“Number of sheep” is a physical abstraction when used to track the number of sheep.[1] “Numbers” is a mathematical abstraction when used to show that you never know when you can split your sheep into equal groups with more than 1 sheep in each (because some numbers are prime and there’s an infinity of primes).
“Forces” is a physical abstraction when used to predict the movement of a pendulum. “Vectors” is a mathematical abstraction when used to prove the hairy ball theorem.
“Programs” is a physical abstraction when used to predict physical computers. “Programs” is a mathematical abstraction when used to prove the insolubility of the halting problem.
“King safety” is a physical abstraction when used to predict one aspect of the chess position (whether the king is likely to get checkmated soon). “King safety” is not a mathematical abstraction when used as a part of an evaluation function, predicting the development of the entire game. Because you can’t use king safety to prove what the evaluation function should be.
“Elastic ball” is a physical abstraction. “Elastic sphere” is a mathematical abstraction when used to prove the Borsuk–Ulam theorem.
Distinction 1B. Physical abstractions refer to groups of objects existing on roughly the same scale and behaving similarly, groups conceptualizable as “singular entities”. Physical abstractions capture properties of those objects which are directly valuable/impactful. Mathematical abstractions do this much more rarely. Their main purpose is to reveal or transfer non-trivial facts about objects.
Referents of “animals”, “minerals”, “cars” exist on roughly the same scale and behave similarly. “Animalness” of an object makes it valuable and impactful. “Animals” can be thought of as a sort of singular entity. Referents of “vectors”, “functions”, “metric spaces” exist all over the place. “Vectorness” of an object doesn’t make it valuable or impactful. Both gravity and noses can be “vectors”, but you can’t think of “gravity + noses” as a coherent entity. Mathematical abstractions are too abstract, they bundle too much dissimilar stuff.[2]
This is a reframing of your Take 4 and suffers from the same problem (e.g. “animals” also exist in books and “real animals + book animals” is not a coherent entity). Though maybe it makes a stronger case that the problem is merely an edge case.[3]
Distinction 1C. Physical abstractions are inside the same causal system, they shape each other by adding causally linked details to each other. Physical abstractions are implicitly reflected in each other, like jigsaw puzzle pieces or the Indra’s net. For example, trees shape animals and animals shape trees. Mathematical abstractions can be combined into a single system too, but they won’t have the same interconnectedness. (This is a variation on Takes 1 & 3 and feels related to the well-abstractibility prior.)
Distinction 1D. Physical abstractions wouldn’t be useful if there were too many of them and they were too difficult to recognize. For example, if you wanted to get some berries in a forest, but there were too many types (with wildly different properties) which were too hard to recognize, then knowing the types would be practically useless. Therefore, there should be something limiting the diversity and hardness-of-recognition of important physical abstractions. Meanwhile, in math you can create infinities of unrecognizable objects (e.g. uncomputable numbers and unpredictable programs). 1D is related to 1B and 1C.
***
Adjectives wouldn’t be useful if the contents of your visual field (and other perceptual fields) didn’t tend to follow certain “composition rules”, such as “different color patterns tend to correspond to different physical abstractions”. Moreover, it’s probable that composition rules themselves wouldn’t be useful if we didn’t have “meta composition rules” to decide which composition rules to apply. This is related to truesight. Some of my previous comments on the topic: one, two. “Adjectives” and “(meta) composition rules” are different types of abstractions, the same way “mathematical abstractions” and “proofs & situations-where-proofs-are-applicable” are different. Technically independent, but the former wouldn’t be useful without the latter.
Distinction/definition 2A. An adjective doesn’t reveal non-trivial facts and is far less predictive than a physical (noun) abstraction. An adjective doesn’t reference a coherent group of objects (e.g. consider the set of all “red” objects or the set of all “big” objects). A composition rule reveals non-trivial statistical facts and references a group of arbitrary situations (which can be split into subgroups of arbitrary objects close in spacetime). Ditto a meta composition rule.
I guess we can keep defining abstraction types. For example, why are verbs useful? Partially for the same reason adjectives are useful (certain actions (transformations of the world) make physical abstractions easier to recognize). And partially because certain actions are important regardless of what does them. For example, if something is moving towards you, it’s very important regardless of whether it’s an animal, a rock, liquid or gas.
A big inconvenience is that the definition of an abstraction type might depend on arbitrary other abstractions. Maybe the most complex philosophical reasoning is about discovering new abstraction types, customized for a specific problem you’re solving / a specific level of reality you’re analyzing.
If that’s correct and we can’t directly classify all abstraction types, what can we do? I hope we can study how mismatch between abstraction types leads to misalignment. Studying what makes abstractions entangled with our universe (OP’s theme) is a special case of that. In the pessimistic case, even the special case would require finding a set of abstraction types which can express any abstraction type (akin to Turing completeness).[5]
***
In the last section I’ll try to explain the most ambitious hope about all this. Those are the most unfinished thoughts.
One topic I’m interested in is something like “emergence (of meaning) in abstractions”. When an abstraction feels qualitatively different from the things it’s abstracting over or the things it’s composed of. Or when an instance of a general abstraction feels qualitatively different from all other instances. Put differently, sometimes it feels like the definition of an abstraction feels much more important than the aggregation of things fitting the definition—as if we’re making a creative leap from examples to the general theme. This might be related to semantic frames and to holism vs. molecularism in general (“can a word be defined independently from the entirety of the language?”).
I can’t give a good example, so here’s a bunch of average ones: “aggression” (more than “dangerous directed actions”), “happiness” (more than a type of pleasure), “kindness” (more than a type of helpfulness), “corrigibility” (ditto), “shit” (more than unpleasant/useless matter), “running” (more than a “fast sequence of actions in a direction”), “murder” (more than a type of destruction), “general intelligence” or “sapience” (more than a sum of specific skills and feelings), “scary” (doesn’t feel like a generalization of particular scary things or scary features), “kafkaesque” or “inconvenient” in general, “cheesy” or “cheap” in general, “harm” or “trauma” (more than a type of lasting disutility), “fragility” (more than a type of instability), “irony” (more than a type of contradiction), “massive” (more than “very big”). Feels like lots of words in natural language have an extra informal oomph. Metaphors (“love is war”, “job is a jail”, “price is rising”) might be one of the main drives behind it. Even if all this is unnatural, there’s gotta be some general psychological mechanism by which concepts get wrapped in extra layers of meaning which feel so intuitive, right?
Maybe my main idea (“analyze abstractions through things which make them useful”) could help to analyze this. For example, “massive” means more than “very big” because “massive” is useful in more ways than a generic adjective. It helps to quickly recognize objects (like generic adjectives do) but also helps to quickly predict important outcomes (like generic verbs do). If you meet a massive bear, you’re unlikely to beat it. If you encounter a massive mountain, you’re unlikely to cross it. If you have a massive debt, you’re unlikely to pay it off.
If we’re lucky, this type of analysis generalizes really well and we can interpret (almost) any thought like this. If we’re even more lucky, this type of analysis naturally leads to a more specific version of ARC-style correlation analysis (it should be plausible because types of abstractions correspond to types of correlations).
If you have N sheep, you can make a prediction that you’ll most likely have N sheep at a later time. But it’s a trivial combination of each sheep’s object permanence.
Mathematical abstractions are made useful by proofs and situations-where-proofs-are-applicable. Proofs don’t refer to any physical objects at all. Situations-where-proofs-are-applicable, like mathematical abstractions themselves, refer to very incoherent groups of physical objects.
Also, maybe “computations about animals + computations about book animals” is a much more coherent entity. But then we need to define the notion of “scale” for computations.
The quote doesn’t actually help to differentiate the physical and mathematical abstractions, but Holmesian deduction in general is relevant to 1C.
Like what you already considered here: